[{"data":1,"prerenderedAt":2911},["ShallowReactive",2],{"evidence-index":3,"evidence-doc-novikov-self-consistency-principle":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":225,"body":316,"description":226,"extension":2904,"field":17,"keywords":2905,"meta":2906,"navigation":2907,"order":232,"path":224,"seo":2908,"stem":2909,"__hash__":2910},"evidence\u002Fevidence\u002Fnovikov-self-consistency-principle.md",{"type":317,"value":318,"toc":2883},"minimark",[319,323,327,337,345,350,353,360,367,374,377,381,391,402,405,409,424,427,509,601,695,850,857,861,868,878,901,908,911,918,935,952,959,963,966,969,976,984,990,994,997,1002,1014,1018,1026,1030,1037,1041,1048,1052,1055,1059,1275,1672,1798,1801,1818,1822,1834,1960,2379,2439,2454,2458,2466,2470,2473,2536,2544,2548,2551,2554,2557,2562,2566,2586,2602,2618,2635,2650,2663,2679,2689,2705,2720,2737,2754,2771,2787,2804,2820,2836,2850,2866],[320,321,225],"h1",{"id":322},"novikov-self-consistency-principle",[324,325,326],"p",{},"Imagine a billiard ball rolling toward the mouth of a wormhole. The wormhole is a time machine: whatever enters one mouth leaves the other mouth a little earlier. The ball is aimed so that, when it emerges in the past, it will strike its own younger self and knock it off course. But if the younger ball is knocked away, it never enters the wormhole. Then there is no older ball to do the knocking. Then the younger ball is not knocked away, and enters the wormhole after all.",[324,328,329,330,336],{},"The physicist Joseph Polchinski posed this puzzle to Kip Thorne's group around 1990, and it became the test case for their analysis. ",[331,332,335],"sup",{"className":333},[334],"c-cite","[2]"," It is the grandfather paradox with the human drama removed: no free will, no murder, only a ball obeying Newton's laws. That was the point. If time travel produces contradictions even for billiard balls, then the problem lies in physics, not in psychology.",[324,338,339,340,344],{},"What Thorne's students found was surprising. The paradox has a way out, and it requires no force to stop the ball. The ball simply does something different from what the puzzle assumed. That escape became the test case for the ",[341,342,343],"strong",{},"Novikov self-consistency principle"," and for competing quantum models of time travel.",[346,347,349],"h2",{"id":348},"one-history-not-two","One History, Not Two",[324,351,352],{},"The grandfather paradox rests on a hidden assumption: that there is a first version of history, in which the traveler sets out, and a second version, in which the traveler has changed things. Stories about time travel often work this way. The traveler changes the past and the present rearranges itself.",[324,354,355,356,359],{},"Physics does not describe history in versions. In relativity, spacetime is one four-dimensional object, and each event in it happens once (see ",[357,358,245],"a",{"href":244},"). If the traveler visits 1950, then the traveler was always in 1950. Whatever the traveler does there was already part of what happened in 1950, including whatever caused the traveler's own existence.",[324,361,362,363,366],{},"On this view the question \"can I change the past?\" has an easy answer: no, because there is only one past, and it already contains your visit. The harder question is different: ",[341,364,365],{},"which histories are possible?"," The self-consistency principle answers that only those histories occur that fit together consistently around every time loop.",[324,368,369,370],{},"The philosopher David Lewis gave a helpful example in 1976. A time traveler, Tim, goes back to kill his grandfather before his father is conceived. Tim is a good shot and has a working rifle. In one sense he can kill his grandfather: he has the skill and the means. In another sense he cannot: his grandfather is known to have survived. Lewis argued that these two claims are not in conflict. They answer different questions. What actually happens is that Tim fails for some ordinary reason, perhaps a noise, a slip, or a change of heart. ",[331,371,373],{"className":372},[334],"[4,5]",[324,375,376],{},"Nothing supernatural prevents Tim from succeeding. His failure is just one of the many events in a consistent history.",[346,378,380],{"id":379},"historical-development","Historical Development",[324,382,383,384,386,387],{},"The idea is associated with Igor Novikov, a Russian astrophysicist who argued in the 1980s that any physics permitting time travel must also restrict events to self-consistent histories. At almost the same time, Thorne's group at Caltech was studying traversable wormholes and showing how they could be turned into time machines (see ",[357,385,215],{"href":214},"). ",[331,388,390],{"className":389},[334],"[16]",[324,392,393,394,397,398],{},"The two lines of work met in a 1990 paper by John Friedman, Michael Morris, Igor Novikov, Fernando Echeverria, Gunnar Klinkhammer, Kip Thorne and Ulvi Yurtsever. They proposed that, if spacetimes with closed timelike curves are allowed, the laws of physics should be supplemented by a ",[341,395,396],{},"principle of self-consistency",". In their formulation, a local solution to the equations of physics can occur in the real universe only if it can be extended to a global solution that is well defined throughout spacetime. ",[331,399,401],{"className":400},[334],"[1]",[324,403,404],{},"That wording deserves attention. It does not introduce a new force. It says that the equations of physics, which are normally solved step by step forward in time, must in a time-machine spacetime be solved as a whole. A local event is allowed only if it belongs to a complete, consistent history.",[346,406,408],{"id":407},"how-it-works-solving-history-as-a-whole","How It Works: Solving History as a Whole",[324,410,411,412,415,416,419,420,423],{},"In ordinary physics you can predict the future from the present. Give the positions and velocities of all particles now, and the laws tell you what comes next. Physicists call the set of starting information ",[341,413,414],{},"initial data",", and the task of computing the future from it the ",[341,417,418],{},"Cauchy problem",". This is also the core of physical ",[357,421,422],{"href":256},"determinism",".",[324,425,426],{},"A time machine breaks this procedure. Something can arrive from the future and influence the present. You cannot compute the future from the present alone, because the present depends in part on the future.",[324,428,429,430,476,477,508],{},"A simple way to picture the resulting condition uses a single number. Suppose some quantity ",[431,432,435,457],"span",{"className":433},[434],"katex",[431,436,439],{"className":437},[438],"katex-mathml",[440,441,443],"math",{"xmlns":442},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[444,445,446,453],"semantics",{},[447,448,449],"mrow",{},[450,451,452],"mi",{},"x",[454,455,452],"annotation",{"encoding":456},"application\u002Fx-tex",[431,458,462],{"className":459,"ariaHidden":461},[460],"katex-html","true",[431,463,466,471],{"className":464},[465],"base",[431,467],{"className":468,"style":470},[469],"strut","height:0.4306em;",[431,472,452],{"className":473},[474,475],"mord","mathnormal"," describes what emerges from the past mouth of a wormhole: a ball's direction, say. What goes into the future mouth later depends on what happened in between, including the effect of whatever emerged. Call that dependence ",[431,478,480,494],{"className":479},[434],[431,481,483],{"className":482},[438],[440,484,485],{"xmlns":442},[444,486,487,492],{},[447,488,489],{},[450,490,491],{},"F",[454,493,491],{"encoding":456},[431,495,497],{"className":496,"ariaHidden":461},[460],[431,498,500,504],{"className":499},[465],[431,501],{"className":502,"style":503},[469],"height:0.6833em;",[431,505,491],{"className":506,"style":507},[474,475],"margin-right:0.1389em;",". Since what enters the future mouth is exactly what emerges from the past mouth, consistency requires",[510,511,513],"equation",{"label":512},"Self-consistency as a fixed point",[431,514,517],{"className":515},[516],"katex-display",[431,518,520,553],{"className":519},[434],[431,521,523],{"className":522},[438],[440,524,526],{"xmlns":442,"display":525},"block",[444,527,528,550],{},[447,529,530,532,536,538,542,544,547],{},[450,531,452],{},[533,534,535],"mo",{},"=",[450,537,491],{},[533,539,541],{"stretchy":540},"false","(",[450,543,452],{},[533,545,546],{"stretchy":540},")",[450,548,423],{"mathvariant":549},"normal",[454,551,552],{"encoding":456},"x = F(x).",[431,554,556,577],{"className":555,"ariaHidden":461},[460],[431,557,559,562,565,570,574],{"className":558},[465],[431,560],{"className":561,"style":470},[469],[431,563,452],{"className":564},[474,475],[431,566],{"className":567,"style":569},[568],"mspace","margin-right:0.2778em;",[431,571,535],{"className":572},[573],"mrel",[431,575],{"className":576,"style":569},[568],[431,578,580,584,587,591,594,598],{"className":579},[465],[431,581],{"className":582,"style":583},[469],"height:1em;vertical-align:-0.25em;",[431,585,491],{"className":586,"style":507},[474,475],[431,588,541],{"className":589},[590],"mopen",[431,592,452],{"className":593},[474,475],[431,595,546],{"className":596},[597],"mclose",[431,599,423],{"className":600},[474],[324,602,603,604,632,633,661,662,665,666,694],{},"Here ",[431,605,607,620],{"className":606},[434],[431,608,610],{"className":609},[438],[440,611,612],{"xmlns":442},[444,613,614,618],{},[447,615,616],{},[450,617,452],{},[454,619,452],{"encoding":456},[431,621,623],{"className":622,"ariaHidden":461},[460],[431,624,626,629],{"className":625},[465],[431,627],{"className":628,"style":470},[469],[431,630,452],{"className":631},[474,475]," is the state that travels around the loop, and ",[431,634,636,649],{"className":635},[434],[431,637,639],{"className":638},[438],[440,640,641],{"xmlns":442},[444,642,643,647],{},[447,644,645],{},[450,646,491],{},[454,648,491],{"encoding":456},[431,650,652],{"className":651,"ariaHidden":461},[460],[431,653,655,658],{"className":654},[465],[431,656],{"className":657,"style":503},[469],[431,659,491],{"className":660,"style":507},[474,475]," is the ordinary physics that acts on it along the way. A consistent history is a ",[341,663,664],{},"fixed point"," of ",[431,667,669,682],{"className":668},[434],[431,670,672],{"className":671},[438],[440,673,674],{"xmlns":442},[444,675,676,680],{},[447,677,678],{},[450,679,491],{},[454,681,491],{"encoding":456},[431,683,685],{"className":684,"ariaHidden":461},[460],[431,686,688,691],{"className":687},[465],[431,689],{"className":690,"style":503},[469],[431,692,491],{"className":693,"style":507},[474,475],": a value that physics carries back to itself.",[324,696,697,698,726,727,755,756,816,817,845,846],{},"This reframes the paradox. The question \"what if the ball knocks itself away?\" becomes \"does ",[431,699,701,714],{"className":700},[434],[431,702,704],{"className":703},[438],[440,705,706],{"xmlns":442},[444,707,708,712],{},[447,709,710],{},[450,711,491],{},[454,713,491],{"encoding":456},[431,715,717],{"className":716,"ariaHidden":461},[460],[431,718,720,723],{"className":719},[465],[431,721],{"className":722,"style":503},[469],[431,724,491],{"className":725,"style":507},[474,475]," have a fixed point?\" Mathematics has general tools for that question. If ",[431,728,730,743],{"className":729},[434],[431,731,733],{"className":732},[438],[440,734,735],{"xmlns":442},[444,736,737,741],{},[447,738,739],{},[450,740,452],{},[454,742,452],{"encoding":456},[431,744,746],{"className":745,"ariaHidden":461},[460],[431,747,749,752],{"className":748},[465],[431,750],{"className":751,"style":470},[469],[431,753,452],{"className":754},[474,475]," ranges over a closed interval, such as ",[431,757,759,787],{"className":758},[434],[431,760,762],{"className":761},[438],[440,763,764],{"xmlns":442},[444,765,766,784],{},[447,767,768,771,775,778,781],{},[533,769,770],{"stretchy":540},"[",[772,773,774],"mn",{},"0",[533,776,777],{"separator":461},",",[772,779,780],{},"1",[533,782,783],{"stretchy":540},"]",[454,785,786],{"encoding":456},"[0,1]",[431,788,790],{"className":789,"ariaHidden":461},[460],[431,791,793,796,799,802,806,810,813],{"className":792},[465],[431,794],{"className":795,"style":583},[469],[431,797,770],{"className":798},[590],[431,800,774],{"className":801},[474],[431,803,777],{"className":804},[805],"mpunct",[431,807],{"className":808,"style":809},[568],"margin-right:0.1667em;",[431,811,780],{"className":812},[474],[431,814,783],{"className":815},[597],", and ",[431,818,820,833],{"className":819},[434],[431,821,823],{"className":822},[438],[440,824,825],{"xmlns":442},[444,826,827,831],{},[447,828,829],{},[450,830,491],{},[454,832,491],{"encoding":456},[431,834,836],{"className":835,"ariaHidden":461},[460],[431,837,839,842],{"className":838},[465],[431,840],{"className":841,"style":503},[469],[431,843,491],{"className":844,"style":507},[474,475]," is a continuous map from that interval back into itself, a basic theorem of topology guarantees at least one solution. The Stanford Encyclopedia of Philosophy illustrates this with a device that sends a photograph back in time to be altered: a continuous family of outcomes, such as shades of gray, guarantees at least one consistent photograph. ",[331,847,849],{"className":848},[334],"[6]",[324,851,852,853,856],{},"The same analysis also shows where trouble could hide. If the possible states form a space in which continuous maps need not have fixed points, a consistent solution is not guaranteed. ",[331,854,849],{"className":855},[334]," So self-consistency is not automatic mathematics. For each physical system, one has to check.",[346,858,860],{"id":859},"the-billiard-ball-test","The Billiard Ball Test",[324,862,863,864,867],{},"Echeverria, Klinkhammer and Thorne carried out that check for the billiard ball in 1991. ",[331,865,335],{"className":866},[334]," Their model was deliberately simple: a hard, classical ball moving slowly through ordinary space containing a wormhole time machine, able to collide with itself.",[324,869,870,871,874,875],{},"They framed the question precisely. For each initial trajectory, meaning how the ball is launched before anything emerges from the wormhole, they defined its ",[341,872,873],{},"multiplicity",": the number of self-consistent histories starting with that trajectory. ",[331,876,335],{"className":877},[334],[879,880,881,889,895],"ul",{},[882,883,884,885,888],"li",{},"Multiplicity ",[341,886,887],{},"zero"," would mean a true paradox: launching the ball that way leads to no consistent history at all.",[882,890,884,891,894],{},[341,892,893],{},"one"," would mean the ordinary situation: the future is uniquely determined.",[882,896,884,897,900],{},[341,898,899],{},"greater than one"," would mean several different consistent histories begin the same way.",[324,902,903,904,907],{},"Consider Polchinski's dangerous trajectory, aimed so that the older ball would knock the younger one entirely out of the wormhole's path. The researchers found consistent solutions in which the collision is gentler than the paradox assumes. The older ball emerges and delivers a ",[341,905,906],{},"glancing blow",". That blow deflects the younger ball only slightly, so it still enters the wormhole, but along a slightly altered path. That altered path is precisely the one that makes it emerge and deliver the same glancing blow.",[324,909,910],{},"The ball does not knock itself away. It nudges itself into the history that produces the nudge.",[324,912,913,914],{},"Two findings stood out. ",[331,915,917],{"className":916},[334],"[2,6]",[919,920,921,928],"ol",{},[882,922,923,924,927],{},"For the wide range of initial trajectories they studied, they found ",[341,925,926],{},"no"," case with multiplicity zero. Every trajectory they examined, including the paradoxical-looking ones, had at least one consistent continuation.",[882,929,930,931,934],{},"The opposite problem appeared instead. Many trajectories had ",[341,932,933],{},"more than one"," consistent continuation, and some had infinitely many. The ball might take zero, one, two, or more trips through the wormhole, and each could be consistent.",[324,936,937,938,942,943,947,948],{},"Later work widened the test. Novikov examined other systems with self-interaction and again found self-consistent solutions. ",[331,939,941],{"className":940},[334],"[3]"," Mikheeva and Novikov treated an inelastic ball, which loses some energy as heat in each collision, and reported that generic initial conditions still have self-consistent solutions. ",[331,944,946],{"className":945},[334],"[7]"," For fields rather than balls, Friedman and Morris proved that the simplest wave equation has well-defined solutions for suitable initial data on a class of wormhole spacetimes. ",[331,949,951],{"className":950},[334],"[19]",[324,953,954,955,958],{},"So the classical lesson was not what the paradox predicted. ",[341,956,957],{},"The worry was too few histories. The finding was too many."," Physics loses the ability to predict a unique outcome, which is arguably the more interesting problem.",[346,960,962],{"id":961},"why-no-special-force-is-needed","Why No Special Force Is Needed",[324,964,965],{},"A common reading of the Novikov principle imagines an agency that stops the traveler: guns jamming for mysterious reasons, brakes failing, the universe conspiring. The billiard-ball analysis shows why this picture is misleading.",[324,967,968],{},"In the consistent billiard histories, every event is ordinary. The ball follows Newton's laws at every moment. It collides with an ordinary ball. Nothing acts on it except that collision. The only unusual feature is global: the history closes on itself. The consistency is not imposed moment by moment. It is a property of the complete solution, just as a standing wave on a guitar string is not held in shape by an extra force but by the fact that only certain wave patterns fit between the fixed ends.",[324,970,971,972,975],{},"This is also why Friedman and colleagues described self-consistency as a principle added to the laws rather than a new law of motion. ",[331,973,401],{"className":974},[334]," In practice it acts as a selection rule: the equations permit many local behaviors, and the loop keeps those that fit.",[324,977,978,979,983],{},"Lewis made the same point for people. ",[331,980,982],{"className":981},[334],"[4]"," Tim's failure is guaranteed only in the sense that the history in which he fails is the one that exists. At the level of local events, it is an ordinary failure with an ordinary cause.",[324,985,986,989],{},[341,987,988],{},"Status:"," mathematical and conceptual result for simple classical systems. It is not a demonstrated law of nature, because no time machine is known to exist.",[346,991,993],{"id":992},"where-the-principle-is-contested","Where the Principle Is Contested",[324,995,996],{},"The self-consistency principle is widely accepted as the natural classical approach, but several objections remain serious.",[998,999,1001],"h3",{"id":1000},"strange-constraints-on-the-past","Strange constraints on the past",[324,1003,1004,1005,1008,1009,1013],{},"If consistent solutions exist only for certain initial conditions, then the existence of a time machine in the future would restrict what can happen long before it is built. Philosophers of physics note that such constraints are unlike ordinary laws: they are non-local and depend on the global shape of spacetime. ",[331,1006,849],{"className":1007},[334]," Rama and Sen argued in a 1994 preprint that some macroscopic setups have no consistent classical solution at all, and that excluding them by fiat would imply odd influences reaching far into the past. ",[331,1010,1012],{"className":1011},[334],"[8]"," Their conclusion was that time machines of that kind should not exist. The claim remains a preprint argument rather than an accepted result.",[998,1015,1017],{"id":1016},"does-geometry-adjust-too","Does geometry adjust too?",[324,1019,1020,1021,1025],{},"Krasnikov distinguished between spacetimes with a fixed, given geometry and full General Relativity, in which geometry itself obeys equations. He constructed an explicit example of a paradox of the first kind and argued that in theories where geometry is governed only by differential equations and initial data, paradoxes do not arise. ",[331,1022,1024],{"className":1023},[334],"[9]"," One reading of this work is that a paradoxical setup would simply not produce the time machine it seems to need.",[998,1027,1029],{"id":1028},"loss-of-predictability","Loss of predictability",[324,1031,1032,1033,1036],{},"The billiard results suggest that, even without contradiction, classical physics near a time machine does not choose a unique history. ",[331,1034,335],{"className":1035},[334]," Some physicists hoped that quantum mechanics, by assigning probabilities to different histories, would resolve this. That hope helped motivate the quantum models below.",[998,1038,1040],{"id":1039},"is-self-consistency-a-principle-or-a-consequence","Is self-consistency a principle or a consequence?",[324,1042,1043,1044],{},"Some authors treat self-consistency as an extra postulate; others see it as nothing more than the requirement that the equations of physics have a solution. The distinction matters philosophically, because a postulate can be questioned. Earman, Smeenk and Wüthrich give a detailed analysis of how global constraints of this kind fit, or fail to fit, the usual understanding of physical law. ",[331,1045,1047],{"className":1046},[334],"[20]",[346,1049,1051],{"id":1050},"quantum-alternatives","Quantum Alternatives",[324,1053,1054],{},"Quantum mechanics changes the problem, because a quantum state can be a superposition or a mixture of classical possibilities. Two main models have been developed.",[998,1056,1058],{"id":1057},"deutschs-model-1991","Deutsch's model (1991)",[324,1060,1061,1062,1066,1067,1165,1166,1243,1244,1274],{},"David Deutsch proposed treating the part of a system that travels around the loop as a quantum system whose state must be consistent in a statistical sense. ",[331,1063,1065],{"className":1064},[334],"[10]"," Let ",[431,1068,1070,1092],{"className":1069},[434],[431,1071,1073],{"className":1072},[438],[440,1074,1075],{"xmlns":442},[444,1076,1077,1089],{},[447,1078,1079],{},[1080,1081,1082,1085],"msub",{},[450,1083,1084],{},"ρ",[1086,1087,1088],"mtext",{},"CR",[454,1090,1091],{"encoding":456},"\\rho_{\\text{CR}}",[431,1093,1095],{"className":1094,"ariaHidden":461},[460],[431,1096,1098,1102],{"className":1097},[465],[431,1099],{"className":1100,"style":1101},[469],"height:0.625em;vertical-align:-0.1944em;",[431,1103,1105,1108],{"className":1104},[474],[431,1106,1084],{"className":1107},[474,475],[431,1109,1112],{"className":1110},[1111],"msupsub",[431,1113,1117,1156],{"className":1114},[1115,1116],"vlist-t","vlist-t2",[431,1118,1121,1151],{"className":1119},[1120],"vlist-r",[431,1122,1126],{"className":1123,"style":1125},[1124],"vlist","height:0.3283em;",[431,1127,1129,1134],{"style":1128},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[431,1130],{"className":1131,"style":1133},[1132],"pstrut","height:2.7em;",[431,1135,1141],{"className":1136},[1137,1138,1139,1140],"sizing","reset-size6","size3","mtight",[431,1142,1144],{"className":1143},[474,1140],[431,1145,1148],{"className":1146},[474,1147,1140],"text",[431,1149,1088],{"className":1150},[474,1140],[431,1152,1155],{"className":1153},[1154],"vlist-s","​",[431,1157,1159],{"className":1158},[1120],[431,1160,1163],{"className":1161,"style":1162},[1124],"height:0.15em;",[431,1164],{}," be the state of the ordinary system that stays in normal time (chronology-respecting), ",[431,1167,1169,1188],{"className":1168},[434],[431,1170,1172],{"className":1171},[438],[440,1173,1174],{"xmlns":442},[444,1175,1176,1185],{},[447,1177,1178],{},[1080,1179,1180,1182],{},[450,1181,1084],{},[1086,1183,1184],{},"CTC",[454,1186,1187],{"encoding":456},"\\rho_{\\text{CTC}}",[431,1189,1191],{"className":1190,"ariaHidden":461},[460],[431,1192,1194,1197],{"className":1193},[465],[431,1195],{"className":1196,"style":1101},[469],[431,1198,1200,1203],{"className":1199},[474],[431,1201,1084],{"className":1202},[474,475],[431,1204,1206],{"className":1205},[1111],[431,1207,1209,1235],{"className":1208},[1115,1116],[431,1210,1212,1232],{"className":1211},[1120],[431,1213,1215],{"className":1214,"style":1125},[1124],[431,1216,1217,1220],{"style":1128},[431,1218],{"className":1219,"style":1133},[1132],[431,1221,1223],{"className":1222},[1137,1138,1139,1140],[431,1224,1226],{"className":1225},[474,1140],[431,1227,1229],{"className":1228},[474,1147,1140],[431,1230,1184],{"className":1231},[474,1140],[431,1233,1155],{"className":1234},[1154],[431,1236,1238],{"className":1237},[1120],[431,1239,1241],{"className":1240,"style":1162},[1124],[431,1242],{}," the state of the system on the loop, and ",[431,1245,1247,1261],{"className":1246},[434],[431,1248,1250],{"className":1249},[438],[440,1251,1252],{"xmlns":442},[444,1253,1254,1259],{},[447,1255,1256],{},[450,1257,1258],{},"U",[454,1260,1258],{"encoding":456},[431,1262,1264],{"className":1263,"ariaHidden":461},[460],[431,1265,1267,1270],{"className":1266},[465],[431,1268],{"className":1269,"style":503},[469],[431,1271,1258],{"className":1272,"style":1273},[474,475],"margin-right:0.109em;"," the quantum interaction between them. Deutsch's consistency condition is",[510,1276,1278],{"label":1277},"Deutsch consistency condition",[431,1279,1281],{"className":1280},[516],[431,1282,1284,1361],{"className":1283},[434],[431,1285,1287],{"className":1286},[438],[440,1288,1289],{"xmlns":442,"display":525},[444,1290,1291,1358],{},[447,1292,1293,1299,1301,1313,1316,1356],{},[1080,1294,1295,1297],{},[450,1296,1084],{},[1086,1298,1184],{},[533,1300,535],{},[1080,1302,1303,1311],{},[447,1304,1305,1308],{},[450,1306,1307],{"mathvariant":549},"Tr",[533,1309,1310],{},"⁡",[1086,1312,1088],{},[1086,1314,1315],{}," ⁣",[447,1317,1318,1320,1323,1325,1346,1354],{},[533,1319,770],{"fence":461},[1086,1321,1322],{}," ",[450,1324,1258],{},[447,1326,1327,1329,1335,1338,1344],{},[533,1328,541],{"fence":461},[1080,1330,1331,1333],{},[450,1332,1084],{},[1086,1334,1088],{},[533,1336,1337],{},"⊗",[1080,1339,1340,1342],{},[450,1341,1084],{},[1086,1343,1184],{},[533,1345,546],{"fence":461},[1347,1348,1349,1351],"msup",{},[450,1350,1258],{},[533,1352,1353],{},"†",[533,1355,783],{"fence":461},[450,1357,423],{"mathvariant":549},[454,1359,1360],{"encoding":456},"\\rho_{\\text{CTC}}\n= \\operatorname{Tr}_{\\text{CR}}\\!\\left[\\,U\\left(\\rho_{\\text{CR}}\\otimes\\rho_{\\text{CTC}}\\right)U^\\dagger\\right].",[431,1362,1364,1425],{"className":1363,"ariaHidden":461},[460],[431,1365,1367,1370,1416,1419,1422],{"className":1366},[465],[431,1368],{"className":1369,"style":1101},[469],[431,1371,1373,1376],{"className":1372},[474],[431,1374,1084],{"className":1375},[474,475],[431,1377,1379],{"className":1378},[1111],[431,1380,1382,1408],{"className":1381},[1115,1116],[431,1383,1385,1405],{"className":1384},[1120],[431,1386,1388],{"className":1387,"style":1125},[1124],[431,1389,1390,1393],{"style":1128},[431,1391],{"className":1392,"style":1133},[1132],[431,1394,1396],{"className":1395},[1137,1138,1139,1140],[431,1397,1399],{"className":1398},[474,1140],[431,1400,1402],{"className":1401},[474,1147,1140],[431,1403,1184],{"className":1404},[474,1140],[431,1406,1155],{"className":1407},[1154],[431,1409,1411],{"className":1410},[1120],[431,1412,1414],{"className":1413,"style":1162},[1124],[431,1415],{},[431,1417],{"className":1418,"style":569},[568],[431,1420,535],{"className":1421},[573],[431,1423],{"className":1424,"style":569},[568],[431,1426,1428,1432,1484,1488,1491,1666,1669],{"className":1427},[465],[431,1429],{"className":1430,"style":1431},[469],"height:1.2491em;vertical-align:-0.35em;",[431,1433,1436,1443],{"className":1434},[1435],"mop",[431,1437,1439],{"className":1438},[1435],[431,1440,1307],{"className":1441},[474,1442],"mathrm",[431,1444,1446],{"className":1445},[1111],[431,1447,1449,1476],{"className":1448},[1115,1116],[431,1450,1452,1473],{"className":1451},[1120],[431,1453,1455],{"className":1454,"style":1125},[1124],[431,1456,1458,1461],{"style":1457},"top:-2.55em;margin-right:0.05em;",[431,1459],{"className":1460,"style":1133},[1132],[431,1462,1464],{"className":1463},[1137,1138,1139,1140],[431,1465,1467],{"className":1466},[474,1140],[431,1468,1470],{"className":1469},[474,1147,1140],[431,1471,1088],{"className":1472},[474,1140],[431,1474,1155],{"className":1475},[1154],[431,1477,1479],{"className":1478},[1120],[431,1480,1482],{"className":1481,"style":1162},[1124],[431,1483],{},[431,1485],{"className":1486,"style":1487},[568],"margin-right:-0.1667em;",[431,1489],{"className":1490,"style":809},[568],[431,1492,1495,1505,1508,1511,1514,1626,1629,1660],{"className":1493},[1494],"minner",[431,1496,1500],{"className":1497,"style":1499},[590,1498],"delimcenter","top:0em;",[431,1501,770],{"className":1502},[1503,1504],"delimsizing","size1",[431,1506],{"className":1507,"style":809},[568],[431,1509,1258],{"className":1510,"style":1273},[474,475],[431,1512],{"className":1513,"style":809},[568],[431,1515,1517,1520,1566,1570,1574,1577,1623],{"className":1516},[1494],[431,1518,541],{"className":1519,"style":1499},[590,1498],[431,1521,1523,1526],{"className":1522},[474],[431,1524,1084],{"className":1525},[474,475],[431,1527,1529],{"className":1528},[1111],[431,1530,1532,1558],{"className":1531},[1115,1116],[431,1533,1535,1555],{"className":1534},[1120],[431,1536,1538],{"className":1537,"style":1125},[1124],[431,1539,1540,1543],{"style":1128},[431,1541],{"className":1542,"style":1133},[1132],[431,1544,1546],{"className":1545},[1137,1138,1139,1140],[431,1547,1549],{"className":1548},[474,1140],[431,1550,1552],{"className":1551},[474,1147,1140],[431,1553,1088],{"className":1554},[474,1140],[431,1556,1155],{"className":1557},[1154],[431,1559,1561],{"className":1560},[1120],[431,1562,1564],{"className":1563,"style":1162},[1124],[431,1565],{},[431,1567],{"className":1568,"style":1569},[568],"margin-right:0.2222em;",[431,1571,1337],{"className":1572},[1573],"mbin",[431,1575],{"className":1576,"style":1569},[568],[431,1578,1580,1583],{"className":1579},[474],[431,1581,1084],{"className":1582},[474,475],[431,1584,1586],{"className":1585},[1111],[431,1587,1589,1615],{"className":1588},[1115,1116],[431,1590,1592,1612],{"className":1591},[1120],[431,1593,1595],{"className":1594,"style":1125},[1124],[431,1596,1597,1600],{"style":1128},[431,1598],{"className":1599,"style":1133},[1132],[431,1601,1603],{"className":1602},[1137,1138,1139,1140],[431,1604,1606],{"className":1605},[474,1140],[431,1607,1609],{"className":1608},[474,1147,1140],[431,1610,1184],{"className":1611},[474,1140],[431,1613,1155],{"className":1614},[1154],[431,1616,1618],{"className":1617},[1120],[431,1619,1621],{"className":1620,"style":1162},[1124],[431,1622],{},[431,1624,546],{"className":1625,"style":1499},[597,1498],[431,1627],{"className":1628,"style":809},[568],[431,1630,1632,1635],{"className":1631},[474],[431,1633,1258],{"className":1634,"style":1273},[474,475],[431,1636,1638],{"className":1637},[1111],[431,1639,1641],{"className":1640},[1115],[431,1642,1644],{"className":1643},[1120],[431,1645,1648],{"className":1646,"style":1647},[1124],"height:0.8991em;",[431,1649,1651,1654],{"style":1650},"top:-3.113em;margin-right:0.05em;",[431,1652],{"className":1653,"style":1133},[1132],[431,1655,1657],{"className":1656},[1137,1138,1139,1140],[431,1658,1353],{"className":1659},[1573,1140],[431,1661,1663],{"className":1662,"style":1499},[597,1498],[431,1664,783],{"className":1665},[1503,1504],[431,1667],{"className":1668,"style":809},[568],[431,1670,423],{"className":1671},[474],[324,1673,603,1674,1704,1705,1789,1790,1793,1794],{},[431,1675,1677,1691],{"className":1676},[434],[431,1678,1680],{"className":1679},[438],[440,1681,1682],{"xmlns":442},[444,1683,1684,1688],{},[447,1685,1686],{},[533,1687,1337],{},[454,1689,1690],{"encoding":456},"\\otimes",[431,1692,1694],{"className":1693,"ariaHidden":461},[460],[431,1695,1697,1701],{"className":1696},[465],[431,1698],{"className":1699,"style":1700},[469],"height:0.6667em;vertical-align:-0.0833em;",[431,1702,1337],{"className":1703},[474]," combines the two systems, and ",[431,1706,1708,1730],{"className":1707},[434],[431,1709,1711],{"className":1710},[438],[440,1712,1713],{"xmlns":442},[444,1714,1715,1727],{},[447,1716,1717],{},[1080,1718,1719,1725],{},[447,1720,1721,1723],{},[450,1722,1307],{"mathvariant":549},[533,1724,1310],{},[1086,1726,1088],{},[454,1728,1729],{"encoding":456},"\\operatorname{Tr}_{\\text{CR}}",[431,1731,1733],{"className":1732,"ariaHidden":461},[460],[431,1734,1736,1740],{"className":1735},[465],[431,1737],{"className":1738,"style":1739},[469],"height:0.8333em;vertical-align:-0.15em;",[431,1741,1743,1749],{"className":1742},[1435],[431,1744,1746],{"className":1745},[1435],[431,1747,1307],{"className":1748},[474,1442],[431,1750,1752],{"className":1751},[1111],[431,1753,1755,1781],{"className":1754},[1115,1116],[431,1756,1758,1778],{"className":1757},[1120],[431,1759,1761],{"className":1760,"style":1125},[1124],[431,1762,1763,1766],{"style":1457},[431,1764],{"className":1765,"style":1133},[1132],[431,1767,1769],{"className":1768},[1137,1138,1139,1140],[431,1770,1772],{"className":1771},[474,1140],[431,1773,1775],{"className":1774},[474,1147,1140],[431,1776,1088],{"className":1777},[474,1140],[431,1779,1155],{"className":1780},[1154],[431,1782,1784],{"className":1783},[1120],[431,1785,1787],{"className":1786,"style":1162},[1124],[431,1788],{}," (a partial trace) discards the chronology-respecting part to leave only the loop's state. In words: whatever state goes back in time must equal the state that came out of the past. Deutsch showed that this equation ",[341,1791,1792],{},"always"," has at least one solution, whatever the initial state and interaction. ",[331,1795,1797],{"className":1796},[334],"[10,6]",[324,1799,1800],{},"Take the grandfather paradox in its simplest form. A single bit travels around the loop, and the interaction flips it: whatever emerges as 0 goes back as 1, and vice versa. Classically there is no consistent value. In Deutsch's model there is: an equal mixture of 0 and 1. Deutsch interpreted this in terms of many worlds. The traveler arrives from a history in which the grandfather lived and enters one in which he dies.",[324,1802,1803,1804,1807,1808,1812,1813,1817],{},"The price is high. Deutsch's model is ",[341,1805,1806],{},"nonlinear",", unlike ordinary quantum mechanics, and it produces startling computational power: Aaronson and Watrous proved that computers using Deutsch-type loops could efficiently solve every problem in the class PSPACE. ",[331,1809,1811],{"className":1810},[334],"[13]"," Bennett and colleagues argued that some claimed Deutsch-model effects depend on treating mixed states in a way that is ambiguous when the input is entangled with other systems. ",[331,1814,1816],{"className":1815},[334],"[14]"," Deutsch's model also gives up the single consistent history that the Novikov principle describes.",[998,1819,1821],{"id":1820},"lloyds-post-selected-model-2011","Lloyd's post-selected model (2011)",[324,1823,1824,1825,1828,1829,1833],{},"Seth Lloyd and colleagues proposed a different model built from quantum teleportation combined with ",[341,1826,1827],{},"post-selection",", keeping only runs of an experiment in which a particular measurement outcome occurs. ",[331,1830,1832],{"className":1831},[334],"[11,12]"," The loop is modeled as a teleportation channel that sends a state backward, and only histories consistent with that channel are kept.",[324,1835,1836,1837,1959],{},"In this model, the effective operation on the ordinary system is determined by the partial trace of the interaction over the loop, ",[431,1838,1840,1873],{"className":1839},[434],[431,1841,1843],{"className":1842},[438],[440,1844,1845],{"xmlns":442},[444,1846,1847,1870],{},[447,1848,1849,1852,1854,1864,1866,1868],{},[450,1850,1851],{},"C",[533,1853,535],{},[1080,1855,1856,1862],{},[447,1857,1858,1860],{},[450,1859,1307],{"mathvariant":549},[533,1861,1310],{},[1086,1863,1184],{},[533,1865,541],{"stretchy":540},[450,1867,1258],{},[533,1869,546],{"stretchy":540},[454,1871,1872],{"encoding":456},"C = \\operatorname{Tr}_{\\text{CTC}}(U)",[431,1874,1876,1895],{"className":1875,"ariaHidden":461},[460],[431,1877,1879,1882,1886,1889,1892],{"className":1878},[465],[431,1880],{"className":1881,"style":503},[469],[431,1883,1851],{"className":1884,"style":1885},[474,475],"margin-right:0.0715em;",[431,1887],{"className":1888,"style":569},[568],[431,1890,535],{"className":1891},[573],[431,1893],{"className":1894,"style":569},[568],[431,1896,1898,1901,1950,1953,1956],{"className":1897},[465],[431,1899],{"className":1900,"style":583},[469],[431,1902,1904,1910],{"className":1903},[1435],[431,1905,1907],{"className":1906},[1435],[431,1908,1307],{"className":1909},[474,1442],[431,1911,1913],{"className":1912},[1111],[431,1914,1916,1942],{"className":1915},[1115,1116],[431,1917,1919,1939],{"className":1918},[1120],[431,1920,1922],{"className":1921,"style":1125},[1124],[431,1923,1924,1927],{"style":1457},[431,1925],{"className":1926,"style":1133},[1132],[431,1928,1930],{"className":1929},[1137,1138,1139,1140],[431,1931,1933],{"className":1932},[474,1140],[431,1934,1936],{"className":1935},[474,1147,1140],[431,1937,1184],{"className":1938},[474,1140],[431,1940,1155],{"className":1941},[1154],[431,1943,1945],{"className":1944},[1120],[431,1946,1948],{"className":1947,"style":1162},[1124],[431,1949],{},[431,1951,541],{"className":1952},[590],[431,1954,1258],{"className":1955,"style":1273},[474,475],[431,1957,546],{"className":1958},[597],", and the output is renormalized:",[431,1961,1963],{"className":1962},[516],[431,1964,1966,2045],{"className":1965},[434],[431,1967,1969],{"className":1968},[438],[440,1970,1971],{"xmlns":442,"display":525},[444,1972,1973,2042],{},[447,1974,1975,1982,1984,2040],{},[1080,1976,1977,1979],{},[450,1978,1084],{},[1086,1980,1981],{},"out",[533,1983,535],{},[1985,1986,1987,2008],"mfrac",{},[447,1988,1989,1991,1993,2000,2002],{},[450,1990,1851],{},[1086,1992,1322],{},[1080,1994,1995,1997],{},[450,1996,1084],{},[1086,1998,1999],{},"in",[1086,2001,1322],{},[1347,2003,2004,2006],{},[450,2005,1851],{},[533,2007,1353],{},[447,2009,2010,2012,2014,2016],{},[450,2011,1307],{"mathvariant":549},[533,2013,1310],{},[1086,2015,1315],{},[447,2017,2018,2020,2022,2024,2030,2032,2038],{},[533,2019,541],{"fence":461},[450,2021,1851],{},[1086,2023,1322],{},[1080,2025,2026,2028],{},[450,2027,1084],{},[1086,2029,1999],{},[1086,2031,1322],{},[1347,2033,2034,2036],{},[450,2035,1851],{},[533,2037,1353],{},[533,2039,546],{"fence":461},[450,2041,423],{"mathvariant":549},[454,2043,2044],{"encoding":456},"\\rho_{\\text{out}} = \\frac{C\\,\\rho_{\\text{in}}\\,C^\\dagger}{\\operatorname{Tr}\\!\\left(C\\,\\rho_{\\text{in}}\\,C^\\dagger\\right)} .",[431,2046,2048,2110],{"className":2047,"ariaHidden":461},[460],[431,2049,2051,2054,2101,2104,2107],{"className":2050},[465],[431,2052],{"className":2053,"style":1101},[469],[431,2055,2057,2060],{"className":2056},[474],[431,2058,1084],{"className":2059},[474,475],[431,2061,2063],{"className":2062},[1111],[431,2064,2066,2093],{"className":2065},[1115,1116],[431,2067,2069,2090],{"className":2068},[1120],[431,2070,2073],{"className":2071,"style":2072},[1124],"height:0.2806em;",[431,2074,2075,2078],{"style":1128},[431,2076],{"className":2077,"style":1133},[1132],[431,2079,2081],{"className":2080},[1137,1138,1139,1140],[431,2082,2084],{"className":2083},[474,1140],[431,2085,2087],{"className":2086},[474,1147,1140],[431,2088,1981],{"className":2089},[474,1140],[431,2091,1155],{"className":2092},[1154],[431,2094,2096],{"className":2095},[1120],[431,2097,2099],{"className":2098,"style":1162},[1124],[431,2100],{},[431,2102],{"className":2103,"style":569},[568],[431,2105,535],{"className":2106},[573],[431,2108],{"className":2109,"style":569},[568],[431,2111,2113,2117,2376],{"className":2112},[465],[431,2114],{"className":2115,"style":2116},[469],"height:2.4621em;vertical-align:-0.936em;",[431,2118,2120,2124,2373],{"className":2119},[474],[431,2121],{"className":2122},[590,2123],"nulldelimiter",[431,2125,2127],{"className":2126},[1985],[431,2128,2130,2364],{"className":2129},[1115,1116],[431,2131,2133,2361],{"className":2132},[1120],[431,2134,2137,2255,2266],{"className":2135,"style":2136},[1124],"height:1.5261em;",[431,2138,2140,2144],{"style":2139},"top:-2.314em;",[431,2141],{"className":2142,"style":2143},[1132],"height:3em;",[431,2145,2147,2153,2156,2159],{"className":2146},[474],[431,2148,2150],{"className":2149},[1435],[431,2151,1307],{"className":2152},[474,1442],[431,2154],{"className":2155,"style":1487},[568],[431,2157],{"className":2158,"style":809},[568],[431,2160,2162,2165,2168,2171,2218,2221,2252],{"className":2161},[1494],[431,2163,541],{"className":2164,"style":1499},[590,1498],[431,2166,1851],{"className":2167,"style":1885},[474,475],[431,2169],{"className":2170,"style":809},[568],[431,2172,2174,2177],{"className":2173},[474],[431,2175,1084],{"className":2176},[474,475],[431,2178,2180],{"className":2179},[1111],[431,2181,2183,2210],{"className":2182},[1115,1116],[431,2184,2186,2207],{"className":2185},[1120],[431,2187,2190],{"className":2188,"style":2189},[1124],"height:0.3175em;",[431,2191,2192,2195],{"style":1128},[431,2193],{"className":2194,"style":1133},[1132],[431,2196,2198],{"className":2197},[1137,1138,1139,1140],[431,2199,2201],{"className":2200},[474,1140],[431,2202,2204],{"className":2203},[474,1147,1140],[431,2205,1999],{"className":2206},[474,1140],[431,2208,1155],{"className":2209},[1154],[431,2211,2213],{"className":2212},[1120],[431,2214,2216],{"className":2215,"style":1162},[1124],[431,2217],{},[431,2219],{"className":2220,"style":809},[568],[431,2222,2224,2227],{"className":2223},[474],[431,2225,1851],{"className":2226,"style":1885},[474,475],[431,2228,2230],{"className":2229},[1111],[431,2231,2233],{"className":2232},[1115],[431,2234,2236],{"className":2235},[1120],[431,2237,2240],{"className":2238,"style":2239},[1124],"height:0.7751em;",[431,2241,2243,2246],{"style":2242},"top:-2.989em;margin-right:0.05em;",[431,2244],{"className":2245,"style":1133},[1132],[431,2247,2249],{"className":2248},[1137,1138,1139,1140],[431,2250,1353],{"className":2251},[1573,1140],[431,2253,546],{"className":2254,"style":1499},[597,1498],[431,2256,2258,2261],{"style":2257},"top:-3.23em;",[431,2259],{"className":2260,"style":2143},[1132],[431,2262],{"className":2263,"style":2265},[2264],"frac-line","border-bottom-width:0.04em;",[431,2267,2269,2272],{"style":2268},"top:-3.677em;",[431,2270],{"className":2271,"style":2143},[1132],[431,2273,2275,2278,2281,2327,2330],{"className":2274},[474],[431,2276,1851],{"className":2277,"style":1885},[474,475],[431,2279],{"className":2280,"style":809},[568],[431,2282,2284,2287],{"className":2283},[474],[431,2285,1084],{"className":2286},[474,475],[431,2288,2290],{"className":2289},[1111],[431,2291,2293,2319],{"className":2292},[1115,1116],[431,2294,2296,2316],{"className":2295},[1120],[431,2297,2299],{"className":2298,"style":2189},[1124],[431,2300,2301,2304],{"style":1128},[431,2302],{"className":2303,"style":1133},[1132],[431,2305,2307],{"className":2306},[1137,1138,1139,1140],[431,2308,2310],{"className":2309},[474,1140],[431,2311,2313],{"className":2312},[474,1147,1140],[431,2314,1999],{"className":2315},[474,1140],[431,2317,1155],{"className":2318},[1154],[431,2320,2322],{"className":2321},[1120],[431,2323,2325],{"className":2324,"style":1162},[1124],[431,2326],{},[431,2328],{"className":2329,"style":809},[568],[431,2331,2333,2336],{"className":2332},[474],[431,2334,1851],{"className":2335,"style":1885},[474,475],[431,2337,2339],{"className":2338},[1111],[431,2340,2342],{"className":2341},[1115],[431,2343,2345],{"className":2344},[1120],[431,2346,2349],{"className":2347,"style":2348},[1124],"height:0.8491em;",[431,2350,2352,2355],{"style":2351},"top:-3.063em;margin-right:0.05em;",[431,2353],{"className":2354,"style":1133},[1132],[431,2356,2358],{"className":2357},[1137,1138,1139,1140],[431,2359,1353],{"className":2360},[1573,1140],[431,2362,1155],{"className":2363},[1154],[431,2365,2367],{"className":2366},[1120],[431,2368,2371],{"className":2369,"style":2370},[1124],"height:0.936em;",[431,2372],{},[431,2374],{"className":2375},[597,2123],[431,2377,423],{"className":2378},[474],[324,2380,2381,2382,2434,2435],{},"For the bit-flipping grandfather loop, ",[431,2383,2385,2403],{"className":2384},[434],[431,2386,2388],{"className":2387},[438],[440,2389,2390],{"xmlns":442},[444,2391,2392,2400],{},[447,2393,2394,2396,2398],{},[450,2395,1851],{},[533,2397,535],{},[772,2399,774],{},[454,2401,2402],{"encoding":456},"C = 0",[431,2404,2406,2424],{"className":2405,"ariaHidden":461},[460],[431,2407,2409,2412,2415,2418,2421],{"className":2408},[465],[431,2410],{"className":2411,"style":503},[469],[431,2413,1851],{"className":2414,"style":1885},[474,475],[431,2416],{"className":2417,"style":569},[568],[431,2419,535],{"className":2420},[573],[431,2422],{"className":2423,"style":569},[568],[431,2425,2427,2431],{"className":2426},[465],[431,2428],{"className":2429,"style":2430},[469],"height:0.6444em;",[431,2432,774],{"className":2433},[474],": the paradoxical history has zero probability amplitude and simply does not occur. When the setup includes some small chance of a non-paradoxical outcome, such as a gun that can misfire, renormalization raises that outcome to certainty. The authors describe their model as a natural quantum version of the Novikov principle: only self-consistent sequences of events happen. ",[331,2436,2438],{"className":2437},[334],"[11]",[324,2440,2441,2442,2445,2446,2450,2451],{},"Lloyd and colleagues tested the consistency of their scheme in a photon experiment that simulates a post-selected loop. In the \"quantum gun\" version, the paradoxical outcome, a photon removing its own past self, was not observed within experimental uncertainty. ",[331,2443,2438],{"className":2444},[334]," Ringbauer and colleagues later simulated Deutsch-type dynamics with photons. ",[331,2447,2449],{"className":2448},[334],"[15]"," These experiments simulate the mathematics of CTC models in ordinary laboratories. ",[341,2452,2453],{},"They do not send anything back in time.",[998,2455,2457],{"id":2456},"many-histories","Many histories",[324,2459,2460,2461,2465],{},"A third family of ideas allows the past to change by sending the traveler into a different history, using branching spacetimes or multiple copies of spacetime. Hauser and Shoshany analyzed such models mathematically and showed that, under certain assumptions, histories can form finite cycles. ",[331,2462,2464],{"className":2463},[334],"[17]"," This abandons the single-history picture that Novikov's principle assumes. It avoids paradox by giving up the idea that \"the past\" names one thing.",[346,2467,2469],{"id":2468},"evidence-and-limitations","Evidence and Limitations",[324,2471,2472],{},"The certainty levels here should be kept apart.",[879,2474,2475,2485,2495,2503,2513,2522,2527],{},[882,2476,2477,2480,2481],{},[341,2478,2479],{},"Established fact:"," No closed timelike curve or time machine has been observed. ",[331,2482,2484],{"className":2483},[334],"[18]",[882,2486,2487,2490,2491],{},[341,2488,2489],{},"Mathematical result:"," In simple classical models of wormhole time machines, consistent histories exist for every initial condition examined, often in large numbers. ",[331,2492,2494],{"className":2493},[334],"[2,3,7]",[882,2496,2497,2499,2500],{},[341,2498,2489],{}," Deutsch's quantum model always admits a consistent state. ",[331,2501,1065],{"className":2502},[334],[882,2504,2505,2508,2509],{},[341,2506,2507],{},"Laboratory simulation:"," Photonic experiments reproduce the predictions of CTC models without any real time travel. ",[331,2510,2512],{"className":2511},[334],"[11,15]",[882,2514,2515,2518,2519],{},[341,2516,2517],{},"Open question:"," Whether self-consistency holds for all physical systems, including complex macroscopic ones. ",[331,2520,1012],{"className":2521},[334],[882,2523,2524,2526],{},[341,2525,2517],{}," Which model, if any, nature would follow. The classical principle, Deutsch's model, Lloyd's model and multiple-history models make different predictions, and no real CTC is available to test them.",[882,2528,2529,2532,2533],{},[341,2530,2531],{},"Philosophical interpretation:"," Lewis's analysis of \"can\" and \"cannot\" explains why the principle need not deny ordinary agency. It is a conceptual argument, not a physical result. ",[331,2534,373],{"className":2535},[334],[324,2537,2538,2539,386,2541],{},"Most physicists also take seriously the possibility that the question never arises in practice, because the laws of physics may prevent time machines from forming (see Hawking's chronology protection conjecture, discussed under ",[357,2540,215],{"href":214},[331,2542,2484],{"className":2543},[334],[346,2545,2547],{"id":2546},"the-question-changes","The Question Changes",[324,2549,2550],{},"The Novikov principle matters because it changes the question asked of time travel. The popular question is whether the past can be changed. Relativity answers that there is only one spacetime, and it already contains every visit. The better question is which self-consistent histories exist, how many there are, and how nature would choose among them.",[324,2552,2553],{},"That shift has useful consequences. It shows that time travel paradoxes are problems about the existence and uniqueness of solutions, a familiar kind of mathematical question. It explains why the classical difficulty turned out to be too many consistent histories rather than none. And it gave quantum theorists a precise target: a model that explains which consistent history happens, and with what probability.",[324,2555,2556],{},"Simple systems in time-machine spacetimes admit consistent histories without a special force policing them. The past in those histories can be visited but never revised; ordinary physics supplies the consistency. No one knows whether this works for every system, which quantum description is right, or whether nature allows the loops in the first place.",[324,2558,2559,2560,423],{},"The same logic reaches its strangest form when a consistent loop produces something with no origin at all. That is the subject of the ",[357,2561,235],{"href":234},[320,2563,2565],{"id":2564},"references","References",[324,2567,2568,2571,2572,2576,2577,2580,2581],{},[341,2569,401,2570],{}," Friedman, J., Morris, M. S., Novikov, I. D., Echeverria, F., Klinkhammer, G., Thorne, K. S., & Yurtsever, U. (1990)."," \"Cauchy problem in spacetimes with closed timelike curves.\" ",[2573,2574,2575],"em",{},"Physical Review D",", ",[341,2578,2579],{},"42",", 1915–1930.\n",[357,2582,2583],{"href":2583,"rel":2584},"https:\u002F\u002Fdoi.org\u002F10.1103\u002FPhysRevD.42.1915",[2585],"nofollow",[324,2587,2588,2591,2592,2576,2594,2597,2598],{},[341,2589,335,2590],{}," Echeverria, F., Klinkhammer, G., & Thorne, K. S. (1991)."," \"Billiard balls in wormhole spacetimes with closed timelike curves: Classical theory.\" ",[2573,2593,2575],{},[341,2595,2596],{},"44",", 1077–1099.\n",[357,2599,2600],{"href":2600,"rel":2601},"https:\u002F\u002Fdoi.org\u002F10.1103\u002FPhysRevD.44.1077",[2585],[324,2603,2604,2607,2608,2576,2610,2613,2614],{},[341,2605,941,2606],{}," Novikov, I. D. (1992)."," \"Time machine and self-consistent evolution in problems with self-interaction.\" ",[2573,2609,2575],{},[341,2611,2612],{},"45",", 1989–1994.\n",[357,2615,2616],{"href":2616,"rel":2617},"https:\u002F\u002Fdoi.org\u002F10.1103\u002FPhysRevD.45.1989",[2585],[324,2619,2620,2623,2624,2576,2627,2630,2631],{},[341,2621,982,2622],{}," Lewis, D. (1976)."," \"The Paradoxes of Time Travel.\" ",[2573,2625,2626],{},"American Philosophical Quarterly",[341,2628,2629],{},"13","(2), 145–152.\n",[357,2632,2633],{"href":2633,"rel":2634},"https:\u002F\u002Fwww.jstor.org\u002Fstable\u002F20009616",[2585],[324,2636,2637,2641,2642,2645,2646],{},[341,2638,2639,2640],{},"[5]"," Smith, N. J. J."," \"Time Travel.\" ",[2573,2643,2644],{},"Stanford Encyclopedia of Philosophy"," (first published 2013; revised 2024).\n",[357,2647,2648],{"href":2648,"rel":2649},"https:\u002F\u002Fplato.stanford.edu\u002Fentries\u002Ftime-travel\u002F",[2585],[324,2651,2652,2655,2656,2658,2659],{},[341,2653,849,2654],{}," Smeenk, C., Arntzenius, F., & Maudlin, T."," \"Time Travel and Modern Physics.\" ",[2573,2657,2644],{}," (first published 2000; revised 2023).\n",[357,2660,2661],{"href":2661,"rel":2662},"https:\u002F\u002Fplato.stanford.edu\u002Fentries\u002Ftime-travel-phys\u002F",[2585],[324,2664,2665,2668,2669,2576,2671,2674,2675],{},[341,2666,946,2667],{}," Mikheeva, E. V., & Novikov, I. D. (1993)."," \"Inelastic billiard ball in a spacetime with a time machine.\" ",[2573,2670,2575],{},[341,2672,2673],{},"47",", 1432–1436.\n",[357,2676,2677],{"href":2677,"rel":2678},"https:\u002F\u002Fdoi.org\u002F10.1103\u002FPhysRevD.47.1432",[2585],[324,2680,2681,2684,2685],{},[341,2682,1012,2683],{}," Rama, S. K., & Sen, S. (1994)."," \"Inconsistent Physics in the Presence of Time Machines.\" arXiv preprint gr-qc\u002F9410031.\n",[357,2686,2687],{"href":2687,"rel":2688},"https:\u002F\u002Farxiv.org\u002Fabs\u002Fgr-qc\u002F9410031",[2585],[324,2690,2691,2694,2695,2576,2697,2700,2701],{},[341,2692,1024,2693],{}," Krasnikov, S. (2002)."," \"Time travel paradox.\" ",[2573,2696,2575],{},[341,2698,2699],{},"65",", 064013.\n",[357,2702,2703],{"href":2703,"rel":2704},"https:\u002F\u002Fdoi.org\u002F10.1103\u002FPhysRevD.65.064013",[2585],[324,2706,2707,2710,2711,2576,2713,2715,2716],{},[341,2708,1065,2709],{}," Deutsch, D. (1991)."," \"Quantum mechanics near closed timelike lines.\" ",[2573,2712,2575],{},[341,2714,2596],{},", 3197–3217.\n",[357,2717,2718],{"href":2718,"rel":2719},"https:\u002F\u002Fdoi.org\u002F10.1103\u002FPhysRevD.44.3197",[2585],[324,2721,2722,2725,2726,2576,2729,2732,2733],{},[341,2723,2438,2724],{}," Lloyd, S., Maccone, L., Garcia-Patron, R., Giovannetti, V., Shikano, Y., Pirandola, S., Rozema, L. A., Darabi, A., Soudagar, Y., Shalm, L. K., & Steinberg, A. M. (2011)."," \"Closed Timelike Curves via Postselection: Theory and Experimental Test of Consistency.\" ",[2573,2727,2728],{},"Physical Review Letters",[341,2730,2731],{},"106",", 040403.\n",[357,2734,2735],{"href":2735,"rel":2736},"https:\u002F\u002Fdoi.org\u002F10.1103\u002FPhysRevLett.106.040403",[2585],[324,2738,2739,2743,2744,2576,2746,2749,2750],{},[341,2740,2741,2742],{},"[12]"," Lloyd, S., Maccone, L., Garcia-Patron, R., Giovannetti, V., & Shikano, Y. (2011)."," \"Quantum mechanics of time travel through post-selected teleportation.\" ",[2573,2745,2575],{},[341,2747,2748],{},"84",", 025007.\n",[357,2751,2752],{"href":2752,"rel":2753},"https:\u002F\u002Fdoi.org\u002F10.1103\u002FPhysRevD.84.025007",[2585],[324,2755,2756,2759,2760,2576,2763,2766,2767],{},[341,2757,1811,2758],{}," Aaronson, S., & Watrous, J. (2009)."," \"Closed timelike curves make quantum and classical computing equivalent.\" ",[2573,2761,2762],{},"Proceedings of the Royal Society A",[341,2764,2765],{},"465",", 631–647.\n",[357,2768,2769],{"href":2769,"rel":2770},"https:\u002F\u002Fdoi.org\u002F10.1098\u002Frspa.2008.0350",[2585],[324,2772,2773,2776,2777,2576,2779,2782,2783],{},[341,2774,1816,2775],{}," Bennett, C. H., Leung, D., Smith, G., & Smolin, J. A. (2009)."," \"Can Closed Timelike Curves or Nonlinear Quantum Mechanics Improve Quantum State Discrimination or Help Solve Hard Problems?\" ",[2573,2778,2728],{},[341,2780,2781],{},"103",", 170502.\n",[357,2784,2785],{"href":2785,"rel":2786},"https:\u002F\u002Fdoi.org\u002F10.1103\u002FPhysRevLett.103.170502",[2585],[324,2788,2789,2792,2793,2576,2796,2799,2800],{},[341,2790,2449,2791],{}," Ringbauer, M., Broome, M. A., Myers, C. R., White, A. G., & Ralph, T. C. (2014)."," \"Experimental simulation of closed timelike curves.\" ",[2573,2794,2795],{},"Nature Communications",[341,2797,2798],{},"5",", 4145.\n",[357,2801,2802],{"href":2802,"rel":2803},"https:\u002F\u002Fdoi.org\u002F10.1038\u002Fncomms5145",[2585],[324,2805,2806,2809,2810,2576,2812,2815,2816],{},[341,2807,390,2808],{}," Morris, M. S., Thorne, K. S., & Yurtsever, U. (1988)."," \"Wormholes, Time Machines, and the Weak Energy Condition.\" ",[2573,2811,2728],{},[341,2813,2814],{},"61",", 1446–1449.\n",[357,2817,2818],{"href":2818,"rel":2819},"https:\u002F\u002Fdoi.org\u002F10.1103\u002FPhysRevLett.61.1446",[2585],[324,2821,2822,2825,2826,2576,2828,2831,2832],{},[341,2823,2464,2824],{}," Hauser, J., & Shoshany, B. (2020)."," \"Time travel paradoxes and multiple histories.\" ",[2573,2827,2575],{},[341,2829,2830],{},"102",", 064062.\n",[357,2833,2834],{"href":2834,"rel":2835},"https:\u002F\u002Fdoi.org\u002F10.1103\u002FPhysRevD.102.064062",[2585],[324,2837,2838,2841,2842,2845,2846],{},[341,2839,2484,2840],{}," Shoshany, B. (2019)."," \"Lectures on faster-than-light travel and time travel.\" ",[2573,2843,2844],{},"SciPost Physics Lecture Notes",", 10.\n",[357,2847,2848],{"href":2848,"rel":2849},"https:\u002F\u002Fdoi.org\u002F10.21468\u002FSciPostPhysLectNotes.10",[2585],[324,2851,2852,2855,2856,2576,2858,2861,2862],{},[341,2853,951,2854],{}," Friedman, J., & Morris, M. S. (1991)."," \"The Cauchy problem for the scalar wave equation is well defined on a class of spacetimes with closed timelike curves.\" ",[2573,2857,2728],{},[341,2859,2860],{},"66",", 401–404.\n",[357,2863,2864],{"href":2864,"rel":2865},"https:\u002F\u002Fdoi.org\u002F10.1103\u002FPhysRevLett.66.401",[2585],[324,2867,2868,2871,2872,2576,2875,2878,2879],{},[341,2869,1047,2870],{}," Earman, J., Smeenk, C., & Wüthrich, C. (2009)."," \"Do the laws of physics forbid the operation of time machines?\" ",[2573,2873,2874],{},"Synthese",[341,2876,2877],{},"169","(1), 91–124.\n",[357,2880,2881],{"href":2881,"rel":2882},"https:\u002F\u002Fdoi.org\u002F10.1007\u002Fs11229-008-9338-2",[2585],{"title":2884,"searchDepth":83,"depth":83,"links":2885},"",[2886,2887,2888,2889,2890,2891,2897,2902,2903],{"id":348,"depth":83,"text":349},{"id":379,"depth":83,"text":380},{"id":407,"depth":83,"text":408},{"id":859,"depth":83,"text":860},{"id":961,"depth":83,"text":962},{"id":992,"depth":83,"text":993,"children":2892},[2893,2894,2895,2896],{"id":1000,"depth":131,"text":1001},{"id":1016,"depth":131,"text":1017},{"id":1028,"depth":131,"text":1029},{"id":1039,"depth":131,"text":1040},{"id":1050,"depth":83,"text":1051,"children":2898},[2899,2900,2901],{"id":1057,"depth":131,"text":1058},{"id":1820,"depth":131,"text":1821},{"id":2456,"depth":131,"text":2457},{"id":2468,"depth":83,"text":2469},{"id":2546,"depth":83,"text":2547},"md",[218,228,229,230,209,231],{},true,{"title":225,"description":226},"evidence\u002Fnovikov-self-consistency-principle","eIWerjvtaWhdBK6U8f44ZHO15BMsBbPpAE_WKKhaxKQ",1789530370343]