Finite Gravitational Phase Space, Unitary Evolution, and the Black Hole Information Paradox in Effective Field Theory

Finite Gravitational Phase Space, Unitary Evolution, and the Black Hole Information Paradox in Effective Field Theory | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Finite Gravitational Phase Space, Unitary Evolution, and the Black Hole Information Paradox in Effective Field Theory Farrukh A. Chishtie This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9110073/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 8 You are reading this latest preprint version Abstract We argue that the effective field theory (EFT) nature of four-dimensional general relativity, as established within the USMEG-EFT framework through canonical quantization, renormalization group analysis, and BRST symmetry considerations~\cite{Chishtie2025CJP,Chishtie2025PRD,Chishtie2026Symmetry}, combined with the Bekenstein entropy bound and the implications of Penrose's singularity theorems, implies a finite-dimensional physically accessible Hilbert space for gravitational degrees of freedom. Within this Hilbert space unitary evolution is guaranteed, and the Page curve follows from Page's theorem~\cite{Page1993}. We provide two explicit, verifiable calculations that ground this argument. First, combining the Bekenstein-Hawking entropy $S_{\rm BH}(M) = 4\pi G_N M^{2}/(\hbar c),k_{B}$ with the Hawking evaporation rate $dM/dt = -\hbar c^{4}/(15360\pi G_{N}^{2}M^{2})$, we derive the Page time at which the entanglement entropy of the radiation peaks: corresponding to $M_{\rm Page} = M_{0}/!\sqrt{2}$. For a solar-mass black hole this gives $t_{\rm Page}(M_\odot)\approx 1.355\times 10^{67}$~yr, a concrete and parameter-free prediction of the framework. Second, using the Parikh--Wilczek mutual information and that the entropy integral from $t=0$ to $t_{\rm Page}$ equals exactly $S_{\rm BH,0}/2$, providing independent dynamical confirmation of unitarity. The framework receives observational support from LIGO-Virgo-KAGRA confirmation of exactly two graviton polarization states, a necessary structural consequence of the USMEG-EFT constraint algebra. We discuss the relationship to the island formula program and derive explicit bounds on configurational entropy consistent with holographic bounds. Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Reviewers agreed at journal 19 May, 2026 Reviews received at journal 20 Apr, 2026 Reviewers agreed at journal 04 Apr, 2026 Reviewers agreed at journal 03 Apr, 2026 Reviewers invited by journal 17 Mar, 2026 Editor assigned by journal 16 Mar, 2026 Submission checks completed at journal 16 Mar, 2026 First submitted to journal 13 Mar, 2026 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-9110073","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":607547795,"identity":"a4ee6162-7c5b-472b-bb66-ec90b8e1d72d","order_by":0,"name":"Farrukh A. 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Within this Hilbert space unitary\nevolution is guaranteed, and the Page curve follows from Page's\ntheorem~\\cite{Page1993}.\nWe provide two explicit, verifiable calculations that ground this argument.\nFirst, combining the Bekenstein-Hawking entropy $S_{\\rm BH}(M)\n= 4\\pi G_N M^{2}/(\\hbar c)\\,k_{B}$ with the Hawking evaporation\nrate $dM/dt = -\\hbar c^{4}/(15360\\pi G_{N}^{2}M^{2})$, we derive the\nPage time at which the entanglement entropy of the radiation peaks:\ncorresponding to $M_{\\rm Page} = M_{0}/\\!\\sqrt{2}$.\nFor a solar-mass black hole this gives\n$t_{\\rm Page}(M_\\odot)\\approx 1.355\\times 10^{67}$~yr, a\nconcrete and parameter-free prediction of the framework.\nSecond, using the Parikh--Wilczek mutual information\nand that the entropy integral from $t=0$ to $t_{\\rm Page}$ equals\nexactly $S_{\\rm BH,0}/2$, providing independent dynamical confirmation\nof unitarity.\nThe framework receives observational support from LIGO-Virgo-KAGRA\nconfirmation of exactly two graviton polarization states, a\nnecessary structural consequence of the USMEG-EFT constraint algebra.\nWe discuss the relationship to the island formula program and derive \nexplicit bounds on configurational entropy consistent with holographic bounds.","manuscriptTitle":"Finite Gravitational Phase Space, Unitary Evolution, and the \nBlack Hole Information Paradox in Effective Field Theory","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-03-19 19:11:49","doi":"10.21203/rs.3.rs-9110073/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"reviewerAgreed","content":"185293074102311787556755322651959397475","date":"2026-05-19T06:03:28+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-04-21T03:21:23+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"43067132135785007452624058466171899890","date":"2026-04-04T12:56:07+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"28358502042423307351089046043689099486","date":"2026-04-03T12:01:27+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2026-03-17T09:46:45+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2026-03-16T07:09:19+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2026-03-16T07:09:00+00:00","index":"","fulltext":""},{"type":"submitted","content":"General Relativity and Gravitation","date":"2026-03-13T04:07:02+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"general-relativity-and-gravitation","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"gerg","sideBox":"Learn more about [General Relativity and Gravitation](http://link.springer.com/journal/10714)","snPcode":"10714","submissionUrl":"https://submission.nature.com/new-submission/10714/3","title":"General Relativity and Gravitation","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"1d45e023-81d1-4d20-b015-228d6502cc2e","owner":[],"postedDate":"March 19th, 2026","published":true,"recentEditorialEvents":[{"type":"reviewerAgreed","content":"185293074102311787556755322651959397475","date":"2026-05-19T06:03:28+00:00","index":34,"fulltext":""}],"rejectedJournal":[],"revision":"","amendment":"","status":"under-review","subjectAreas":[],"tags":[],"updatedAt":"2026-03-19T19:11:49+00:00","versionOfRecord":[],"versionCreatedAt":"2026-03-19 19:11:49","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-9110073","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-9110073","identity":"rs-9110073","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}