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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)
Sponsoring Org.:
USDOE Office of Science (SC), Advanced Scientific Computing Research (SC-21)
OSTI Identifier:
1566869
Report Number(s):
BNL-212128-2019-JAAM
Journal ID: ISSN 0264-9381
Grant/Contract Number:  
SC0012704
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

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. doi: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. doi:10.1088/1361-6382/ab377f.
@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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font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2016-10-01">October 2016</span></small> </h2> <ul id="references-list" class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#7cb342;"> Sanches, Fabio; Weinberg, Sean J.</span> </li> <li> Physical Review D, Vol. 94, Issue 8</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1103/PhysRevD.94.084034" class="text-muted" target="_blank" rel="noopener noreferrer">10.1103/PhysRevD.94.084034<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div><div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1103/PhysRevD.95.086002" target="_blank" rel="noopener noreferrer" class="name">Toward a holographic theory for general spacetimes<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2017-04-01">April 2017</span></small> </h2> <ul id="references-list" class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#7cb342;"> Nomura, Yasunori; Salzetta, Nico; Sanches, Fabio</span> </li> <li> Physical Review D, Vol. 95, Issue 8</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1103/PhysRevD.95.086002" class="text-muted" target="_blank" rel="noopener noreferrer">10.1103/PhysRevD.95.086002<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div><div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1103/PhysRevD.95.024031" target="_blank" rel="noopener noreferrer" class="name">Space from Hilbert space: Recovering geometry from bulk entanglement<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2017-01-01">January 2017</span></small> </h2> <ul id="references-list" class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#7cb342;"> Cao, ChunJun; Carroll, Sean M.; Michalakis, Spyridon</span> </li> <li> Physical Review D, Vol. 95, Issue 2</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1103/PhysRevD.95.024031" class="text-muted" target="_blank" rel="noopener noreferrer">10.1103/PhysRevD.95.024031<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div><div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1103/PhysRevD.97.086003" target="_blank" rel="noopener noreferrer" class="name">Bulk entanglement gravity without a boundary: Towards finding Einstein’s equation in Hilbert space<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2018-04-01">April 2018</span></small> </h2> <ul id="references-list" class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#7cb342;"> Cao, ChunJun; Carroll, Sean M.</span> </li> <li> Physical Review D, Vol. 97, Issue 8</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1103/PhysRevD.97.086003" class="text-muted" target="_blank" rel="noopener noreferrer">10.1103/PhysRevD.97.086003<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div><div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1007/JHEP01(2015)073" target="_blank" rel="noopener noreferrer" class="name">Quantum extremal surfaces: holographic entanglement entropy beyond the classical regime<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2015-01-01">January 2015</span></small> </h2> <ul id="references-list" class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#7cb342;"> Engelhardt, Netta; Wall, Aron C.</span> </li> <li> Journal of High Energy Physics, Vol. 2015, Issue 1</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1007/JHEP01(2015)073" class="text-muted" target="_blank" rel="noopener noreferrer">10.1007/JHEP01(2015)073<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div></div> <ul class="pagination"></ul> </div> </div> <div class="col-sm-3 order-sm-3"> <ul class="nav nav-stacked"> <li class="active"><a href="" class="reference-type-filter tab-nav" data-tab="biblio-references" data-filter="type" data-pattern="*"><span class="fa fa-angle-right"></span> All References</a></li> <li class="small" style="margin-left:.75em; 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float:none;">[ × clear filter / sort ]</a> </div> </form> </div> </div> </div> </section> <section id="biblio-related" class="tab-content tab-content-sec " data-tab="biblio"> <div class="row"> <div class="col-sm-9 order-sm-9"> <section id="biblio-similar" class="tab-content tab-content-sec active" data-tab="related"> <div class="padding"> <p class="lead text-muted" style="font-size: 18px; margin-top:0px;">Similar records in 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="0" /><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">DOI: <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" >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="1" /><div class="item-info"> <h2 class="title" itemprop="name headline"><a href="/pages/biblio/1498657-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">DOI: <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="1498657" data-product-type="Journal Article" data-product-subtype="AM" >10.1103/physrevd.97.086013</a></span></li> <li class="pure-menu-item"><span class="item-info-ftlink"><a class="misc fulltext-link " href="/pages/servlets/purl/1498657" title="Link to document media" target="_blank" rel="noopener" data-ostiid="1498657" 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="2" /><div class="item-info"> <h2 class="title" itemprop="name headline"><a href="/pages/biblio/1459238-pulling-boundary-bulk" itemprop="url">Pulling the boundary into the bulk</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">Nomura, Yasunori</span> ; <span class="author">Rath, Pratik</span> ; <span class="author">Salzetta, Nico</span> <span class="text-muted pubdata"> - Physical Review D</span> </span> </div> <div class="abstract">Motivated by the ability to consistently apply the Ryu-Takayanagi prescription for general convex surfaces and the relationship between entanglement and geometry in tensor networks, we introduce a novel, covariant bulk object - the holographic slice. The holographic slice is found by considering the continual removal of short-range information in a boundary state. It thus provides a natural interpretation as the bulk dual of a series of coarse-grained holographic states. The slice possesses many desirable properties that provide consistency checks for its boundary interpretation. These include the monotonicity of both area and entanglement entropy, uniqueness, and the inability to probe beyond<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> late-time black hole horizons. Additionally, the holographic slice illuminates physics behind entanglement shadows, as minimal-area extremal surfaces anchored to a coarse-grained boundary may probe entanglement shadows. This lets the slice flow through shadows. To aid in developing intuition for these slices, many explicit examples of holographic slices are investigated. Finally, the relationship to tensor networks and renormalization (particularly in AdS/CFT) is discussed.</span><a href='#' onclick='$(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 12<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">DOI: <a class="misc doi-link " href="https://doi.org/10.1103/PhysRevD.98.026010" target="_blank" rel="noopener" title="Link to document DOI" data-ostiid="1459238" data-product-type="Journal Article" data-product-subtype="PA" >10.1103/PhysRevD.98.026010</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/1506336-pulling-boundary-bulk" itemprop="url">Pulling the boundary into the bulk</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">Nomura, Yasunori</span> ; <span class="author">Rath, Pratik</span> ; <span class="author">Salzetta, Nico</span> <span class="text-muted pubdata"> - Physical Review D</span> </span> </div> <div class="abstract">Motivated by the ability to consistently apply the Ryu-Takayanagi prescription for general convex surfaces and the relationship between entanglement and geometry in tensor networks, we introduce a novel, covariant bulk object - the holographic slice. The holographic slice is found by considering the continual removal of short-range information in a boundary state. It thus provides a natural interpretation as the bulk dual of a series of coarse-grained holographic states. The slice possesses many desirable properties that provide consistency checks for its boundary interpretation. These include the monotonicity of both area and entanglement entropy, uniqueness, and the inability to probe beyond<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> late-time black hole horizons. Additionally, the holographic slice illuminates physics behind entanglement shadows, as minimal-area extremal surfaces anchored to a coarse-grained boundary may probe entanglement shadows. This lets the slice flow through shadows. To aid in developing intuition for these slices, many explicit examples of holographic slices are investigated. Finally, the relationship to tensor networks and renormalization (particularly in AdS/CFT) is discussed.</span><a href='#' onclick='$(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 12<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">DOI: <a class="misc doi-link " href="https://doi.org/10.1103/PhysRevD.98.026010" target="_blank" rel="noopener" title="Link to document DOI" data-ostiid="1506336" data-product-type="Journal Article" data-product-subtype="AM" >10.1103/PhysRevD.98.026010</a></span></li> <li class="pure-menu-item"><span class="item-info-ftlink"><a class="misc fulltext-link " href="/pages/servlets/purl/1506336" title="Link to document media" target="_blank" rel="noopener" data-ostiid="1506336" 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="4" /><div class="item-info"> <h2 class="title" itemprop="name headline"><a href="/pages/biblio/1183697-integral-geometry-holography" itemprop="url">Integral geometry and holography</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">Czech, Bartlomiej</span> ; <span class="author">Lamprou, Lampros</span> ; <span class="author">McCandlish, Samuel</span> ; <span class="author">...</span> <span class="text-muted pubdata"> - Journal of High Energy Physics (Online)</span> </span> </div> <div class="abstract">We present a mathematical framework which underlies the connection between information theory and the bulk spacetime in the AdS <sub>3</sub>/CFT <sub>2</sub> correspondence. A key concept is <i>kinematic</i> space: an auxiliary Lorentzian geometry whose metric is defined in terms of conditional mutual informations and which organizes the entanglement pattern of a CFT state. When the field theory has a holographic dual obeying the Ryu-Takayanagi proposal, kinematic space has a direct geometric meaning: it is the space of bulk geodesics studied in integral geometry. Lengths of bulk curves are computed by kinematic volumes, giving a precise entropic interpretation of the length of<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> any bulk curve. We explain how basic geometric concepts -- points, distances and angles -- are reflected in kinematic space, allowing one to reconstruct a large class of spatial bulk geometries from boundary entanglement entropies. In this way, kinematic space translates between information theoretic and geometric descriptions of a CFT state. As an example, we discuss in detail the static slice of AdS <sub>3</sub> whose kinematic space is two-dimensional de Sitter space.</span><a href='#' onclick='$(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 21<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">DOI: <a class="misc doi-link " href="https://doi.org/10.1007/JHEP10(2015)175" target="_blank" rel="noopener" title="Link to document DOI" data-ostiid="1183697" data-product-type="Journal Article" data-product-subtype="AM" >10.1007/JHEP10(2015)175</a></span></li> <li class="pure-menu-item"><span class="item-info-ftlink"><a class="misc fulltext-link " href="/pages/servlets/purl/1183697" title="Link to document media" target="_blank" rel="noopener" data-ostiid="1183697" 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="fa fa-facebook" style=""></span></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="fa fa-twitter" style=""></span></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="fa fa-youtube-play" style=""></span></a></li> </ul> </div> </div> </div> </div> </footer> <link href="/pages/css/pages.fonts.191210.1608.css" rel="stylesheet"> <script src="/pages/js/pages.191210.1608.js"></script><noscript></noscript> <script defer src="/pages/js/pages.biblio.191210.1608.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.191210.1608 --> </html>