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Title: Towards bulk metric reconstruction from extremal area variations

Abstract

The Ryu–Takayanagi and Hubeny–Rangamani–Takayanagi formulae suggest that bulk geometry emerges from the entanglement structure of the boundary theory. Using these formulae, we build on a result of Alexakis, Balehowsky, and Nachman to show that in four bulk dimensions, the entanglement entropies of boundary regions of disk topology uniquely fix the bulk metric in any region foliated by the corresponding HRT surfaces. More generally, for a bulk of any dimension , knowledge of the (variations of the) areas of two-dimensional boundary-anchored extremal surfaces of disk topology uniquely fixes the bulk metric wherever these surfaces reach. This result is covariant and not reliant on any symmetry assumptions; its applicability thus includes regions of strong dynamical gravity such as the early-time interior of black holes formed from collapse. While we only show uniqueness of the metric, the approach we present provides a clear path towards an explicit spacetime metric reconstruction.

Authors:
 [1]; ORCiD logo [2]; ORCiD logo [3]; ORCiD logo [4]
  1. Univ. of California, Berkeley, CA (United States); Brookhaven National Lab. (BNL), Upton, NY (United States)
  2. California Inst. of Technology (CalTech), Pasadena, CA (United States); Univ. of Maryland, College Park, MD (United States)
  3. McGill Univ., Montreal, QC (Canada)
  4. Arizona State Univ., Tempe, AZ (United States)
Publication Date:
Research Org.:
Brookhaven National Lab. (BNL), Upton, NY (United States); Arizona State Univ., Tempe, AZ (United States)
Sponsoring Org.:
USDOE Office of Science (SC), Advanced Scientific Computing Research (SC-21); USDOE Office of Science (SC), High Energy Physics (HEP)
OSTI Identifier:
1566869
Alternate Identifier(s):
OSTI ID: 1597022
Report Number(s):
BNL-212128-2019-JAAM
Journal ID: ISSN 0264-9381; TRN: US2001006
Grant/Contract Number:  
SC0012704; SC0019470
Resource Type:
Accepted Manuscript
Journal Name:
Classical and Quantum Gravity
Additional Journal Information:
Journal Volume: 36; Journal Issue: 18; Journal ID: ISSN 0264-9381
Publisher:
IOP Publishing
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; AdS/CFT; bulk reconstruction; holographic entanglement entropy; boundary rigidity; 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS

Citation Formats

Bao, Ning, Cao, ChunJun, Fischetti, Sebastian, and Keeler, Cynthia. Towards bulk metric reconstruction from extremal area variations. United States: N. p., 2019. Web. doi:10.1088/1361-6382/ab377f.
Bao, Ning, Cao, ChunJun, Fischetti, Sebastian, & Keeler, Cynthia. Towards bulk metric reconstruction from extremal area variations. United States. https://doi.org/10.1088/1361-6382/ab377f
Bao, Ning, Cao, ChunJun, Fischetti, Sebastian, and Keeler, Cynthia. Tue . "Towards bulk metric reconstruction from extremal area variations". United States. https://doi.org/10.1088/1361-6382/ab377f. https://www.osti.gov/servlets/purl/1566869.
@article{osti_1566869,
title = {Towards bulk metric reconstruction from extremal area variations},
author = {Bao, Ning and Cao, ChunJun and Fischetti, Sebastian and Keeler, Cynthia},
abstractNote = {The Ryu–Takayanagi and Hubeny–Rangamani–Takayanagi formulae suggest that bulk geometry emerges from the entanglement structure of the boundary theory. Using these formulae, we build on a result of Alexakis, Balehowsky, and Nachman to show that in four bulk dimensions, the entanglement entropies of boundary regions of disk topology uniquely fix the bulk metric in any region foliated by the corresponding HRT surfaces. More generally, for a bulk of any dimension , knowledge of the (variations of the) areas of two-dimensional boundary-anchored extremal surfaces of disk topology uniquely fixes the bulk metric wherever these surfaces reach. This result is covariant and not reliant on any symmetry assumptions; its applicability thus includes regions of strong dynamical gravity such as the early-time interior of black holes formed from collapse. While we only show uniqueness of the metric, the approach we present provides a clear path towards an explicit spacetime metric reconstruction.},
doi = {10.1088/1361-6382/ab377f},
journal = {Classical and Quantum Gravity},
number = 18,
volume = 36,
place = {United States},
year = {2019},
month = {8}
}

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margin-top:0px;">Similar Records in DOE PAGES and OSTI.GOV collections:</p> <aside> <ul class="item-list" itemscope itemtype="http://schema.org/ItemList" style="padding-left:0; list-style-type: none;"> <li> <div class="article item document" itemprop="itemListElement" itemscope itemtype="http://schema.org/WebPage"><meta itemprop="position" content="1" /><div class="item-info"> <h2 class="title" itemprop="name headline"><a href="/pages/biblio/1764602-more-bulk-from-extremal-area-variations" itemprop="url">More of the bulk from extremal area variations</a></h2> <div class="metadata"> <small class="text-muted" style="text-transform:uppercase;display:block;line-height:2.5em;">Journal Article</small><span class="authors"> <span class="author">Bao, Ning</span> ; <span class="author">Cao, ChunJun</span> ; <span class="author">Fischetti, Sebastian</span> ; <span class="author">...</span> <span class="text-muted pubdata"> - Classical and Quantum Gravity</span> </span> </div> <div class="abstract">It was shown recently in (Bao N et al 2019 Class. Quantum Grav. 36 185002), building on work of Alexakis, Balehowksy, and Nachman (Alexakis S et al 2017 arXiv:1711.09379), that the geometry of (some portion of) a manifold with boundary is uniquely fixed by the areas of a foliation of two-dimensional disk-shaped surfaces anchored to the boundary. In the context of AdS/CFT, this implies that (a portion of) a four-dimensional bulk geometry can be fixed uniquely from the entanglement entropies of disk-shaped boundary regions, subject to several constraints. Here, we loosen some of these constraints, in particular allowing for the<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> bulk foliation of extremal surfaces to be local and removing the constraint of disk topology; these generalizations ensure uniqueness of more of the deep bulk geometry by allowing for e.g. surfaces anchored on disconnected asymptotic boundaries, or HRT surfaces past a phase transition. We also explore in more depth the generality of the local foliation requirement, showing that even in a highly dynamical geometry like AdS-Vaidya it is satisfied.</span><a href='#' onclick='$(this).hide().prev().hide().prev().show();return false;' style='margin-left:10px;display:none;'>« less</a></div><div class="metadata-links small clearfix text-muted" style="margin-top:15px;"> <div class="pure-menu pure-menu-horizontal pull-right" style="width:unset;"> <ul class="pure-menu-list"> <li class="pure-menu-item"><span class="item-info-ftlink"><a class="misc doi-link " href="https://doi.org/10.1088/1361-6382/abcfd0" target="_blank" rel="noopener" title="Link to document DOI" data-ostiid="1764602" data-product-type="Journal Article" data-product-subtype="AM" >https://doi.org/10.1088/1361-6382/abcfd0</a></span></li> <li class="pure-menu-item"><span class="item-info-ftlink"><a class="misc fulltext-link " href="/pages/servlets/purl/1764602" title="Link to document media" target="_blank" rel="noopener" data-ostiid="1764602" data-product-type="Journal Article" data-product-subtype="AM" >Full Text Available</a></span></li> </ul> </div> </div> </div> <div class="clearfix"></div> </div> </li> <li> <div class="article item document" itemprop="itemListElement" itemscope itemtype="http://schema.org/WebPage"><meta itemprop="position" content="3" /><div class="item-info"> <h2 class="title" itemprop="name headline"><a href="/pages/biblio/1434555-new-constraints-holographic-entropy-from-maximin-go-theorem" itemprop="url">New constraints for holographic entropy from maximin: A no-go theorem</a></h2> <div class="metadata"> <small class="text-muted" style="text-transform:uppercase;display:block;line-height:2.5em;">Journal Article</small><span class="authors"> <span class="author">Rota, Massimiliano</span> ; <span class="author">Weinberg, Sean J.</span> <span class="text-muted pubdata"> - Physical Review D</span> </span> </div> <div class="abstract">The Ryu-Takayanagi (RT) formula for static spacetimes arising in the AdS/CFT correspondence satisfies inequalities that are not yet proven in the case of the Rangamani-Hubeny-Takayanagi (HRT) formula, which applies to general dynamical spacetimes. Wall’s maximin construction is the only known technique for extending inequalities of holographic entanglement entropy from the static to dynamical case. We show that this method currently has no further utility when dealing with inequalities for five or fewer regions. Despite this negative result, we propose the validity of one new inequality for covariant holographic entanglement entropy for five regions. This inequality, while not maximin provable, is<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> much weaker than many of the inequalities satisfied by the RT formula and should therefore be easier to prove. If it is valid, then there is strong evidence that holographic entanglement entropy plays a role in general spacetimes including those that arise in cosmology. Our new inequality is obtained by the assumption that the HRT formula satisfies every known balanced inequality obeyed by the Shannon entropies of classical probability distributions. This is a property that the RT formula has been shown to possess and which has been previously conjectured to hold for quantum mechanics in general.</span><a href='#' onclick='$(this).hide().prev().hide().prev().show();return false;' style='margin-left:10px;display:none;'>« less</a></div><div class="metadata-links small clearfix text-muted" style="margin-top:15px;"> <span class="fa fa-book text-muted" aria-hidden="true"></span> Cited by 3<div class="pure-menu pure-menu-horizontal pull-right" style="width:unset;"> <ul class="pure-menu-list"> <li class="pure-menu-item"><span class="item-info-ftlink"><a class="misc doi-link " href="https://doi.org/10.1103/PhysRevD.97.086013" target="_blank" rel="noopener" title="Link to document DOI" data-ostiid="1434555" data-product-type="Journal Article" data-product-subtype="PA" >https://doi.org/10.1103/PhysRevD.97.086013</a></span></li> </ul> </div> </div> </div> <div class="clearfix"></div> </div> </li> <li> <div class="article item document" itemprop="itemListElement" itemscope itemtype="http://schema.org/WebPage"><meta itemprop="position" content="4" /><div class="item-info"> <h2 class="title" itemprop="name headline"><a href="/pages/biblio/1803476-flat-entanglement-spectra-fixed-area-states-quantum-gravity" itemprop="url">Flat entanglement spectra in fixed-area states of quantum gravity</a></h2> <div class="metadata"> <small class="text-muted" style="text-transform:uppercase;display:block;line-height:2.5em;">Journal Article</small><span class="authors"> <span class="author">Dong, Xi</span> ; <span class="author">Harlow, Daniel</span> ; <span class="author">Marolf, Donald</span> <span class="text-muted pubdata"> - Journal of High Energy Physics (Online)</span> </span> </div> <div class="abstract">We use the Einstein-Hilbert gravitational path integral to investigate gravitational entanglement at leading order O(1/G). We argue that semiclassical states prepared by a Euclidean path integral have the property that projecting them onto a subspace in which the Ryu-Takayanagi or Hubeny-Rangamani-Takayanagi surface has definite area gives a state with a flat entanglement spectrum at this order in gravitational perturbation theory. This means that the reduced density matrix can be approximated as proportional to the identity to the extent that its Renyi entropies Sn are independent of n at this order. The n-dependence of Sn in more general states then arises<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> from sums over the RT/HRT-area, which are generally dominated by different values of this area for each n. This provides a simple picture of gravitational entanglement, bolsters the connection between holographic systems and tensor network models, clarifies the bulk interpretation of algebraic centers which arise in the quantum errorcorrecting description of holography, and strengthens the connection between bulk and boundary modular Hamiltonians described by Jafferis, Lewkowycz, Maldacena, and Suh.</span><a href='#' onclick='$(this).hide().prev().hide().prev().show();return false;' style='margin-left:10px;display:none;'>« less</a></div><div class="metadata-links small clearfix text-muted" style="margin-top:15px;"> <div class="pure-menu pure-menu-horizontal pull-right" style="width:unset;"> <ul class="pure-menu-list"> <li class="pure-menu-item"><span class="item-info-ftlink"><a class="misc doi-link " href="https://doi.org/10.1007/jhep10(2019)240" target="_blank" rel="noopener" title="Link to document DOI" data-ostiid="1803476" data-product-type="Journal Article" data-product-subtype="AM" >https://doi.org/10.1007/jhep10(2019)240</a></span></li> <li class="pure-menu-item"><span class="item-info-ftlink"><a class="misc fulltext-link " href="/pages/servlets/purl/1803476" title="Link to document media" target="_blank" rel="noopener" data-ostiid="1803476" data-product-type="Journal Article" data-product-subtype="AM" >Full Text Available</a></span></li> </ul> </div> </div> </div> <div class="clearfix"></div> </div> </li> <li> <div class="article item document" itemprop="itemListElement" itemscope itemtype="http://schema.org/WebPage"><meta itemprop="position" content="5" /><div class="item-info"> <h2 class="title" itemprop="name headline"><a href="/pages/biblio/1853139-effective-entropy-quantum-fields-coupled-gravity" itemprop="url">Effective entropy of quantum fields coupled with gravity</a></h2> <div class="metadata"> <small class="text-muted" style="text-transform:uppercase;display:block;line-height:2.5em;">Journal Article</small><span class="authors"> <span class="author">Dong, Xi</span> ; <span class="author">Qi, Xiao-Liang</span> ; <span class="author">Shangnan, Zhou</span> ; <span class="author">...</span> <span class="text-muted pubdata"> - Journal of High Energy Physics (Online)</span> </span> </div> <div class="abstract">Entanglement entropy, or von Neumann entropy, quantifies the amount of uncertainty of a quantum state. For quantum fields in curved space, entanglement entropy of the quantum field theory degrees of freedom is well-defined for a fixed background geometry. In this paper, we propose a generalization of the quantum field theory entanglement entropy by including dynamical gravity. The generalized quantity named effective entropy, and its Renyi entropy generalizations, are defined by analytic continuation of a replica calculation. The replicated theory is defined as a gravitational path integral with multiple copies of the original boundary conditions, with a co-dimension-2 brane at the<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> boundary of region we are studying. We discuss different approaches to define the region in a gauge invariant way, and show that the effective entropy satisfies the quantum extremal surface formula. When the quantum fields carry a significant amount of entanglement, the quantum extremal surface can have a topology transition, after which an entanglement island region appears. Our result generalizes the Hubeny-Rangamani-Takayanagi formula of holographic entropy (with quantum corrections) to general geometries without asymptotic AdS boundary, and provides a more solid framework for addressing problems such as the Page curve of evaporating black holes in asymptotic flat spacetime. We apply the formula to two example systems, a closed two-dimensional universe and a four-dimensional maximally extended Schwarzchild black hole. We discuss the analog of the effective entropy in random tensor network models, which provides more concrete understanding of quantum information properties in general dynamical geometries. We show that, in absence of a large boundary like in AdS space case, it is essential to introduce ancilla that couples to the original system, in order for correctly characterizing quantum states and correlation functions in the random tensor network. Using the superdensity operator formalism, we study the system with ancilla and show how quantum information in the entanglement island can be reconstructed in a state-dependent and observer-dependent map. We study the closed universe (without spatial boundary) case and discuss how it is related to open universe.</span><a href='#' onclick='$(this).hide().prev().hide().prev().show();return false;' style='margin-left:10px;display:none;'>« less</a></div><div class="metadata-links small clearfix text-muted" style="margin-top:15px;"> <div class="pure-menu pure-menu-horizontal pull-right" style="width:unset;"> <ul class="pure-menu-list"> <li class="pure-menu-item"><span class="item-info-ftlink"><a class="misc doi-link " href="https://doi.org/10.1007/jhep10(2020)052" target="_blank" rel="noopener" title="Link to document DOI" data-ostiid="1853139" data-product-type="Journal Article" data-product-subtype="AM" >https://doi.org/10.1007/jhep10(2020)052</a></span></li> <li class="pure-menu-item"><span class="item-info-ftlink"><a class="misc fulltext-link " href="/pages/servlets/purl/1853139" title="Link to document media" target="_blank" rel="noopener" data-ostiid="1853139" data-product-type="Journal Article" data-product-subtype="AM" >Full Text Available</a></span></li> </ul> </div> </div> </div> <div class="clearfix"></div> </div> </li> <li> <div class="article item document" itemprop="itemListElement" itemscope itemtype="http://schema.org/WebPage"><meta itemprop="position" content="6" /><div class="item-info"> <h2 class="title" itemprop="name headline"><a href="/pages/biblio/1800260-dual-non-extremal-area-differential-entropy-higher-dimensions" itemprop="url">The dual of non-extremal area: differential entropy in higher dimensions</a></h2> <div class="metadata"> <small class="text-muted" style="text-transform:uppercase;display:block;line-height:2.5em;">Journal Article</small><span class="authors"> <span class="author">Balasubramanian, Vijay</span> ; <span class="author">Rabideau, Charles</span> <span class="text-muted pubdata"> - Journal of High Energy Physics (Online)</span> </span> </div> <div class="abstract">The Ryu-Takayanagi formula relates entanglement entropy in a field theory to the area of extremal surfaces anchored to the boundary of a dual AdS space. It is interesting to ask if there is also an information theoretic interpretation of the areas of non-extremal surfaces that are not necessarily boundary-anchored. In general, the physics outside such surfaces is associated to observers restricted to a time-strip in the dual boundary field theory. When the latter is two-dimensional, it is known that thedifferential entropyassociated to the strip computes the length of the dual bulk curve, and has an interpretation in terms of the<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> information cost in Bell pairs of restoring correlations inaccessible to observers in the strip. A general realization of this formalism in higher dimensions is unknown. We first prove a no-go theorem eliminating candidate expressions for higher dimensional differential entropy based on entropic c-theorems. Then we propose a new formula in terms of an integral of shape derivatives of the entanglement entropy of ball shaped regions. Our proposal stems from the physical requirement that differential entropy must be locally finite and conformally invariant. Demanding cancelation of the well-known UV divergences of entanglement entropy in field theory guides us to our conjecture, which we test for surfaces inAdS<sub>4</sub>. Our results suggest a candidate c-function for field theories in arbitrary dimensions.</span><a href='#' onclick='$(this).hide().prev().hide().prev().show();return false;' style='margin-left:10px;display:none;'>« less</a></div><div class="metadata-links small clearfix text-muted" style="margin-top:15px;"> <div class="pure-menu pure-menu-horizontal pull-right" style="width:unset;"> <ul class="pure-menu-list"> <li class="pure-menu-item"><span class="item-info-ftlink"><a class="misc doi-link " href="https://doi.org/10.1007/jhep09(2020)051" target="_blank" rel="noopener" title="Link to document DOI" data-ostiid="1800260" data-product-type="Journal Article" data-product-subtype="AM" >https://doi.org/10.1007/jhep09(2020)051</a></span></li> <li class="pure-menu-item"><span class="item-info-ftlink"><a class="misc fulltext-link " href="/pages/servlets/purl/1800260" title="Link to document media" target="_blank" rel="noopener" data-ostiid="1800260" data-product-type="Journal Article" data-product-subtype="AM" >Full Text Available</a></span></li> </ul> </div> </div> </div> <div class="clearfix"></div> </div> </li> </ul> </aside> </div> </section> </div> <div class="col-sm-3 order-sm-3"> <ul class="nav nav-stacked"> <li class="active"><a class="tab-nav disabled" data-tab="related" style="color: #636c72 !important; opacity: 1;"><span class="fa fa-angle-right"></span> Similar Records</a></li> </ul> </div> </div> </section> </div></div> </div> </div> </section> <footer class="" style="background-color:#f9f9f9; /* padding-top: 0.5rem; */"> <div class="footer-minor"> <div class="container"> <hr class="footer-separator" /> <div class="text-center" style="margin-top:1.25rem;"> <div class="pure-menu pure-menu-horizontal"> <ul class="pure-menu-list" id="footer-org-menu"> <li class="pure-menu-item"> <a href="https://energy.gov" target="_blank" rel="noopener noreferrer"> <img src="data:image/gif;base64,R0lGODlhAQABAIAAAP///wAAACH5BAEAAAAALAAAAAABAAEAAAICRAEAOw==" class="sprite sprite-footer-us-doe-min" alt="U.S. Department of Energy" /> </a> </li> <li class="pure-menu-item"> <a href="https://www.energy.gov/science/office-science" target="_blank" rel="noopener noreferrer"> <img src="data:image/gif;base64,R0lGODlhAQABAIAAAP///wAAACH5BAEAAAAALAAAAAABAAEAAAICRAEAOw==" class="sprite sprite-footer-office-of-science-min" alt="Office of Science" /> </a> </li> <li class="pure-menu-item"> <a href="/"> <img src="data:image/gif;base64,R0lGODlhAQABAIAAAP///wAAACH5BAEAAAAALAAAAAABAAEAAAICRAEAOw==" class="sprite sprite-footer-osti-min" alt="Office of Scientific and Technical Information" /> </a> </li> </ul> </div> </div> <div class="text-center small" style="margin-top:0.5em;margin-bottom:2.0rem;"> <div class="pure-menu pure-menu-horizontal"> <ul class="pure-menu-list"> <li class="pure-menu-item"><a href="/disclaim" class="pure-menu-link"><span class="fa fa-institution"></span> Website Policies <span class="hidden-xs">/ Important Links</span></a></li> <li class="pure-menu-item"><a href="/pages/contact" class="pure-menu-link"><span class="fa fa-comments-o"></span> Contact Us</a></li> <li class="d-block d-md-none"></li> <li class="pure-menu-item"><a href="https://www.facebook.com/ostigov" target="_blank" rel="noopener noreferrer" class="pure-menu-link social"><span class="sr-only">Facebook</span><em class="fa fa-facebook" style=""></em></a></li> <li class="pure-menu-item"><a href="https://twitter.com/OSTIgov" target="_blank" rel="noopener noreferrer" class="pure-menu-link social"><span class="sr-only">Twitter</span><em class="fa fa-twitter" style=""></em></a></li> <li class="pure-menu-item"><a href="https://www.youtube.com/user/ostigov" target="_blank" rel="noopener noreferrer" class="pure-menu-link social"><span class="sr-only">YouTube</span><em class="fa fa-youtube-play" style=""></em></a></li> </ul> </div> </div> </div> </div> </footer> <link href="/pages/css/pages.fonts.220701.1625.css" rel="stylesheet"> <script src="/pages/js/pages.220701.1625.js"></script><noscript></noscript> <script defer src="/pages/js/pages.biblio.220701.1625.js"></script><noscript></noscript> <script defer src="/pages/js/lity.js"></script><noscript></noscript> <script async�type="text/javascript" src="/pages/js/Universal-Federated-Analytics-Min.js?agency=DOE" id="_fed_an_ua_tag"></script><noscript></noscript> </body> <!-- DOE PAGES v.220701.1625 --> </html>