[{"data":1,"prerenderedAt":9185},["ShallowReactive",2],{"evidence-index":3,"evidence-doc-kaluza-klein-reduction":314},[4,19,31,44,58,71,84,95,108,120,132,143,154,168,178,189,202,213,223,233,243,255,266,277,289,303],{"path":5,"title":6,"description":7,"keywords":8,"field":17,"order":18},"\u002Fevidence\u002Fmaxwells-equations","Maxwell's Equations","Four equations that joined electricity, magnetism and light, and set physics on the road to relativity.",[9,10,11,12,13,14,15,16],"electromagnetism","electric fields","magnetic fields","electromagnetic waves","light","Gauss law","Faraday law","Ampere law","Physical",1,{"path":20,"title":21,"description":22,"keywords":23,"field":30,"order":18},"\u002Fevidence\u002Fnumerology-origin","Numerology Origin","How ancient number symbolism, built to describe cosmic order and interpret sacred texts, became modern personality numerology.",[24,25,26,27,28,29],"number symbolism","Pythagoras","gematria","sacred numbers","divination","history of mathematics","Historical",{"path":32,"title":33,"description":34,"keywords":35,"field":43,"order":18},"\u002Fevidence\u002Forigin-of-life","Origin of Life","What chemistry, geology and genomics currently tell us about how non-living matter became the first living cells.",[36,37,38,39,40,41,42],"abiogenesis","RNA world","Miller-Urey experiment","prebiotic chemistry","hydrothermal vents","protocells","LUCA","Biological",{"path":45,"title":46,"description":47,"keywords":48,"field":57,"order":18},"\u002Fevidence\u002Fsuffering","Suffering","How Siddhartha Gautama tested both luxury and severe self-denial, and what his diagnosis of suffering became.",[49,50,51,52,53,54,55,56],"Buddhism","Buddha","Siddhartha Gautama","dukkha","Four Noble Truths","Middle Way","Eightfold Path","craving","Cultural",{"path":59,"title":60,"description":61,"keywords":62,"field":70,"order":18},"\u002Fevidence\u002Fthe-evolution-of-trust","The Evolution of Trust","How the prisoner's dilemma, Axelrod's tournaments, and later models explain when cooperation beats selfishness, and when it does not.",[63,64,65,66,67,68,69],"game theory","prisoners dilemma","cooperation","Robert Axelrod","tit for tat","reciprocity","evolutionary dynamics","Social",{"path":72,"title":73,"description":74,"keywords":75,"field":43,"order":83},"\u002Fevidence\u002Fplacebo","Placebo","How an inert treatment can change pain, symptoms and brain chemistry, why it works, and where its power ends.",[76,77,78,79,80,81,82],"placebo effect","nocebo","expectation","conditioning","pain relief","clinical trials","endogenous opioids",2,{"path":85,"title":86,"description":87,"keywords":88,"field":70,"order":83},"\u002Fevidence\u002Fpuer-aeternus","Puer Aeternus","The Jungian \"eternal youth\" archetype, what it describes, what empirical psychology can and cannot say about it, and where such people thrive or struggle.",[89,90,91,92,93,94],"Carl Jung","Jungian psychology","eternal youth","archetype","Peter Pan","adulthood",{"path":96,"title":97,"description":98,"keywords":99,"field":57,"order":83},"\u002Fevidence\u002Ftaoism","Taoism","What the Dao, wu wei and naturalness mean, and why Daoist texts separate caring for something from clinging to it.",[100,101,102,103,104,105,106,107],"Daoism","Laozi","Zhuangzi","Daodejing","Tao Te Ching","wu wei","yin and yang","attachment",{"path":109,"title":110,"description":111,"keywords":112,"field":17,"order":83},"\u002Fevidence\u002Ftime-dilation","Time Dilation","Why moving clocks and clocks deeper in gravity tick slower, and how the twin paradox resolves.",[113,114,115,116,117,118,119],"special relativity","general relativity","twin paradox","Lorentz factor","proper time","atomic clocks","GPS",{"path":121,"title":122,"description":123,"keywords":124,"field":17,"order":131},"\u002Fevidence\u002Fhiggs-field","Higgs Field","How an invisible field filling all of space gives mass to fundamental particles, and how it was found in 2012.",[125,126,127,128,129,130],"Higgs boson","particle physics","Standard Model","CERN","Large Hadron Collider","symmetry breaking",3,{"path":133,"title":134,"description":135,"keywords":136,"field":43,"order":131},"\u002Fevidence\u002Fmeditation","Meditation","What meditation trains, how it seems to work in the brain, and how strong the evidence really is.",[137,138,139,140,141,142],"mindfulness","attention","neuroscience","brain","stress","contemplative practice",{"path":144,"title":145,"description":146,"keywords":147,"field":57,"order":131},"\u002Fevidence\u002Fthe-god-of-the-gaps","The God of the Gaps","Why explaining unknowns by appeal to God keeps losing ground, and why many theologians reject the move themselves.",[148,149,150,151,152,153],"theology","religion and science","argument from ignorance","naturalism","cosmological argument","fine-tuning",{"path":155,"title":156,"description":157,"keywords":158,"field":43,"order":167},"\u002Fevidence\u002Fif-the-world-was-vegan","If the World Was Vegan","A scenario analysis of whether global agriculture could feed everyone without animals, and what it would mean for land, soil and ecosystems.",[159,160,161,162,163,164,165,166],"veganism","plant-based diet","agriculture","livestock","food security","soil fertility","biodiversity","nutrition",4,{"path":169,"title":170,"description":171,"keywords":172,"field":17,"order":167},"\u002Fevidence\u002Fkaluza-klein-reduction","Kaluza–Klein Reduction","How one extra, curled-up dimension turns Einstein's gravity into gravity plus electromagnetism, and why we have not seen it.",[173,174,175,9,176,177],"extra dimensions","compactification","gravity","unification","charge quantization",{"path":179,"title":180,"description":181,"keywords":182,"field":17,"order":188},"\u002Fevidence\u002F13-dimensional-models","13-Dimensional Models","Where the number 13 comes from in theoretical physics, why a universe with two time dimensions is mathematically possible, and why none of it is confirmed.",[183,184,185,186,173,187,174],"string theory","M-theory","F-theory","S-theory","two-time physics",5,{"path":190,"title":191,"description":192,"keywords":193,"field":57,"order":188},"\u002Fevidence\u002F1984","1984","What Orwell's Nineteen Eighty-Four is about, why he wrote it, and why rival governments have banned or challenged it as propaganda for the other side.",[194,195,196,197,198,199,200,201],"George Orwell","Nineteen Eighty-Four","dystopia","totalitarianism","surveillance","Big Brother","Newspeak","censorship",{"path":203,"title":204,"description":205,"keywords":206,"field":17,"order":212},"\u002Fevidence\u002Ftachyons","Tachyons","Hypothetical particles that always move faster than light, why they would carry messages into the past, and why modern physics reads them as a sign of instability instead.",[207,208,209,210,211,113],"faster than light","superluminal","causality","imaginary mass","instability",6,{"path":214,"title":215,"description":216,"keywords":217,"field":17,"order":222},"\u002Fevidence\u002Fclosed-timelike-curves","Closed Timelike Curves","How Einstein's equations permit paths through spacetime that return to their own past, and why no one has ever found one.",[218,114,209,219,220,221],"time travel","wormholes","Godel","chronology protection",7,{"path":224,"title":225,"description":226,"keywords":227,"field":17,"order":232},"\u002Fevidence\u002Fnovikov-self-consistency-principle","Novikov Self-Consistency Principle","Why a journey into the past might never change it, and why that need not require any force preventing change.",[218,228,229,230,209,231],"grandfather paradox","causal loops","closed timelike curves","consistency",8,{"path":234,"title":235,"description":236,"keywords":237,"field":17,"order":242},"\u002Fevidence\u002Fbootstrap-paradox","Bootstrap Paradox","Causal loops in which an object or idea exists only because it was carried into its own past, and whether physics can tolerate them.",[218,238,239,209,240,241],"causal loop","information paradox","entropy","self-consistency",9,{"path":244,"title":245,"description":246,"keywords":247,"field":17,"order":254},"\u002Fevidence\u002Fblock-universe","Block Universe","The view, motivated by relativity, that past, present and future all exist and time does not flow.",[248,249,250,251,252,253],"eternalism","spacetime","relativity","simultaneity","presentism","arrow of time",10,{"path":256,"title":257,"description":258,"keywords":259,"field":17,"order":265},"\u002Fevidence\u002Fdeterminism","Determinism","Whether the present fixes the entire future, what physics says about it, and why free will may survive either way.",[260,209,261,262,263,264],"free will","Laplace demon","compatibilism","chaos","quantum mechanics",11,{"path":267,"title":268,"description":269,"keywords":270,"field":17,"order":276},"\u002Fevidence\u002Fbig-bounce","Big Bounce","The idea that our universe began not from nothing but from the collapse of a previous one, and what physics can and cannot say about it.",[271,272,273,274,240,275],"cosmology","Big Bang","cyclic universe","loop quantum cosmology","eternal return",12,{"path":278,"title":279,"description":280,"keywords":281,"field":17,"order":288},"\u002Fevidence\u002Fgolden-ratio","Golden Ratio","Why one number, the golden ratio, keeps turning up in sunflowers, and why many famous claims about it are myths.",[282,283,284,285,286,287],"phi","Fibonacci","phyllotaxis","spirals","irrational numbers","geometry",13,{"path":290,"title":291,"description":292,"keywords":293,"field":17,"order":302},"\u002Fevidence\u002Fsha-256-encryption","SHA-256 Hashing","How the SHA-256 hash function works, why it cannot be run backwards, and what reversing it would actually require.",[294,295,296,297,298,299,300,301],"cryptography","hash function","SHA256","encryption","preimage resistance","collisions","Bitcoin","brute force",14,{"path":304,"title":305,"description":306,"keywords":307,"field":17,"order":313},"\u002Fevidence\u002Fgrovers-algorithm","Grover's Algorithm","How a quantum computer can search N possibilities in about √N steps, and why that is both remarkable and far less than popular accounts claim.",[308,309,310,311,312,294],"quantum computing","quantum search","qubits","amplitude amplification","oracle",15,{"id":315,"title":170,"body":316,"description":171,"extension":9178,"field":17,"keywords":9179,"meta":9180,"navigation":9181,"order":167,"path":169,"seo":9182,"stem":9183,"__hash__":9184},"evidence\u002Fevidence\u002Fkaluza-klein-reduction.md",{"type":317,"value":318,"toc":9150},"minimark",[319,323,327,339,353,360,365,370,373,376,380,387,390,406,409,413,533,537,540,595,598,1018,1029,1033,1037,1347,1907,1910,2257,2322,2326,2329,2805,2941,3877,3880,4240,4276,4582,4586,4618,4929,5123,5445,5452,5733,6016,6020,6151,6407,6546,6550,6632,6670,6895,7037,7051,7061,7065,7068,7072,7192,7196,7202,7266,7622,7979,8069,8158,8167,8173,8177,8181,8374,8378,8447,8451,8462,8466,8469,8473,8555,8559,8562,8568,8574,8654,8657,8661,8692,8712,8731,8755,8771,8790,8813,8838,8858,8882,8905,8996],[320,321,170],"h1",{"id":322},"kaluzaklein-reduction",[324,325,326],"p",{},"Around 1920 there were two great field theories in physics, and they looked nothing alike.",[324,328,329,330,334,335,338],{},"Einstein's General Relativity, then only four years old, described gravity as the curvature of spacetime. Maxwell's electromagnetism, half a century older, described electric and magnetic forces as a field living ",[331,332,333],"em",{},"inside"," spacetime (see ",[336,337,6],"a",{"href":5},"). One was geometry. The other was something placed on top of geometry.",[324,340,341,342,346,347],{},"A little-known mathematician in Königsberg, Theodor Kaluza, noticed something odd. If you take Einstein's equations and simply write them in ",[343,344,345],"strong",{},"five"," dimensions instead of four, and then ask what a four-dimensional observer would see, Maxwell's equations fall out of the extra pieces of the geometry. Electromagnetism appears not as something added to spacetime, but as gravity in a direction we cannot see. ",[348,349,352],"sup",{"className":350},[351],"c-cite","[1]",[324,354,355,356,359],{},"That observation, and the machinery built around it, is called ",[343,357,358],{},"Kaluza–Klein reduction",". It failed as a final theory of nature, yet it became one of the most reused ideas in theoretical physics.",[361,362,364],"h2",{"id":363},"the-hidden-direction","The Hidden Direction",[366,367,369],"h3",{"id":368},"a-garden-hose-seen-from-far-away","A garden hose seen from far away",[324,371,372],{},"The standard picture is a garden hose. From across a lawn it looks like a line: one-dimensional. Up close, an ant walking on it discovers a second direction, around the circumference. That second direction is small and closes on itself.",[324,374,375],{},"Kaluza–Klein theory proposes that every point of our three-dimensional space might be like the hose: there could be an extra direction, so small and so tightly curled that nothing we normally do can move along it.",[366,377,379],{"id":378},"what-the-extra-direction-does","What the extra direction does",[324,381,382,383,386],{},"Now imagine a particle moving in five dimensions. It can have momentum in the usual three directions, and it can also have momentum ",[331,384,385],{},"around"," the small circle.",[324,388,389],{},"To a four-dimensional observer who cannot resolve the circle, that hidden momentum does not look like motion. It looks like two other things:",[391,392,393,400],"ul",{},[394,395,396,399],"li",{},[343,397,398],{},"mass",", because energy of motion around the circle is energy the particle carries even while it sits still in ordinary space;",[394,401,402,405],{},[343,403,404],{},"electric charge",", because the geometry that mixes the circle with ordinary directions behaves exactly like an electromagnetic field, and the hidden momentum is what that field couples to.",[324,407,408],{},"In this picture, charge is not a separate substance. It is motion in a direction you cannot see.",[366,410,412],{"id":411},"why-a-circle-gives-steps","Why a circle gives steps",[324,414,415,416,524,525,528,529],{},"Because the circle closes on itself, a quantum wave travelling around it must fit a whole number of wavelengths into the circumference, just like a vibrating guitar string must fit a whole number of half-wavelengths between its ends. So the hidden momentum comes in steps: ",[417,418,421,467],"span",{"className":419},[420],"katex",[417,422,425],{"className":423},[424],"katex-mathml",[426,427,429],"math",{"xmlns":428},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[430,431,432,462],"semantics",{},[433,434,435,439,444,447,449,452,454,457,459],"mrow",{},[436,437,438],"mn",{},"0",[440,441,443],"mo",{"separator":442},"true",",",[436,445,446],{},"1",[440,448,443],{"separator":442},[436,450,451],{},"2",[440,453,443],{"separator":442},[436,455,456],{},"3",[440,458,443],{"separator":442},[440,460,461],{},"…",[463,464,466],"annotation",{"encoding":465},"application\u002Fx-tex","0, 1, 2, 3, \\dots",[417,468,471],{"className":469,"ariaHidden":442},[470],"katex-html",[417,472,475,480,484,488,493,496,499,502,505,508,511,514,517,520],{"className":473},[474],"base",[417,476],{"className":477,"style":479},[478],"strut","height:0.8389em;vertical-align:-0.1944em;",[417,481,438],{"className":482},[483],"mord",[417,485,443],{"className":486},[487],"mpunct",[417,489],{"className":490,"style":492},[491],"mspace","margin-right:0.1667em;",[417,494,446],{"className":495},[483],[417,497,443],{"className":498},[487],[417,500],{"className":501,"style":492},[491],[417,503,451],{"className":504},[483],[417,506,443],{"className":507},[487],[417,509],{"className":510,"style":492},[491],[417,512,456],{"className":513},[483],[417,515,443],{"className":516},[487],[417,518],{"className":519,"style":492},[491],[417,521,461],{"className":522},[523],"minner"," units. That means both the extra mass and the charge come in ",[343,526,527],{},"discrete steps",". This was Oskar Klein's contribution in 1926, and it offered, for the first time, a geometric reason why electric charge is quantized. ",[348,530,532],{"className":531},[351],"[2,3]",[361,534,536],{"id":535},"how-it-works","How It Works",[324,538,539],{},"The procedure has three stages. Each one is simple on its own.",[541,542,543,549,589],"ol",{},[394,544,545,548],{},[343,546,547],{},"Start with a higher-dimensional theory."," Take pure gravity — Einstein's equations — in five spacetime dimensions, with no electromagnetism put in by hand.",[394,550,551,554,555,588],{},[343,552,553],{},"Compactify."," Assume the fifth dimension is a circle of radius ",[417,556,558,573],{"className":557},[420],[417,559,561],{"className":560},[424],[426,562,563],{"xmlns":428},[430,564,565,571],{},[433,566,567],{},[568,569,570],"mi",{},"R",[463,572,570],{"encoding":465},[417,574,576],{"className":575,"ariaHidden":442},[470],[417,577,579,583],{"className":578},[474],[417,580],{"className":581,"style":582},[478],"height:0.6833em;",[417,584,570],{"className":585,"style":587},[483,586],"mathnormal","margin-right:0.0077em;",", much smaller than anything we can probe.",[394,590,591,594],{},[343,592,593],{},"Reduce."," Expand every field in harmonics around the circle (like a Fourier series) and integrate over the circle. What is left is a four-dimensional theory, but with more fields than you started with.",[324,596,597],{},"The surprise is in which fields appear. The five-dimensional metric has 15 independent components. A four-dimensional observer sorts them into three groups:",[599,600,601,617],"table",{},[602,603,604],"thead",{},[605,606,607,611,614],"tr",{},[608,609,610],"th",{},"Piece of the 5D metric",[608,612,613],{},"Number of components",[608,615,616],{},"What a 4D observer sees",[618,619,620,733,900],"tbody",{},[605,621,622,727,730],{},[623,624,625,726],"td",{},[417,626,628,654],{"className":627},[420],[417,629,631],{"className":630},[424],[426,632,633],{"xmlns":428},[430,634,635,651],{},[433,636,637],{},[638,639,640,643],"msub",{},[568,641,642],{},"g",[433,644,645,648],{},[568,646,647],{},"μ",[568,649,650],{},"ν",[463,652,653],{"encoding":465},"g_{\\mu\\nu}",[417,655,657],{"className":656,"ariaHidden":442},[470],[417,658,660,664],{"className":659},[474],[417,661],{"className":662,"style":663},[478],"height:0.7167em;vertical-align:-0.2861em;",[417,665,667,671],{"className":666},[483],[417,668,642],{"className":669,"style":670},[483,586],"margin-right:0.0359em;",[417,672,675],{"className":673},[674],"msupsub",[417,676,680,717],{"className":677},[678,679],"vlist-t","vlist-t2",[417,681,684,712],{"className":682},[683],"vlist-r",[417,685,689],{"className":686,"style":688},[687],"vlist","height:0.1514em;",[417,690,692,697],{"style":691},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[417,693],{"className":694,"style":696},[695],"pstrut","height:2.7em;",[417,698,704],{"className":699},[700,701,702,703],"sizing","reset-size6","size3","mtight",[417,705,707],{"className":706},[483,703],[417,708,711],{"className":709,"style":710},[483,586,703],"margin-right:0.0637em;","μν",[417,713,716],{"className":714},[715],"vlist-s","​",[417,718,720],{"className":719},[683],[417,721,724],{"className":722,"style":723},[687],"height:0.2861em;",[417,725],{}," (the 4×4 block)",[623,728,729],{},"10",[623,731,732],{},"ordinary gravity (the graviton)",[605,734,735,820,823],{},[623,736,737,819],{},[417,738,740,763],{"className":739},[420],[417,741,743],{"className":742},[424],[426,744,745],{"xmlns":428},[430,746,747,760],{},[433,748,749],{},[638,750,751,753],{},[568,752,642],{},[433,754,755,757],{},[568,756,647],{},[436,758,759],{},"5",[463,761,762],{"encoding":465},"g_{\\mu 5}",[417,764,766],{"className":765,"ariaHidden":442},[470],[417,767,769,772],{"className":768},[474],[417,770],{"className":771,"style":663},[478],[417,773,775,778],{"className":774},[483],[417,776,642],{"className":777,"style":670},[483,586],[417,779,781],{"className":780},[674],[417,782,784,811],{"className":783},[678,679],[417,785,787,808],{"className":786},[683],[417,788,791],{"className":789,"style":790},[687],"height:0.3011em;",[417,792,793,796],{"style":691},[417,794],{"className":795,"style":696},[695],[417,797,799],{"className":798},[700,701,702,703],[417,800,802,805],{"className":801},[483,703],[417,803,647],{"className":804},[483,586,703],[417,806,759],{"className":807},[483,703],[417,809,716],{"className":810},[715],[417,812,814],{"className":813},[683],[417,815,817],{"className":816,"style":723},[687],[417,818],{}," (the mixed column)",[623,821,822],{},"4",[623,824,825,826,899],{},"a vector field ",[417,827,829,848],{"className":828},[420],[417,830,832],{"className":831},[424],[426,833,834],{"xmlns":428},[430,835,836,845],{},[433,837,838],{},[638,839,840,843],{},[568,841,842],{},"A",[568,844,647],{},[463,846,847],{"encoding":465},"A_\\mu",[417,849,851],{"className":850,"ariaHidden":442},[470],[417,852,854,858],{"className":853},[474],[417,855],{"className":856,"style":857},[478],"height:0.9694em;vertical-align:-0.2861em;",[417,859,861,864],{"className":860},[483],[417,862,842],{"className":863},[483,586],[417,865,867],{"className":866},[674],[417,868,870,891],{"className":869},[678,679],[417,871,873,888],{"className":872},[683],[417,874,876],{"className":875,"style":688},[687],[417,877,879,882],{"style":878},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[417,880],{"className":881,"style":696},[695],[417,883,885],{"className":884},[700,701,702,703],[417,886,647],{"className":887},[483,586,703],[417,889,716],{"className":890},[715],[417,892,894],{"className":893},[683],[417,895,897],{"className":896,"style":723},[687],[417,898],{}," — electromagnetism (the photon)",[605,901,902,981,983],{},[623,903,904,980],{},[417,905,907,926],{"className":906},[420],[417,908,910],{"className":909},[424],[426,911,912],{"xmlns":428},[430,913,914,923],{},[433,915,916],{},[638,917,918,920],{},[568,919,642],{},[436,921,922],{},"55",[463,924,925],{"encoding":465},"g_{55}",[417,927,929],{"className":928,"ariaHidden":442},[470],[417,930,932,936],{"className":931},[474],[417,933],{"className":934,"style":935},[478],"height:0.625em;vertical-align:-0.1944em;",[417,937,939,942],{"className":938},[483],[417,940,642],{"className":941,"style":670},[483,586],[417,943,945],{"className":944},[674],[417,946,948,971],{"className":947},[678,679],[417,949,951,968],{"className":950},[683],[417,952,954],{"className":953,"style":790},[687],[417,955,956,959],{"style":691},[417,957],{"className":958,"style":696},[695],[417,960,962],{"className":961},[700,701,702,703],[417,963,965],{"className":964},[483,703],[417,966,922],{"className":967},[483,703],[417,969,716],{"className":970},[715],[417,972,974],{"className":973},[683],[417,975,978],{"className":976,"style":977},[687],"height:0.15em;",[417,979],{}," (the corner)",[623,982,446],{},[623,984,985,986,1017],{},"a scalar field ",[417,987,989,1004],{"className":988},[420],[417,990,992],{"className":991},[424],[426,993,994],{"xmlns":428},[430,995,996,1001],{},[433,997,998],{},[568,999,1000],{},"ϕ",[463,1002,1003],{"encoding":465},"\\phi",[417,1005,1007],{"className":1006,"ariaHidden":442},[470],[417,1008,1010,1014],{"className":1009},[474],[417,1011],{"className":1012,"style":1013},[478],"height:0.8889em;vertical-align:-0.1944em;",[417,1015,1000],{"className":1016},[483,586]," — the size of the circle",[324,1019,1020,1021,1024,1025,1028],{},"The scalar goes by several names — the ",[343,1022,1023],{},"dilaton",", the ",[343,1026,1027],{},"radion",", or in older literature the Jordan–Thiry scalar. It measures how large the extra dimension is at each point of ordinary spacetime.",[361,1030,1032],{"id":1031},"the-formal-description","The Formal Description",[366,1034,1036],{"id":1035},"the-metric-ansatz","The metric ansatz",[324,1038,1039,1040,1103,1104,1197,1198,1227,1228,1256,1257,1346],{},"Write the five coordinates as ",[417,1041,1043,1063],{"className":1042},[420],[417,1044,1046],{"className":1045},[424],[426,1047,1048],{"xmlns":428},[430,1049,1050,1060],{},[433,1051,1052],{},[1053,1054,1055,1058],"msup",{},[568,1056,1057],{},"x",[568,1059,647],{},[463,1061,1062],{"encoding":465},"x^\\mu",[417,1064,1066],{"className":1065,"ariaHidden":442},[470],[417,1067,1069,1073],{"className":1068},[474],[417,1070],{"className":1071,"style":1072},[478],"height:0.6644em;",[417,1074,1076,1079],{"className":1075},[483],[417,1077,1057],{"className":1078},[483,586],[417,1080,1082],{"className":1081},[674],[417,1083,1085],{"className":1084},[678],[417,1086,1088],{"className":1087},[683],[417,1089,1091],{"className":1090,"style":1072},[687],[417,1092,1094,1097],{"style":1093},"top:-3.063em;margin-right:0.05em;",[417,1095],{"className":1096,"style":696},[695],[417,1098,1100],{"className":1099},[700,701,702,703],[417,1101,647],{"className":1102},[483,586,703]," (",[417,1105,1107,1138],{"className":1106},[420],[417,1108,1110],{"className":1109},[424],[426,1111,1112],{"xmlns":428},[430,1113,1114,1135],{},[433,1115,1116,1118,1121,1123,1125,1127,1129,1131,1133],{},[568,1117,647],{},[440,1119,1120],{},"=",[436,1122,438],{},[440,1124,443],{"separator":442},[436,1126,446],{},[440,1128,443],{"separator":442},[436,1130,451],{},[440,1132,443],{"separator":442},[436,1134,456],{},[463,1136,1137],{"encoding":465},"\\mu = 0,1,2,3",[417,1139,1141,1161],{"className":1140,"ariaHidden":442},[470],[417,1142,1144,1147,1150,1154,1158],{"className":1143},[474],[417,1145],{"className":1146,"style":935},[478],[417,1148,647],{"className":1149},[483,586],[417,1151],{"className":1152,"style":1153},[491],"margin-right:0.2778em;",[417,1155,1120],{"className":1156},[1157],"mrel",[417,1159],{"className":1160,"style":1153},[491],[417,1162,1164,1167,1170,1173,1176,1179,1182,1185,1188,1191,1194],{"className":1163},[474],[417,1165],{"className":1166,"style":479},[478],[417,1168,438],{"className":1169},[483],[417,1171,443],{"className":1172},[487],[417,1174],{"className":1175,"style":492},[491],[417,1177,446],{"className":1178},[483],[417,1180,443],{"className":1181},[487],[417,1183],{"className":1184,"style":492},[491],[417,1186,451],{"className":1187},[483],[417,1189,443],{"className":1190},[487],[417,1192],{"className":1193,"style":492},[491],[417,1195,456],{"className":1196},[483],") for ordinary spacetime and ",[417,1199,1201,1215],{"className":1200},[420],[417,1202,1204],{"className":1203},[424],[426,1205,1206],{"xmlns":428},[430,1207,1208,1213],{},[433,1209,1210],{},[568,1211,1212],{},"y",[463,1214,1212],{"encoding":465},[417,1216,1218],{"className":1217,"ariaHidden":442},[470],[417,1219,1221,1224],{"className":1220},[474],[417,1222],{"className":1223,"style":935},[478],[417,1225,1212],{"className":1226,"style":670},[483,586]," for the extra one, with ",[417,1229,1231,1244],{"className":1230},[420],[417,1232,1234],{"className":1233},[424],[426,1235,1236],{"xmlns":428},[430,1237,1238,1242],{},[433,1239,1240],{},[568,1241,1212],{},[463,1243,1212],{"encoding":465},[417,1245,1247],{"className":1246,"ariaHidden":442},[470],[417,1248,1250,1253],{"className":1249},[474],[417,1251],{"className":1252,"style":935},[478],[417,1254,1212],{"className":1255,"style":670},[483,586]," identified periodically: ",[417,1258,1260,1289],{"className":1259},[420],[417,1261,1263],{"className":1262},[424],[426,1264,1265],{"xmlns":428},[430,1266,1267,1286],{},[433,1268,1269,1271,1274,1276,1279,1281,1284],{},[568,1270,1212],{},[440,1272,1273],{},"∼",[568,1275,1212],{},[440,1277,1278],{},"+",[436,1280,451],{},[568,1282,1283],{},"π",[568,1285,570],{},[463,1287,1288],{"encoding":465},"y \\sim y + 2\\pi R",[417,1290,1292,1310,1331],{"className":1291,"ariaHidden":442},[470],[417,1293,1295,1298,1301,1304,1307],{"className":1294},[474],[417,1296],{"className":1297,"style":935},[478],[417,1299,1212],{"className":1300,"style":670},[483,586],[417,1302],{"className":1303,"style":1153},[491],[417,1305,1273],{"className":1306},[1157],[417,1308],{"className":1309,"style":1153},[491],[417,1311,1313,1317,1320,1324,1328],{"className":1312},[474],[417,1314],{"className":1315,"style":1316},[478],"height:0.7778em;vertical-align:-0.1944em;",[417,1318,1212],{"className":1319,"style":670},[483,586],[417,1321],{"className":1322,"style":1323},[491],"margin-right:0.2222em;",[417,1325,1278],{"className":1326},[1327],"mbin",[417,1329],{"className":1330,"style":1323},[491],[417,1332,1334,1337,1340,1343],{"className":1333},[474],[417,1335],{"className":1336,"style":582},[478],[417,1338,451],{"className":1339},[483],[417,1341,1283],{"className":1342,"style":670},[483,586],[417,1344,570],{"className":1345,"style":587},[483,586],". The general five-dimensional line element can then be written as",[1348,1349,1351],"equation",{"label":1350},"Kaluza–Klein metric",[417,1352,1355],{"className":1353},[1354],"katex-display",[417,1356,1358,1484],{"className":1357},[420],[417,1359,1361],{"className":1360},[424],[426,1362,1364],{"xmlns":428,"display":1363},"block",[430,1365,1366,1481],{},[433,1367,1368,1371,1384,1386,1396,1400,1402,1405,1409,1411,1417,1419,1425,1427,1433,1435,1437,1439,1441,1444,1446,1448,1450,1453,1459,1461,1463,1465,1467,1469,1475],{},[568,1369,1370],{},"d",[1053,1372,1373,1382],{},[1374,1375,1376,1379],"mover",{"accent":442},[568,1377,1378],{},"s",[440,1380,1381],{},"^",[436,1383,451],{},[440,1385,1120],{},[638,1387,1388,1390],{},[568,1389,642],{},[433,1391,1392,1394],{},[568,1393,647],{},[568,1395,650],{},[440,1397,1399],{"stretchy":1398},"false","(",[568,1401,1057],{},[440,1403,1404],{"stretchy":1398},")",[1406,1407,1408],"mtext",{}," ",[568,1410,1370],{},[1053,1412,1413,1415],{},[568,1414,1057],{},[568,1416,647],{},[568,1418,1370],{},[1053,1420,1421,1423],{},[568,1422,1057],{},[568,1424,650],{},[440,1426,1278],{},[1053,1428,1429,1431],{},[568,1430,1000],{},[436,1432,451],{},[440,1434,1399],{"stretchy":1398},[568,1436,1057],{},[440,1438,1404],{"stretchy":1398},[1406,1440,1408],{},[440,1442,1399],{"fence":1398,"stretchy":442,"minsize":1443,"maxsize":1443},"1.2em",[568,1445,1370],{},[568,1447,1212],{},[440,1449,1278],{},[568,1451,1452],{},"κ",[638,1454,1455,1457],{},[568,1456,842],{},[568,1458,647],{},[440,1460,1399],{"stretchy":1398},[568,1462,1057],{},[440,1464,1404],{"stretchy":1398},[1406,1466,1408],{},[568,1468,1370],{},[1053,1470,1471,1473],{},[568,1472,1057],{},[568,1474,647],{},[1053,1476,1477,1479],{},[440,1478,1404],{"fence":1398,"stretchy":442,"minsize":1443,"maxsize":1443},[436,1480,451],{},[463,1482,1483],{"encoding":465},"d\\hat{s}^2 = g_{\\mu\\nu}(x)\\,dx^\\mu dx^\\nu + \\phi^2(x)\\,\\big(dy + \\kappa A_\\mu(x)\\,dx^\\mu\\big)^2",[417,1485,1487,1570,1708,1779],{"className":1486,"ariaHidden":442},[470],[417,1488,1490,1494,1497,1561,1564,1567],{"className":1489},[474],[417,1491],{"className":1492,"style":1493},[478],"height:0.8641em;",[417,1495,1370],{"className":1496},[483,586],[417,1498,1500,1537],{"className":1499},[483],[417,1501,1504],{"className":1502},[483,1503],"accent",[417,1505,1507],{"className":1506},[678],[417,1508,1510],{"className":1509},[683],[417,1511,1514,1524],{"className":1512,"style":1513},[687],"height:0.6944em;",[417,1515,1517,1521],{"style":1516},"top:-3em;",[417,1518],{"className":1519,"style":1520},[695],"height:3em;",[417,1522,1378],{"className":1523},[483,586],[417,1525,1526,1529],{"style":1516},[417,1527],{"className":1528,"style":1520},[695],[417,1530,1534],{"className":1531,"style":1533},[1532],"accent-body","left:-0.1944em;",[417,1535,1381],{"className":1536},[483],[417,1538,1540],{"className":1539},[674],[417,1541,1543],{"className":1542},[678],[417,1544,1546],{"className":1545},[683],[417,1547,1549],{"className":1548,"style":1493},[687],[417,1550,1552,1555],{"style":1551},"top:-3.113em;margin-right:0.05em;",[417,1553],{"className":1554,"style":696},[695],[417,1556,1558],{"className":1557},[700,701,702,703],[417,1559,451],{"className":1560},[483,703],[417,1562],{"className":1563,"style":1153},[491],[417,1565,1120],{"className":1566},[1157],[417,1568],{"className":1569,"style":1153},[491],[417,1571,1573,1577,1620,1624,1627,1631,1634,1637,1667,1670,1699,1702,1705],{"className":1572},[474],[417,1574],{"className":1575,"style":1576},[478],"height:1.0361em;vertical-align:-0.2861em;",[417,1578,1580,1583],{"className":1579},[483],[417,1581,642],{"className":1582,"style":670},[483,586],[417,1584,1586],{"className":1585},[674],[417,1587,1589,1612],{"className":1588},[678,679],[417,1590,1592,1609],{"className":1591},[683],[417,1593,1595],{"className":1594,"style":688},[687],[417,1596,1597,1600],{"style":691},[417,1598],{"className":1599,"style":696},[695],[417,1601,1603],{"className":1602},[700,701,702,703],[417,1604,1606],{"className":1605},[483,703],[417,1607,711],{"className":1608,"style":710},[483,586,703],[417,1610,716],{"className":1611},[715],[417,1613,1615],{"className":1614},[683],[417,1616,1618],{"className":1617,"style":723},[687],[417,1619],{},[417,1621,1399],{"className":1622},[1623],"mopen",[417,1625,1057],{"className":1626},[483,586],[417,1628,1404],{"className":1629},[1630],"mclose",[417,1632],{"className":1633,"style":492},[491],[417,1635,1370],{"className":1636},[483,586],[417,1638,1640,1643],{"className":1639},[483],[417,1641,1057],{"className":1642},[483,586],[417,1644,1646],{"className":1645},[674],[417,1647,1649],{"className":1648},[678],[417,1650,1652],{"className":1651},[683],[417,1653,1656],{"className":1654,"style":1655},[687],"height:0.7144em;",[417,1657,1658,1661],{"style":1551},[417,1659],{"className":1660,"style":696},[695],[417,1662,1664],{"className":1663},[700,701,702,703],[417,1665,647],{"className":1666},[483,586,703],[417,1668,1370],{"className":1669},[483,586],[417,1671,1673,1676],{"className":1672},[483],[417,1674,1057],{"className":1675},[483,586],[417,1677,1679],{"className":1678},[674],[417,1680,1682],{"className":1681},[678],[417,1683,1685],{"className":1684},[683],[417,1686,1688],{"className":1687,"style":1655},[687],[417,1689,1690,1693],{"style":1551},[417,1691],{"className":1692,"style":696},[695],[417,1694,1696],{"className":1695},[700,701,702,703],[417,1697,650],{"className":1698,"style":710},[483,586,703],[417,1700],{"className":1701,"style":1323},[491],[417,1703,1278],{"className":1704},[1327],[417,1706],{"className":1707,"style":1323},[491],[417,1709,1711,1715,1744,1747,1750,1753,1756,1764,1767,1770,1773,1776],{"className":1710},[474],[417,1712],{"className":1713,"style":1714},[478],"height:1.2141em;vertical-align:-0.35em;",[417,1716,1718,1721],{"className":1717},[483],[417,1719,1000],{"className":1720},[483,586],[417,1722,1724],{"className":1723},[674],[417,1725,1727],{"className":1726},[678],[417,1728,1730],{"className":1729},[683],[417,1731,1733],{"className":1732,"style":1493},[687],[417,1734,1735,1738],{"style":1551},[417,1736],{"className":1737,"style":696},[695],[417,1739,1741],{"className":1740},[700,701,702,703],[417,1742,451],{"className":1743},[483,703],[417,1745,1399],{"className":1746},[1623],[417,1748,1057],{"className":1749},[483,586],[417,1751,1404],{"className":1752},[1630],[417,1754],{"className":1755,"style":492},[491],[417,1757,1759],{"className":1758},[483],[417,1760,1399],{"className":1761},[1762,1763],"delimsizing","size1",[417,1765,1370],{"className":1766},[483,586],[417,1768,1212],{"className":1769,"style":670},[483,586],[417,1771],{"className":1772,"style":1323},[491],[417,1774,1278],{"className":1775},[1327],[417,1777],{"className":1778,"style":1323},[491],[417,1780,1782,1786,1789,1829,1832,1835,1838,1841,1844,1873],{"className":1781},[474],[417,1783],{"className":1784,"style":1785},[478],"height:1.404em;vertical-align:-0.35em;",[417,1787,1452],{"className":1788},[483,586],[417,1790,1792,1795],{"className":1791},[483],[417,1793,842],{"className":1794},[483,586],[417,1796,1798],{"className":1797},[674],[417,1799,1801,1821],{"className":1800},[678,679],[417,1802,1804,1818],{"className":1803},[683],[417,1805,1807],{"className":1806,"style":688},[687],[417,1808,1809,1812],{"style":878},[417,1810],{"className":1811,"style":696},[695],[417,1813,1815],{"className":1814},[700,701,702,703],[417,1816,647],{"className":1817},[483,586,703],[417,1819,716],{"className":1820},[715],[417,1822,1824],{"className":1823},[683],[417,1825,1827],{"className":1826,"style":723},[687],[417,1828],{},[417,1830,1399],{"className":1831},[1623],[417,1833,1057],{"className":1834},[483,586],[417,1836,1404],{"className":1837},[1630],[417,1839],{"className":1840,"style":492},[491],[417,1842,1370],{"className":1843},[483,586],[417,1845,1847,1850],{"className":1846},[483],[417,1848,1057],{"className":1849},[483,586],[417,1851,1853],{"className":1852},[674],[417,1854,1856],{"className":1855},[678],[417,1857,1859],{"className":1858},[683],[417,1860,1862],{"className":1861,"style":1655},[687],[417,1863,1864,1867],{"style":1551},[417,1865],{"className":1866,"style":696},[695],[417,1868,1870],{"className":1869},[700,701,702,703],[417,1871,647],{"className":1872},[483,586,703],[417,1874,1876,1882],{"className":1875},[483],[417,1877,1879],{"className":1878},[483],[417,1880,1404],{"className":1881},[1762,1763],[417,1883,1885],{"className":1884},[674],[417,1886,1888],{"className":1887},[678],[417,1889,1891],{"className":1890},[683],[417,1892,1895],{"className":1893,"style":1894},[687],"height:1.054em;",[417,1896,1898,1901],{"style":1897},"top:-3.3029em;margin-right:0.05em;",[417,1899],{"className":1900,"style":696},[695],[417,1902,1904],{"className":1903},[700,701,702,703],[417,1905,451],{"className":1906},[483,703],[324,1908,1909],{},"Here:",[391,1911,1912,2012,2091,2163,2194],{},[394,1913,1914,2011],{},[417,1915,1917,1941],{"className":1916},[420],[417,1918,1920],{"className":1919},[424],[426,1921,1922],{"xmlns":428},[430,1923,1924,1938],{},[433,1925,1926,1928],{},[568,1927,1370],{},[1053,1929,1930,1936],{},[1374,1931,1932,1934],{"accent":442},[568,1933,1378],{},[440,1935,1381],{},[436,1937,451],{},[463,1939,1940],{"encoding":465},"d\\hat{s}^2",[417,1942,1944],{"className":1943,"ariaHidden":442},[470],[417,1945,1947,1951,1954],{"className":1946},[474],[417,1948],{"className":1949,"style":1950},[478],"height:0.8141em;",[417,1952,1370],{"className":1953},[483,586],[417,1955,1957,1988],{"className":1956},[483],[417,1958,1960],{"className":1959},[483,1503],[417,1961,1963],{"className":1962},[678],[417,1964,1966],{"className":1965},[683],[417,1967,1969,1977],{"className":1968,"style":1513},[687],[417,1970,1971,1974],{"style":1516},[417,1972],{"className":1973,"style":1520},[695],[417,1975,1378],{"className":1976},[483,586],[417,1978,1979,1982],{"style":1516},[417,1980],{"className":1981,"style":1520},[695],[417,1983,1985],{"className":1984,"style":1533},[1532],[417,1986,1381],{"className":1987},[483],[417,1989,1991],{"className":1990},[674],[417,1992,1994],{"className":1993},[678],[417,1995,1997],{"className":1996},[683],[417,1998,2000],{"className":1999,"style":1950},[687],[417,2001,2002,2005],{"style":1093},[417,2003],{"className":2004,"style":696},[695],[417,2006,2008],{"className":2007},[700,701,702,703],[417,2009,451],{"className":2010},[483,703]," is the five-dimensional interval (hats mark 5D quantities);",[394,2013,2014,2090],{},[417,2015,2017,2038],{"className":2016},[420],[417,2018,2020],{"className":2019},[424],[426,2021,2022],{"xmlns":428},[430,2023,2024,2036],{},[433,2025,2026],{},[638,2027,2028,2030],{},[568,2029,642],{},[433,2031,2032,2034],{},[568,2033,647],{},[568,2035,650],{},[463,2037,653],{"encoding":465},[417,2039,2041],{"className":2040,"ariaHidden":442},[470],[417,2042,2044,2047],{"className":2043},[474],[417,2045],{"className":2046,"style":663},[478],[417,2048,2050,2053],{"className":2049},[483],[417,2051,642],{"className":2052,"style":670},[483,586],[417,2054,2056],{"className":2055},[674],[417,2057,2059,2082],{"className":2058},[678,679],[417,2060,2062,2079],{"className":2061},[683],[417,2063,2065],{"className":2064,"style":688},[687],[417,2066,2067,2070],{"style":691},[417,2068],{"className":2069,"style":696},[695],[417,2071,2073],{"className":2072},[700,701,702,703],[417,2074,2076],{"className":2075},[483,703],[417,2077,711],{"className":2078,"style":710},[483,586,703],[417,2080,716],{"className":2081},[715],[417,2083,2085],{"className":2084},[683],[417,2086,2088],{"className":2087,"style":723},[687],[417,2089],{}," is the ordinary four-dimensional metric of General Relativity;",[394,2092,2093,2162],{},[417,2094,2096,2113],{"className":2095},[420],[417,2097,2099],{"className":2098},[424],[426,2100,2101],{"xmlns":428},[430,2102,2103,2111],{},[433,2104,2105],{},[638,2106,2107,2109],{},[568,2108,842],{},[568,2110,647],{},[463,2112,847],{"encoding":465},[417,2114,2116],{"className":2115,"ariaHidden":442},[470],[417,2117,2119,2122],{"className":2118},[474],[417,2120],{"className":2121,"style":857},[478],[417,2123,2125,2128],{"className":2124},[483],[417,2126,842],{"className":2127},[483,586],[417,2129,2131],{"className":2130},[674],[417,2132,2134,2154],{"className":2133},[678,679],[417,2135,2137,2151],{"className":2136},[683],[417,2138,2140],{"className":2139,"style":688},[687],[417,2141,2142,2145],{"style":878},[417,2143],{"className":2144,"style":696},[695],[417,2146,2148],{"className":2147},[700,701,702,703],[417,2149,647],{"className":2150},[483,586,703],[417,2152,716],{"className":2153},[715],[417,2155,2157],{"className":2156},[683],[417,2158,2160],{"className":2159,"style":723},[687],[417,2161],{}," is a four-component field that will turn out to be the electromagnetic potential;",[394,2164,2165,2193],{},[417,2166,2168,2181],{"className":2167},[420],[417,2169,2171],{"className":2170},[424],[426,2172,2173],{"xmlns":428},[430,2174,2175,2179],{},[433,2176,2177],{},[568,2178,1000],{},[463,2180,1003],{"encoding":465},[417,2182,2184],{"className":2183,"ariaHidden":442},[470],[417,2185,2187,2190],{"className":2186},[474],[417,2188],{"className":2189,"style":1013},[478],[417,2191,1000],{"className":2192},[483,586]," is the scalar that sets the local size of the circle;",[394,2195,2196,2226,2227,2256],{},[417,2197,2199,2213],{"className":2198},[420],[417,2200,2202],{"className":2201},[424],[426,2203,2204],{"xmlns":428},[430,2205,2206,2210],{},[433,2207,2208],{},[568,2209,1452],{},[463,2211,2212],{"encoding":465},"\\kappa",[417,2214,2216],{"className":2215,"ariaHidden":442},[470],[417,2217,2219,2223],{"className":2218},[474],[417,2220],{"className":2221,"style":2222},[478],"height:0.4306em;",[417,2224,1452],{"className":2225},[483,586]," is a constant that fixes units, related to Newton's constant ",[417,2228,2230,2244],{"className":2229},[420],[417,2231,2233],{"className":2232},[424],[426,2234,2235],{"xmlns":428},[430,2236,2237,2242],{},[433,2238,2239],{},[568,2240,2241],{},"G",[463,2243,2241],{"encoding":465},[417,2245,2247],{"className":2246,"ariaHidden":442},[470],[417,2248,2250,2253],{"className":2249},[474],[417,2251],{"className":2252,"style":582},[478],[417,2254,2241],{"className":2255},[483,586],".",[324,2258,2259,2260,2263,2264,2292,2293,2321],{},"Kaluza's key assumption, the ",[343,2261,2262],{},"cylinder condition",", is that none of these fields depend on ",[417,2265,2267,2280],{"className":2266},[420],[417,2268,2270],{"className":2269},[424],[426,2271,2272],{"xmlns":428},[430,2273,2274,2278],{},[433,2275,2276],{},[568,2277,1212],{},[463,2279,1212],{"encoding":465},[417,2281,2283],{"className":2282,"ariaHidden":442},[470],[417,2284,2286,2289],{"className":2285},[474],[417,2287],{"className":2288,"style":935},[478],[417,2290,1212],{"className":2291,"style":670},[483,586],". Klein later reinterpreted this as the statement that we only see the lowest, ",[417,2294,2296,2309],{"className":2295},[420],[417,2297,2299],{"className":2298},[424],[426,2300,2301],{"xmlns":428},[430,2302,2303,2307],{},[433,2304,2305],{},[568,2306,1212],{},[463,2308,1212],{"encoding":465},[417,2310,2312],{"className":2311,"ariaHidden":442},[470],[417,2313,2315,2318],{"className":2314},[474],[417,2316],{"className":2317,"style":935},[478],[417,2319,1212],{"className":2320,"style":670},[483,586],"-independent harmonic of each field.",[366,2323,2325],{"id":2324},"reducing-the-action","Reducing the action",[324,2327,2328],{},"Einstein's theory in five dimensions is defined by the Einstein–Hilbert action",[417,2330,2332],{"className":2331},[1354],[417,2333,2335,2418],{"className":2334},[420],[417,2336,2338],{"className":2337},[424],[426,2339,2340],{"xmlns":428,"display":1363},[430,2341,2342,2415],{},[433,2343,2344,2351,2353,2371,2374,2380,2382,2384,2386,2388,2390,2404,2407,2413],{},[638,2345,2346,2349],{},[568,2347,2348],{},"S",[436,2350,759],{},[440,2352,1120],{},[2354,2355,2356,2358],"mfrac",{},[436,2357,446],{},[433,2359,2360,2363,2365],{},[436,2361,2362],{},"16",[568,2364,1283],{},[638,2366,2367,2369],{},[568,2368,2241],{},[436,2370,759],{},[440,2372,2373],{},"∫",[1053,2375,2376,2378],{},[568,2377,1370],{},[436,2379,822],{},[568,2381,1057],{},[1406,2383,1408],{},[568,2385,1370],{},[568,2387,1212],{},[1406,2389,1408],{},[2391,2392,2393],"msqrt",{},[433,2394,2395,2398],{},[440,2396,2397],{},"−",[1374,2399,2400,2402],{"accent":442},[568,2401,642],{},[440,2403,1381],{},[1406,2405,2406],{},"  ",[1374,2408,2409,2411],{"accent":442},[568,2410,570],{},[440,2412,1381],{},[440,2414,443],{"separator":442},[463,2416,2417],{"encoding":465},"S_5 = \\frac{1}{16\\pi G_5}\\int d^4x\\,dy\\,\\sqrt{-\\hat g}\\;\\hat R ,",[417,2419,2421,2479],{"className":2420,"ariaHidden":442},[470],[417,2422,2424,2428,2470,2473,2476],{"className":2423},[474],[417,2425],{"className":2426,"style":2427},[478],"height:0.8333em;vertical-align:-0.15em;",[417,2429,2431,2435],{"className":2430},[483],[417,2432,2348],{"className":2433,"style":2434},[483,586],"margin-right:0.0576em;",[417,2436,2438],{"className":2437},[674],[417,2439,2441,2462],{"className":2440},[678,679],[417,2442,2444,2459],{"className":2443},[683],[417,2445,2447],{"className":2446,"style":790},[687],[417,2448,2450,2453],{"style":2449},"top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;",[417,2451],{"className":2452,"style":696},[695],[417,2454,2456],{"className":2455},[700,701,702,703],[417,2457,759],{"className":2458},[483,703],[417,2460,716],{"className":2461},[715],[417,2463,2465],{"className":2464},[683],[417,2466,2468],{"className":2467,"style":977},[687],[417,2469],{},[417,2471],{"className":2472,"style":1153},[491],[417,2474,1120],{"className":2475},[1157],[417,2477],{"className":2478,"style":1153},[491],[417,2480,2482,2486,2599,2602,2609,2612,2641,2644,2647,2650,2653,2656,2765,2768,2802],{"className":2481},[474],[417,2483],{"className":2484,"style":2485},[478],"height:2.2222em;vertical-align:-0.8622em;",[417,2487,2489,2493,2596],{"className":2488},[483],[417,2490],{"className":2491},[1623,2492],"nulldelimiter",[417,2494,2496],{"className":2495},[2354],[417,2497,2499,2587],{"className":2498},[678,679],[417,2500,2502,2584],{"className":2501},[683],[417,2503,2506,2561,2572],{"className":2504,"style":2505},[687],"height:1.3214em;",[417,2507,2509,2512],{"style":2508},"top:-2.314em;",[417,2510],{"className":2511,"style":1520},[695],[417,2513,2515,2518,2521],{"className":2514},[483],[417,2516,2362],{"className":2517},[483],[417,2519,1283],{"className":2520,"style":670},[483,586],[417,2522,2524,2527],{"className":2523},[483],[417,2525,2241],{"className":2526},[483,586],[417,2528,2530],{"className":2529},[674],[417,2531,2533,2553],{"className":2532},[678,679],[417,2534,2536,2550],{"className":2535},[683],[417,2537,2539],{"className":2538,"style":790},[687],[417,2540,2541,2544],{"style":878},[417,2542],{"className":2543,"style":696},[695],[417,2545,2547],{"className":2546},[700,701,702,703],[417,2548,759],{"className":2549},[483,703],[417,2551,716],{"className":2552},[715],[417,2554,2556],{"className":2555},[683],[417,2557,2559],{"className":2558,"style":977},[687],[417,2560],{},[417,2562,2564,2567],{"style":2563},"top:-3.23em;",[417,2565],{"className":2566,"style":1520},[695],[417,2568],{"className":2569,"style":2571},[2570],"frac-line","border-bottom-width:0.04em;",[417,2573,2575,2578],{"style":2574},"top:-3.677em;",[417,2576],{"className":2577,"style":1520},[695],[417,2579,2581],{"className":2580},[483],[417,2582,446],{"className":2583},[483],[417,2585,716],{"className":2586},[715],[417,2588,2590],{"className":2589},[683],[417,2591,2594],{"className":2592,"style":2593},[687],"height:0.836em;",[417,2595],{},[417,2597],{"className":2598},[1630,2492],[417,2600],{"className":2601,"style":492},[491],[417,2603,2373],{"className":2604,"style":2608},[2605,2606,2607],"mop","op-symbol","large-op","margin-right:0.4445em;position:relative;top:-0.0011em;",[417,2610],{"className":2611,"style":492},[491],[417,2613,2615,2618],{"className":2614},[483],[417,2616,1370],{"className":2617},[483,586],[417,2619,2621],{"className":2620},[674],[417,2622,2624],{"className":2623},[678],[417,2625,2627],{"className":2626},[683],[417,2628,2630],{"className":2629,"style":1493},[687],[417,2631,2632,2635],{"style":1551},[417,2633],{"className":2634,"style":696},[695],[417,2636,2638],{"className":2637},[700,701,702,703],[417,2639,822],{"className":2640},[483,703],[417,2642,1057],{"className":2643},[483,586],[417,2645],{"className":2646,"style":492},[491],[417,2648,1370],{"className":2649},[483,586],[417,2651,1212],{"className":2652,"style":670},[483,586],[417,2654],{"className":2655,"style":492},[491],[417,2657,2660],{"className":2658},[483,2659],"sqrt",[417,2661,2663,2756],{"className":2662},[678,679],[417,2664,2666,2753],{"className":2665},[683],[417,2667,2670,2730],{"className":2668,"style":2669},[687],"height:0.9839em;",[417,2671,2675,2679],{"className":2672,"style":2674},[2673],"svg-align","top:-3.2em;",[417,2676],{"className":2677,"style":2678},[695],"height:3.2em;",[417,2680,2683,2686],{"className":2681,"style":2682},[483],"padding-left:1em;",[417,2684,2397],{"className":2685},[483],[417,2687,2689],{"className":2688},[483,1503],[417,2690,2692,2721],{"className":2691},[678,679],[417,2693,2695,2718],{"className":2694},[683],[417,2696,2698,2706],{"className":2697,"style":1513},[687],[417,2699,2700,2703],{"style":1516},[417,2701],{"className":2702,"style":1520},[695],[417,2704,642],{"className":2705,"style":670},[483,586],[417,2707,2708,2711],{"style":1516},[417,2709],{"className":2710,"style":1520},[695],[417,2712,2715],{"className":2713,"style":2714},[1532],"left:-0.2222em;",[417,2716,1381],{"className":2717},[483],[417,2719,716],{"className":2720},[715],[417,2722,2724],{"className":2723},[683],[417,2725,2728],{"className":2726,"style":2727},[687],"height:0.1944em;",[417,2729],{},[417,2731,2733,2736],{"style":2732},"top:-2.9439em;",[417,2734],{"className":2735,"style":2678},[695],[417,2737,2741],{"className":2738,"style":2740},[2739],"hide-tail","min-width:1.02em;height:1.28em;",[2742,2743,2749],"svg",{"xmlns":2744,"width":2745,"height":2746,"viewBox":2747,"preserveAspectRatio":2748},"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg","400em","1.28em","0 0 400000 1296","xMinYMin slice",[2750,2751],"path",{"d":2752},"M263,681c0.7,0,18,39.7,52,119\nc34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120\nc340,-704.7,510.7,-1060.3,512,-1067\nl0 -0\nc4.7,-7.3,11,-11,19,-11\nH40000v40H1012.3\ns-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232\nc-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1\ns-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26\nc-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z\nM1001 80h400000v40h-400000z",[417,2754,716],{"className":2755},[715],[417,2757,2759],{"className":2758},[683],[417,2760,2763],{"className":2761,"style":2762},[687],"height:0.2561em;",[417,2764],{},[417,2766],{"className":2767,"style":1153},[491],[417,2769,2771],{"className":2770},[483,1503],[417,2772,2774],{"className":2773},[678],[417,2775,2777],{"className":2776},[683],[417,2778,2781,2789],{"className":2779,"style":2780},[687],"height:0.9468em;",[417,2782,2783,2786],{"style":1516},[417,2784],{"className":2785,"style":1520},[695],[417,2787,570],{"className":2788,"style":587},[483,586],[417,2790,2792,2795],{"style":2791},"top:-3.2523em;",[417,2793],{"className":2794,"style":1520},[695],[417,2796,2799],{"className":2797,"style":2798},[1532],"left:-0.1667em;",[417,2800,1381],{"className":2801},[483],[417,2803,443],{"className":2804},[487],[324,2806,2807,2808,2869,2870,2940],{},"where ",[417,2809,2811,2829],{"className":2810},[420],[417,2812,2814],{"className":2813},[424],[426,2815,2816],{"xmlns":428},[430,2817,2818,2826],{},[433,2819,2820],{},[1374,2821,2822,2824],{"accent":442},[568,2823,570],{},[440,2825,1381],{},[463,2827,2828],{"encoding":465},"\\hat R",[417,2830,2832],{"className":2831,"ariaHidden":442},[470],[417,2833,2835,2838],{"className":2834},[474],[417,2836],{"className":2837,"style":2780},[478],[417,2839,2841],{"className":2840},[483,1503],[417,2842,2844],{"className":2843},[678],[417,2845,2847],{"className":2846},[683],[417,2848,2850,2858],{"className":2849,"style":2780},[687],[417,2851,2852,2855],{"style":1516},[417,2853],{"className":2854,"style":1520},[695],[417,2856,570],{"className":2857,"style":587},[483,586],[417,2859,2860,2863],{"style":2791},[417,2861],{"className":2862,"style":1520},[695],[417,2864,2866],{"className":2865,"style":2798},[1532],[417,2867,1381],{"className":2868},[483]," is the five-dimensional curvature scalar and ",[417,2871,2873,2891],{"className":2872},[420],[417,2874,2876],{"className":2875},[424],[426,2877,2878],{"xmlns":428},[430,2879,2880,2888],{},[433,2881,2882],{},[638,2883,2884,2886],{},[568,2885,2241],{},[436,2887,759],{},[463,2889,2890],{"encoding":465},"G_5",[417,2892,2894],{"className":2893,"ariaHidden":442},[470],[417,2895,2897,2900],{"className":2896},[474],[417,2898],{"className":2899,"style":2427},[478],[417,2901,2903,2906],{"className":2902},[483],[417,2904,2241],{"className":2905},[483,586],[417,2907,2909],{"className":2908},[674],[417,2910,2912,2932],{"className":2911},[678,679],[417,2913,2915,2929],{"className":2914},[683],[417,2916,2918],{"className":2917,"style":790},[687],[417,2919,2920,2923],{"style":878},[417,2921],{"className":2922,"style":696},[695],[417,2924,2926],{"className":2925},[700,701,702,703],[417,2927,759],{"className":2928},[483,703],[417,2930,716],{"className":2931},[715],[417,2933,2935],{"className":2934},[683],[417,2936,2938],{"className":2937,"style":977},[687],[417,2939],{}," is the five-dimensional gravitational constant. Substituting the ansatz and integrating over the circle gives, up to a total derivative,",[1348,2942,2944],{"label":2943},"Reduced 4D action",[417,2945,2947],{"className":2946},[1354],[417,2948,2950,3112],{"className":2949},[420],[417,2951,2953],{"className":2952},[424],[426,2954,2955],{"xmlns":428,"display":1363},[430,2956,2957,3109],{},[433,2958,2959,2965,2967,2989,2991,2997,2999,3001,3009,3011,3013,3064,3066,3069,3079,3081,3089,3095,3097,3103],{},[638,2960,2961,2963],{},[568,2962,2348],{},[436,2964,822],{},[440,2966,1120],{},[2354,2968,2969,2977],{},[433,2970,2971,2973,2975],{},[436,2972,451],{},[568,2974,1283],{},[568,2976,570],{},[433,2978,2979,2981,2983],{},[436,2980,2362],{},[568,2982,1283],{},[638,2984,2985,2987],{},[568,2986,2241],{},[436,2988,759],{},[440,2990,2373],{},[1053,2992,2993,2995],{},[568,2994,1370],{},[436,2996,822],{},[568,2998,1057],{},[1406,3000,1408],{},[2391,3002,3003],{},[433,3004,3005,3007],{},[440,3006,2397],{},[568,3008,642],{},[1406,3010,2406],{},[568,3012,1000],{},[433,3014,3015,3017,3019,3021,3039,3041,3052,3062],{},[440,3016,1399],{"fence":442},[568,3018,570],{},[440,3020,2397],{},[2354,3022,3023,3037],{},[433,3024,3025,3031],{},[1053,3026,3027,3029],{},[568,3028,1452],{},[436,3030,451],{},[1053,3032,3033,3035],{},[568,3034,1000],{},[436,3036,451],{},[436,3038,822],{},[1406,3040,1408],{},[638,3042,3043,3046],{},[568,3044,3045],{},"F",[433,3047,3048,3050],{},[568,3049,647],{},[568,3051,650],{},[1053,3053,3054,3056],{},[568,3055,3045],{},[433,3057,3058,3060],{},[568,3059,647],{},[568,3061,650],{},[440,3063,1404],{"fence":442},[440,3065,443],{"separator":442},[491,3067],{"width":3068},"2em",[638,3070,3071,3073],{},[568,3072,3045],{},[433,3074,3075,3077],{},[568,3076,647],{},[568,3078,650],{},[440,3080,1120],{},[638,3082,3083,3087],{},[568,3084,3086],{"mathvariant":3085},"normal","∂",[568,3088,647],{},[638,3090,3091,3093],{},[568,3092,842],{},[568,3094,650],{},[440,3096,2397],{},[638,3098,3099,3101],{},[568,3100,3086],{"mathvariant":3085},[568,3102,650],{},[638,3104,3105,3107],{},[568,3106,842],{},[568,3108,647],{},[463,3110,3111],{"encoding":465},"S_4 = \\frac{2\\pi R}{16\\pi G_5}\\int d^4x\\,\\sqrt{-g}\\;\\phi\\left(R - \\frac{\\kappa^2\\phi^2}{4}\\,F_{\\mu\\nu}F^{\\mu\\nu}\\right),\n\\qquad F_{\\mu\\nu} = \\partial_\\mu A_\\nu - \\partial_\\nu A_\\mu",[417,3113,3115,3170,3693,3791],{"className":3114,"ariaHidden":442},[470],[417,3116,3118,3121,3161,3164,3167],{"className":3117},[474],[417,3119],{"className":3120,"style":2427},[478],[417,3122,3124,3127],{"className":3123},[483],[417,3125,2348],{"className":3126,"style":2434},[483,586],[417,3128,3130],{"className":3129},[674],[417,3131,3133,3153],{"className":3132},[678,679],[417,3134,3136,3150],{"className":3135},[683],[417,3137,3139],{"className":3138,"style":790},[687],[417,3140,3141,3144],{"style":2449},[417,3142],{"className":3143,"style":696},[695],[417,3145,3147],{"className":3146},[700,701,702,703],[417,3148,822],{"className":3149},[483,703],[417,3151,716],{"className":3152},[715],[417,3154,3156],{"className":3155},[683],[417,3157,3159],{"className":3158,"style":977},[687],[417,3160],{},[417,3162],{"className":3163,"style":1153},[491],[417,3165,1120],{"className":3166},[1157],[417,3168],{"className":3169,"style":1153},[491],[417,3171,3173,3177,3289,3292,3295,3298,3327,3330,3333,3391,3394,3397,3400,3628,3631,3634,3638,3641,3684,3687,3690],{"className":3172},[474],[417,3174],{"className":3175,"style":3176},[478],"height:2.4411em;vertical-align:-0.95em;",[417,3178,3180,3183,3286],{"className":3179},[483],[417,3181],{"className":3182},[1623,2492],[417,3184,3186],{"className":3185},[2354],[417,3187,3189,3278],{"className":3188},[678,679],[417,3190,3192,3275],{"className":3191},[683],[417,3193,3196,3250,3258],{"className":3194,"style":3195},[687],"height:1.3603em;",[417,3197,3198,3201],{"style":2508},[417,3199],{"className":3200,"style":1520},[695],[417,3202,3204,3207,3210],{"className":3203},[483],[417,3205,2362],{"className":3206},[483],[417,3208,1283],{"className":3209,"style":670},[483,586],[417,3211,3213,3216],{"className":3212},[483],[417,3214,2241],{"className":3215},[483,586],[417,3217,3219],{"className":3218},[674],[417,3220,3222,3242],{"className":3221},[678,679],[417,3223,3225,3239],{"className":3224},[683],[417,3226,3228],{"className":3227,"style":790},[687],[417,3229,3230,3233],{"style":878},[417,3231],{"className":3232,"style":696},[695],[417,3234,3236],{"className":3235},[700,701,702,703],[417,3237,759],{"className":3238},[483,703],[417,3240,716],{"className":3241},[715],[417,3243,3245],{"className":3244},[683],[417,3246,3248],{"className":3247,"style":977},[687],[417,3249],{},[417,3251,3252,3255],{"style":2563},[417,3253],{"className":3254,"style":1520},[695],[417,3256],{"className":3257,"style":2571},[2570],[417,3259,3260,3263],{"style":2574},[417,3261],{"className":3262,"style":1520},[695],[417,3264,3266,3269,3272],{"className":3265},[483],[417,3267,451],{"className":3268},[483],[417,3270,1283],{"className":3271,"style":670},[483,586],[417,3273,570],{"className":3274,"style":587},[483,586],[417,3276,716],{"className":3277},[715],[417,3279,3281],{"className":3280},[683],[417,3282,3284],{"className":3283,"style":2593},[687],[417,3285],{},[417,3287],{"className":3288},[1630,2492],[417,3290],{"className":3291,"style":492},[491],[417,3293,2373],{"className":3294,"style":2608},[2605,2606,2607],[417,3296],{"className":3297,"style":492},[491],[417,3299,3301,3304],{"className":3300},[483],[417,3302,1370],{"className":3303},[483,586],[417,3305,3307],{"className":3306},[674],[417,3308,3310],{"className":3309},[678],[417,3311,3313],{"className":3312},[683],[417,3314,3316],{"className":3315,"style":1493},[687],[417,3317,3318,3321],{"style":1551},[417,3319],{"className":3320,"style":696},[695],[417,3322,3324],{"className":3323},[700,701,702,703],[417,3325,822],{"className":3326},[483,703],[417,3328,1057],{"className":3329},[483,586],[417,3331],{"className":3332,"style":492},[491],[417,3334,3336],{"className":3335},[483,2659],[417,3337,3339,3382],{"className":3338},[678,679],[417,3340,3342,3379],{"className":3341},[683],[417,3343,3346,3362],{"className":3344,"style":3345},[687],"height:0.8283em;",[417,3347,3349,3352],{"className":3348,"style":1516},[2673],[417,3350],{"className":3351,"style":1520},[695],[417,3353,3356,3359],{"className":3354,"style":3355},[483],"padding-left:0.833em;",[417,3357,2397],{"className":3358},[483],[417,3360,642],{"className":3361,"style":670},[483,586],[417,3363,3365,3368],{"style":3364},"top:-2.7883em;",[417,3366],{"className":3367,"style":1520},[695],[417,3369,3372],{"className":3370,"style":3371},[2739],"min-width:0.853em;height:1.08em;",[2742,3373,3376],{"xmlns":2744,"width":2745,"height":3374,"viewBox":3375,"preserveAspectRatio":2748},"1.08em","0 0 400000 1080",[2750,3377],{"d":3378},"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 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assName":3689},[1157],[417,3691],{"className":3692,"style":1153},[491],[417,3694,3696,3700,3742,3782,3785,3788],{"className":3695},[474],[417,3697],{"className":3698,"style":3699},[478],"height:0.9805em;vertical-align:-0.2861em;",[417,3701,3703,3707],{"className":3702},[483],[417,3704,3086],{"className":3705,"style":3706},[483],"margin-right:0.0556em;",[417,3708,3710],{"className":3709},[674],[417,3711,3713,3734],{"className":3712},[678,679],[417,3714,3716,3731],{"className":3715},[683],[417,3717,3719],{"className":3718,"style":688},[687],[417,3720,3722,3725],{"style":3721},"top:-2.55em;margin-left:-0.0556em;margin-right:0.05em;",[417,3723],{"className":3724,"style":696},[695],[417,3726,3728],{"className":3727},[700,701,702,703],[417,3729,647],{"className":3730},[483,586,703],[417,3732,716],{"className":3733},[715],[417,3735,3737],{"className":3736},[683],[417,3738,3740],{"className":3739,"style":723},[687],[417,3741],{},[417,3743,3745,3748],{"className":3744},[483],[417,3746,842],{"className":3747},[483,586],[417,3749,3751],{"className":3750},[674],[417,3752,3754,3774],{"className":3753},[678,679],[417,3755,3757,3771],{"className":3756},[683],[417,3758,3760],{"className":3759,"style":688},[687],[417,3761,3762,3765],{"style":878},[417,3763],{"className":3764,"style":696},[695],[417,3766,3768],{"className":3767},[700,701,702,703],[417,3769,650],{"className":3770,"style":710},[483,586,703],[417,3772,716],{"className":3773},[715],[417,3775,3777],{"className":3776},[683],[417,3778,3780],{"className":3779,"style":977},[687],[417,3781],{},[417,3783],{"className":3784,"style":1323},[491],[417,3786,2397],{"className":3787},[1327],[417,3789],{"className":3790,"style":1323},[491],[417,3792,3794,3797,3837],{"className":3793},[474],[417,3795],{"className":3796,"style":3699},[478],[417,3798,3800,3803],{"className":3799},[483],[417,3801,3086],{"className":3802,"style":3706},[483],[417,3804,3806],{"className":3805},[674],[417,3807,3809,3829],{"className":3808},[678,679],[417,3810,3812,3826],{"className":3811},[683],[417,3813,3815],{"className":3814,"style":688},[687],[417,3816,3817,3820],{"style":3721},[417,3818],{"className":3819,"style":696},[695],[417,3821,3823],{"className":3822},[700,701,702,703],[417,3824,650],{"className":3825,"style":710},[483,586,703],[417,3827,716],{"className":3828},[715],[417,3830,3832],{"className":3831},[683],[417,3833,3835],{"className":3834,"style":977},[687],[417,3836],{},[417,3838,3840,3843],{"className":3839},[483],[417,3841,842],{"className":3842},[483,586],[417,3844,3846],{"className":3845},[674],[417,3847,3849,3869],{"className":3848},[678,679],[417,3850,3852,3866],{"className":3851},[683],[417,3853,3855],{"className":3854,"style":688},[687],[417,3856,3857,3860],{"style":878},[417,3858],{"className":3859,"style":696},[695],[417,3861,3863],{"className":3862},[700,701,702,703],[417,3864,647],{"className":3865},[483,586,703],[417,3867,716],{"className":3868},[715],[417,3870,3872],{"className":3871},[683],[417,3873,3875],{"className":3874,"style":723},[687],[417,3876],{},[324,3878,3879],{},"Read term by term:",[391,3881,3882,3916,4041,4072],{},[394,3883,3884,3912,3913,2256],{},[417,3885,3887,3900],{"className":3886},[420],[417,3888,3890],{"className":3889},[424],[426,3891,3892],{"xmlns":428},[430,3893,3894,3898],{},[433,3895,3896],{},[568,3897,570],{},[463,3899,570],{"encoding":465},[417,3901,3903],{"className":3902,"ariaHidden":442},[470],[417,3904,3906,3909],{"className":3905},[474],[417,3907],{"className":3908,"style":582},[478],[417,3910,570],{"className":3911,"style":587},[483,586]," is the four-dimensional curvature: this is ",[343,3914,3915],{},"General Relativity",[394,3917,3918,4037,4038,2256],{},[417,3919,3921,3953],{"className":3920},[420],[417,3922,3924],{"className":3923},[424],[426,3925,3926],{"xmlns":428},[430,3927,3928,3950],{},[433,3929,3930,3940],{},[638,3931,3932,3934],{},[568,3933,3045],{},[433,3935,3936,3938],{},[568,3937,647],{},[568,3939,650],{},[1053,3941,3942,3944],{},[568,3943,3045],{},[433,3945,3946,3948],{},[568,3947,647],{},[568,3949,650],{},[463,3951,3952],{"encoding":465},"F_{\\mu\\nu}F^{\\mu\\nu}",[417,3954,3956],{"className":3955,"ariaHidden":442},[470],[417,3957,3959,3962,4005],{"className":3958},[474],[417,3960],{"className":3961,"style":857},[478],[417,3963,3965,3968],{"className":3964},[483],[417,3966,3045],{"className":3967,"style":3551},[483,586],[417,3969,3971],{"className":3970},[674],[417,3972,3974,3997],{"className":3973},[678,679],[417,3975,3977,3994],{"className":3976},[683],[417,3978,3980],{"className":3979,"style":688},[687],[417,3981,3982,3985],{"style":3566},[417,3983],{"className":3984,"style":696},[695],[417,3986,3988],{"className":3987},[700,701,702,703],[417,3989,3991],{"className":3990},[483,703],[417,3992,711],{"className":3993,"style":710},[483,586,703],[417,3995,716],{"className":3996},[715],[417,3998,4000],{"className":3999},[683],[417,4001,4003],{"className":4002,"style":723},[687],[417,4004],{},[417,4006,4008,4011],{"className":4007},[483],[417,4009,3045],{"className":4010,"style":3551},[483,586],[417,4012,4014],{"className":4013},[674],[417,4015,4017],{"className":4016},[678],[417,4018,4020],{"className":4019},[683],[417,4021,4023],{"className":4022,"style":1072},[687],[417,4024,4025,4028],{"style":1093},[417,4026],{"className":4027,"style":696},[695],[417,4029,4031],{"className":4030},[700,701,702,703],[417,4032,4034],{"className":4033},[483,703],[417,4035,711],{"className":4036,"style":710},[483,586,703]," is the standard Lagrangian of the electromagnetic field: this is ",[343,4039,4040],{},"Maxwell's theory",[394,4042,4043,4071],{},[417,4044,4046,4059],{"className":4045},[420],[417,4047,4049],{"className":4048},[424],[426,4050,4051],{"xmlns":428},[430,4052,4053,4057],{},[433,4054,4055],{},[568,4056,1000],{},[463,4058,1003],{"encoding":465},[417,4060,4062],{"className":4061,"ariaHidden":442},[470],[417,4063,4065,4068],{"className":4064},[474],[417,4066],{"className":4067,"style":1013},[478],[417,4069,1000],{"className":4070},[483,586]," multiplies both, acting as a field-dependent strength of gravity and of electromagnetism.",[394,4073,4074,4075,2256],{},"The prefactor shows that four-dimensional gravity is weaker than five-dimensional gravity by the size of the circle: ",[417,4076,4078,4117],{"className":4077},[420],[417,4079,4081],{"className":4080},[424],[426,4082,4083],{"xmlns":428},[430,4084,4085,4114],{},[433,4086,4087,4093,4095,4101,4104,4106,4108,4110,4112],{},[638,4088,4089,4091],{},[568,4090,2241],{},[436,4092,822],{},[440,4094,1120],{},[638,4096,4097,4099],{},[568,4098,2241],{},[436,4100,759],{},[568,4102,4103],{"mathvariant":3085},"\u002F",[440,4105,1399],{"stretchy":1398},[436,4107,451],{},[568,4109,1283],{},[568,4111,570],{},[440,4113,1404],{"stretchy":1398},[463,4115,4116],{"encoding":465},"G_4 = G_5 \u002F (2\\pi R)",[417,4118,4120,4175],{"className":4119,"ariaHidden":442},[470],[417,4121,4123,4126,4166,4169,4172],{"className":4122},[474],[417,4124],{"className":4125,"style":2427},[478],[417,4127,4129,4132],{"className":4128},[483],[417,4130,2241],{"className":4131},[483,586],[417,4133,4135],{"className":4134},[674],[417,4136,4138,4158],{"className":4137},[678,679],[417,4139,4141,4155],{"className":4140},[683],[417,4142,4144],{"className":4143,"style":790},[687],[417,4145,4146,4149],{"style":878},[417,4147],{"className":4148,"style":696},[695],[417,4150,4152],{"className":4151},[700,701,702,703],[417,4153,822],{"className":4154},[483,703],[417,4156,716],{"className":4157},[715],[417,4159,4161],{"className":4160},[683],[417,4162,4164],{"className":4163,"style":977},[687],[417,4165],{},[417,4167],{"className":4168,"style":1153},[491],[417,4170,1120],{"className":4171},[1157],[417,4173],{"className":4174,"style":1153},[491],[417,4176,4178,4182,4222,4225,4228,4231,4234,4237],{"className":4177},[474],[417,4179],{"className":4180,"style":4181},[478],"height:1em;vertical-align:-0.25em;",[417,4183,4185,4188],{"className":4184},[483],[417,4186,2241],{"className":4187},[483,586],[417,4189,4191],{"className":4190},[674],[417,4192,4194,4214],{"className":4193},[678,679],[417,4195,4197,4211],{"className":4196},[683],[417,4198,4200],{"className":4199,"style":790},[687],[417,4201,4202,4205],{"style":878},[417,4203],{"className":4204,"style":696},[695],[417,4206,4208],{"className":4207},[700,701,702,703],[417,4209,759],{"className":4210},[483,703],[417,4212,716],{"className":4213},[715],[417,4215,4217],{"className":4216},[683],[417,4218,4220],{"className":4219,"style":977},[687],[417,4221],{},[417,4223,4103],{"className":4224},[483],[417,4226,1399],{"className":4227},[1623],[417,4229,451],{"className":4230},[483],[417,4232,1283],{"className":4233,"style":670},[483,586],[417,4235,570],{"className":4236,"style":587},[483,586],[417,4238,1404],{"className":4239},[1630],[324,4241,4242,4243,4271,4272],{},"If ",[417,4244,4246,4259],{"className":4245},[420],[417,4247,4249],{"className":4248},[424],[426,4250,4251],{"xmlns":428},[430,4252,4253,4257],{},[433,4254,4255],{},[568,4256,1000],{},[463,4258,1003],{"encoding":465},[417,4260,4262],{"className":4261,"ariaHidden":442},[470],[417,4263,4265,4268],{"className":4264},[474],[417,4266],{"className":4267,"style":1013},[478],[417,4269,1000],{"className":4270},[483,586]," is held fixed at 1, the result is exactly Einstein–Maxwell theory. Nothing electromagnetic was put in. It was already inside five-dimensional geometry. ",[348,4273,4275],{"className":4274},[351],"[1,4]",[324,4277,4278,4279,4379,4380,4581],{},"The symmetry of electromagnetism also has a geometric origin. Shifting the extra coordinate by an amount that depends on position, ",[417,4280,4282,4316],{"className":4281},[420],[417,4283,4285],{"className":4284},[424],[426,4286,4287],{"xmlns":428},[430,4288,4289,4313],{},[433,4290,4291,4293,4296,4298,4300,4302,4304,4307,4309,4311],{},[568,4292,1212],{},[440,4294,4295],{},"→",[568,4297,1212],{},[440,4299,1278],{},[568,4301,1452],{},[1406,4303,1408],{},[568,4305,4306],{},"λ",[440,4308,1399],{"stretchy":1398},[568,4310,1057],{},[440,4312,1404],{"stretchy":1398},[463,4314,4315],{"encoding":465},"y \\to y + \\kappa\\,\\lambda(x)",[417,4317,4319,4337,4355],{"className":4318,"ariaHidden":442},[470],[417,4320,4322,4325,4328,4331,4334],{"className":4321},[474],[417,4323],{"className":4324,"style":935},[478],[417,4326,1212],{"className":4327,"style":670},[483,586],[417,4329],{"className":4330,"style":1153},[491],[417,4332,4295],{"className":4333},[1157],[417,4335],{"className":4336,"style":1153},[491],[417,4338,4340,4343,4346,4349,4352],{"className":4339},[474],[417,4341],{"className":4342,"style":1316},[478],[417,4344,1212],{"className":4345,"style":670},[483,586],[417,4347],{"className":4348,"style":1323},[491],[417,4350,1278],{"className":4351},[1327],[417,4353],{"className":4354,"style":1323},[491],[417,4356,4358,4361,4364,4367,4370,4373,4376],{"className":4357},[474],[417,4359],{"className":4360,"style":4181},[478],[417,4362,1452],{"className":4363},[483,586],[417,4365],{"className":4366,"style":492},[491],[417,4368,4306],{"className":4369},[483,586],[417,4371,1399],{"className":4372},[1623],[417,4374,1057],{"className":4375},[483,586],[417,4377,1404],{"className":4378},[1630],", changes ",[417,4381,4383,4419],{"className":4382},[420],[417,4384,4386],{"className":4385},[424],[426,4387,4388],{"xmlns":428},[430,4389,4390,4416],{},[433,4391,4392,4398,4400,4406,4408,4414],{},[638,4393,4394,4396],{},[568,4395,842],{},[568,4397,647],{},[440,4399,4295],{},[638,4401,4402,4404],{},[568,4403,842],{},[568,4405,647],{},[440,4407,2397],{},[638,4409,4410,4412],{},[568,4411,3086],{"mathvariant":3085},[568,4413,647],{},[568,4415,4306],{},[463,4417,4418],{"encoding":465},"A_\\mu \\to A_\\mu - \\partial_\\mu \\lambda",[417,4420,4422,4477,4532],{"className":4421,"ariaHidden":442},[470],[417,4423,4425,4428,4468,4471,4474],{"className":4424},[474],[417,4426],{"className":4427,"style":857},[478],[417,4429,4431,4434],{"className":4430},[483],[417,4432,842],{"className":4433},[483,586],[417,4435,4437],{"className":4436},[674],[417,4438,4440,4460],{"className":4439},[678,679],[417,4441,4443,4457],{"className":4442},[683],[417,4444,4446],{"className":4445,"style":688},[687],[417,4447,4448,4451],{"style":878},[417,4449],{"className":4450,"style":696},[695],[417,4452,4454],{"className":4453},[700,701,702,703],[417,4455,647],{"className":4456},[483,586,703],[417,4458,716],{"className":4459},[715],[417,4461,4463],{"className":4462},[683],[417,4464,4466],{"className":4465,"style":723},[687],[417,4467],{},[417,4469],{"className":4470,"style":1153},[491],[417,4472,4295],{"className":4473},[1157],[417,4475],{"className":4476,"style":1153},[491],[417,4478,4480,4483,4523,4526,4529],{"className":4479},[474],[417,4481],{"className":4482,"style":857},[478],[417,4484,4486,4489],{"className":4485},[483],[417,4487,842],{"className":4488},[483,586],[417,4490,4492],{"className":4491},[674],[417,4493,4495,4515],{"className":4494},[678,679],[417,4496,4498,4512],{"className":4497},[683],[417,4499,4501],{"className":4500,"style":688},[687],[417,4502,4503,4506],{"style":878},[417,4504],{"className":4505,"style":696},[695],[417,4507,4509],{"className":4508},[700,701,702,703],[417,4510,647],{"className":4511},[483,586,703],[417,4513,716],{"className":4514},[715],[417,4516,4518],{"className":4517},[683],[417,4519,4521],{"className":4520,"style":723},[687],[417,4522],{},[417,4524],{"className":4525,"style":1323},[491],[417,4527,2397],{"className":4528},[1327],[417,4530],{"className":4531,"style":1323},[491],[417,4533,4535,4538,4578],{"className":4534},[474],[417,4536],{"className":4537,"style":3699},[478],[417,4539,4541,4544],{"className":4540},[483],[417,4542,3086],{"className":4543,"style":3706},[483],[417,4545,4547],{"className":4546},[674],[417,4548,4550,4570],{"className":4549},[678,679],[417,4551,4553,4567],{"className":4552},[683],[417,4554,4556],{"className":4555,"style":688},[687],[417,4557,4558,4561],{"style":3721},[417,4559],{"className":4560,"style":696},[695],[417,4562,4564],{"className":4563},[700,701,702,703],[417,4565,647],{"className":4566},[483,586,703],[417,4568,716],{"className":4569},[715],[417,4571,4573],{"className":4572},[683],[417,4574,4576],{"className":4575,"style":723},[687],[417,4577],{},[417,4579,4306],{"className":4580},[483,586],". That is precisely the gauge freedom of the electromagnetic potential. In this language, electromagnetic gauge symmetry is just the freedom to relabel points around the circle.",[366,4583,4585],{"id":4584},"the-kaluzaklein-tower","The Kaluza–Klein tower",[324,4587,4588,4589,4617],{},"Now let the fields depend on ",[417,4590,4592,4605],{"className":4591},[420],[417,4593,4595],{"className":4594},[424],[426,4596,4597],{"xmlns":428},[430,4598,4599,4603],{},[433,4600,4601],{},[568,4602,1212],{},[463,4604,1212],{"encoding":465},[417,4606,4608],{"className":4607,"ariaHidden":442},[470],[417,4609,4611,4614],{"className":4610},[474],[417,4612],{"className":4613,"style":935},[478],[417,4615,1212],{"className":4616,"style":670},[483,586],". Any field on a circle can be expanded in a Fourier series:",[417,4619,4621],{"className":4620},[1354],[417,4622,4624,4705],{"className":4623},[420],[417,4625,4627],{"className":4626},[424],[426,4628,4629],{"xmlns":428,"display":1363},[430,4630,4631,4702],{},[433,4632,4633,4636,4638,4640,4642,4644,4646,4648,4668,4674,4676,4678,4680,4682,4700],{},[568,4634,4635],{"mathvariant":3085},"Φ",[440,4637,1399],{"stretchy":1398},[568,4639,1057],{},[440,4641,443],{"separator":442},[568,4643,1212],{},[440,4645,1404],{"stretchy":1398},[440,4647,1120],{},[4649,4650,4651,4654,4666],"munderover",{},[440,4652,4653],{},"∑",[433,4655,4656,4659,4661,4663],{},[568,4657,4658],{},"n",[440,4660,1120],{},[440,4662,2397],{},[568,4664,4665],{"mathvariant":3085},"∞",[568,4667,4665],{"mathvariant":3085},[638,4669,4670,4672],{},[568,4671,4635],{"mathvariant":3085},[568,4673,4658],{},[440,4675,1399],{"stretchy":1398},[568,4677,1057],{},[440,4679,1404],{"stretchy":1398},[1406,4681,1408],{},[1053,4683,4684,4687],{},[568,4685,4686],{},"e",[433,4688,4689,4692,4694,4696,4698],{},[568,4690,4691],{},"i",[568,4693,4658],{},[568,4695,1212],{},[568,4697,4103],{"mathvariant":3085},[568,4699,570],{},[568,4701,2256],{"mathvariant":3085},[463,4703,4704],{"encoding":465},"\\Phi(x,y) = \\sum_{n=-\\infty}^{\\infty} \\Phi_n(x)\\,e^{i n y \u002F R}.",[417,4706,4708,4744],{"className":4707,"ariaHidden":442},[470],[417,4709,4711,4714,4717,4720,4723,4726,4729,4732,4735,4738,4741],{"className":4710},[474],[417,4712],{"className":4713,"style":4181},[478],[417,4715,4635],{"className":4716},[483],[417,4718,1399],{"className":4719},[1623],[417,4721,1057],{"className":4722},[483,586],[417,4724,443],{"className":4725},[487],[417,4727],{"className":4728,"style":492},[491],[417,4730,1212],{"className":4731,"style":670},[483,586],[417,4733,1404],{"className":4734},[1630],[417,4736],{"className":4737,"style":1153},[491],[417,4739,1120],{"className":4740},[1157],[417,4742],{"className":4743,"style":1153},[491],[417,4745,4747,4751,4828,4831,4871,4874,4877,4880,4883,4926],{"className":4746},[474],[417,4748],{"className":4749,"style":4750},[478],"height:2.9597em;vertical-align:-1.3083em;",[417,4752,4755],{"className":4753},[2605,4754],"op-limits",[417,4756,4758,4819],{"className":4757},[678,679],[417,4759,4761,4816],{"className":4760},[683],[417,4762,4765,4790,4801],{"className":4763,"style":4764},[687],"height:1.6514em;",[417,4766,4768,4772],{"style":4767},"top:-1.9em;margin-left:0em;",[417,4769],{"className":4770,"style":4771},[695],"height:3.05em;",[417,4773,4775],{"className":4774},[700,701,702,703],[417,4776,4778,4781,4784,4787],{"className":4777},[483,703],[417,4779,4658],{"className":4780},[483,586,703],[417,4782,1120],{"className":4783},[1157,703],[417,4785,2397],{"className":4786},[483,703],[417,4788,4665],{"className":4789},[483,703],[417,4791,4793,4796],{"style":4792},"top:-3.05em;",[417,4794],{"className":4795,"style":4771},[695],[417,4797,4798],{},[417,4799,4653],{"className":4800},[2605,2606,2607],[417,4802,4804,4807],{"style":4803},"top:-4.3em;margin-left:0em;",[417,4805],{"className":4806,"style":4771},[695],[417,4808,4810],{"className":4809},[700,701,702,703],[417,4811,4813],{"className":4812},[483,703],[417,4814,4665],{"className":4815},[483,703],[417,4817,716],{"className":4818},[715],[417,4820,4822],{"className":4821},[683],[417,4823,4826],{"className":4824,"style":4825},[687],"height:1.3083em;",[417,4827],{},[417,4829],{"className":4830,"style":492},[491],[417,4832,4834,4837],{"className":4833},[483],[417,4835,4635],{"className":4836},[483],[417,4838,4840],{"className":4839},[674],[417,4841,4843,4863],{"className":4842},[678,679],[417,4844,4846,4860],{"className":4845},[683],[417,4847,4849],{"className":4848,"style":688},[687],[417,4850,4851,4854],{"style":878},[417,4852],{"className":4853,"style":696},[695],[417,4855,4857],{"className":4856},[700,701,702,703],[417,4858,4658],{"className":4859},[483,586,703],[417,4861,716],{"className":4862},[715],[417,4864,4866],{"className":4865},[683],[417,4867,4869],{"className":4868,"style":977},[687],[417,4870],{},[417,4872,1399],{"className":4873},[1623],[417,4875,1057],{"className":4876},[483,586],[417,4878,1404],{"className":4879},[1630],[417,4881],{"className":4882,"style":492},[491],[417,4884,4886,4889],{"className":4885},[483],[417,4887,4686],{"className":4888},[483,586],[417,4890,4892],{"className":4891},[674],[417,4893,4895],{"className":4894},[678],[417,4896,4898],{"className":4897},[683],[417,4899,4902],{"className":4900,"style":4901},[687],"height:0.938em;",[417,4903,4904,4907],{"style":1551},[417,4905],{"className":4906,"style":696},[695],[417,4908,4910],{"className":4909},[700,701,702,703],[417,4911,4913,4917,4920,4923],{"className":4912},[483,703],[417,4914,4916],{"className":4915},[483,586,703],"in",[417,4918,1212],{"className":4919,"style":670},[483,586,703],[417,4921,4103],{"className":4922},[483,703],[417,4924,570],{"className":4925,"style":587},[483,586,703],[417,4927,2256],{"className":4928},[483],[324,4930,4931,4932,5031,5032,5093,5094,5122],{},"A field that is massless in five dimensions obeys ",[417,4933,4935,4962],{"className":4934},[420],[417,4936,4938],{"className":4937},[424],[426,4939,4940],{"xmlns":428},[430,4941,4942,4959],{},[433,4943,4944,4951,4953,4955,4957],{},[1374,4945,4946,4949],{"accent":442},[568,4947,4948],{"mathvariant":3085},"□",[440,4950,1381],{},[1406,4952,1408],{},[568,4954,4635],{"mathvariant":3085},[440,4956,1120],{},[436,4958,438],{},[463,4960,4961],{"encoding":465},"\\hat\\Box\\,\\Phi = 0",[417,4963,4965,5021],{"className":4964,"ariaHidden":442},[470],[417,4966,4968,4972,5006,5009,5012,5015,5018],{"className":4967},[474],[417,4969],{"className":4970,"style":4971},[478],"height:0.9384em;",[417,4973,4975],{"className":4974},[483,1503],[417,4976,4978],{"className":4977},[678],[417,4979,4981],{"className":4980},[683],[417,4982,4984,4993],{"className":4983,"style":4971},[687],[417,4985,4986,4989],{"style":1516},[417,4987],{"className":4988,"style":1520},[695],[417,4990,4948],{"className":4991},[483,4992],"amsrm",[417,4994,4996,4999],{"style":4995},"top:-3.244em;",[417,4997],{"className":4998,"style":1520},[695],[417,5000,5003],{"className":5001,"style":5002},[1532],"left:-0.25em;",[417,5004,1381],{"className":5005},[483],[417,5007],{"className":5008,"style":492},[491],[417,5010,4635],{"className":5011},[483],[417,5013],{"className":5014,"style":1153},[491],[417,5016,1120],{"className":5017},[1157],[417,5019],{"className":5020,"style":1153},[491],[417,5022,5024,5028],{"className":5023},[474],[417,5025],{"className":5026,"style":5027},[478],"height:0.6444em;",[417,5029,438],{"className":5030},[483],", where ",[417,5033,5035,5053],{"className":5034},[420],[417,5036,5038],{"className":5037},[424],[426,5039,5040],{"xmlns":428},[430,5041,5042,5050],{},[433,5043,5044],{},[1374,5045,5046,5048],{"accent":442},[568,5047,4948],{"mathvariant":3085},[440,5049,1381],{},[463,5051,5052],{"encoding":465},"\\hat\\Box",[417,5054,5056],{"className":5055,"ariaHidden":442},[470],[417,5057,5059,5062],{"className":5058},[474],[417,5060],{"className":5061,"style":4971},[478],[417,5063,5065],{"className":5064},[483,1503],[417,5066,5068],{"className":5067},[678],[417,5069,5071],{"className":5070},[683],[417,5072,5074,5082],{"className":5073,"style":4971},[687],[417,5075,5076,5079],{"style":1516},[417,5077],{"className":5078,"style":1520},[695],[417,5080,4948],{"className":5081},[483,4992],[417,5083,5084,5087],{"style":4995},[417,5085],{"className":5086,"style":1520},[695],[417,5088,5090],{"className":5089,"style":5002},[1532],[417,5091,1381],{"className":5092},[483]," is the five-dimensional wave operator. Splitting it into the four-dimensional part and the ",[417,5095,5097,5110],{"className":5096},[420],[417,5098,5100],{"className":5099},[424],[426,5101,5102],{"xmlns":428},[430,5103,5104,5108],{},[433,5105,5106],{},[568,5107,1212],{},[463,5109,1212],{"encoding":465},[417,5111,5113],{"className":5112,"ariaHidden":442},[470],[417,5114,5116,5119],{"className":5115},[474],[417,5117],{"className":5118,"style":935},[478],[417,5120,1212],{"className":5121,"style":670},[483,586]," part gives, for each mode,",[417,5124,5126],{"className":5125},[1354],[417,5127,5129,5186],{"className":5128},[420],[417,5130,5132],{"className":5131},[424],[426,5133,5134],{"xmlns":428,"display":1363},[430,5135,5136,5183],{},[433,5137,5138,5166,5172,5174,5176,5178,5180],{},[433,5139,5140,5142,5148,5150,5164],{},[440,5141,1399],{"fence":442},[638,5143,5144,5146],{},[568,5145,4948],{"mathvariant":3085},[436,5147,822],{},[440,5149,2397],{},[2354,5151,5152,5158],{},[1053,5153,5154,5156],{},[568,5155,4658],{},[436,5157,451],{},[1053,5159,5160,5162],{},[568,5161,570],{},[436,5163,451],{},[440,5165,1404],{"fence":442},[638,5167,5168,5170],{},[568,5169,4635],{"mathvariant":3085},[568,5171,4658],{},[440,5173,1399],{"stretchy":1398},[568,5175,1057],{},[440,5177,1404],{"stretchy":1398},[440,5179,1120],{},[436,5181,5182],{},"0.",[463,5184,5185],{"encoding":465},"\\left(\\Box_4 - \\frac{n^2}{R^2}\\right)\\Phi_n(x) = 0 .",[417,5187,5189,5436],{"className":5188,"ariaHidden":442},[470],[417,5190,5192,5195,5375,5378,5418,5421,5424,5427,5430,5433],{"className":5191},[474],[417,5193],{"className":5194,"style":3176},[478],[417,5196,5198,5204,5244,5247,5250,5253,5369],{"className":5197},[523],[417,5199,5201],{"className":5200,"style":3407},[1623,3406],[417,5202,1399],{"className":5203},[1762,702],[417,5205,5207,5210],{"className":5206},[483],[417,5208,4948],{"className":5209},[483,4992],[417,5211,5213],{"className":5212},[674],[417,5214,5216,5236],{"className":5215},[678,679],[417,5217,5219,5233],{"className":5218},[683],[417,5220,5222],{"className":5221,"style":790},[687],[417,5223,5224,5227],{"style":878},[417,5225],{"className":5226,"style":696},[695],[417,5228,5230],{"className":5229},[700,701,702,703],[417,5231,822],{"className":5232},[483,703],[417,5234,716],{"className":5235},[715],[417,5237,5239],{"className":5238},[683],[417,5240,5242],{"className":5241,"style":977},[687],[417,5243],{},[417,5245],{"className":5246,"style":1323},[491],[417,5248,2397],{"className":5249},[1327],[417,5251],{"className":5252,"style":1323},[491],[417,5254,5256,5259,5366],{"className":5255},[483],[417,5257],{"className":5258},[1623,2492],[417,5260,5262],{"className":5261},[2354],[417,5263,5265,5358],{"className":5264},[678,679],[417,5266,5268,5355],{"className":5267},[683],[417,5269,5271,5310,5318],{"className":5270,"style":3441},[687],[417,5272,5273,5276],{"style":2508},[417,5274],{"className":5275,"style":1520},[695],[417,5277,5279],{"className":5278},[483],[417,5280,5282,5285],{"className":5281},[483],[417,5283,570],{"className":5284,"style":587},[483,586],[417,5286,5288],{"className":5287},[674],[417,5289,5291],{"className":5290},[678],[417,5292,5294],{"className":5293},[683],[417,5295,5298],{"className":5296,"style":5297},[687],"height:0.7401em;",[417,5299,5301,5304],{"style":5300},"top:-2.989em;margin-right:0.05em;",[417,5302],{"className":5303,"style":696},[695],[417,5305,5307],{"className":5306},[700,701,702,703],[417,5308,451],{"className":5309},[483,703],[417,5311,5312,5315],{"style":2563},[417,5313],{"className":5314,"style":1520},[695],[417,5316],{"className":5317,"style":2571},[2570],[417,5319,5320,5323],{"style":2574},[417,5321],{"className":5322,"style":1520},[695],[417,5324,5326],{"className":5325},[483],[417,5327,5329,5332],{"className":5328},[483],[417,5330,4658],{"className":5331},[483,586],[417,5333,5335],{"className":5334},[674],[417,5336,5338],{"className":5337},[678],[417,5339,5341],{"className":5340},[683],[417,5342,5344],{"className":5343,"style":1950},[687],[417,5345,5346,5349],{"style":1093},[417,5347],{"className":5348,"style":696},[695],[417,5350,5352],{"className":5351},[700,701,702,703],[417,5353,451],{"className":5354},[483,703],[417,5356,716],{"className":5357},[715],[417,5359,5361],{"className":5360},[683],[417,5362,5364],{"className":5363,"style":3536},[687],[417,5365],{},[417,5367],{"className":5368},[1630,2492],[417,5370,5372],{"className":5371,"style":3407},[1630,3406],[417,5373,1404],{"className":5374},[1762,702],[417,5376],{"className":5377,"style":492},[491],[417,5379,5381,5384],{"className":5380},[483],[417,5382,4635],{"className":5383},[483],[417,5385,5387],{"className":5386},[674],[417,5388,5390,5410],{"className":5389},[678,679],[417,5391,5393,5407],{"className":5392},[683],[417,5394,5396],{"className":5395,"style":688},[687],[417,5397,5398,5401],{"style":878},[417,5399],{"className":5400,"style":696},[695],[417,5402,5404],{"className":5403},[700,701,702,703],[417,5405,4658],{"className":5406},[483,586,703],[417,5408,716],{"className":5409},[715],[417,5411,5413],{"className":5412},[683],[417,5414,5416],{"className":5415,"style":977},[687],[417,5417],{},[417,5419,1399],{"className":5420},[1623],[417,5422,1057],{"className":5423},[483,586],[417,5425,1404],{"className":5426},[1630],[417,5428],{"className":5429,"style":1153},[491],[417,5431,1120],{"className":5432},[1157],[417,5434],{"className":5435,"style":1153},[491],[417,5437,5439,5442],{"className":5438},[474],[417,5440],{"className":5441,"style":5027},[478],[417,5443,5182],{"className":5444},[483],[324,5446,5447,5448,5451],{},"This is the wave equation of a particle with mass. So one five-dimensional field becomes an infinite family of four-dimensional particles, the ",[343,5449,5450],{},"Kaluza–Klein tower",", with masses",[1348,5453,5455],{"label":5454},"Kaluza–Klein mass tower",[417,5456,5458],{"className":5457},[1354],[417,5459,5461,5529],{"className":5460},[420],[417,5462,5464],{"className":5463},[424],[426,5465,5466],{"xmlns":428,"display":1363},[430,5467,5468,5526],{},[433,5469,5470,5477,5479,5504,5506,5508,5510,5512,5514,5516,5518,5520,5522,5524],{},[638,5471,5472,5475],{},[568,5473,5474],{},"m",[568,5476,4658],{},[440,5478,1120],{},[2354,5480,5481,5495],{},[433,5482,5483,5486,5488,5490,5492],{},[568,5484,5485],{"mathvariant":3085},"∣",[568,5487,4658],{},[568,5489,5485],{"mathvariant":3085},[1406,5491,1408],{},[568,5493,5494],{"mathvariant":3085},"ℏ",[433,5496,5497,5499,5501],{},[568,5498,570],{},[1406,5500,1408],{},[568,5502,5503],{},"c",[440,5505,443],{"separator":442},[491,5507],{"width":3068},[568,5509,4658],{},[440,5511,1120],{},[436,5513,438],{},[440,5515,443],{"separator":442},[436,5517,446],{},[440,5519,443],{"separator":442},[436,5521,451],{},[440,5523,443],{"separator":442},[440,5525,461],{},[463,5527,5528],{"encoding":465},"m_n = \\frac{|n|\\,\\hbar}{R\\,c}, \\qquad n = 0, 1, 2, \\dots",[417,5530,5532,5588,5697],{"className":5531,"ariaHidden":442},[470],[417,5533,5535,5539,5579,5582,5585],{"className":5534},[474],[417,5536],{"className":5537,"style":5538},[478],"height:0.5806em;vertical-align:-0.15em;",[417,5540,5542,5545],{"className":5541},[483],[417,5543,5474],{"className":5544},[483,586],[417,5546,5548],{"className":5547},[674],[417,5549,5551,5571],{"className":5550},[678,679],[417,5552,5554,5568],{"className":5553},[683],[417,5555,5557],{"className":5556,"style":688},[687],[417,5558,5559,5562],{"style":878},[417,5560],{"className":5561,"style":696},[695],[417,5563,5565],{"className":5564},[700,701,702,703],[417,5566,4658],{"className":5567},[483,586,703],[417,5569,716],{"className":5570},[715],[417,5572,5574],{"className":5573},[683],[417,5575,5577],{"className":5576,"style":977},[687],[417,5578],{},[417,5580],{"className":5581,"style":1153},[491],[417,5583,1120],{"className":5584},[1157],[417,5586],{"className":5587,"style":1153},[491],[417,5589,5591,5595,5676,5679,5682,5685,5688,5691,5694],{"className":5590},[474],[417,5592],{"className":5593,"style":5594},[478],"height:2.113em;vertical-align:-0.686em;",[417,5596,5598,5601,5673],{"className":5597},[483],[417,5599],{"className":5600},[1623,2492],[417,5602,5604],{"className":5603},[2354],[417,5605,5607,5665],{"className":5606},[678,679],[417,5608,5610,5662],{"className":5609},[683],[417,5611,5614,5631,5639],{"className":5612,"style":5613},[687],"height:1.427em;",[417,5615,5616,5619],{"style":2508},[417,5617],{"className":5618,"style":1520},[695],[417,5620,5622,5625,5628],{"className":5621},[483],[417,5623,570],{"className":5624,"style":587},[483,586],[417,5626],{"className":5627,"style":492},[491],[417,5629,5503],{"className":5630},[483,586],[417,5632,5633,5636],{"style":2563},[417,5634],{"className":5635,"style":1520},[695],[417,5637],{"className":5638,"style":2571},[2570],[417,5640,5641,5644],{"style":2574},[417,5642],{"className":5643,"style":1520},[695],[417,5645,5647,5650,5653,5656,5659],{"className":5646},[483],[417,5648,5485],{"className":5649},[483],[417,5651,4658],{"className":5652},[483,586],[417,5654,5485],{"className":5655},[483],[417,5657],{"className":5658,"style":492},[491],[417,5660,5494],{"className":5661},[483],[417,5663,716],{"className":5664},[715],[417,5666,5668],{"className":5667},[683],[417,5669,5671],{"className":5670,"style":3536},[687],[417,5672],{},[417,5674],{"className":5675},[1630,2492],[417,5677,443],{"className":5678},[487],[417,5680],{"className":5681,"style":3637},[491],[417,5683],{"className":5684,"style":492},[491],[417,5686,4658],{"className":5687},[483,586],[417,5689],{"className":5690,"style":1153},[491],[417,5692,1120],{"className":5693},[1157],[417,5695],{"className":5696,"style":1153},[491],[417,5698,5700,5703,5706,5709,5712,5715,5718,5721,5724,5727,5730],{"className":5699},[474],[417,5701],{"className":5702,"style":479},[478],[417,5704,438],{"className":5705},[483],[417,5707,443],{"className":5708},[487],[417,5710],{"className":5711,"style":492},[491],[417,5713,446],{"className":5714},[483],[417,5716,443],{"className":5717},[487],[417,5719],{"className":5720,"style":492},[491],[417,5722,451],{"className":5723},[483],[417,5725,443],{"className":5726},[487],[417,5728],{"className":5729,"style":492},[491],[417,5731,461],{"className":5732},[523],[324,5734,5735,5736,5810,5811,5921,5922,5973,5974,6011,6012],{},"In natural units (",[417,5737,5739,5761],{"className":5738},[420],[417,5740,5742],{"className":5741},[424],[426,5743,5744],{"xmlns":428},[430,5745,5746,5758],{},[433,5747,5748,5750,5752,5754,5756],{},[568,5749,5494],{"mathvariant":3085},[440,5751,1120],{},[568,5753,5503],{},[440,5755,1120],{},[436,5757,446],{},[463,5759,5760],{"encoding":465},"\\hbar = c = 1",[417,5762,5764,5783,5801],{"className":5763,"ariaHidden":442},[470],[417,5765,5767,5771,5774,5777,5780],{"className":5766},[474],[417,5768],{"className":5769,"style":5770},[478],"height:0.6889em;",[417,5772,5494],{"className":5773},[483],[417,5775],{"className":5776,"style":1153},[491],[417,5778,1120],{"className":5779},[1157],[417,5781],{"className":5782,"style":1153},[491],[417,5784,5786,5789,5792,5795,5798],{"className":5785},[474],[417,5787],{"className":5788,"style":2222},[478],[417,5790,5503],{"className":5791},[483,586],[417,5793],{"className":5794,"style":1153},[491],[417,5796,1120],{"className":5797},[1157],[417,5799],{"className":5800,"style":1153},[491],[417,5802,5804,5807],{"className":5803},[474],[417,5805],{"className":5806,"style":5027},[478],[417,5808,446],{"className":5809},[483],") this is simply ",[417,5812,5814,5844],{"className":5813},[420],[417,5815,5817],{"className":5816},[424],[426,5818,5819],{"xmlns":428},[430,5820,5821,5841],{},[433,5822,5823,5829,5831,5833,5835,5837,5839],{},[638,5824,5825,5827],{},[568,5826,5474],{},[568,5828,4658],{},[440,5830,1120],{},[568,5832,5485],{"mathvariant":3085},[568,5834,4658],{},[568,5836,5485],{"mathvariant":3085},[568,5838,4103],{"mathvariant":3085},[568,5840,570],{},[463,5842,5843],{"encoding":465},"m_n = |n|\u002FR",[417,5845,5847,5902],{"className":5846,"ariaHidden":442},[470],[417,5848,5850,5853,5893,5896,5899],{"className":5849},[474],[417,5851],{"className":5852,"style":5538},[478],[417,5854,5856,5859],{"className":5855},[483],[417,5857,5474],{"className":5858},[483,586],[417,5860,5862],{"className":5861},[674],[417,5863,5865,5885],{"className":5864},[678,679],[417,5866,5868,5882],{"className":5867},[683],[417,5869,5871],{"className":5870,"style":688},[687],[417,5872,5873,5876],{"style":878},[417,5874],{"className":5875,"style":696},[695],[417,5877,5879],{"className":5878},[700,701,702,703],[417,5880,4658],{"className":5881},[483,586,703],[417,5883,716],{"className":5884},[715],[417,5886,5888],{"className":5887},[683],[417,5889,5891],{"className":5890,"style":977},[687],[417,5892],{},[417,5894],{"className":5895,"style":1153},[491],[417,5897,1120],{"className":5898},[1157],[417,5900],{"className":5901,"style":1153},[491],[417,5903,5905,5908,5911,5914,5918],{"className":5904},[474],[417,5906],{"className":5907,"style":4181},[478],[417,5909,5485],{"className":5910},[483],[417,5912,4658],{"className":5913},[483,586],[417,5915,5917],{"className":5916},[483],"∣\u002F",[417,5919,570],{"className":5920,"style":587},[483,586],". The ",[417,5923,5925,5943],{"className":5924},[420],[417,5926,5928],{"className":5927},[424],[426,5929,5930],{"xmlns":428},[430,5931,5932,5940],{},[433,5933,5934,5936,5938],{},[568,5935,4658],{},[440,5937,1120],{},[436,5939,438],{},[463,5941,5942],{"encoding":465},"n = 0",[417,5944,5946,5964],{"className":5945,"ariaHidden":442},[470],[417,5947,5949,5952,5955,5958,5961],{"className":5948},[474],[417,5950],{"className":5951,"style":2222},[478],[417,5953,4658],{"className":5954},[483,586],[417,5956],{"className":5957,"style":1153},[491],[417,5959,1120],{"className":5960},[1157],[417,5962],{"className":5963,"style":1153},[491],[417,5965,5967,5970],{"className":5966},[474],[417,5968],{"className":5969,"style":5027},[478],[417,5971,438],{"className":5972},[483]," mode is massless and is the field we see at low energy. The others are heavier and heavier copies, spaced by ",[417,5975,5977,5995],{"className":5976},[420],[417,5978,5980],{"className":5979},[424],[426,5981,5982],{"xmlns":428},[430,5983,5984,5992],{},[433,5985,5986,5988,5990],{},[436,5987,446],{},[568,5989,4103],{"mathvariant":3085},[568,5991,570],{},[463,5993,5994],{"encoding":465},"1\u002FR",[417,5996,5998],{"className":5997,"ariaHidden":442},[470],[417,5999,6001,6004,6008],{"className":6000},[474],[417,6002],{"className":6003,"style":4181},[478],[417,6005,6007],{"className":6006},[483],"1\u002F",[417,6009,570],{"className":6010,"style":587},[483,586],". A small circle means widely spaced, very heavy copies — which is why they would be invisible at everyday energies. ",[348,6013,6015],{"className":6014},[351],"[4,5]",[366,6017,6019],{"id":6018},"charge-quantization","Charge quantization",[324,6021,6022,6023,6051,6052,6121,6122,6150],{},"The same integer ",[417,6024,6026,6039],{"className":6025},[420],[417,6027,6029],{"className":6028},[424],[426,6030,6031],{"xmlns":428},[430,6032,6033,6037],{},[433,6034,6035],{},[568,6036,4658],{},[463,6038,4658],{"encoding":465},[417,6040,6042],{"className":6041,"ariaHidden":442},[470],[417,6043,6045,6048],{"className":6044},[474],[417,6046],{"className":6047,"style":2222},[478],[417,6049,4658],{"className":6050},[483,586]," also controls how strongly the mode couples to ",[417,6053,6055,6072],{"className":6054},[420],[417,6056,6058],{"className":6057},[424],[426,6059,6060],{"xmlns":428},[430,6061,6062,6070],{},[433,6063,6064],{},[638,6065,6066,6068],{},[568,6067,842],{},[568,6069,647],{},[463,6071,847],{"encoding":465},[417,6073,6075],{"className":6074,"ariaHidden":442},[470],[417,6076,6078,6081],{"className":6077},[474],[417,6079],{"className":6080,"style":857},[478],[417,6082,6084,6087],{"className":6083},[483],[417,6085,842],{"className":6086},[483,586],[417,6088,6090],{"className":6089},[674],[417,6091,6093,6113],{"className":6092},[678,679],[417,6094,6096,6110],{"className":6095},[683],[417,6097,6099],{"className":6098,"style":688},[687],[417,6100,6101,6104],{"style":878},[417,6102],{"className":6103,"style":696},[695],[417,6105,6107],{"className":6106},[700,701,702,703],[417,6108,647],{"className":6109},[483,586,703],[417,6111,716],{"className":6112},[715],[417,6114,6116],{"className":6115},[683],[417,6117,6119],{"className":6118,"style":723},[687],[417,6120],{},". With the normalization in which the Maxwell term has its standard form, the charge of mode ",[417,6123,6125,6138],{"className":6124},[420],[417,6126,6128],{"className":6127},[424],[426,6129,6130],{"xmlns":428},[430,6131,6132,6136],{},[433,6133,6134],{},[568,6135,4658],{},[463,6137,4658],{"encoding":465},[417,6139,6141],{"className":6140,"ariaHidden":442},[470],[417,6142,6144,6147],{"className":6143},[474],[417,6145],{"className":6146,"style":2222},[478],[417,6148,4658],{"className":6149},[483,586]," is",[417,6152,6154],{"className":6153},[1354],[417,6155,6157,6208],{"className":6156},[420],[417,6158,6160],{"className":6159},[424],[426,6161,6162],{"xmlns":428,"display":1363},[430,6163,6164,6205],{},[433,6165,6166,6173,6175,6177,6179],{},[638,6167,6168,6171],{},[568,6169,6170],{},"q",[568,6172,4658],{},[440,6174,1120],{},[568,6176,4658],{},[1406,6178,1408],{},[2354,6180,6181,6197],{},[433,6182,6183,6193,6195],{},[2391,6184,6185],{},[433,6186,6187,6189,6191],{},[436,6188,2362],{},[568,6190,1283],{},[568,6192,2241],{},[1406,6194,2406],{},[568,6196,5494],{"mathvariant":3085},[433,6198,6199,6201,6203],{},[568,6200,5503],{},[1406,6202,1408],{},[568,6204,570],{},[463,6206,6207],{"encoding":465},"q_n = n\\,\\frac{\\sqrt{16\\pi G}\\;\\hbar}{c\\,R}",[417,6209,6211,6266],{"className":6210,"ariaHidden":442},[470],[417,6212,6214,6217,6257,6260,6263],{"className":6213},[474],[417,6215],{"className":6216,"style":935},[478],[417,6218,6220,6223],{"className":6219},[483],[417,6221,6170],{"className":6222,"style":670},[483,586],[417,6224,6226],{"className":6225},[674],[417,6227,6229,6249],{"className":6228},[678,679],[417,6230,6232,6246],{"className":6231},[683],[417,6233,6235],{"className":6234,"style":688},[687],[417,6236,6237,6240],{"style":691},[417,6238],{"className":6239,"style":696},[695],[417,6241,6243],{"className":6242},[700,701,702,703],[417,6244,4658],{"className":6245},[483,586,703],[417,6247,716],{"className":6248},[715],[417,6250,6252],{"className":6251},[683],[417,6253,6255],{"className":6254,"style":977},[687],[417,6256],{},[417,6258],{"className":6259,"style":1153},[491],[417,6261,1120],{"className":6262},[1157],[417,6264],{"className":6265,"style":1153},[491],[417,6267,6269,6273,6276,6279],{"className":6268},[474],[417,6270],{"className":6271,"style":6272},[478],"height:2.2897em;vertical-align:-0.686em;",[417,6274,4658],{"className":6275},[483,586],[417,6277],{"className":6278,"style":492},[491],[417,6280,6282,6285,6404],{"className":6281},[483],[417,6283],{"className":6284},[1623,2492],[417,6286,6288],{"className":6287},[2354],[417,6289,6291,6396],{"className":6290},[678,679],[417,6292,6294,6393],{"className":6293},[683],[417,6295,6298,6315,6323],{"className":6296,"style":6297},[687],"height:1.6037em;",[417,6299,6300,6303],{"style":2508},[417,6301],{"className":6302,"style":1520},[695],[417,6304,6306,6309,6312],{"className":6305},[483],[417,6307,5503],{"className":6308},[483,586],[417,6310],{"className":6311,"style":492},[491],[417,6313,570],{"className":6314,"style":587},[483,586],[417,6316,6317,6320],{"style":2563},[417,6318],{"className":6319,"style":1520},[695],[417,6321],{"className":6322,"style":2571},[2570],[417,6324,6325,6328],{"style":2574},[417,6326],{"className":6327,"style":1520},[695],[417,6329,6331,6387,6390],{"className":6330},[483],[417,6332,6334],{"className":6333},[483,2659],[417,6335,6337,6378],{"className":6336},[678,679],[417,6338,6340,6375],{"className":6339},[683],[417,6341,6344,6362],{"className":6342,"style":6343},[687],"height:0.9267em;",[417,6345,6347,6350],{"className":6346,"style":1516},[2673],[417,6348],{"className":6349,"style":1520},[695],[417,6351,6353,6356,6359],{"className":6352,"style":3355},[483],[417,6354,2362],{"className":6355},[483],[417,6357,1283],{"className":6358,"style":670},[483,586],[417,6360,2241],{"className":6361},[483,586],[417,6363,6365,6368],{"style":6364},"top:-2.8867em;",[417,6366],{"className":6367,"style":1520},[695],[417,6369,6371],{"className":6370,"style":3371},[2739],[2742,6372,6373],{"xmlns":2744,"width":2745,"height":3374,"viewBox":3375,"preserveAspectRatio":2748},[2750,6374],{"d":3378},[417,6376,716],{"className":6377},[715],[417,6379,6381],{"className":6380},[683],[417,6382,6385],{"className":6383,"style":6384},[687],"height:0.1133em;",[417,6386],{},[417,6388],{"className":6389,"style":1153},[491],[417,6391,5494],{"className":6392},[483],[417,6394,716],{"className":6395},[715],[417,6397,6399],{"className":6398},[683],[417,6400,6402],{"className":6401,"style":3536},[687],[417,6403],{},[417,6405],{"className":6406},[1630,2492],[324,6408,6409,6410,6438,6439,6467,6468,6541,6542],{},"(in Gaussian units), where ",[417,6411,6413,6426],{"className":6412},[420],[417,6414,6416],{"className":6415},[424],[426,6417,6418],{"xmlns":428},[430,6419,6420,6424],{},[433,6421,6422],{},[568,6423,2241],{},[463,6425,2241],{"encoding":465},[417,6427,6429],{"className":6428,"ariaHidden":442},[470],[417,6430,6432,6435],{"className":6431},[474],[417,6433],{"className":6434,"style":582},[478],[417,6436,2241],{"className":6437},[483,586]," is the ordinary Newton constant. Charge comes in integer multiples of a basic unit set by the size of the circle. Klein turned this around: if the basic unit is the electron's charge ",[417,6440,6442,6455],{"className":6441},[420],[417,6443,6445],{"className":6444},[424],[426,6446,6447],{"xmlns":428},[430,6448,6449,6453],{},[433,6450,6451],{},[568,6452,4686],{},[463,6454,4686],{"encoding":465},[417,6456,6458],{"className":6457,"ariaHidden":442},[470],[417,6459,6461,6464],{"className":6460},[474],[417,6462],{"className":6463,"style":2222},[478],[417,6465,4686],{"className":6466},[483,586],", the circle's circumference must be about ",[417,6469,6471,6494],{"className":6470},[420],[417,6472,6474],{"className":6473},[424],[426,6475,6476],{"xmlns":428},[430,6477,6478,6491],{},[433,6479,6480],{},[1053,6481,6482,6484],{},[436,6483,729],{},[433,6485,6486,6488],{},[440,6487,2397],{},[436,6489,6490],{},"30",[463,6492,6493],{"encoding":465},"10^{-30}",[417,6495,6497],{"className":6496,"ariaHidden":442},[470],[417,6498,6500,6503,6506],{"className":6499},[474],[417,6501],{"className":6502,"style":1950},[478],[417,6504,446],{"className":6505},[483],[417,6507,6509,6512],{"className":6508},[483],[417,6510,438],{"className":6511},[483],[417,6513,6515],{"className":6514},[674],[417,6516,6518],{"className":6517},[678],[417,6519,6521],{"className":6520},[683],[417,6522,6524],{"className":6523,"style":1950},[687],[417,6525,6526,6529],{"style":1093},[417,6527],{"className":6528,"style":696},[695],[417,6530,6532],{"className":6531},[700,701,702,703],[417,6533,6535,6538],{"className":6534},[483,703],[417,6536,2397],{"className":6537},[483,703],[417,6539,6490],{"className":6540},[483,703]," cm, some seventeen orders of magnitude smaller than a proton. ",[348,6543,6545],{"className":6544},[351],"[2,3,4]",[361,6547,6549],{"id":6548},"historical-development","Historical Development",[324,6551,6552,6555,6556,6628,6629],{},[343,6553,6554],{},"Kaluza (1921)."," Kaluza's paper, \"On the Unification Problem in Physics,\" was presented to the Prussian Academy of Sciences by Einstein and published in its proceedings in 1921. It showed that five-dimensional gravity with the cylinder condition contains both Einstein's and Maxwell's equations. Kaluza treated the fifth dimension as a formal device and fixed ",[417,6557,6559,6576],{"className":6558},[420],[417,6560,6562],{"className":6561},[424],[426,6563,6564],{"xmlns":428},[430,6565,6566,6574],{},[433,6567,6568],{},[638,6569,6570,6572],{},[568,6571,642],{},[436,6573,922],{},[463,6575,925],{"encoding":465},[417,6577,6579],{"className":6578,"ariaHidden":442},[470],[417,6580,6582,6585],{"className":6581},[474],[417,6583],{"className":6584,"style":935},[478],[417,6586,6588,6591],{"className":6587},[483],[417,6589,642],{"className":6590,"style":670},[483,586],[417,6592,6594],{"className":6593},[674],[417,6595,6597,6620],{"className":6596},[678,679],[417,6598,6600,6617],{"className":6599},[683],[417,6601,6603],{"className":6602,"style":790},[687],[417,6604,6605,6608],{"style":691},[417,6606],{"className":6607,"style":696},[695],[417,6609,6611],{"className":6610},[700,701,702,703],[417,6612,6614],{"className":6613},[483,703],[417,6615,922],{"className":6616},[483,703],[417,6618,716],{"className":6619},[715],[417,6621,6623],{"className":6622},[683],[417,6624,6626],{"className":6625,"style":977},[687],[417,6627],{}," to a constant. ",[348,6630,4275],{"className":6631},[351],[324,6633,6634,6637,6638,6666,6667],{},[343,6635,6636],{},"Klein (1926)."," Oskar Klein, working in the new quantum mechanics, gave the fifth dimension a physical form: a tiny circle. He showed that quantum waves on the circle explain why charge comes in discrete units, and estimated the circle's size from the value of ",[417,6639,6641,6654],{"className":6640},[420],[417,6642,6644],{"className":6643},[424],[426,6645,6646],{"xmlns":428},[430,6647,6648,6652],{},[433,6649,6650],{},[568,6651,4686],{},[463,6653,4686],{"encoding":465},[417,6655,6657],{"className":6656,"ariaHidden":442},[470],[417,6658,6660,6663],{"className":6659},[474],[417,6661],{"className":6662,"style":2222},[478],[417,6664,4686],{"className":6665},[483,586],". ",[348,6668,532],{"className":6669},[351],[324,6671,6672,6675,6676,6748,6749,6890,6891],{},[343,6673,6674],{},"The scalar is noticed (1940s–1950s)."," Setting ",[417,6677,6679,6696],{"className":6678},[420],[417,6680,6682],{"className":6681},[424],[426,6683,6684],{"xmlns":428},[430,6685,6686,6694],{},[433,6687,6688],{},[638,6689,6690,6692],{},[568,6691,642],{},[436,6693,922],{},[463,6695,925],{"encoding":465},[417,6697,6699],{"className":6698,"ariaHidden":442},[470],[417,6700,6702,6705],{"className":6701},[474],[417,6703],{"className":6704,"style":935},[478],[417,6706,6708,6711],{"className":6707},[483],[417,6709,642],{"className":6710,"style":670},[483,586],[417,6712,6714],{"className":6713},[674],[417,6715,6717,6740],{"className":6716},[678,679],[417,6718,6720,6737],{"className":6719},[683],[417,6721,6723],{"className":6722,"style":790},[687],[417,6724,6725,6728],{"style":691},[417,6726],{"className":6727,"style":696},[695],[417,6729,6731],{"className":6730},[700,701,702,703],[417,6732,6734],{"className":6733},[483,703],[417,6735,922],{"className":6736},[483,703],[417,6738,716],{"className":6739},[715],[417,6741,6743],{"className":6742},[683],[417,6744,6746],{"className":6745,"style":977},[687],[417,6747],{}," to a constant turned out to be inconsistent with the full field equations unless ",[417,6750,6752,6788],{"className":6751},[420],[417,6753,6755],{"className":6754},[424],[426,6756,6757],{"xmlns":428},[430,6758,6759,6785],{},[433,6760,6761,6771,6781,6783],{},[638,6762,6763,6765],{},[568,6764,3045],{},[433,6766,6767,6769],{},[568,6768,647],{},[568,6770,650],{},[1053,6772,6773,6775],{},[568,6774,3045],{},[433,6776,6777,6779],{},[568,6778,647],{},[568,6780,650],{},[440,6782,1120],{},[436,6784,438],{},[463,6786,6787],{"encoding":465},"F_{\\mu\\nu}F^{\\mu\\nu} = 0",[417,6789,6791,6881],{"className":6790,"ariaHidden":442},[470],[417,6792,6794,6797,6840,6872,6875,6878],{"className":6793},[474],[417,6795],{"className":6796,"style":857},[478],[417,6798,6800,6803],{"className":6799},[483],[417,6801,3045],{"className":6802,"style":3551},[483,586],[417,6804,6806],{"className":6805},[674],[417,6807,6809,6832],{"className":6808},[678,679],[417,6810,6812,6829],{"className":6811},[683],[417,6813,6815],{"className":6814,"style":688},[687],[417,6816,6817,6820],{"style":3566},[417,6818],{"className":6819,"style":696},[695],[417,6821,6823],{"className":6822},[700,701,702,703],[417,6824,6826],{"className":6825},[483,703],[417,6827,711],{"className":6828,"style":710},[483,586,703],[417,6830,716],{"className":6831},[715],[417,6833,6835],{"className":6834},[683],[417,6836,6838],{"className":6837,"style":723},[687],[417,6839],{},[417,6841,6843,6846],{"className":6842},[483],[417,6844,3045],{"className":6845,"style":3551},[483,586],[417,6847,6849],{"className":6848},[674],[417,6850,6852],{"className":6851},[678],[417,6853,6855],{"className":6854},[683],[417,6856,6858],{"className":6857,"style":1072},[687],[417,6859,6860,6863],{"style":1093},[417,6861],{"className":6862,"style":696},[695],[417,6864,6866],{"className":6865},[700,701,702,703],[417,6867,6869],{"className":6868},[483,703],[417,6870,711],{"className":6871,"style":710},[483,586,703],[417,6873],{"className":6874,"style":1153},[491],[417,6876,1120],{"className":6877},[1157],[417,6879],{"className":6880,"style":1153},[491],[417,6882,6884,6887],{"className":6883},[474],[417,6885],{"className":6886,"style":5027},[478],[417,6888,438],{"className":6889},[483]," everywhere. Jordan, Thiry and others restored the scalar as a genuine dynamical field, which is why it is sometimes called the Jordan–Thiry scalar. ",[348,6892,6894],{"className":6893},[351],"[4]",[324,6896,6897,6900,6901,7032,7033],{},[343,6898,6899],{},"Non-abelian generalizations (1960s–1980s)."," Replacing the circle with a more complicated compact space yields more complicated gauge fields. Compactifying on a space whose symmetries form a non-abelian group produces Yang–Mills fields of the type used in the Standard Model. In 1981 Edward Witten showed that eleven is the minimum total number of dimensions in which the symmetry group ",[417,6902,6904,6950],{"className":6903},[420],[417,6905,6907],{"className":6906},[424],[426,6908,6909],{"xmlns":428},[430,6910,6911,6947],{},[433,6912,6913,6915,6918,6920,6922,6924,6927,6929,6931,6933,6935,6937,6939,6941,6943,6945],{},[568,6914,2348],{},[568,6916,6917],{},"U",[440,6919,1399],{"stretchy":1398},[436,6921,456],{},[440,6923,1404],{"stretchy":1398},[440,6925,6926],{},"×",[568,6928,2348],{},[568,6930,6917],{},[440,6932,1399],{"stretchy":1398},[436,6934,451],{},[440,6936,1404],{"stretchy":1398},[440,6938,6926],{},[568,6940,6917],{},[440,6942,1399],{"stretchy":1398},[436,6944,446],{},[440,6946,1404],{"stretchy":1398},[463,6948,6949],{"encoding":465},"SU(3)\\times SU(2)\\times U(1)",[417,6951,6953,6984,7014],{"className":6952,"ariaHidden":442},[470],[417,6954,6956,6959,6962,6966,6969,6972,6975,6978,6981],{"className":6955},[474],[417,6957],{"className":6958,"style":4181},[478],[417,6960,2348],{"className":6961,"style":2434},[483,586],[417,6963,6917],{"className":6964,"style":6965},[483,586],"margin-right:0.109em;",[417,6967,1399],{"className":6968},[1623],[417,6970,456],{"className":6971},[483],[417,6973,1404],{"className":6974},[1630],[417,6976],{"className":6977,"style":1323},[491],[417,6979,6926],{"className":6980},[1327],[417,6982],{"className":6983,"style":1323},[491],[417,6985,6987,6990,6993,6996,6999,7002,7005,7008,7011],{"className":6986},[474],[417,6988],{"className":6989,"style":4181},[478],[417,6991,2348],{"className":6992,"style":2434},[483,586],[417,6994,6917],{"className":6995,"style":6965},[483,586],[417,6997,1399],{"className":6998},[1623],[417,7000,451],{"className":7001},[483],[417,7003,1404],{"className":7004},[1630],[417,7006],{"className":7007,"style":1323},[491],[417,7009,6926],{"className":7010},[1327],[417,7012],{"className":7013,"style":1323},[491],[417,7015,7017,7020,7023,7026,7029],{"className":7016},[474],[417,7018],{"className":7019,"style":4181},[478],[417,7021,6917],{"className":7022,"style":6965},[483,586],[417,7024,1399],{"className":7025},[1623],[417,7027,446],{"className":7028},[483],[417,7030,1404],{"className":7031},[1630]," of the Standard Model can arise this way — which intriguingly matches the maximum dimension allowed for supergravity. ",[348,7034,7036],{"className":7035},[351],"[6]",[324,7038,7039,7042,7043,7047,7048,7050],{},[343,7040,7041],{},"Strings and M-theory (1980s–1990s)."," Superstring theory is consistent in ten dimensions. To describe a four-dimensional world, six dimensions must be compactified, and Kaluza–Klein reduction is the standard tool for working out what a four-dimensional observer sees. In 1995 Witten argued that one type of ten-dimensional string theory (type IIA) is itself the Kaluza–Klein reduction of an eleven-dimensional theory, now called M-theory, on a circle whose radius grows with the string coupling. The Kaluza–Klein modes of the eleventh dimension correspond to known objects in the string theory. ",[348,7044,7046],{"className":7045},[351],"[7]"," (For the higher-dimensional and extra-time proposals built on this, see ",[336,7049,180],{"href":179},".)",[324,7052,7053,7056,7057],{},[343,7054,7055],{},"Large and warped extra dimensions (1998–1999)."," Arkani-Hamed, Dimopoulos and Dvali (ADD) proposed that extra dimensions could be as large as a millimetre if only gravity travels through them, while ordinary matter is confined to a four-dimensional \"brane\". Randall and Sundrum proposed a single warped extra dimension. Both ideas were attempts to explain why gravity is so much weaker than the other forces, and both predicted Kaluza–Klein gravitons that could be seen at colliders. ",[348,7058,7060],{"className":7059},[351],"[8,9]",[361,7062,7064],{"id":7063},"evidence","Evidence",[324,7066,7067],{},"It helps to separate what is mathematically established from what is known about nature.",[366,7069,7071],{"id":7070},"established-mathematics","Established mathematics",[324,7073,7074,7075,7078,7079,7082,7083,7191],{},"That five-dimensional General Relativity on a circle reduces to four-dimensional gravity, electromagnetism and a scalar is a ",[343,7076,7077],{},"mathematical fact",". It is a derivation, not a hypothesis, and it is reproduced in standard reviews. ",[348,7080,6894],{"className":7081},[351]," The same is true of the tower formula ",[417,7084,7086,7115],{"className":7085},[420],[417,7087,7089],{"className":7088},[424],[426,7090,7091],{"xmlns":428},[430,7092,7093,7113],{},[433,7094,7095,7101,7103,7105,7107,7109,7111],{},[638,7096,7097,7099],{},[568,7098,5474],{},[568,7100,4658],{},[440,7102,1120],{},[568,7104,5485],{"mathvariant":3085},[568,7106,4658],{},[568,7108,5485],{"mathvariant":3085},[568,7110,4103],{"mathvariant":3085},[568,7112,570],{},[463,7114,5843],{"encoding":465},[417,7116,7118,7173],{"className":7117,"ariaHidden":442},[470],[417,7119,7121,7124,7164,7167,7170],{"className":7120},[474],[417,7122],{"className":7123,"style":5538},[478],[417,7125,7127,7130],{"className":7126},[483],[417,7128,5474],{"className":7129},[483,586],[417,7131,7133],{"className":7132},[674],[417,7134,7136,7156],{"className":7135},[678,679],[417,7137,7139,7153],{"className":7138},[683],[417,7140,7142],{"className":7141,"style":688},[687],[417,7143,7144,7147],{"style":878},[417,7145],{"className":7146,"style":696},[695],[417,7148,7150],{"className":7149},[700,701,702,703],[417,7151,4658],{"className":7152},[483,586,703],[417,7154,716],{"className":7155},[715],[417,7157,7159],{"className":7158},[683],[417,7160,7162],{"className":7161,"style":977},[687],[417,7163],{},[417,7165],{"className":7166,"style":1153},[491],[417,7168,1120],{"className":7169},[1157],[417,7171],{"className":7172,"style":1153},[491],[417,7174,7176,7179,7182,7185,7188],{"className":7175},[474],[417,7177],{"className":7178,"style":4181},[478],[417,7180,5485],{"className":7181},[483],[417,7183,4658],{"className":7184},[483,586],[417,7186,5917],{"className":7187},[483],[417,7189,570],{"className":7190,"style":587},[483,586]," for fields on a circle.",[366,7193,7195],{"id":7194},"what-is-known-about-nature","What is known about nature",[324,7197,7198,7201],{},[343,7199,7200],{},"No Kaluza–Klein particle has ever been observed, and no extra dimension has been detected."," Every experiment so far is consistent with a world of three space dimensions and one time, down to the smallest distances tested. The results are limits.",[324,7203,7204,7207,7208,7236,7237,7265],{},[343,7205,7206],{},"Short-range gravity tests."," If gravity leaks into extra dimensions of size ",[417,7209,7211,7224],{"className":7210},[420],[417,7212,7214],{"className":7213},[424],[426,7215,7216],{"xmlns":428},[430,7217,7218,7222],{},[433,7219,7220],{},[568,7221,570],{},[463,7223,570],{"encoding":465},[417,7225,7227],{"className":7226,"ariaHidden":442},[470],[417,7228,7230,7233],{"className":7229},[474],[417,7231],{"className":7232,"style":582},[478],[417,7234,570],{"className":7235,"style":587},[483,586],", Newton's inverse-square law should change at distances shorter than about ",[417,7238,7240,7253],{"className":7239},[420],[417,7241,7243],{"className":7242},[424],[426,7244,7245],{"xmlns":428},[430,7246,7247,7251],{},[433,7248,7249],{},[568,7250,570],{},[463,7252,570],{"encoding":465},[417,7254,7256],{"className":7255,"ariaHidden":442},[470],[417,7257,7259,7262],{"className":7258},[474],[417,7260],{"className":7261,"style":582},[478],[417,7263,570],{"className":7264,"style":587},[483,586],". Such deviations are usually written as",[417,7267,7269],{"className":7268},[1354],[417,7270,7272,7351],{"className":7271},[420],[417,7273,7275],{"className":7274},[424],[426,7276,7277],{"xmlns":428,"display":1363},[430,7278,7279,7348],{},[433,7280,7281,7284,7286,7289,7291,7293,7295,7317,7346],{},[568,7282,7283],{},"V",[440,7285,1399],{"stretchy":1398},[568,7287,7288],{},"r",[440,7290,1404],{"stretchy":1398},[440,7292,1120],{},[440,7294,2397],{},[2354,7296,7297,7315],{},[433,7298,7299,7301,7303,7309],{},[568,7300,2241],{},[1406,7302,1408],{},[638,7304,7305,7307],{},[568,7306,5474],{},[436,7308,446],{},[638,7310,7311,7313],{},[568,7312,5474],{},[436,7314,451],{},[568,7316,7288],{},[433,7318,7319,7321,7323,7325,7328,7330,7344],{},[440,7320,1399],{"fence":442},[436,7322,446],{},[440,7324,1278],{},[568,7326,7327],{},"α",[1406,7329,1408],{},[1053,7331,7332,7334],{},[568,7333,4686],{},[433,7335,7336,7338,7340,7342],{},[440,7337,2397],{},[568,7339,7288],{},[568,7341,4103],{"mathvariant":3085},[568,7343,4306],{},[440,7345,1404],{"fence":442},[440,7347,443],{"separator":442},[463,7349,7350],{"encoding":465},"V(r) = -\\frac{G\\,m_1 m_2}{r}\\left(1 + \\alpha\\, e^{-r\u002F\\lambda}\\right),",[417,7352,7354,7382],{"className":7353,"ariaHidden":442},[470],[417,7355,7357,7360,7363,7366,7370,7373,7376,7379],{"className":7356},[474],[417,7358],{"className":7359,"style":4181},[478],[417,7361,7283],{"className":7362,"style":1323},[483,586],[417,7364,1399],{"className":7365},[1623],[417,7367,7288],{"className":7368,"style":7369},[483,586],"margin-right:0.0278em;",[417,7371,1404],{"className":7372},[1630],[417,7374],{"className":7375,"style":1153},[491],[417,7377,1120],{"className":7378},[1157],[417,7380],{"className":7381,"style":1153},[491],[417,7383,7385,7389,7392,7537,7540,7616,7619],{"className":7384},[474],[417,7386],{"className":7387,"style":7388},[478],"height:2.0463em;vertical-align:-0.686em;",[417,7390,2397],{"className":7391},[483],[417,7393,7395,7398,7534],{"className":7394},[483],[417,7396],{"className":7397},[1623,2492],[417,7399,7401],{"className":7400},[2354],[417,7402,7404,7526],{"className":7403},[678,679],[417,7405,7407,7523],{"className":7406},[683],[417,7408,7410,7421,7429],{"className":7409,"style":3195},[687],[417,7411,7412,7415],{"style":2508},[417,7413],{"className":7414,"style":1520},[695],[417,7416,7418],{"className":7417},[483],[417,7419,7288],{"className":7420,"style":7369},[483,586],[417,7422,7423,7426],{"style":2563},[417,7424],{"className":7425,"style":1520},[695],[417,7427],{"className":7428,"style":2571},[2570],[417,7430,7431,7434],{"style":2574},[417,7432],{"className":7433,"style":1520},[695],[417,7435,7437,7440,7443,7483],{"className":7436},[483],[417,7438,2241],{"className":7439},[483,586],[417,7441],{"className":7442,"style":492},[491],[417,7444,7446,7449],{"className":7445},[483],[417,7447,5474],{"className":7448},[483,586],[417,7450,7452],{"className":7451},[674],[417,7453,7455,7475],{"className":7454},[678,679],[417,7456,7458,7472],{"className":7457},[683],[417,7459,7461],{"className":7460,"style":790},[687],[417,7462,7463,7466],{"style":878},[417,7464],{"className":7465,"style":696},[695],[417,7467,7469],{"className":7468},[700,701,702,703],[417,7470,446],{"className":7471},[483,703],[417,7473,716],{"className":7474},[715],[417,7476,7478],{"className":7477},[683],[417,7479,7481],{"className":7480,"style":977},[687],[417,7482],{},[417,7484,7486,7489],{"className":7485},[483],[417,7487,5474],{"className":7488},[483,586],[417,7490,7492],{"className":7491},[674],[417,7493,7495,7515],{"className":7494},[678,679],[417,7496,7498,7512],{"className":7497},[683],[417,7499,7501],{"className":7500,"style":790},[687],[417,7502,7503,7506],{"style":878},[417,7504],{"className":7505,"style":696},[695],[417,7507,7509],{"className":7508},[700,701,702,703],[417,7510,451],{"className":7511},[483,703],[417,7513,716],{"className":7514},[715],[417,7516,7518],{"className":7517},[683],[417,7519,7521],{"className":7520,"style":977},[687],[417,7522],{},[417,7524,716],{"className":7525},[715],[417,7527,7529],{"className":7528},[683],[417,7530,7532],{"className":7531,"style":3536},[687],[417,7533],{},[417,7535],{"className":7536},[1630,2492],[417,7538],{"className":7539,"style":492},[491],[417,7541,7543,7550,7553,7556,7559,7562,7566,7569,7610],{"className":7542},[523],[417,7544,7546],{"className":7545,"style":3407},[1623,3406],[417,7547,1399],{"className":7548},[1762,7549],"size2",[417,7551,446],{"className":7552},[483],[417,7554],{"className":7555,"style":1323},[491],[417,7557,1278],{"className":7558},[1327],[417,7560],{"className":7561,"style":1323},[491],[417,7563,7327],{"className":7564,"style":7565},[483,586],"margin-right:0.0037em;",[417,7567],{"className":7568,"style":492},[491],[417,7570,7572,7575],{"className":7571},[483],[417,7573,4686],{"className":7574},[483,586],[417,7576,7578],{"className":7577},[674],[417,7579,7581],{"className":7580},[678],[417,7582,7584],{"className":7583},[683],[417,7585,7587],{"className":7586,"style":4901},[687],[417,7588,7589,7592],{"style":1551},[417,7590],{"className":7591,"style":696},[695],[417,7593,7595],{"className":7594},[700,701,702,703],[417,7596,7598,7601,7604,7607],{"className":7597},[483,703],[417,7599,2397],{"className":7600},[483,703],[417,7602,7288],{"className":7603,"style":7369},[483,586,703],[417,7605,4103],{"className":7606},[483,703],[417,7608,4306],{"className":7609},[483,586,703],[417,7611,7613],{"className":7612,"style":3407},[1630,3406],[417,7614,1404],{"className":7615},[1762,7549],[417,7617],{"className":7618,"style":492},[491],[417,7620,443],{"className":7621},[487],[324,7623,2807,7624,7652,7653,7777,7778,7806,7807,7836,7837,7866,7867,7923,7924,6666,7975],{},[417,7625,7627,7640],{"className":7626},[420],[417,7628,7630],{"className":7629},[424],[426,7631,7632],{"xmlns":428},[430,7633,7634,7638],{},[433,7635,7636],{},[568,7637,7283],{},[463,7639,7283],{"encoding":465},[417,7641,7643],{"className":7642,"ariaHidden":442},[470],[417,7644,7646,7649],{"className":7645},[474],[417,7647],{"className":7648,"style":582},[478],[417,7650,7283],{"className":7651,"style":1323},[483,586]," is the potential energy of two masses ",[417,7654,7656,7682],{"className":7655},[420],[417,7657,7659],{"className":7658},[424],[426,7660,7661],{"xmlns":428},[430,7662,7663,7679],{},[433,7664,7665,7671,7673],{},[638,7666,7667,7669],{},[568,7668,5474],{},[436,7670,446],{},[440,7672,443],{"separator":442},[638,7674,7675,7677],{},[568,7676,5474],{},[436,7678,451],{},[463,7680,7681],{"encoding":465},"m_1, m_2",[417,7683,7685],{"className":7684,"ariaHidden":442},[470],[417,7686,7688,7691,7731,7734,7737],{"className":7687},[474],[417,7689],{"className":7690,"style":935},[478],[417,7692,7694,7697],{"className":7693},[483],[417,7695,5474],{"className":7696},[483,586],[417,7698,7700],{"className":7699},[674],[417,7701,7703,7723],{"className":7702},[678,679],[417,7704,7706,7720],{"className":7705},[683],[417,7707,7709],{"className":7708,"style":790},[687],[417,7710,7711,7714],{"style":878},[417,7712],{"className":7713,"style":696},[695],[417,7715,7717],{"className":7716},[700,701,702,703],[417,7718,446],{"className":7719},[483,703],[417,7721,716],{"className":7722},[715],[417,7724,7726],{"className":7725},[683],[417,7727,7729],{"className":7728,"style":977},[687],[417,7730],{},[417,7732,443],{"className":7733},[487],[417,7735],{"className":7736,"style":492},[491],[417,7738,7740,7743],{"className":7739},[483],[417,7741,5474],{"className":7742},[483,586],[417,7744,7746],{"className":7745},[674],[417,7747,7749,7769],{"className":7748},[678,679],[417,7750,7752,7766],{"className":7751},[683],[417,7753,7755],{"className":7754,"style":790},[687],[417,7756,7757,7760],{"style":878},[417,7758],{"className":7759,"style":696},[695],[417,7761,7763],{"className":7762},[700,701,702,703],[417,7764,451],{"className":7765},[483,703],[417,7767,716],{"className":7768},[715],[417,7770,7772],{"className":7771},[683],[417,7773,7775],{"className":7774,"style":977},[687],[417,7776],{}," at separation ",[417,7779,7781,7794],{"className":7780},[420],[417,7782,7784],{"className":7783},[424],[426,7785,7786],{"xmlns":428},[430,7787,7788,7792],{},[433,7789,7790],{},[568,7791,7288],{},[463,7793,7288],{"encoding":465},[417,7795,7797],{"className":7796,"ariaHidden":442},[470],[417,7798,7800,7803],{"className":7799},[474],[417,7801],{"className":7802,"style":2222},[478],[417,7804,7288],{"className":7805,"style":7369},[483,586],", ",[417,7808,7810,7824],{"className":7809},[420],[417,7811,7813],{"className":7812},[424],[426,7814,7815],{"xmlns":428},[430,7816,7817,7821],{},[433,7818,7819],{},[568,7820,7327],{},[463,7822,7823],{"encoding":465},"\\alpha",[417,7825,7827],{"className":7826,"ariaHidden":442},[470],[417,7828,7830,7833],{"className":7829},[474],[417,7831],{"className":7832,"style":2222},[478],[417,7834,7327],{"className":7835,"style":7565},[483,586]," measures the strength of the extra force, and ",[417,7838,7840,7854],{"className":7839},[420],[417,7841,7843],{"className":7842},[424],[426,7844,7845],{"xmlns":428},[430,7846,7847,7851],{},[433,7848,7849],{},[568,7850,4306],{},[463,7852,7853],{"encoding":465},"\\lambda",[417,7855,7857],{"className":7856,"ariaHidden":442},[470],[417,7858,7860,7863],{"className":7859},[474],[417,7861],{"className":7862,"style":1513},[478],[417,7864,4306],{"className":7865},[483,586]," its range. For two extra dimensions shaped like a torus, Kaluza–Klein gravitons give ",[417,7868,7870,7892],{"className":7869},[420],[417,7871,7873],{"className":7872},[424],[426,7874,7875],{"xmlns":428},[430,7876,7877,7889],{},[433,7878,7879,7881,7883,7885,7887],{},[568,7880,7327],{},[440,7882,1120],{},[436,7884,2362],{},[568,7886,4103],{"mathvariant":3085},[436,7888,456],{},[463,7890,7891],{"encoding":465},"\\alpha = 16\u002F3",[417,7893,7895,7913],{"className":7894,"ariaHidden":442},[470],[417,7896,7898,7901,7904,7907,7910],{"className":7897},[474],[417,7899],{"className":7900,"style":2222},[478],[417,7902,7327],{"className":7903,"style":7565},[483,586],[417,7905],{"className":7906,"style":1153},[491],[417,7908,1120],{"className":7909},[1157],[417,7911],{"className":7912,"style":1153},[491],[417,7914,7916,7919],{"className":7915},[474],[417,7917],{"className":7918,"style":4181},[478],[417,7920,7922],{"className":7921},[483],"16\u002F3"," and ",[417,7925,7927,7945],{"className":7926},[420],[417,7928,7930],{"className":7929},[424],[426,7931,7932],{"xmlns":428},[430,7933,7934,7942],{},[433,7935,7936,7938,7940],{},[568,7937,4306],{},[440,7939,1120],{},[568,7941,570],{},[463,7943,7944],{"encoding":465},"\\lambda = R",[417,7946,7948,7966],{"className":7947,"ariaHidden":442},[470],[417,7949,7951,7954,7957,7960,7963],{"className":7950},[474],[417,7952],{"className":7953,"style":1513},[478],[417,7955,4306],{"className":7956},[483,586],[417,7958],{"className":7959,"style":1153},[491],[417,7961,1120],{"className":7962},[1157],[417,7964],{"className":7965,"style":1153},[491],[417,7967,7969,7972],{"className":7968},[474],[417,7970],{"className":7971,"style":582},[478],[417,7973,570],{"className":7974,"style":587},[483,586],[348,7976,7978],{"className":7977},[351],"[10]",[324,7980,7981,7982,7986,7987,7991,7992,8065,8066],{},"The Eöt-Wash group at the University of Washington uses torsion balances — delicate rotating pendulums — to measure gravity between test masses separated by less than a tenth of a millimetre. In 2007 they found the inverse-square law holding down to 56 micrometres and concluded that an extra dimension must be smaller than about 44 μm. ",[348,7983,7985],{"className":7984},[351],"[11]"," A 2020 experiment probed separations down to 52 μm and found Newtonian gravity fitting the data perfectly, excluding gravitational-strength extra forces with ranges above 38.6 μm. ",[348,7988,7990],{"className":7989},[351],"[12]"," The Particle Data Group summary quotes ",[417,7993,7995,8021],{"className":7994},[420],[417,7996,7998],{"className":7997},[424],[426,7999,8000],{"xmlns":428},[430,8001,8002,8018],{},[433,8003,8004,8006,8009,8011,8014,8016],{},[568,8005,570],{},[440,8007,8008],{},"\u003C",[436,8010,6490],{},[1406,8012,8013],{}," ",[568,8015,647],{},[1406,8017,5474],{},[463,8019,8020],{"encoding":465},"R \u003C 30\\ \\mu\\text{m}",[417,8022,8024,8043],{"className":8023,"ariaHidden":442},[470],[417,8025,8027,8031,8034,8037,8040],{"className":8026},[474],[417,8028],{"className":8029,"style":8030},[478],"height:0.7224em;vertical-align:-0.0391em;",[417,8032,570],{"className":8033,"style":587},[483,586],[417,8035],{"className":8036,"style":1153},[491],[417,8038,8008],{"className":8039},[1157],[417,8041],{"className":8042,"style":1153},[491],[417,8044,8046,8049,8052,8055,8058],{"className":8045},[474],[417,8047],{"className":8048,"style":479},[478],[417,8050,6490],{"className":8051},[483],[417,8053,8013],{"className":8054},[491],[417,8056,647],{"className":8057},[483,586],[417,8059,8062],{"className":8060},[483,8061],"text",[417,8063,5474],{"className":8064},[483]," for two large extra dimensions. ",[348,8067,7978],{"className":8068},[351],[324,8070,8071,8074,8075,8149,8150,8154,8155],{},[343,8072,8073],{},"Collider searches."," At the Large Hadron Collider, ADD-type gravitons would escape the detector, leaving events with an energetic jet and \"missing\" momentum. ATLAS found no excess in its full 13 TeV dataset, pushing the fundamental higher-dimensional gravity scale ",[417,8076,8078,8098],{"className":8077},[420],[417,8079,8081],{"className":8080},[424],[426,8082,8083],{"xmlns":428},[430,8084,8085,8095],{},[433,8086,8087],{},[638,8088,8089,8092],{},[568,8090,8091],{},"M",[568,8093,8094],{},"D",[463,8096,8097],{"encoding":465},"M_D",[417,8099,8101],{"className":8100,"ariaHidden":442},[470],[417,8102,8104,8107],{"className":8103},[474],[417,8105],{"className":8106,"style":2427},[478],[417,8108,8110,8113],{"className":8109},[483],[417,8111,8091],{"className":8112,"style":6965},[483,586],[417,8114,8116],{"className":8115},[674],[417,8117,8119,8141],{"className":8118},[678,679],[417,8120,8122,8138],{"className":8121},[683],[417,8123,8126],{"className":8124,"style":8125},[687],"height:0.3283em;",[417,8127,8129,8132],{"style":8128},"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;",[417,8130],{"className":8131,"style":696},[695],[417,8133,8135],{"className":8134},[700,701,702,703],[417,8136,8094],{"className":8137,"style":7369},[483,586,703],[417,8139,716],{"className":8140},[715],[417,8142,8144],{"className":8143},[683],[417,8145,8147],{"className":8146,"style":977},[687],[417,8148],{}," above roughly 6–11 TeV depending on the number of extra dimensions. ",[348,8151,8153],{"className":8152},[351],"[13,10]"," Searches for Randall–Sundrum Kaluza–Klein gravitons, which would appear as bumps in the mass spectra of photon or lepton pairs, have excluded the lightest such graviton below about 4.5 TeV for a benchmark coupling. ",[348,8156,7978],{"className":8157},[351],[324,8159,8160,8163,8164],{},[343,8161,8162],{},"Astrophysics."," Light Kaluza–Klein gravitons would carry energy out of supernovae. The observed neutrino burst from Supernova 1987A therefore gives constraints on large extra dimensions that are, for two dimensions, even stronger than the collider limits. ",[348,8165,7978],{"className":8166},[351],[324,8168,8169,8172],{},[343,8170,8171],{},"Summary of the evidence:"," the original Kaluza–Klein picture, with a circle near the Planck length, is untestable with any foreseeable technology. The newer \"large\" versions made testable predictions, and so far those predictions have not been seen.",[361,8174,8176],{"id":8175},"limitations-and-open-problems","Limitations and Open Problems",[366,8178,8180],{"id":8179},"the-masses-are-wrong","The masses are wrong",[324,8182,8183,8184,8236,8237,8303,8304,8370,8371],{},"Klein's own numbers expose the first problem. If the circle is small enough to give the electron's charge, the first charged Kaluza–Klein mode has a mass ",[417,8185,8187,8211],{"className":8186},[420],[417,8188,8190],{"className":8189},[424],[426,8191,8192],{"xmlns":428},[430,8193,8194,8208],{},[433,8195,8196,8198,8200,8202,8204,8206],{},[568,8197,5494],{"mathvariant":3085},[568,8199,4103],{"mathvariant":3085},[440,8201,1399],{"stretchy":1398},[568,8203,570],{},[568,8205,5503],{},[440,8207,1404],{"stretchy":1398},[463,8209,8210],{"encoding":465},"\\hbar\u002F(Rc)",[417,8212,8214],{"className":8213,"ariaHidden":442},[470],[417,8215,8217,8220,8224,8227,8230,8233],{"className":8216},[474],[417,8218],{"className":8219,"style":4181},[478],[417,8221,8223],{"className":8222},[483],"ℏ\u002F",[417,8225,1399],{"className":8226},[1623],[417,8228,570],{"className":8229,"style":587},[483,586],[417,8231,5503],{"className":8232},[483,586],[417,8234,1404],{"className":8235},[1630]," of order ",[417,8238,8240,8259],{"className":8239},[420],[417,8241,8243],{"className":8242},[424],[426,8244,8245],{"xmlns":428},[430,8246,8247,8256],{},[433,8248,8249],{},[1053,8250,8251,8253],{},[436,8252,729],{},[436,8254,8255],{},"17",[463,8257,8258],{"encoding":465},"10^{17}",[417,8260,8262],{"className":8261,"ariaHidden":442},[470],[417,8263,8265,8268,8271],{"className":8264},[474],[417,8266],{"className":8267,"style":1950},[478],[417,8269,446],{"className":8270},[483],[417,8272,8274,8277],{"className":8273},[483],[417,8275,438],{"className":8276},[483],[417,8278,8280],{"className":8279},[674],[417,8281,8283],{"className":8282},[678],[417,8284,8286],{"className":8285},[683],[417,8287,8289],{"className":8288,"style":1950},[687],[417,8290,8291,8294],{"style":1093},[417,8292],{"className":8293,"style":696},[695],[417,8295,8297],{"className":8296},[700,701,702,703],[417,8298,8300],{"className":8299},[483,703],[417,8301,8255],{"className":8302},[483,703]," GeV — about ",[417,8305,8307,8326],{"className":8306},[420],[417,8308,8310],{"className":8309},[424],[426,8311,8312],{"xmlns":428},[430,8313,8314,8323],{},[433,8315,8316],{},[1053,8317,8318,8320],{},[436,8319,729],{},[436,8321,8322],{},"20",[463,8324,8325],{"encoding":465},"10^{20}",[417,8327,8329],{"className":8328,"ariaHidden":442},[470],[417,8330,8332,8335,8338],{"className":8331},[474],[417,8333],{"className":8334,"style":1950},[478],[417,8336,446],{"className":8337},[483],[417,8339,8341,8344],{"className":8340},[483],[417,8342,438],{"className":8343},[483],[417,8345,8347],{"className":8346},[674],[417,8348,8350],{"className":8349},[678],[417,8351,8353],{"className":8352},[683],[417,8354,8356],{"className":8355,"style":1950},[687],[417,8357,8358,8361],{"style":1093},[417,8359],{"className":8360,"style":696},[695],[417,8362,8364],{"className":8363},[700,701,702,703],[417,8365,8367],{"className":8366},[483,703],[417,8368,8322],{"className":8369},[483,703]," times heavier than the electron. The simplest version of the theory cannot contain the electron as a charged Kaluza–Klein mode. ",[348,8372,6894],{"className":8373},[351],[366,8375,8377],{"id":8376},"the-scalar-will-not-sit-still","The scalar will not sit still",[324,8379,8380,8381,8409,8410,8438,8439,8442,8443],{},"The scalar ",[417,8382,8384,8397],{"className":8383},[420],[417,8385,8387],{"className":8386},[424],[426,8388,8389],{"xmlns":428},[430,8390,8391,8395],{},[433,8392,8393],{},[568,8394,1000],{},[463,8396,1003],{"encoding":465},[417,8398,8400],{"className":8399,"ariaHidden":442},[470],[417,8401,8403,8406],{"className":8402},[474],[417,8404],{"className":8405,"style":1013},[478],[417,8407,1000],{"className":8408},[483,586]," is massless in the simplest reduction. A massless scalar coupled to matter would produce a long-range \"fifth force\" and would make the effective strengths of gravity and electromagnetism vary from place to place. Precision tests of gravity and of the constancy of fundamental constants strongly restrict such effects. A realistic model must therefore give ",[417,8411,8413,8426],{"className":8412},[420],[417,8414,8416],{"className":8415},[424],[426,8417,8418],{"xmlns":428},[430,8419,8420,8424],{},[433,8421,8422],{},[568,8423,1000],{},[463,8425,1003],{"encoding":465},[417,8427,8429],{"className":8428,"ariaHidden":442},[470],[417,8430,8432,8435],{"className":8431},[474],[417,8433],{"className":8434,"style":1013},[478],[417,8436,1000],{"className":8437},[483,586]," a mass, fixing the size of the extra dimension. This task, called ",[343,8440,8441],{},"moduli stabilization"," in string theory, remains one of the hardest problems in the field. ",[348,8444,8446],{"className":8445},[351],"[4,10]",[366,8448,8450],{"id":8449},"chirality","Chirality",[324,8452,8453,8454,8457,8458,8461],{},"The Standard Model is ",[343,8455,8456],{},"chiral",": the weak force treats left-handed and right-handed particles differently. Pure Kaluza–Klein theories on smooth compact spaces have great difficulty producing this asymmetry. Witten's 1981 analysis found that the Standard Model gauge group could be obtained, but the right fermion quantum numbers could not. ",[348,8459,7036],{"className":8460},[351]," String theory addresses this by adding ingredients beyond pure geometry — gauge fields already present in ten dimensions, singular compact spaces, and branes — so Kaluza–Klein reduction survives as a tool rather than as the whole explanation.",[366,8463,8465],{"id":8464},"which-extra-space","Which extra space?",[324,8467,8468],{},"Once one extra dimension is allowed, nothing obvious fixes how many there are or what shape they take. String theory offers an enormous number of candidate six-dimensional shapes, and no known principle selects the one that describes our world. This weakens the predictive power of the approach: many observations can be matched after choosing the shape, which is different from predicting them in advance.",[366,8470,8472],{"id":8471},"what-would-count-as-a-discovery","What would count as a discovery",[324,8474,8475,8476,8554],{},"The idea is falsifiable in its large-dimension versions. A clear deviation from the inverse-square law at short range, with the right strength, or a regularly spaced series of heavy resonances at a collider — copies of a known particle at masses ",[417,8477,8479,8509],{"className":8478},[420],[417,8480,8482],{"className":8481},[424],[426,8483,8484],{"xmlns":428},[430,8485,8486,8506],{},[433,8487,8488,8490,8492,8494,8496,8498,8500,8502,8504],{},[568,8489,5474],{},[440,8491,443],{"separator":442},[436,8493,451],{},[568,8495,5474],{},[440,8497,443],{"separator":442},[436,8499,456],{},[568,8501,5474],{},[440,8503,443],{"separator":442},[440,8505,461],{},[463,8507,8508],{"encoding":465},"m, 2m, 3m, \\dots",[417,8510,8512],{"className":8511,"ariaHidden":442},[470],[417,8513,8515,8518,8521,8524,8527,8530,8533,8536,8539,8542,8545,8548,8551],{"className":8514},[474],[417,8516],{"className":8517,"style":479},[478],[417,8519,5474],{"className":8520},[483,586],[417,8522,443],{"className":8523},[487],[417,8525],{"className":8526,"style":492},[491],[417,8528,451],{"className":8529},[483],[417,8531,5474],{"className":8532},[483,586],[417,8534,443],{"className":8535},[487],[417,8537],{"className":8538,"style":492},[491],[417,8540,456],{"className":8541},[483],[417,8543,5474],{"className":8544},[483,586],[417,8546,443],{"className":8547},[487],[417,8549],{"className":8550,"style":492},[491],[417,8552,461],{"className":8553},[523]," above a threshold — would be strong evidence. The absence of both, so far, is why large extra dimensions are now tightly constrained.",[361,8556,8558],{"id":8557},"a-failed-theorys-long-afterlife","A Failed Theory's Long Afterlife",[324,8560,8561],{},"Kaluza–Klein reduction matters for three separate reasons, and they should not be confused.",[324,8563,8564,8567],{},[343,8565,8566],{},"As an established mathematical result",", it shows that the forces of nature can, in principle, be different aspects of one geometry. Electromagnetism and its gauge symmetry can be derived from gravity in one more dimension. That is a real and lasting insight into how gauge theories are built.",[324,8569,8570,8573],{},[343,8571,8572],{},"As a working tool",", it is used constantly. Any theory in more than four dimensions — supergravity, string theory, M-theory, brane-world models — relies on Kaluza–Klein reduction to connect its equations to the four-dimensional world. It even appears in condensed-matter physics and in ordinary engineering problems where one direction of a system is much smaller than the others.",[324,8575,8576,8579,8580,8653],{},[343,8577,8578],{},"As a claim about nature",", it remains unconfirmed. Our universe is consistent with three space dimensions and one time down to tens of micrometres in gravity and to scales of roughly ",[417,8581,8583,8606],{"className":8582},[420],[417,8584,8586],{"className":8585},[424],[426,8587,8588],{"xmlns":428},[430,8589,8590,8603],{},[433,8591,8592],{},[1053,8593,8594,8596],{},[436,8595,729],{},[433,8597,8598,8600],{},[440,8599,2397],{},[436,8601,8602],{},"19",[463,8604,8605],{"encoding":465},"10^{-19}",[417,8607,8609],{"className":8608,"ariaHidden":442},[470],[417,8610,8612,8615,8618],{"className":8611},[474],[417,8613],{"className":8614,"style":1950},[478],[417,8616,446],{"className":8617},[483],[417,8619,8621,8624],{"className":8620},[483],[417,8622,438],{"className":8623},[483],[417,8625,8627],{"className":8626},[674],[417,8628,8630],{"className":8629},[678],[417,8631,8633],{"className":8632},[683],[417,8634,8636],{"className":8635,"style":1950},[687],[417,8637,8638,8641],{"style":1093},[417,8639],{"className":8640,"style":696},[695],[417,8642,8644],{"className":8643},[700,701,702,703],[417,8645,8647,8650],{"className":8646},[483,703],[417,8648,2397],{"className":8649},[483,703],[417,8651,8602],{"className":8652},[483,703]," m in particle collisions. Extra dimensions may exist below those scales, or they may not exist at all.",[324,8655,8656],{},"Five-dimensional geometry really does contain Maxwell's theory, and higher-dimensional physics can make forces appear as the shapes of hidden dimensions. The unresolved part is the physical one: whether our universe has any such dimension to hide.",[320,8658,8660],{"id":8659},"references","References",[324,8662,8663,8666,8667,8670,8671,7806,8674,8677,8678,8681,8686,8687],{},[343,8664,352,8665],{}," Kaluza, Th. (1921; English translation 2018)."," \"On the Unification Problem in Physics.\" Original: ",[331,8668,8669],{},"Sitzungsberichte der Preussischen Akademie der Wissenschaften"," (Math. Phys.), 966–972. Translation: ",[331,8672,8673],{},"International Journal of Modern Physics D",[343,8675,8676],{},"27","(14), 1870001.",[8679,8680],"br",{},[336,8682,8683],{"href":8683,"rel":8684},"https:\u002F\u002Fdoi.org\u002F10.1142\u002FS0218271818700017",[8685],"nofollow"," · ",[336,8688,8691],{"href":8689,"rel":8690},"https:\u002F\u002Farxiv.org\u002Fabs\u002F1803.08616",[8685],"arXiv:1803.08616",[324,8693,8694,8698,8699,7806,8702,8705,8706,8708],{},[343,8695,8696,8697],{},"[2]"," Klein, O. (1926)."," \"Quantentheorie und fünfdimensionale Relativitätstheorie.\" ",[331,8700,8701],{},"Zeitschrift für Physik",[343,8703,8704],{},"37","(12), 895–906.",[8679,8707],{},[336,8709,8710],{"href":8710,"rel":8711},"https:\u002F\u002Fdoi.org\u002F10.1007\u002FBF01397481",[8685],[324,8713,8714,8717,8718,7806,8721,8724,8725,8727],{},[343,8715,8716,8697],{},"[3]"," \"The Atomicity of Electricity as a Quantum Theory Law.\" ",[331,8719,8720],{},"Nature",[343,8722,8723],{},"118",", 516.",[8679,8726],{},[336,8728,8729],{"href":8729,"rel":8730},"https:\u002F\u002Fdoi.org\u002F10.1038\u002F118516a0",[8685],[324,8732,8733,8736,8737,7806,8740,8743,8744,8746,8686,8750],{},[343,8734,6894,8735],{}," Overduin, J. M., & Wesson, P. S. (1997)."," \"Kaluza–Klein Gravity.\" ",[331,8738,8739],{},"Physics Reports",[343,8741,8742],{},"283",", 303–378.",[8679,8745],{},[336,8747,8748],{"href":8748,"rel":8749},"https:\u002F\u002Fdoi.org\u002F10.1016\u002FS0370-1573(96)00046-4",[8685],[336,8751,8754],{"href":8752,"rel":8753},"https:\u002F\u002Farxiv.org\u002Fabs\u002Fgr-qc\u002F9805018",[8685],"arXiv:gr-qc\u002F9805018",[324,8756,8757,8761,8762,2256,8765,8767],{},[343,8758,8759,8760],{},"[5]"," Duff, M. J. (1994)."," \"Kaluza–Klein Theory in Perspective.\" Talk at the Oskar Klein Centenary Symposium. ",[331,8763,8764],{},"arXiv preprint",[8679,8766],{},[336,8768,8769],{"href":8769,"rel":8770},"https:\u002F\u002Farxiv.org\u002Fabs\u002Fhep-th\u002F9410046",[8685],[324,8772,8773,8776,8777,7806,8780,8783,8784,8786],{},[343,8774,7036,8775],{}," Witten, E. (1981)."," \"Search for a Realistic Kaluza–Klein Theory.\" ",[331,8778,8779],{},"Nuclear Physics B",[343,8781,8782],{},"186",", 412–428.",[8679,8785],{},[336,8787,8788],{"href":8788,"rel":8789},"https:\u002F\u002Fdoi.org\u002F10.1016\u002F0550-3213(81)90021-3",[8685],[324,8791,8792,8795,8796,7806,8798,8801,8802,8804,8686,8808],{},[343,8793,7046,8794],{}," Witten, E. (1995)."," \"String Theory Dynamics in Various Dimensions.\" ",[331,8797,8779],{},[343,8799,8800],{},"443",", 85–126.",[8679,8803],{},[336,8805,8806],{"href":8806,"rel":8807},"https:\u002F\u002Fdoi.org\u002F10.1016\u002F0550-3213(95)00158-O",[8685],[336,8809,8812],{"href":8810,"rel":8811},"https:\u002F\u002Farxiv.org\u002Fabs\u002Fhep-th\u002F9503124",[8685],"arXiv:hep-th\u002F9503124",[324,8814,8815,8819,8820,7806,8823,8826,8827,8829,8686,8833],{},[343,8816,8817,8818],{},"[8]"," Arkani-Hamed, N., Dimopoulos, S., & Dvali, G. (1998)."," \"The Hierarchy Problem and New Dimensions at a Millimeter.\" ",[331,8821,8822],{},"Physics Letters B",[343,8824,8825],{},"429",", 263–272.",[8679,8828],{},[336,8830,8831],{"href":8831,"rel":8832},"https:\u002F\u002Fdoi.org\u002F10.1016\u002FS0370-2693(98)00466-3",[8685],[336,8834,8837],{"href":8835,"rel":8836},"https:\u002F\u002Farxiv.org\u002Fabs\u002Fhep-ph\u002F9803315",[8685],"arXiv:hep-ph\u002F9803315",[324,8839,8840,8844,8845,7806,8848,8851,8852,8854],{},[343,8841,8842,8843],{},"[9]"," Randall, L., & Sundrum, R. (1999)."," \"Large Mass Hierarchy from a Small Extra Dimension.\" ",[331,8846,8847],{},"Physical Review Letters",[343,8849,8850],{},"83",", 3370–3373.",[8679,8853],{},[336,8855,8856],{"href":8856,"rel":8857},"https:\u002F\u002Fdoi.org\u002F10.1103\u002FPhysRevLett.83.3370",[8685],[324,8859,8860,8863,8864,7806,8867,8870,8871,8873,8686,8877],{},[343,8861,7978,8862],{}," Demiragli, Z., & Pomarol, A. (2024)."," \"Extra Dimensions.\" In Navas, S., et al. (Particle Data Group), \"Review of Particle Physics.\" ",[331,8865,8866],{},"Physical Review D",[343,8868,8869],{},"110",", 030001.",[8679,8872],{},[336,8874,8875],{"href":8875,"rel":8876},"https:\u002F\u002Fdoi.org\u002F10.1103\u002FPhysRevD.110.030001",[8685],[336,8878,8881],{"href":8879,"rel":8880},"https:\u002F\u002Fpdg.lbl.gov\u002F2024\u002Freviews\u002Frpp2024-rev-extra-dimensions.pdf",[8685],"Chapter PDF",[324,8883,8884,8887,8888,7806,8890,8893,8894,8896,8686,8900],{},[343,8885,7985,8886],{}," Kapner, D. J., Cook, T. S., Adelberger, E. G., Gundlach, J. H., Heckel, B. R., Hoyle, C. D., & Swanson, H. E. (2007)."," \"Tests of the Gravitational Inverse-Square Law below the Dark-Energy Length Scale.\" ",[331,8889,8847],{},[343,8891,8892],{},"98",", 021101.",[8679,8895],{},[336,8897,8898],{"href":8898,"rel":8899},"https:\u002F\u002Fdoi.org\u002F10.1103\u002FPhysRevLett.98.021101",[8685],[336,8901,8904],{"href":8902,"rel":8903},"https:\u002F\u002Farxiv.org\u002Fabs\u002Fhep-ph\u002F0611184",[8685],"arXiv:hep-ph\u002F0611184",[324,8906,8907,8910,8911,8978,8979,7806,8981,8984,8985,8987,8686,8991],{},[343,8908,7990,8909],{}," Lee, J. G., Adelberger, E. G., Cook, T. S., Fleischer, S. M., & Heckel, B. R. (2020)."," \"New Test of the Gravitational ",[417,8912,8914,8936],{"className":8913},[420],[417,8915,8917],{"className":8916},[424],[426,8918,8919],{"xmlns":428},[430,8920,8921,8933],{},[433,8922,8923,8925,8927],{},[436,8924,446],{},[568,8926,4103],{"mathvariant":3085},[1053,8928,8929,8931],{},[568,8930,7288],{},[436,8932,451],{},[463,8934,8935],{"encoding":465},"1\u002Fr^2",[417,8937,8939],{"className":8938,"ariaHidden":442},[470],[417,8940,8942,8946,8949],{"className":8941},[474],[417,8943],{"className":8944,"style":8945},[478],"height:1.0641em;vertical-align:-0.25em;",[417,8947,6007],{"className":8948},[483],[417,8950,8952,8955],{"className":8951},[483],[417,8953,7288],{"className":8954,"style":7369},[483,586],[417,8956,8958],{"className":8957},[674],[417,8959,8961],{"className":8960},[678],[417,8962,8964],{"className":8963},[683],[417,8965,8967],{"className":8966,"style":1950},[687],[417,8968,8969,8972],{"style":1093},[417,8970],{"className":8971,"style":696},[695],[417,8973,8975],{"className":8974},[700,701,702,703],[417,8976,451],{"className":8977},[483,703]," Law at Separations down to 52 μm.\" ",[331,8980,8847],{},[343,8982,8983],{},"124",", 101101.",[8679,8986],{},[336,8988,8989],{"href":8989,"rel":8990},"https:\u002F\u002Fdoi.org\u002F10.1103\u002FPhysRevLett.124.101101",[8685],[336,8992,8995],{"href":8993,"rel":8994},"https:\u002F\u002Farxiv.org\u002Fabs\u002F2002.11761",[8685],"arXiv:2002.11761",[324,8997,8998,9002,9003,9034,9035,9137,9138,7806,9140,9143,9144,9146],{},[343,8999,9000,9001],{},"[13]"," ATLAS Collaboration (2021)."," \"Search for new phenomena in events with an energetic jet and missing transverse momentum in ",[417,9004,9006,9022],{"className":9005},[420],[417,9007,9009],{"className":9008},[424],[426,9010,9011],{"xmlns":428},[430,9012,9013,9019],{},[433,9014,9015,9017],{},[568,9016,324],{},[568,9018,324],{},[463,9020,9021],{"encoding":465},"pp",[417,9023,9025],{"className":9024,"ariaHidden":442},[470],[417,9026,9028,9031],{"className":9027},[474],[417,9029],{"className":9030,"style":935},[478],[417,9032,9021],{"className":9033},[483,586]," collisions at ",[417,9036,9038,9059],{"className":9037},[420],[417,9039,9041],{"className":9040},[424],[426,9042,9043],{"xmlns":428},[430,9044,9045,9056],{},[433,9046,9047,9051,9053],{},[2391,9048,9049],{},[568,9050,1378],{},[440,9052,1120],{},[436,9054,9055],{},"13",[463,9057,9058],{"encoding":465},"\\sqrt{s} = 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TeV with the ATLAS detector.\" ",[331,9139,8866],{},[343,9141,9142],{},"103",", 112006.",[8679,9145],{},[336,9147,9148],{"href":9148,"rel":9149},"https:\u002F\u002Fdoi.org\u002F10.1103\u002FPhysRevD.103.112006",[8685],{"title":9151,"searchDepth":83,"depth":83,"links":9152},"",[9153,9158,9159,9165,9166,9170,9177],{"id":363,"depth":83,"text":364,"children":9154},[9155,9156,9157],{"id":368,"depth":131,"text":369},{"id":378,"depth":131,"text":379},{"id":411,"depth":131,"text":412},{"id":535,"depth":83,"text":536},{"id":1031,"depth":83,"text":1032,"children":9160},[9161,9162,9163,9164],{"id":1035,"depth":131,"text":1036},{"id":2324,"depth":131,"text":2325},{"id":4584,"depth":131,"text":4585},{"id":6018,"depth":131,"text":6019},{"id":6548,"depth":83,"text":6549},{"id":7063,"depth":83,"text":7064,"children":9167},[9168,9169],{"id":7070,"depth":131,"text":7071},{"id":7194,"depth":131,"text":7195},{"id":8175,"depth":83,"text":8176,"children":9171},[9172,9173,9174,9175,9176],{"id":8179,"depth":131,"text":8180},{"id":8376,"depth":131,"text":8377},{"id":8449,"depth":131,"text":8450},{"id":8464,"depth":131,"text":8465},{"id":8471,"depth":131,"text":8472},{"id":8557,"depth":83,"text":8558},"md",[173,174,175,9,176,177],{},true,{"title":170,"description":171},"evidence\u002Fkaluza-klein-reduction","xV9WGoVDrkm_lHQQMnuPQu19ZfAyL9ht_1tOYXwX6DY",1789530370340]