Decoding Quantum Gravity Information with Black Hole Accretion Disk

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} div.type-section h2 { font-size: 20px; line-height: 26px; font-weight: 300; } div.type-section h3 { margin-left: 15px; margin-bottom: 0px; font-weight: 300; } .journal-tabs .tab-title.active a { } </style> <link rel="stylesheet" href="https://pub.mdpi-res.com/assets/css/slick.css?f38b2db10e01b157?1732615622"> <meta name="title" content="Decoding Quantum Gravity Information with Black Hole Accretion Disk"> <meta name="description" content="Integrating loop quantum gravity with classical gravitational collapse models offers an effective solution to the black hole singularity problem and predicts the formation of a white hole in the later stages of collapse. Furthermore, the quantum extension of Kruskal spacetime indicates that white holes may convey information about earlier companion black holes. Photons emitted from the accretion disks of these companion black holes enter the black hole, traverse the highly quantum region, and then re-emerge from white holes in our universe. This process enables us to observe images of the companion black holes’ accretion disks, providing insights into quantum gravity. In our study, we successfully obtained these accretion disk images. Our results indicate that these accretion disk images are confined within a circle with a radius equal to the critical impact parameter, while traditional accretion disk images are typically located outside this circle. As the observational angle increases, the accretion disk images transition from a ring shape to a shell-like shape. Furthermore, the positional and width characteristics of these accretion disk images are opposite to those of traditional accretion disk images. These findings provide valuable references for astronomical observations aimed at validating the investigated quantum gravity model." > <link rel="image_src" href="https://pub.mdpi-res.com/img/journals/universe-logo.png?8600e93ff98dbf14" > <meta name="dc.title" content="Decoding Quantum Gravity Information with Black Hole Accretion Disk"> <meta name="dc.creator" content="Lei You"> <meta name="dc.creator" content="Yu-Hang Feng"> <meta name="dc.creator" content="Rui-Bo Wang"> <meta name="dc.creator" content="Xian-Ru Hu"> <meta name="dc.creator" content="Jian-Bo Deng"> <meta name="dc.type" content="Article"> <meta name="dc.source" content="Universe 2024, Vol. 10, Page 393"> <meta name="dc.date" content="2024-10-11"> <meta name ="dc.identifier" content="10.3390/universe10100393"> <meta name="dc.publisher" content="Multidisciplinary Digital Publishing Institute"> <meta name="dc.rights" content="http://creativecommons.org/licenses/by/3.0/"> <meta name="dc.format" content="application/pdf" > <meta name="dc.language" content="en" > <meta name="dc.description" content="Integrating loop quantum gravity with classical gravitational collapse models offers an effective solution to the black hole singularity problem and predicts the formation of a white hole in the later stages of collapse. Furthermore, the quantum extension of Kruskal spacetime indicates that white holes may convey information about earlier companion black holes. Photons emitted from the accretion disks of these companion black holes enter the black hole, traverse the highly quantum region, and then re-emerge from white holes in our universe. This process enables us to observe images of the companion black holes’ accretion disks, providing insights into quantum gravity. In our study, we successfully obtained these accretion disk images. Our results indicate that these accretion disk images are confined within a circle with a radius equal to the critical impact parameter, while traditional accretion disk images are typically located outside this circle. As the observational angle increases, the accretion disk images transition from a ring shape to a shell-like shape. Furthermore, the positional and width characteristics of these accretion disk images are opposite to those of traditional accretion disk images. These findings provide valuable references for astronomical observations aimed at validating the investigated quantum gravity model." > <meta name="dc.subject" content="loop quantum gravity" > <meta name="dc.subject" content="white hole" > <meta name="dc.subject" content="thin accretion disk" > <meta name ="prism.issn" content="2218-1997"> <meta name ="prism.publicationName" content="Universe"> <meta name ="prism.publicationDate" content="2024-10-11"> <meta name ="prism.volume" content="10"> <meta name ="prism.number" content="10"> <meta name ="prism.section" content="Article" > <meta name ="prism.startingPage" content="393" > <meta name="citation_issn" content="2218-1997"> <meta name="citation_journal_title" content="Universe"> <meta name="citation_publisher" content="Multidisciplinary Digital Publishing Institute"> <meta name="citation_title" content="Decoding Quantum Gravity Information with Black Hole Accretion Disk"> <meta name="citation_publication_date" content="2024/10"> <meta name="citation_online_date" content="2024/10/11"> <meta name="citation_volume" content="10"> <meta name="citation_issue" content="10"> <meta name="citation_firstpage" content="393"> <meta name="citation_author" content="You, Lei"> <meta name="citation_author" content="Feng, Yu-Hang"> <meta name="citation_author" content="Wang, Rui-Bo"> <meta name="citation_author" content="Hu, Xian-Ru"> <meta name="citation_author" content="Deng, Jian-Bo"> <meta name="citation_doi" content="10.3390/universe10100393"> <meta name="citation_id" content="mdpi-universe10100393"> <meta name="citation_abstract_html_url" content="https://www.mdpi.com/2218-1997/10/10/393"> <meta name="citation_pdf_url" content="https://www.mdpi.com/2218-1997/10/10/393/pdf?version=1728630725"> <link rel="alternate" type="application/pdf" title="PDF Full-Text" href="https://www.mdpi.com/2218-1997/10/10/393/pdf?version=1728630725"> <meta name="fulltext_pdf" content="https://www.mdpi.com/2218-1997/10/10/393/pdf?version=1728630725"> <meta name="citation_fulltext_html_url" content="https://www.mdpi.com/2218-1997/10/10/393/htm"> <link rel="alternate" type="text/html" title="HTML Full-Text" href="https://www.mdpi.com/2218-1997/10/10/393/htm"> <meta name="fulltext_html" content="https://www.mdpi.com/2218-1997/10/10/393/htm"> <link rel="alternate" type="text/xml" title="XML Full-Text" href="https://www.mdpi.com/2218-1997/10/10/393/xml"> <meta name="fulltext_xml" content="https://www.mdpi.com/2218-1997/10/10/393/xml"> <meta name="citation_xml_url" content="https://www.mdpi.com/2218-1997/10/10/393/xml"> <meta name="twitter:card" content="summary" /> <meta name="twitter:site" content="@MDPIOpenAccess" /> <meta name="twitter:image" content="https://pub.mdpi-res.com/img/journals/universe-logo-social.png?8600e93ff98dbf14" /> <meta property="fb:app_id" content="131189377574"/> <meta property="og:site_name" content="MDPI"/> <meta property="og:type" content="article"/> <meta property="og:url" content="https://www.mdpi.com/2218-1997/10/10/393" /> <meta property="og:title" content="Decoding Quantum Gravity Information with Black Hole Accretion Disk" /> <meta property="og:description" content="Integrating loop quantum gravity with classical gravitational collapse models offers an effective solution to the black hole singularity problem and predicts the formation of a white hole in the later stages of collapse. Furthermore, the quantum extension of Kruskal spacetime indicates that white holes may convey information about earlier companion black holes. Photons emitted from the accretion disks of these companion black holes enter the black hole, traverse the highly quantum region, and then re-emerge from white holes in our universe. This process enables us to observe images of the companion black holes’ accretion disks, providing insights into quantum gravity. In our study, we successfully obtained these accretion disk images. Our results indicate that these accretion disk images are confined within a circle with a radius equal to the critical impact parameter, while traditional accretion disk images are typically located outside this circle. As the observational angle increases, the accretion disk images transition from a ring shape to a shell-like shape. Furthermore, the positional and width characteristics of these accretion disk images are opposite to those of traditional accretion disk images. These findings provide valuable references for astronomical observations aimed at validating the investigated quantum gravity model." /> <meta property="og:image" content="https://pub.mdpi-res.com/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g001-550.jpg?1728630854" /> <link rel="alternate" type="application/rss+xml" title="MDPI Publishing - Latest articles" href="https://www.mdpi.com/rss"> <meta name="google-site-verification" content="PxTlsg7z2S00aHroktQd57fxygEjMiNHydKn3txhvwY"> <meta name="facebook-domain-verification" content="mcoq8dtq6sb2hf7z29j8w515jjoof7" /> <script id="Cookiebot" data-cfasync="false" src="https://consent.cookiebot.com/uc.js" data-cbid="51491ddd-fe7a-4425-ab39-69c78c55829f" type="text/javascript" async></script>  <script type="text/plain" data-cookieconsent="statistics"> (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); 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href="/cdn-cgi/l/email-protection#3f105c515b125c58561053105a525e5653124f4d504b5a5c4b5650511c0f0f0e0908060f5c0e0a0b5d0b070c060e0a0f0c0f5c0a080e5c0e5b0f5c0a080e5e0e08"><sup><i class="fa fa-envelope-o"></i></sup></a><a href="https://orcid.org/0009-0004-5900-3318" target="_blank" rel="noopener noreferrer"><img src="https://pub.mdpi-res.com/img/design/orcid.png?0465bc3812adeb52?1732615622" title="ORCID" style="position: relative; width: 13px; margin-left: 3px; max-width: 13px !important; height: auto; top: -5px;"></a>, </span><span class="inlineblock "><div class='profile-card-drop' data-dropdown='profile-card-drop13255482' data-options='is_hover:true, hover_timeout:5000'> Yu-Hang Feng</div><div id="profile-card-drop13255482" data-dropdown-content class="f-dropdown content profile-card-content" aria-hidden="true" tabindex="-1"><div class="profile-card__title"><div class="sciprofiles-link" style="display: inline-block"><div class="sciprofiles-link__link"><img class="sciprofiles-link__image" 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href="/cdn-cgi/l/email-protection#69460a070d440a0e004605460c0408000544191b061d0c0a1d0006074a5959585f5e5e58505859595c585c5a5e580b590d595b5c50585b585a595b5c50585d5850"><sup><i class="fa fa-envelope-o"></i></sup></a>, </span><span class="inlineblock "><div class='profile-card-drop' data-dropdown='profile-card-drop13255484' data-options='is_hover:true, hover_timeout:5000'> Xian-Ru Hu</div><div id="profile-card-drop13255484" data-dropdown-content class="f-dropdown content profile-card-content" aria-hidden="true" tabindex="-1"><div class="profile-card__title"><div class="sciprofiles-link" style="display: inline-block"><div class="sciprofiles-link__link"><img class="sciprofiles-link__image" src="/bundles/mdpisciprofileslink/img/unknown-user.png" style="width: auto; height: 16px; border-radius: 50%;"><span class="sciprofiles-link__name">Xian-Ru Hu</span></div></div></div><div class="profile-card__buttons" style="margin-bottom: 10px;"><a 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href="/cdn-cgi/l/email-protection#113e727f753c7276783e7d3e747c70787d3c61637e65747265787e7f32212121202725217021222174212723252129207420202570212021212020257021262170"><sup><i class="fa fa-envelope-o"></i></sup></a><a href="https://orcid.org/0000-0002-0586-6220" target="_blank" rel="noopener noreferrer"><img src="https://pub.mdpi-res.com/img/design/orcid.png?0465bc3812adeb52?1732615622" title="ORCID" style="position: relative; width: 13px; margin-left: 3px; max-width: 13px !important; height: auto; top: -5px;"></a></span> </div> <div class="nrm"></div> <span style="display:block; height:6px;"></span> <div></div> <div style="margin: 5px 0 15px 0;" class="hypothesis_container"> <div class="art-affiliations"> <div class="affiliation "> <div class="affiliation-name ">Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical Physics of Gansu Province, Lanzhou University, Lanzhou 730000, China</div> </div> <div class="affiliation"> <div class="affiliation-item"><sup>*</sup></div> <div class="affiliation-name ">Author to whom correspondence should be addressed. </div> </div> </div> </div> <div class="bib-identity" style="margin-bottom: 10px;"> <em>Universe</em> <b>2024</b>, <em>10</em>(10), 393; <a href="https://doi.org/10.3390/universe10100393">https://doi.org/10.3390/universe10100393</a> </div> <div class="pubhistory" style="font-weight: bold; padding-bottom: 10px;"> <span style="display: inline-block">Submission received: 27 August 2024</span> / <span style="display: inline-block">Revised: 6 October 2024</span> / <span style="display: inline-block">Accepted: 9 October 2024</span> / <span style="display: inline-block">Published: 11 October 2024</span> </div> <div class="belongsTo" style="margin-bottom: 10px;"> (This article belongs to the Special Issue <a href=" /journal/universe/special_issues/48P0F45G72 ">Loop Quantum Gravity and Non-Perturbative Approaches to Quantum Cosmology, Second Edition</a>)<br/> </div> <div class="highlight-box1"> <div 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style="display: inline-block;"> <a href="#" class="button button--color-inversed margin-bottom-10 openpopupgallery UI_BrowseArticleFigures" data-target='article-popup' data-counterslink = "https://www.mdpi.com/2218-1997/10/10/393/browse" >Browse Figures</a> </div> <div id="article-popup" class="popupgallery" style="display: inline; line-height: 200%"> <a href="https://pub.mdpi-res.com/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g001.png?1728630853" title=" <strong>Figure 1</strong><br/> <p>The Penrose diagram of the quantum extended black hole spacetime. The green dotted line represents the orbit of an accretion disk with a radius <span class="html-italic">r</span>. The red and blue lines represent the trajectories of photons emitted from point <math display="inline"><semantics> <msub> <mi>Q</mi> <mi>BH</mi> </msub> </semantics></math> at location <math display="inline"><semantics> <mi mathvariant="normal">A</mi> </semantics></math> and point <math display="inline"><semantics> <msub> <mi>Q</mi> <msup> <mrow> <mi>BH</mi> </mrow> <mo>′</mo> </msup> </msub> </semantics></math> at location <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="normal">A</mi> </mrow> <mo>′</mo> </msup> </semantics></math>, respectively, reaching an observer at infinity, denoted as point <span class="html-italic">q</span>.</p> "> </a> <a href="https://pub.mdpi-res.com/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g002.png?1728630854" title=" <strong>Figure 2</strong><br/> <p>The variation of <math display="inline"><semantics> <msub> <mi>r</mi> <mo>±</mo> </msub> </semantics></math> with <math display="inline"><semantics> <mi>γ</mi> </semantics></math> for different values of <span class="html-italic">M</span>. The solid line represents <math display="inline"><semantics> <msub> <mi>r</mi> <mo>−</mo> </msub> </semantics></math>, and the dashed line represents <math display="inline"><semantics> <msub> <mi>r</mi> <mo>+</mo> </msub> </semantics></math>.</p> "> </a> <a href="https://pub.mdpi-res.com/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g003.png?1728630855" title=" <strong>Figure 3</strong><br/> <p>The variation of <math display="inline"><semantics> <msub> <mi>r</mi> <mrow> <mi>p</mi> <mi>e</mi> </mrow> </msub> </semantics></math> with <span class="html-italic">b</span> is depicted, with the inset illustrating the trajectory connecting the periastron of all photons with <math display="inline"><semantics> <mrow> <mi>b</mi> <mo>&lt;</mo> <msub> <mi>b</mi> <mi>c</mi> </msub> </mrow> </semantics></math> in spacetime.</p> "> </a> <a href="https://pub.mdpi-res.com/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g004.png?1728630856" title=" <strong>Figure 4</strong><br/> <p>The <math display="inline"><semantics> <mrow> <mi>G</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> function in the spacetimes of QBH (<b>left</b>) and Schwarzschild black hole (<b>right</b>). Where <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>0</mn> </msub> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <mo>∞</mo> </mfrac> </mstyle> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mrow> <mi>p</mi> <mi>h</mi> </mrow> </msub> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <msub> <mi>r</mi> <mrow> <mi>p</mi> <mi>h</mi> </mrow> </msub> </mfrac> </mstyle> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mo>±</mo> </msub> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <msub> <mi>r</mi> <mo>±</mo> </msub> </mfrac> </mstyle> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mrow> <mi>p</mi> <mi>e</mi> </mrow> </msub> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <msub> <mi>r</mi> <mrow> <mi>p</mi> <mi>e</mi> </mrow> </msub> </mfrac> </mstyle> </mrow> </semantics></math>.</p> "> </a> <a href="https://pub.mdpi-res.com/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g005.png?1728630857" title=" <strong>Figure 5</strong><br/> <p>The left panel illustrates the photon entering the white hole in spacetime <span class="html-italic">A</span>, while the right panel depicts the photon exiting the black hole in spacetime <span class="html-italic">B</span> (considering the reverse process for clarity). The orange dashed line indicates the trajectory of the photon with <math display="inline"><semantics> <mrow> <mi>b</mi> <mo>=</mo> <msub> <mi>b</mi> <mi>c</mi> </msub> </mrow> </semantics></math>.</p> "> </a> <a href="https://pub.mdpi-res.com/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g006.png?1728630859" title=" <strong>Figure 6</strong><br/> <p>The full trajectory of certain photons with <math display="inline"><semantics> <mrow> <mi>b</mi> <mo>&lt;</mo> <msub> <mi>b</mi> <mi>c</mi> </msub> </mrow> </semantics></math> in quantum-corrected spacetime.</p> "> </a> <a href="https://pub.mdpi-res.com/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g007.png?1728630861" title=" <strong>Figure 7</strong><br/> <p>Observational schematic diagram.</p> "> </a> <a href="https://pub.mdpi-res.com/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g008.png?1728630863" title=" <strong>Figure 8</strong><br/> <p>The variation of <math display="inline"><semantics> <msub> <mi>r</mi> <mrow> <mi>I</mi> <mi>S</mi> <mi>C</mi> <mi>O</mi> </mrow> </msub> </semantics></math> with <math display="inline"><semantics> <mi>γ</mi> </semantics></math> for different values of <span class="html-italic">M</span>.</p> "> </a> <a href="https://pub.mdpi-res.com/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g009.png?1728630864" title=" <strong>Figure 9</strong><br/> <p>The variation of <span class="html-italic">b</span> with <math display="inline"><semantics> <mi>ϕ</mi> </semantics></math> for various values of <span class="html-italic">r</span>.</p> "> </a> <a href="https://pub.mdpi-res.com/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g010.png?1728630867" title=" <strong>Figure 10</strong><br/> <p>The accretion disk image produced by the white hole. The red, orange, green, blue, and purple lines represent the 0th-, 1st-, 2nd-, 3rd-, and 4th-order images, respectively. We set <math display="inline"><semantics> <mrow> <mi>r</mi> <mo>=</mo> <msub> <mi>r</mi> <mi>ISCO</mi> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <msub> <mi>r</mi> <mo>∞</mo> </msub> </semantics></math>.</p> "> </a> <a href="https://pub.mdpi-res.com/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g011.png?1728630868" title=" <strong>Figure 11</strong><br/> <p>A composite image displaying the accretion disk image produced by the white hole (blue) and the accretion disk image generated by the black hole (red). For the black hole’s accretion disk image, we adjust <span class="html-italic">r</span> from <math display="inline"><semantics> <msub> <mi>r</mi> <mo>∞</mo> </msub> </semantics></math> to <math display="inline"><semantics> <mrow> <mn>15</mn> <mi>M</mi> </mrow> </semantics></math>.</p> "> </a> <a href="https://pub.mdpi-res.com/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g012.png?1728630869" title=" <strong>Figure 12</strong><br/> <p>With <math display="inline"><semantics> <mrow> <msub> <mi>θ</mi> <mn>0</mn> </msub> <mo>=</mo> <msup> <mn>0</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, the left panel illustrates the variation of <math display="inline"><semantics> <mrow> <msub> <mi>b</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> </msub> <mo>/</mo> <mi>M</mi> </mrow> </semantics></math> (solid line) and <math display="inline"><semantics> <mrow> <msub> <mi>b</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> <mo>/</mo> <mi>M</mi> </mrow> </semantics></math> (dashed line) with <span class="html-italic">M</span>, while the right panel depicts the variation of <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>b</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>b</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>/</mo> <mi>M</mi> </mrow> </semantics></math> with <span class="html-italic">M</span>. The red, orange, green, blue, and purple lines represent the 0th-, 1st-, 2nd-, 3rd-, and 4th-order images, respectively.</p> "> </a> <a href="https://pub.mdpi-res.com/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g013.png?1728630870" title=" <strong>Figure 13</strong><br/> <p>The variation of <math display="inline"><semantics> <mi>ϵ</mi> </semantics></math> with <math display="inline"><semantics> <mi>γ</mi> </semantics></math> for different values of <span class="html-italic">M</span>.</p> "> </a> <a href="https://pub.mdpi-res.com/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g014.png?1728630871" title=" <strong>Figure 14</strong><br/> <p>With <math display="inline"><semantics> <mrow> <msub> <mi>θ</mi> <mn>0</mn> </msub> <mo>=</mo> <msup> <mn>0</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, the left panel shows the variation of <math display="inline"><semantics> <msub> <mi>F</mi> <mrow> <mi>e</mi> <mi>m</mi> </mrow> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>F</mi> <mrow> <mi>o</mi> <mi>b</mi> <mi>s</mi> </mrow> </msub> </semantics></math> with <span class="html-italic">r</span> for <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> (Schwarzschild black hole) and <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>0.2375</mn> </mrow> </semantics></math>, while the right panel shows the variation of <math display="inline"><semantics> <msub> <mi>F</mi> <mrow> <mi>o</mi> <mi>b</mi> <mi>s</mi> </mrow> </msub> </semantics></math> with <span class="html-italic">b</span> for <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>0.2375</mn> </mrow> </semantics></math>. The red, orange, green, blue, and purple lines represent the 0th-, 1st-, 2nd-, 3rd-, and 4th-order images, respectively.</p> "> </a> <a href="https://pub.mdpi-res.com/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g015.png?1728630872" title=" <strong>Figure 15</strong><br/> <p>The accretion disk images with brightness produced by the white hole, with the panels from left to right corresponding to <math display="inline"><semantics> <mrow> <msub> <mi>θ</mi> <mn>0</mn> </msub> <mo>=</mo> <msup> <mn>0</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msup> <mn>40</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msup> <mn>80</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msup> <mn>88</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, respectively.</p> "> </a> </div> <a class="button button--color-inversed" href="/2218-1997/10/10/393/notes">Versions Notes</a> </div> </div> <div class="responsive-moving-container small hidden" data-id="article-counters" style="margin-top: 15px;"></div> <div class="html-dynamic"> <section> <div class="art-abstract art-abstract-new in-tab hypothesis_container"> <p> <div><section class="html-abstract" id="html-abstract"> <h2 id="html-abstract-title">Abstract</h2><b>:</b> <div class="html-p">Integrating loop quantum gravity with classical gravitational collapse models offers an effective solution to the black hole singularity problem and predicts the formation of a white hole in the later stages of collapse. Furthermore, the quantum extension of Kruskal spacetime indicates that white holes may convey information about earlier companion black holes. Photons emitted from the accretion disks of these companion black holes enter the black hole, traverse the highly quantum region, and then re-emerge from white holes in our universe. This process enables us to observe images of the companion black holes’ accretion disks, providing insights into quantum gravity. In our study, we successfully obtained these accretion disk images. Our results indicate that these accretion disk images are confined within a circle with a radius equal to the critical impact parameter, while traditional accretion disk images are typically located outside this circle. As the observational angle increases, the accretion disk images transition from a ring shape to a shell-like shape. Furthermore, the positional and width characteristics of these accretion disk images are opposite to those of traditional accretion disk images. These findings provide valuable references for astronomical observations aimed at validating the investigated quantum gravity model.</div> </section> <div id="html-keywords"> <div class="html-gwd-group"><div id="html-keywords-title">Keywords: </div><a href="/search?q=loop+quantum+gravity">loop quantum gravity</a>; <a href="/search?q=white+hole">white hole</a>; <a href="/search?q=thin+accretion+disk">thin accretion disk</a></div> <div> </div> </div> </div> </p> </div> </section> </div> <div class="hypothesis_container"> <ul class="menu html-nav" data-prev-node="#html-quick-links-title"> </ul> <div class="html-body"> <section id='sec1-universe-10-00393' type='intro'><h2 data-nested='1'> 1. Introduction</h2><div class='html-p'>General relativity (GR) provides an excellent description of gravity, but the singularity problem remains unresolved [<a href="#B1-universe-10-00393" class="html-bibr">1</a>,<a href="#B2-universe-10-00393" class="html-bibr">2</a>,<a href="#B3-universe-10-00393" class="html-bibr">3</a>]. Physicists have made many attempts to address this issue, with quantizing gravity being a mainstream approach. This effort has led to the development of various quantum gravity (QG) theories [<a href="#B4-universe-10-00393" class="html-bibr">4</a>,<a href="#B5-universe-10-00393" class="html-bibr">5</a>,<a href="#B6-universe-10-00393" class="html-bibr">6</a>,<a href="#B7-universe-10-00393" class="html-bibr">7</a>,<a href="#B8-universe-10-00393" class="html-bibr">8</a>,<a href="#B9-universe-10-00393" class="html-bibr">9</a>]. Experimental validation is crucial for these theories. In 2019, the Event Horizon Telescope (EHT) collaboration released the first image of a black hole shadow [<a href="#B10-universe-10-00393" class="html-bibr">10</a>], providing strong evidence in support of general relativity. Observing black holes may also serve as an effective method for testing the validity of quantum gravity theories.</div><div class='html-p'>Loop Quantum Gravity (LQG) is a quantum gravity theory that seeks to reconcile GR with quantum mechanics. LQG posits that spacetime is quantized at the Planck scale, implying that spacetime is not continuous but consists of discrete loops or “spin networks” [<a href="#B11-universe-10-00393" class="html-bibr">11</a>,<a href="#B12-universe-10-00393" class="html-bibr">12</a>,<a href="#B13-universe-10-00393" class="html-bibr">13</a>,<a href="#B14-universe-10-00393" class="html-bibr">14</a>,<a href="#B15-universe-10-00393" class="html-bibr">15</a>,<a href="#B16-universe-10-00393" class="html-bibr">16</a>,<a href="#B17-universe-10-00393" class="html-bibr">17</a>,<a href="#B18-universe-10-00393" class="html-bibr">18</a>]. This discrete structure of spacetime resolves singularity issues in traditional general relativity and offers new insights into black hole evaporation and the early universe. Specifically, when matter collapses to the Planck scale, the quantum effects of LQG become significant, leading to a bounce in the collapse process that prevents the formation of singularities with infinite density and curvature. This bounce mechanism suggests the existence of a quantum bridge at the center of a black hole, connecting to a new region of spacetime and potentially explaining black hole dynamics [<a href="#B19-universe-10-00393" class="html-bibr">19</a>,<a href="#B20-universe-10-00393" class="html-bibr">20</a>,<a href="#B21-universe-10-00393" class="html-bibr">21</a>,<a href="#B22-universe-10-00393" class="html-bibr">22</a>,<a href="#B23-universe-10-00393" class="html-bibr">23</a>,<a href="#B24-universe-10-00393" class="html-bibr">24</a>].</div><div class='html-p'>In recent years, researchers have integrated LQG effects into classical gravitational collapse models, such as the Oppenheimer–Snyder (OS) model [<a href="#B25-universe-10-00393" class="html-bibr">25</a>], to study the collapse of dust matter in spherically symmetric spacetimes [<a href="#B19-universe-10-00393" class="html-bibr">19</a>,<a href="#B26-universe-10-00393" class="html-bibr">26</a>,<a href="#B27-universe-10-00393" class="html-bibr">27</a>,<a href="#B28-universe-10-00393" class="html-bibr">28</a>,<a href="#B29-universe-10-00393" class="html-bibr">29</a>,<a href="#B30-universe-10-00393" class="html-bibr">30</a>,<a href="#B31-universe-10-00393" class="html-bibr">31</a>,<a href="#B32-universe-10-00393" class="html-bibr">32</a>,<a href="#B33-universe-10-00393" class="html-bibr">33</a>]. Building on this approach, Lewandowski et al. recently derived a modified Schwarzschild black hole solution within the framework of LQG [<a href="#B34-universe-10-00393" class="html-bibr">34</a>]. This model demonstrates that in the initial stages of collapse, the exterior spacetime of the dust sphere is described by a quantum-corrected black hole. In the later stages, the dust sphere transitions from collapse to expansion, eventually forming a white hole. Additionally, considering the concept of companion black holes, photons entering a companion black hole could traverse a highly quantum region and re-emerge from a white hole in our universe [<a href="#B33-universe-10-00393" class="html-bibr">33</a>,<a href="#B35-universe-10-00393" class="html-bibr">35</a>,<a href="#B36-universe-10-00393" class="html-bibr">36</a>]. This process offers a potential opportunity to observe quantum gravity effects.</div><div class='html-p'><a href="#universe-10-00393-f001" class="html-fig">Figure 1</a> illustrates this process. <math display='inline'><semantics> <msub> <mi>Q</mi> <mi>BH</mi> </msub> </semantics></math> represents particles in the accretion disk surrounding a black hole in our universe (A). These particles emit photons that travel along the red geodesic and are detected by an observer at <span class='html-italic'>q</span>, forming a traditional image of the black hole accretion disk. Our focus is on <math display='inline'><semantics> <msub> <mi>Q</mi> <msup> <mrow> <mi>BH</mi> </mrow> <mo>′</mo> </msup> </msub> </semantics></math>, representing particles in the accretion disk surrounding a companion black hole in an earlier universe (<math display='inline'><semantics> <msup> <mrow> <mi mathvariant="normal">A</mi> </mrow> <mo>′</mo> </msup> </semantics></math>). These particles emit photons that travel along the blue geodesic into the black hole, re-emerge from a white hole in our universe, and eventually reach an observer at <span class='html-italic'>q</span>, forming an additional accretion disk image. This additional accretion disk image was first investigated by Zhang et al. [<a href="#B37-universe-10-00393" class="html-bibr">37</a>]. Although Zhang et al. pointed out several limitations in this idea, such as the lack of strong mathematical support for the qualitative aspects, we have further developed their work. We hope our work will inspire and serve as a reference for future research in this area.</div><div class='html-p'>The organization of this paper is as follows: In <a href="#sec2-universe-10-00393" class="html-sec">Section 2</a>, we examine the motion of photons in a quantum-corrected spacetime. In <a href="#sec3-universe-10-00393" class="html-sec">Section 3</a>, we establish the observer’s coordinate system and plot the additional accretion disk image as observed. We then use the Novikov–Thorne model to depict the luminous accretion disk image. Finally, in <a href="#sec4-universe-10-00393" class="html-sec">Section 4</a>, we present a summary and discussion.</div></section><section id='sec2-universe-10-00393' type=''><h2 data-nested='1'> 2. Null Geodesics in the Quantum-Corrected Spacetime</h2><div class='html-p'>For a static, spherically symmetric quantum-corrected black hole (QBH) solution, the line element is given by [<a href="#B34-universe-10-00393" class="html-bibr">34</a>] <div class='html-disp-formula-info' id='FD1-universe-10-00393'> <div class='f'> <math display='block'><semantics> <mrow> <mi>d</mi> <msup> <mi>s</mi> <mn>2</mn> </msup> <mo>=</mo> <mo>−</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <mi>d</mi> <msup> <mi>t</mi> <mn>2</mn> </msup> <mo>+</mo> <mi>f</mi> <msup> <mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>d</mi> <msup> <mi>r</mi> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>r</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <msup> <mi>θ</mi> <mn>2</mn> </msup> <mo>+</mo> <msup> <mo form="prefix">sin</mo> <mn>2</mn> </msup> <mi>θ</mi> <mi>d</mi> <msup> <mi>ϕ</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(1)</label> </div> </div> with <div class='html-disp-formula-info' id='FD2-universe-10-00393'> <div class='f'> <math display='block'><semantics> <mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>2</mn> <mi>M</mi> </mrow> <mi>r</mi> </mfrac> </mstyle> <mo>+</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>β</mi> <msup> <mi>M</mi> <mn>2</mn> </msup> </mrow> <msup> <mi>r</mi> <mn>4</mn> </msup> </mfrac> </mstyle> <mo>,</mo> <mspace width="1.em"/> <mi>β</mi> <mo>=</mo> <mn>16</mn> <msqrt> <mn>3</mn> </msqrt> <mi>π</mi> <msup> <mi>γ</mi> <mn>3</mn> </msup> <msubsup> <mo>ℓ</mo> <mrow> <mi>p</mi> </mrow> <mn>2</mn> </msubsup> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(2)</label> </div> </div> where <math display='inline'><semantics> <mrow> <msub> <mo>ℓ</mo> <mi>p</mi> </msub> <mo>=</mo> <msqrt> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>ћ</mi> <mi>G</mi> </mrow> <msup> <mi>c</mi> <mn>3</mn> </msup> </mfrac> </mstyle> </msqrt> </mrow> </semantics></math> is the Planck length and <math display='inline'><semantics> <mi>γ</mi> </semantics></math> is the Barbero–Immirzi parameter (determined to be <math display='inline'><semantics> <mrow> <mn>0.2375</mn> </mrow> </semantics></math> in [<a href="#B38-universe-10-00393" class="html-bibr">38</a>,<a href="#B39-universe-10-00393" class="html-bibr">39</a>]). We adopt natural units (<math display='inline'><semantics> <mrow> <mi>c</mi> <mo>=</mo> <mi>G</mi> <mo>=</mo> <mi>ћ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>), resulting in <math display='inline'><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mn>16</mn> <msqrt> <mn>3</mn> </msqrt> <mi>π</mi> <msup> <mi>γ</mi> <mn>3</mn> </msup> <mo>≈</mo> <mn>1.16633</mn> </mrow> </semantics></math>. To ensure that <math display='inline'><semantics> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> has real roots, <span class='html-italic'>M</span> must have a minimum value of <math display='inline'><semantics> <mrow> <msub> <mi>M</mi> <mi>min</mi> </msub> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>16</mn> <mi>γ</mi> <msqrt> <mrow> <mi>π</mi> <mi>γ</mi> </mrow> </msqrt> </mrow> <mrow> <mn>3</mn> <msqrt> <mn>3</mn> </msqrt> </mrow> </mfrac> </mstyle> <mo>≈</mo> <mn>0.8314</mn> </mrow> </semantics></math>. By solving <math display='inline'><semantics> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, we find that the event horizon radius <math display='inline'><semantics> <msub> <mi>r</mi> <mo>±</mo> </msub> </semantics></math> of this QBH is <div class='html-disp-formula-info' id='FD3-universe-10-00393'> <div class='f'> <math display='block'><semantics> <mtable displaystyle="true"> <mtr> <mtd columnalign="right"> <msub> <mi>r</mi> <mo>±</mo> </msub> </mtd> <mtd columnalign="left"> <mrow> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mstyle> <mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mi>M</mi> <mo>+</mo> <msqrt> <mn>3</mn> </msqrt> <msqrt> <mrow> <msup> <mfenced separators="" open="(" close=")"> <mn>6</mn> <mi>σ</mi> </mfenced> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </msup> <mo>+</mo> <mn>3</mn> <msup> <mi>M</mi> <mn>2</mn> </msup> <mo>+</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>2</mn> <mo>×</mo> <msup> <mn>6</mn> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </msup> <msup> <mi>M</mi> <mn>2</mn> </msup> <mi>β</mi> </mrow> <msup> <mfenced open="(" close=")"> <mi>σ</mi> </mfenced> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </msup> </mfrac> </mstyle> </mrow> </msqrt> <mo stretchy="false">)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="right"> <mspace width="1.em"/> </mtd> <mtd columnalign="left"> <mrow> <mo>±</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mstyle> <msqrt> <mn>3</mn> </msqrt> <msqrt> <mrow> <mn>6</mn> <msup> <mi>M</mi> <mn>2</mn> </msup> <mo>−</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>2</mn> <mo>×</mo> <msup> <mn>6</mn> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </msup> <msup> <mi>M</mi> <mn>2</mn> </msup> <mi>β</mi> </mrow> <msup> <mfenced open="(" close=")"> <mi>σ</mi> </mfenced> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </msup> </mfrac> </mstyle> <mo>−</mo> <msup> <mfenced separators="" open="(" close=")"> <mn>6</mn> <mi>σ</mi> </mfenced> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </msup> <mo>+</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>6</mn> <msqrt> <mn>3</mn> </msqrt> <msup> <mi>M</mi> <mn>3</mn> </msup> </mrow> <msqrt> <mrow> <msup> <mfenced separators="" open="(" close=")"> <mn>6</mn> <mi>σ</mi> </mfenced> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </msup> <mo>+</mo> <mn>3</mn> <msup> <mi>M</mi> <mn>2</mn> </msup> <mo>+</mo> <mfrac> <mrow> <mn>2</mn> <mo> </mo> <mo>×</mo> <mo> </mo> <msup> <mn>6</mn> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </msup> <msup> <mi>M</mi> <mn>2</mn> </msup> <mi>β</mi> </mrow> <msup> <mfenced open="(" close=")"> <mi>σ</mi> </mfenced> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </msup> </mfrac> </mrow> </msqrt> </mfrac> </mstyle> </mrow> </msqrt> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </semantics></math> </div> <div class='l'> <label >(3)</label> </div> </div> where <math display='inline'><semantics> <mrow> <mi>σ</mi> <mo>=</mo> <mn>9</mn> <msup> <mi>M</mi> <mn>4</mn> </msup> <mi>β</mi> <mo>+</mo> <msqrt> <mn>3</mn> </msqrt> <msqrt> <mrow> <msup> <mi>M</mi> <mn>6</mn> </msup> <mfenced separators="" open="(" close=")"> <mn>27</mn> <msup> <mi>M</mi> <mn>2</mn> </msup> <mo>−</mo> <mn>16</mn> <mi>β</mi> </mfenced> <msup> <mi>β</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> </semantics></math>. <a href="#universe-10-00393-f002" class="html-fig">Figure 2</a> illustrates the variation of <math display='inline'><semantics> <msub> <mi>r</mi> <mo>±</mo> </msub> </semantics></math> with <math display='inline'><semantics> <mi>γ</mi> </semantics></math> for different values of <span class='html-italic'>M</span>. A specific value of <math display='inline'><semantics> <mi>γ</mi> </semantics></math> can be observed where <math display='inline'><semantics> <msub> <mi>r</mi> <mo>+</mo> </msub> </semantics></math> and <math display='inline'><semantics> <msub> <mi>r</mi> <mo>−</mo> </msub> </semantics></math> converge, beyond which no horizon exists. For <math display='inline'><semantics> <mrow> <mi>M</mi> <mo>=</mo> <mn>0.8314</mn> </mrow> </semantics></math>, this critical value of <math display='inline'><semantics> <mi>γ</mi> </semantics></math> is <math display='inline'><semantics> <mrow> <mn>0.2375</mn> </mrow> </semantics></math> (<math display='inline'><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mn>1.16633</mn> </mrow> </semantics></math>). Additionally, LQG corrections result in <math display='inline'><semantics> <msub> <mi>r</mi> <mo>+</mo> </msub> </semantics></math> and <math display='inline'><semantics> <msub> <mi>r</mi> <mo>−</mo> </msub> </semantics></math> always being smaller than the Schwarzschild black hole’s horizon radius. In the subsequent calculations, we set <math display='inline'><semantics> <mrow> <mi>M</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</div><div class='html-p'>Next, we analyze the motion of test particles within the QBH spacetime. The Lagrangian for these test particles is expressed as <div class='html-disp-formula-info' id='FD4-universe-10-00393'> <div class='f'> <math display='block'><semantics> <mrow> <mi mathvariant="script">L</mi> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> <msub> <mi>g</mi> <mrow> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <msup> <mover accent="true"> <mi>x</mi> <mo>˙</mo> </mover> <mi>μ</mi> </msup> <msup> <mover accent="true"> <mi>x</mi> <mo>˙</mo> </mover> <mi>ν</mi> </msup> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> <mfenced separators="" open="(" close=")"> <mo>−</mo> <mi>f</mi> <mfenced open="(" close=")"> <mi>r</mi> </mfenced> <msup> <mover accent="true"> <mi>t</mi> <mo>˙</mo> </mover> <mn>2</mn> </msup> <mo>+</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <mrow> <mi>f</mi> <mfenced open="(" close=")"> <mi>r</mi> </mfenced> </mrow> </mfrac> </mstyle> <msup> <mover accent="true"> <mi>r</mi> <mo>˙</mo> </mover> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>r</mi> <mn>2</mn> </msup> <msup> <mover accent="true"> <mi>θ</mi> <mo>˙</mo> </mover> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>r</mi> <mn>2</mn> </msup> <msup> <mo form="prefix">sin</mo> <mn>2</mn> </msup> <mi>θ</mi> <msup> <mover accent="true"> <mi>ϕ</mi> <mo>˙</mo> </mover> <mn>2</mn> </msup> </mfenced> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(4)</label> </div> </div> where <math display='inline'><semantics> <mrow> <msup> <mover accent="true"> <mi>x</mi> <mo>˙</mo> </mover> <mi>μ</mi> </msup> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mi>μ</mi> </msup> </mrow> <mrow> <mi>d</mi> <mi>λ</mi> </mrow> </mfrac> </mstyle> </mrow> </semantics></math>. For photons, <math display='inline'><semantics> <mi>λ</mi> </semantics></math> represents the affine parameter, while for time-like particles, <math display='inline'><semantics> <mi>λ</mi> </semantics></math> corresponds to the proper time <math display='inline'><semantics> <mi>τ</mi> </semantics></math>. The presence of two killing vector fields, <math display='inline'><semantics> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mo>∂</mo> <mrow> <mo>∂</mo> <mi>t</mi> </mrow> </mfrac> </mstyle> </semantics></math> and <math display='inline'><semantics> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mo>∂</mo> <mrow> <mo>∂</mo> <mi>ϕ</mi> </mrow> </mfrac> </mstyle> </semantics></math>, in a static, spherically symmetric spacetime results in two corresponding conserved quantities for the test particles, <div class='html-disp-formula-info' id='FD5-universe-10-00393'> <div class='f'> <math display='block'><semantics> <mrow> <mi>E</mi> <mo>=</mo> <mo>−</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mo>∂</mo> <mi mathvariant="script">L</mi> </mrow> <mrow> <mo>∂</mo> <mover accent="true"> <mi>t</mi> <mo>˙</mo> </mover> </mrow> </mfrac> </mstyle> <mo>=</mo> <mi>f</mi> <mfenced open="(" close=")"> <mi>r</mi> </mfenced> <mover accent="true"> <mi>t</mi> <mo>˙</mo> </mover> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(5)</label> </div> </div> and <div class='html-disp-formula-info' id='FD6-universe-10-00393'> <div class='f'> <math display='block'><semantics> <mrow> <mi>L</mi> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mo>∂</mo> <mi mathvariant="script">L</mi> </mrow> <mrow> <mo>∂</mo> <mover accent="true"> <mi>ϕ</mi> <mo>˙</mo> </mover> </mrow> </mfrac> </mstyle> <mo>=</mo> <msup> <mi>r</mi> <mn>2</mn> </msup> <mover accent="true"> <mi>ϕ</mi> <mo>˙</mo> </mover> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(6)</label> </div> </div> where <span class='html-italic'>E</span> represents the energy of the test particle, <span class='html-italic'>L</span> denotes its angular momentum, and the impact parameter <span class='html-italic'>b</span> is given by their ratio, <math display='inline'><semantics> <mrow> <mi>b</mi> <mo>:</mo> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mo>|</mo> <mi>L</mi> <mo>|</mo> </mrow> <mi>E</mi> </mfrac> </mstyle> </mrow> </semantics></math>. Note that in Equation (<a href="#FD6-universe-10-00393" class="html-disp-formula">6</a>), the test particle is chosen to move on the equatorial plane, i.e., <math display='inline'><semantics> <mrow> <mi>θ</mi> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> </semantics></math>.</div><div class='html-p'>For a photon, with <math display='inline'><semantics> <mrow> <mi mathvariant="script">L</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, the equation of motion on the equatorial plane is derived by simultaneously solving Equations (<a href="#FD4-universe-10-00393" class="html-disp-formula">4</a>)–(<a href="#FD6-universe-10-00393" class="html-disp-formula">6</a>), and eliminating the affine parameter, resulting in <div class='html-disp-formula-info' id='FD7-universe-10-00393'> <div class='f'> <math display='block'><semantics> <mrow> <mi>G</mi> <mfenced separators="" open="(" close=")"> <mi>u</mi> <mo>,</mo> <mi>b</mi> </mfenced> <mo>:</mo> <mo>=</mo> <msup> <mfenced separators="" open="(" close=")"> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>d</mi> <mi>u</mi> </mrow> <mrow> <mi>d</mi> <mi>ϕ</mi> </mrow> </mfrac> </mstyle> </mfenced> <mn>2</mn> </msup> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <msup> <mi>b</mi> <mn>2</mn> </msup> </mfrac> </mstyle> <mo>−</mo> <msup> <mi>u</mi> <mn>2</mn> </msup> <mi>f</mi> <mfenced separators="" open="(" close=")"> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <mi>u</mi> </mfrac> </mstyle> </mfenced> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(7)</label> </div> </div> where <math display='inline'><semantics> <mrow> <mi>u</mi> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <mi>r</mi> </mfrac> </mstyle> </mrow> </semantics></math>. Circular orbital motion satisfies <math display='inline'><semantics> <mrow> <mi>G</mi> <mfenced separators="" open="(" close=")"> <mi>u</mi> <mo>,</mo> <mi>b</mi> </mfenced> <mo>=</mo> <msub> <mo>∂</mo> <mi>u</mi> </msub> <mi>G</mi> <mfenced separators="" open="(" close=")"> <mi>u</mi> <mo>,</mo> <mi>b</mi> </mfenced> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>. Solving this yields the photon sphere radius <math display='inline'><semantics> <msub> <mi>r</mi> <mrow> <mi>p</mi> <mi>h</mi> </mrow> </msub> </semantics></math> and the corresponding critical impact parameter <math display='inline'><semantics> <msub> <mi>b</mi> <mi>c</mi> </msub> </semantics></math> as <div class='html-disp-formula-info' id='FD8-universe-10-00393'> <div class='f'> <math display='block'><semantics> <mtable displaystyle="true"> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi>r</mi> <mrow> <mi>p</mi> <mi>h</mi> </mrow> </msub> <mo>=</mo> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>3</mn> <mi>M</mi> </mrow> <mn>4</mn> </mfrac> </mstyle> <mo>+</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> <msqrt> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>9</mn> <msup> <mi>M</mi> <mn>2</mn> </msup> </mrow> <mn>4</mn> </mfrac> </mstyle> <mo>+</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>4</mn> <mi>M</mi> <msup> <mi>β</mi> <mrow> <mn>2</mn> <mo>/</mo> <mn>3</mn> </mrow> </msup> </mrow> <mi>η</mi> </mfrac> </mstyle> <mo>+</mo> <mi>M</mi> <msup> <mi>β</mi> <mrow> <mn>1</mn> <mo>/</mo> <mn>3</mn> </mrow> </msup> <mi>η</mi> </mrow> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd/> <mtd columnalign="left"> <mrow> <mo>+</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> <msqrt> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>9</mn> <msup> <mi>M</mi> <mn>2</mn> </msup> </mrow> <mn>2</mn> </mfrac> </mstyle> <mo>−</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>4</mn> <mi>M</mi> <msup> <mi>β</mi> <mrow> <mn>2</mn> <mo>/</mo> <mn>3</mn> </mrow> </msup> </mrow> <mi>η</mi> </mfrac> </mstyle> <mo>−</mo> <msup> <mi>β</mi> <mrow> <mn>1</mn> <mo>/</mo> <mn>3</mn> </mrow> </msup> <mi>η</mi> <mi>M</mi> <mo>+</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>27</mn> <msup> <mi>M</mi> <mn>3</mn> </msup> </mrow> <mrow> <mn>4</mn> <msqrt> <mrow> <mfrac> <mrow> <mn>9</mn> <msup> <mi>M</mi> <mn>2</mn> </msup> </mrow> <mn>4</mn> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>4</mn> <mi>M</mi> <msup> <mi>β</mi> <mrow> <mn>2</mn> <mo>/</mo> <mn>3</mn> </mrow> </msup> </mrow> <mi>η</mi> </mfrac> <mo>+</mo> <msup> <mi>β</mi> <mrow> <mn>1</mn> <mo>/</mo> <mn>3</mn> </mrow> </msup> <mi>η</mi> <mi>M</mi> </mrow> </msqrt> </mrow> </mfrac> </mstyle> </mrow> </msqrt> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </semantics></math> </div> <div class='l'> <label >(8)</label> </div> </div><div class='html-disp-formula-info' id='FD9-universe-10-00393'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>b</mi> <mi>c</mi> </msub> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <msub> <mi>r</mi> <mrow> <mi>p</mi> <mi>h</mi> </mrow> </msub> <msqrt> <mrow> <mi>f</mi> <mfenced separators="" open="(" close=")"> <msub> <mi>r</mi> <mrow> <mi>p</mi> <mi>h</mi> </mrow> </msub> </mfenced> </mrow> </msqrt> </mfrac> </mstyle> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(9)</label> </div> </div> where <math display='inline'><semantics> <mrow> <mi>η</mi> <mo>=</mo> <msup> <mfenced separators="" open="(" close=")"> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>27</mn> <mi>M</mi> <mo>−</mo> <msqrt> <mrow> <mn>729</mn> <msup> <mi>M</mi> <mn>2</mn> </msup> <mo>−</mo> <mn>256</mn> <mi>β</mi> </mrow> </msqrt> </mrow> <mn>2</mn> </mfrac> </mstyle> </mfenced> <mrow> <mn>1</mn> <mo>/</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math>. The periastron radius <math display='inline'><semantics> <msub> <mi>r</mi> <mrow> <mi>p</mi> <mi>e</mi> </mrow> </msub> </semantics></math> of the photon’s motion satisfies <math display='inline'><semantics> <mrow> <mi>G</mi> <mo stretchy="false">(</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <msub> <mi>r</mi> <mrow> <mi>p</mi> <mi>e</mi> </mrow> </msub> </mfrac> </mstyle> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>. In <a href="#universe-10-00393-f003" class="html-fig">Figure 3</a>, we have plotted the variation of <math display='inline'><semantics> <msub> <mi>r</mi> <mrow> <mi>p</mi> <mi>e</mi> </mrow> </msub> </semantics></math> with <span class='html-italic'>b</span>. It is evident that for <math display='inline'><semantics> <mrow> <mi>b</mi> <mo><</mo> <msub> <mi>b</mi> <mi>c</mi> </msub> </mrow> </semantics></math>, <math display='inline'><semantics> <msub> <mi>r</mi> <mrow> <mi>p</mi> <mi>e</mi> </mrow> </msub> </semantics></math> is always less than <math display='inline'><semantics> <msub> <mi>r</mi> <mo>−</mo> </msub> </semantics></math> and undergoes a sudden change at <math display='inline'><semantics> <mrow> <mi>b</mi> <mo>=</mo> <msub> <mi>b</mi> <mi>c</mi> </msub> </mrow> </semantics></math>. The inset shows the trajectory formed by these periastron on the equatorial plane.</div><div class='html-p'><a href="#universe-10-00393-f004" class="html-fig">Figure 4</a>, which illustrates the <math display='inline'><semantics> <mrow> <mi>G</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> function, indicates that photons with <math display='inline'><semantics> <mrow> <mi>b</mi> <mo>⩾</mo> <msub> <mi>b</mi> <mi>c</mi> </msub> </mrow> </semantics></math> in the QBH spacetime move similarly to those in the Schwarzschild spacetime. However, photons with <math display='inline'><semantics> <mrow> <mi>b</mi> <mo><</mo> <msub> <mi>b</mi> <mi>c</mi> </msub> </mrow> </semantics></math> sequentially enter the outer and inner horizons, and then reach their periastron. As seen in <a href="#universe-10-00393-f003" class="html-fig">Figure 3</a>, these periastrons are all located within the inner horizon. According to [<a href="#B37-universe-10-00393" class="html-bibr">37</a>], the radius of the inner horizon <math display='inline'><semantics> <msub> <mi>r</mi> <mo>−</mo> </msub> </semantics></math> is a characteristic radius where the spacetime becomes highly quantized. As mentioned earlier, photons travel through this highly quantum region, disappear at their periastron, and then reappear from a white hole (corresponding to WH in <a href="#universe-10-00393-f001" class="html-fig">Figure 1</a>), continuing their motion along a symmetrical trajectory and eventually moving to infinity. When static observers at infinity (corresponding to q in <a href="#universe-10-00393-f001" class="html-fig">Figure 1</a>) receive these photons, they can obtain information from the companion black hole spacetime (corresponding to <math display='inline'><semantics> <msup> <mrow> <mi mathvariant="normal">A</mi> </mrow> <mo>′</mo> </msup> </semantics></math> in <a href="#universe-10-00393-f001" class="html-fig">Figure 1</a>), such as images of the accretion disk. For photons with <math display='inline'><semantics> <mrow> <mi>b</mi> <mo><</mo> <msub> <mi>b</mi> <mi>c</mi> </msub> </mrow> </semantics></math>, their impact parameter <span class='html-italic'>b</span> remains unchanged. Therefore, the images received by observers at infinity will all be located within the circle of radius <math display='inline'><semantics> <msub> <mi>b</mi> <mi>c</mi> </msub> </semantics></math>. The accretion disk images in the next section visually illustrate this point.</div><div class='html-p'>For convenience in examining the images received by the observer, we treat the path from <math display='inline'><semantics> <msub> <mi>Q</mi> <msup> <mrow> <mi>BH</mi> </mrow> <mo>′</mo> </msup> </msub> </semantics></math> to <span class='html-italic'>q</span> as the path from <span class='html-italic'>q</span> to <math display='inline'><semantics> <msub> <mi>Q</mi> <msup> <mrow> <mi>BH</mi> </mrow> <mo>′</mo> </msup> </msub> </semantics></math>, given the reversibility of the light path. This allows us to set the observer’s position <span class='html-italic'>q</span> as the initial position of the photon, i.e., <math display='inline'><semantics> <mrow> <mi>u</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <mi>ϕ</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>. Therefore, for photons with <math display='inline'><semantics> <mrow> <mi>b</mi> <mo><</mo> <msub> <mi>b</mi> <mi>c</mi> </msub> </mrow> </semantics></math>, the azimuthal angle at <math display='inline'><semantics> <msub> <mi>r</mi> <mi mathvariant="normal">A</mi> </msub> </semantics></math> in region <math display='inline'><semantics> <mi mathvariant="normal">A</mi> </semantics></math> is <div class='html-disp-formula-info' id='FD10-universe-10-00393'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>ϕ</mi> <msub> <mi>r</mi> <mi mathvariant="normal">A</mi> </msub> </msub> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow> <mn>0</mn> </mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <msub> <mi>r</mi> <mi mathvariant="normal">A</mi> </msub> </mfrac> </mstyle> </msubsup> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <msqrt> <mrow> <mi>G</mi> <mfenced separators="" open="(" close=")"> <mi>u</mi> <mo>,</mo> <mi>b</mi> </mfenced> </mrow> </msqrt> </mfrac> </mstyle> <mi>d</mi> <mi>u</mi> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(10)</label> </div> </div> and at <math display='inline'><semantics> <msub> <mi>r</mi> <msup> <mrow> <mi mathvariant="normal">A</mi> </mrow> <mo>′</mo> </msup> </msub> </semantics></math> in region <math display='inline'><semantics> <msup> <mrow> <mi mathvariant="normal">A</mi> </mrow> <mo>′</mo> </msup> </semantics></math> is <div class='html-disp-formula-info' id='FD11-universe-10-00393'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>ϕ</mi> <msub> <mi>r</mi> <msup> <mrow> <mi mathvariant="normal">A</mi> </mrow> <mo>′</mo> </msup> </msub> </msub> <mo>=</mo> <mn>2</mn> <msubsup> <mo>∫</mo> <mrow> <mn>0</mn> </mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <msub> <mi>r</mi> <mrow> <mi>p</mi> <mi>e</mi> </mrow> </msub> </mfrac> </mstyle> </msubsup> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <msqrt> <mrow> <mi>G</mi> <mfenced separators="" open="(" close=")"> <mi>u</mi> <mo>,</mo> <mi>b</mi> </mfenced> </mrow> </msqrt> </mfrac> </mstyle> <mi>d</mi> <mi>u</mi> <mo>−</mo> <msubsup> <mo>∫</mo> <mrow> <mn>0</mn> </mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <msub> <mi>r</mi> <msup> <mrow> <mi mathvariant="normal">A</mi> </mrow> <mo>′</mo> </msup> </msub> </mfrac> </mstyle> </msubsup> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <msqrt> <mrow> <mi>G</mi> <mfenced separators="" open="(" close=")"> <mi>u</mi> <mo>,</mo> <mi>b</mi> </mfenced> </mrow> </msqrt> </mfrac> </mstyle> <mi>d</mi> <mi>u</mi> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(11)</label> </div> </div></div><div class='html-p'>Thus, the position of the photon in spacetime is determined by <span class='html-italic'>r</span> and <math display='inline'><semantics> <msub> <mi>ϕ</mi> <mi>r</mi> </msub> </semantics></math>. Using these coordinates, we have plotted the photon’s trajectory in <a href="#universe-10-00393-f005" class="html-fig">Figure 5</a>. In the figure, the left panel represents spacetime <math display='inline'><semantics> <mi mathvariant="normal">A</mi> </semantics></math>, and the right panel represents spacetime <math display='inline'><semantics> <msup> <mrow> <mi mathvariant="normal">A</mi> </mrow> <mo>′</mo> </msup> </semantics></math>. Photons with <math display='inline'><semantics> <mrow> <mi>b</mi> <mo><</mo> <msub> <mi>b</mi> <mi>c</mi> </msub> </mrow> </semantics></math> (blue lines) on the left panel sequentially enter the outer horizon (boundary of the gray disk) and the inner horizon (boundary of the green disk), disappear at their periastron (spiral dashed line), and then reappear on the right panel, continuing their motion along a trajectory symmetrical to the one on the left. In fact, the trajectories on the left and right panels can be connected to form a complete trajectory, determined by the metric described in Equation (<a href="#FD1-universe-10-00393" class="html-disp-formula">1</a>) (as shown in <a href="#universe-10-00393-f006" class="html-fig">Figure 6</a>). However, according to LQG theory, this complete trajectory is truncated at the periastron, and the segments on either side are assigned different physical meanings.</div></section><section id='sec3-universe-10-00393' type=''><h2 data-nested='1'> 3. Image of a Thin Accretion Disk Produced by a White Hole</h2><section id='sec3dot1-universe-10-00393' type=''><h4 class='html-italic' data-nested='2'> 3.1. Observation Coordinate System</h4><div class='html-p'>To derive the observed image of the accretion disk, we constructed the observational schematic diagram shown in <a href="#universe-10-00393-f007" class="html-fig">Figure 7</a>. In the figure, the coordinate system <math display='inline'><semantics> <mrow> <msup> <mrow> <mi mathvariant="normal">O</mi> </mrow> <mo>′</mo> </msup> <msup> <mrow> <mi mathvariant="normal">X</mi> </mrow> <mo>′</mo> </msup> <msup> <mrow> <mi mathvariant="normal">Y</mi> </mrow> <mo>′</mo> </msup> </mrow> </semantics></math> is positioned on the observer’s reception screen at <math display='inline'><semantics> <mrow> <mi>q</mi> <mfenced separators="" open="(" close=")"> <mo>∞</mo> <mo>,</mo> <msub> <mi>θ</mi> <mn>0</mn> </msub> <mo>,</mo> <mn>0</mn> </mfenced> </mrow> </semantics></math>. An image point on this screen is defined by the radial coordinate <span class='html-italic'>b</span> (the photon’s impact parameter) and the angular coordinate <math display='inline'><semantics> <mi>α</mi> </semantics></math>. The coordinate system <math display='inline'><semantics> <mi>OXYZ</mi> </semantics></math> corresponds to the black hole in spacetime <math display='inline'><semantics> <msup> <mrow> <mi mathvariant="normal">A</mi> </mrow> <mo>′</mo> </msup> </semantics></math> and the white hole in spacetime <math display='inline'><semantics> <mi mathvariant="normal">A</mi> </semantics></math>. We represent the black hole and white hole together for the sake of their independence, which is both logical and convenient. Point <math display='inline'><semantics> <mrow> <msub> <mi>Q</mi> <msup> <mrow> <mi>BH</mi> </mrow> <mo>′</mo> </msup> </msub> <mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math> is located on the orbit of the accretion disk around the black hole in spacetime <math display='inline'><semantics> <msup> <mrow> <mi mathvariant="normal">A</mi> </mrow> <mo>′</mo> </msup> </semantics></math>. Photons emitted from <math display='inline'><semantics> <msub> <mi>Q</mi> <msup> <mrow> <mi>BH</mi> </mrow> <mo>′</mo> </msup> </msub> </semantics></math> traverse the highly quantum region of the black hole, re-emerge from a white hole in spacetime A, travel along the alpha plane, and finally reach <span class='html-italic'>q</span> to form an image. This process is depicted by the red line in <a href="#universe-10-00393-f007" class="html-fig">Figure 7</a>, with the dashed line indicating motion in spacetime <math display='inline'><semantics> <msup> <mrow> <mi mathvariant="normal">A</mi> </mrow> <mo>′</mo> </msup> </semantics></math> and the solid line representing motion in spacetime A. Owing to the intense gravitational field of the black hole, photons emitted from point <math display='inline'><semantics> <msub> <mi>Q</mi> <msup> <mrow> <mi>BH</mi> </mrow> <mo>′</mo> </msup> </msub> </semantics></math> with varying impact parameters <span class='html-italic'>b</span> can reach point <span class='html-italic'>q</span> along the alpha plane. This scenario is illustrated in the lower part of <a href="#universe-10-00393-f007" class="html-fig">Figure 7</a>. The images produced by photons can be categorized into <math display='inline'><semantics> <mrow> <mi>n</mi> <mo> </mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> orders based on the magnitude of the <math display='inline'><semantics> <mi>ϕ</mi> </semantics></math> angle they traverse, with the corresponding <math display='inline'><semantics> <mi>ϕ</mi> </semantics></math> angles for these <span class='html-italic'>n</span> orders given by <div class='html-disp-formula-info' id='FD12-universe-10-00393'> <div class='f'> <math display='block'><semantics> <mrow> <msup> <mi>ϕ</mi> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>θ</mi> <mn>0</mn> </msub> <mo>,</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mfenced separators="" open="{" close=""> <mtable> <mtr> <mtd columnalign="left"> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mstyle> <mn>2</mn> <mi>π</mi> <mo>+</mo> <msup> <mfenced separators="" open="(" close=")"> <mo>−</mo> <mn>1</mn> </mfenced> <mi>n</mi> </msup> <mfenced separators="" open="[" close="]"> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> <mo>+</mo> <mi>arc</mi> <mo form="prefix">tan</mo> <mfenced separators="" open="(" close=")"> <mo form="prefix">tan</mo> <msub> <mi>θ</mi> <mn>0</mn> </msub> <mo form="prefix">sin</mo> <mi>α</mi> </mfenced> </mfenced> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mo>,</mo> <mo> </mo> <mi>when</mi> <mo> </mo> <mi>n</mi> <mo> </mo> <mi>is</mi> <mo> </mo> <mi>even</mi> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mstyle> <mn>2</mn> <mi>π</mi> <mo>+</mo> <msup> <mfenced separators="" open="(" close=")"> <mo>−</mo> <mn>1</mn> </mfenced> <mi>n</mi> </msup> <mfenced separators="" open="[" close="]"> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> <mo>+</mo> <mi>arc</mi> <mo form="prefix">tan</mo> <mfenced separators="" open="(" close=")"> <mo form="prefix">tan</mo> <msub> <mi>θ</mi> <mn>0</mn> </msub> <mo form="prefix">sin</mo> <mi>α</mi> </mfenced> </mfenced> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mo>,</mo> <mo> </mo> <mrow> <mi>when</mi> <mo> </mo> <mi>n</mi> <mo> </mo> <mi>is</mi> <mo> </mo> <mi>odd</mi> </mrow> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow> </semantics></math> </div> <div class='l'> <label >(12)</label> </div> </div></div><div class='html-p'>Substituting <math display='inline'><semantics> <mrow> <msup> <mi>ϕ</mi> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>θ</mi> <mn>0</mn> </msub> <mo>,</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math> into Equation (<a href="#FD11-universe-10-00393" class="html-disp-formula">11</a>) yields <math display='inline'><semantics> <msup> <mi>b</mi> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> </semantics></math>. Together, <math display='inline'><semantics> <msup> <mi>b</mi> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> </semantics></math> and <math display='inline'><semantics> <mi>α</mi> </semantics></math> define an image point on the observer’s reception screen.</div></section><section id='sec3dot2-universe-10-00393' type=''><h4 class='html-italic' data-nested='2'> 3.2. Image of a Thin Accretion Disk</h4><div class='html-p'>Geometrically, the accretion disk is a flat, wide, ring-like structure, theoretically composed of matter accumulated between the innermost stable circular orbit and an outer circular orbit. For simplicity, we assume the accretion disk extends between <math display='inline'><semantics> <mrow> <mi>r</mi> <mo>=</mo> <msub> <mi>r</mi> <mi>ISCO</mi> </msub> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <mi>r</mi> <mo>=</mo> <msub> <mi>r</mi> <mo>∞</mo> </msub> <mo>≡</mo> <mo>∞</mo> </mrow> </semantics></math>. <math display='inline'><semantics> <msub> <mi>r</mi> <mi>ISCO</mi> </msub> </semantics></math> represents the innermost stable circular orbit (ISCO) for timelike particles, which satisfies the condition <math display='inline'><semantics> <mrow> <msub> <mfenced separators="" open="" close="|"> <msup> <mover accent="true"> <mi>r</mi> <mo>˙</mo> </mover> <mn>2</mn> </msup> </mfenced> <msub> <mi>r</mi> <mi>ISCO</mi> </msub> </msub> <mo>=</mo> <msub> <mfenced separators="" open="" close="|"> <mfenced separators="" open="(" close=")"> <msub> <mo>∂</mo> <mi>r</mi> </msub> <msup> <mover accent="true"> <mi>r</mi> <mo>˙</mo> </mover> <mn>2</mn> </msup> </mfenced> </mfenced> <msub> <mi>r</mi> <mi>ISCO</mi> </msub> </msub> <mo>=</mo> <msub> <mfenced separators="" open="" close="|"> <mfenced separators="" open="(" close=")"> <msub> <mo>∂</mo> <mrow> <mi>r</mi> <mo>,</mo> <mi>r</mi> </mrow> </msub> <msup> <mover accent="true"> <mi>r</mi> <mo>˙</mo> </mover> <mn>2</mn> </msup> </mfenced> </mfenced> <msub> <mi>r</mi> <mi>ISCO</mi> </msub> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, where <math display='inline'><semantics> <mrow> <msup> <mover accent="true"> <mi>r</mi> <mo>˙</mo> </mover> <mn>2</mn> </msup> <mo>≡</mo> <msup> <mfenced separators="" open="(" close=")"> <msub> <mo>∂</mo> <mi>τ</mi> </msub> <mi>r</mi> </mfenced> <mn>2</mn> </msup> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <msup> <mi>b</mi> <mn>2</mn> </msup> </mfrac> </mstyle> <mo>−</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>f</mi> <mfenced open="(" close=")"> <mi>r</mi> </mfenced> </mrow> <msup> <mi>r</mi> <mn>2</mn> </msup> </mfrac> </mstyle> <mo>−</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>f</mi> <mfenced open="(" close=")"> <mi>r</mi> </mfenced> </mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> </mfrac> </mstyle> </mrow> </semantics></math>. <a href="#universe-10-00393-f008" class="html-fig">Figure 8</a> illustrates the variation of <math display='inline'><semantics> <msub> <mi>r</mi> <mrow> <mi>I</mi> <mi>S</mi> <mi>C</mi> <mi>O</mi> </mrow> </msub> </semantics></math> with <math display='inline'><semantics> <mi>γ</mi> </semantics></math> for different values of <span class='html-italic'>M</span>. At <math display='inline'><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mi>r</mi> <mrow> <mi>I</mi> <mi>S</mi> <mi>C</mi> <mi>O</mi> </mrow> </msub> <mo>=</mo> <mn>6</mn> </mrow> </semantics></math>, corresponding to the Schwarzschild black hole. As <math display='inline'><semantics> <mi>γ</mi> </semantics></math> increases, <math display='inline'><semantics> <msub> <mi>r</mi> <mrow> <mi>I</mi> <mi>S</mi> <mi>C</mi> <mi>O</mi> </mrow> </msub> </semantics></math> decreases, while as <span class='html-italic'>M</span> increases, <math display='inline'><semantics> <msub> <mi>r</mi> <mrow> <mi>I</mi> <mi>S</mi> <mi>C</mi> <mi>O</mi> </mrow> </msub> </semantics></math> increases. Therefore, LQG corrections always reduce <math display='inline'><semantics> <msub> <mi>r</mi> <mrow> <mi>I</mi> <mi>S</mi> <mi>C</mi> <mi>O</mi> </mrow> </msub> </semantics></math>.</div><div class='html-p'>We trace a photon with <math display='inline'><semantics> <mrow> <mi>b</mi> <mo><</mo> <msub> <mi>b</mi> <mi>c</mi> </msub> </mrow> </semantics></math>, which is emitted from point <span class='html-italic'>q</span>, enters the white hole, re-emerges from the black hole (considering the reverse process for convenience), and finally reaches the accretion disk at radius <span class='html-italic'>r</span>. During this process, the photon undergoes a change in the <math display='inline'><semantics> <mi>ϕ</mi> </semantics></math> angle, denoted as <math display='inline'><semantics> <mrow> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math>. In fact, it corresponds exactly to <math display='inline'><semantics> <mi>ϕ</mi> </semantics></math> in Equation (<a href="#FD11-universe-10-00393" class="html-disp-formula">11</a>). While we could plot the function <math display='inline'><semantics> <mrow> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math>, for the following discussion, we have instead plotted <math display='inline'><semantics> <mrow> <mi>b</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> in <a href="#universe-10-00393-f009" class="html-fig">Figure 9</a>.</div><div class='html-p'>In the diagram, the S-shaped curve with an arrow represents the function <math display='inline'><semantics> <mrow> <mi>α</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math>, which is the inverse of <math display='inline'><semantics> <mrow> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math>, as defined in Equation (<a href="#FD12-universe-10-00393" class="html-disp-formula">12</a>). The red solid line corresponds to the circular orbit at <math display='inline'><semantics> <msub> <mi>r</mi> <mi>ISCO</mi> </msub> </semantics></math>, while the blue solid line corresponds to the circular orbit at <math display='inline'><semantics> <msub> <mi>r</mi> <mo>∞</mo> </msub> </semantics></math>. The dashed line illustrates the scenario where the photon does not traverse the high quantum region or, equivalently, the situation where an observer at infinity in spacetime <math display='inline'><semantics> <msup> <mrow> <mi mathvariant="normal">A</mi> </mrow> <mo>′</mo> </msup> </semantics></math> observes the black hole’s accretion disk, corresponding to the traditional accretion disk image. The five colored strip regions, as determined by Equation (<a href="#FD12-universe-10-00393" class="html-disp-formula">12</a>), represent the range of <math display='inline'><semantics> <mi>ϕ</mi> </semantics></math> angles for different orders.</div><div class='html-p'>Using <a href="#universe-10-00393-f009" class="html-fig">Figure 9</a>: Select an angle <math display='inline'><semantics> <msub> <mi>θ</mi> <mn>0</mn> </msub> </semantics></math> and an order <span class='html-italic'>n</span>, then identify a point on the corresponding S-shaped curve that matches a specific value of <math display='inline'><semantics> <mi>α</mi> </semantics></math>, with <math display='inline'><semantics> <mi>α</mi> </semantics></math> increasing from 0 to <math display='inline'><semantics> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </semantics></math> in the direction indicated by the arrow. Draw a vertical line from this point, perpendicular to the <math display='inline'><semantics> <mi>ϕ</mi> </semantics></math>-axis, to determine the value of <math display='inline'><semantics> <mi>ϕ</mi> </semantics></math>. Then, calculate the <span class='html-italic'>b</span> value at the intersection of this vertical line with either the red or blue curve. After completing these steps, the values of <math display='inline'><semantics> <mi>α</mi> </semantics></math> and <span class='html-italic'>b</span> are obtained, determining the position of the image point on the observer’s reception screen formed by photons emitted from particles on the accretion disk orbit at radii <math display='inline'><semantics> <msub> <mi>r</mi> <mi>ISCO</mi> </msub> </semantics></math> or <math display='inline'><semantics> <msub> <mi>r</mi> <mo>∞</mo> </msub> </semantics></math>. By repeating this process for each <math display='inline'><semantics> <mi>α</mi> </semantics></math> value from 0 to <math display='inline'><semantics> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </semantics></math>, the <span class='html-italic'>n</span>th order image of the accretion disk with radius <span class='html-italic'>r</span> is obtained.</div><div class='html-p'>From <a href="#universe-10-00393-f009" class="html-fig">Figure 9</a>, the following properties of the accretion disk images produced by the white hole can be observed:</div><div class='html-p'><dl class='html-alpha-lower'><dt id=''>(a)</dt><dd><div class='html-p'>All accretion disk images are confined within a circle of radius <math display='inline'><semantics> <msub> <mi>b</mi> <mi>c</mi> </msub> </semantics></math>;</div></dd><dt id=''>(b)</dt><dd><div class='html-p'>Images corresponding to larger orbital radii have smaller <span class='html-italic'>b</span> values and appear on the inner side of the image, which is the opposite of what occurs with traditional accretion disk images;</div></dd><dt id=''>(c)</dt><dd><div class='html-p'>The 2nd-order image of the accretion disk is the widest, whereas the 0th-order image is very narrow. In contrast, in traditional accretion disk images, the 0th-order is the widest. Additionally, compared to traditional accretion disk images, even the widest image produced by the white hole remains relatively narrow. It is important to note that the blue curve plotted corresponds to the circular orbit at <math display='inline'><semantics> <msub> <mi>r</mi> <mo>∞</mo> </msub> </semantics></math>. Since the actual outermost circular orbit of the accretion disk has a finite radius, the disk image will be even narrower;</div></dd><dt id=''>(d)</dt><dd><div class='html-p'>As <math display='inline'><semantics> <msub> <mi>θ</mi> <mn>0</mn> </msub> </semantics></math> increases, the value of <span class='html-italic'>b</span> increases in <math display='inline'><semantics> <mrow> <mi>α</mi> <mo>∈</mo> <mfenced separators="" open="(" close=")"> <mn>0</mn> <mo>,</mo> <mi>π</mi> </mfenced> </mrow> </semantics></math>, while it decreases in <math display='inline'><semantics> <mrow> <mi>α</mi> <mo>∈</mo> <mfenced separators="" open="(" close=")"> <mi>π</mi> <mo>,</mo> <mn>2</mn> <mi>π</mi> </mfenced> </mrow> </semantics></math>. This causes the upper part of the accretion disk image to enlarge and the lower part to shrink, resulting in a shell-like shape. This contrasts sharply with traditional accretion disk images beyond the 1st-order. Additionally, the variation in accretion disk image width with respect to <math display='inline'><semantics> <mi>α</mi> </semantics></math> differs from that of traditional accretion disks. For instance, in the 1st-order image, the traditional accretion disk is widest at <math display='inline'><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> </semantics></math>, whereas the accretion disk image produced by the white hole is widest at <math display='inline'><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> </semantics></math>;</div></dd><dt id=''>(e)</dt><dd><div class='html-p'>Other features of the accretion disk images are generally consistent with those of traditional accretion disk images, including the circular nature of the 0th-order image and the symmetry of the images about the <math display='inline'><semantics> <mi>y</mi> </semantics></math>-axis.</div></dd></dl></div><div class='html-p'><a href="#universe-10-00393-f010" class="html-fig">Figure 10</a> illustrates the accretion disk images, clearly demonstrating the characteristics described above. For <math display='inline'><semantics> <mrow> <msub> <mi>θ</mi> <mn>0</mn> </msub> <mo>=</mo> <msup> <mn>0</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, the corresponding values of <math display='inline'><semantics> <mrow> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> </msub> <mo>/</mo> <mi>M</mi> <mo>,</mo> <msub> <mi>b</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> <mo>/</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> for <math display='inline'><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mo> </mo> <mn>1</mn> <mo>,</mo> <mo> </mo> <mn>2</mn> <mo>,</mo> <mo> </mo> <mn>3</mn> </mrow> </semantics></math>, and 4 are <math display='inline'><semantics> <mrow> <mo stretchy="false">(</mo> <mn>0.50389</mn> <mo>,</mo> <mn>0.547009</mn> <mo stretchy="false">)</mo> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mo stretchy="false">(</mo> <mn>2.17901</mn> <mo>,</mo> <mn>2.41511</mn> <mo stretchy="false">)</mo> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mo stretchy="false">(</mo> <mn>3.74008</mn> <mo>,</mo> <mn>4.01542</mn> <mo stretchy="false">)</mo> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mo stretchy="false">(</mo> <mn>4.62536</mn> <mo>,</mo> <mn>4.7603</mn> <mo stretchy="false">)</mo> </mrow> </semantics></math>, and <math display='inline'><semantics> <mrow> <mo stretchy="false">(</mo> <mn>4.94967</mn> <mo>,</mo> <mn>4.99131</mn> <mo stretchy="false">)</mo> </mrow> </semantics></math>, respectively. Additionally, for better comparison, <a href="#universe-10-00393-f011" class="html-fig">Figure 11</a> combines the traditional black hole accretion disk images (represented by the dashed lines in <a href="#universe-10-00393-f009" class="html-fig">Figure 9</a>) with the accretion disk images produced by the white hole. Note that the red dashed line corresponding to <math display='inline'><semantics> <mrow> <mi>r</mi> <mo>=</mo> <msub> <mi>r</mi> <mo>∞</mo> </msub> </mrow> </semantics></math> was adjusted to <math display='inline'><semantics> <mrow> <mi>r</mi> <mo>=</mo> <mn>15</mn> <mi>M</mi> </mrow> </semantics></math> to enable the plotting of the 0th-order image, which would otherwise be impossible.</div><div class='html-p'><a href="#universe-10-00393-f012" class="html-fig">Figure 12</a> illustrates the variation in edge positions and widths of the first five-order images as a function of <span class='html-italic'>M</span> when <math display='inline'><semantics> <mrow> <msub> <mi>θ</mi> <mn>0</mn> </msub> <mo>=</mo> <msup> <mn>0</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>. It is evident that the 2nd-order image is the widest, while the 0th-order image is the narrowest in most cases. As the order <span class='html-italic'>n</span> increases, the positions of these images cluster around <math display='inline'><semantics> <msub> <mi>b</mi> <mi>c</mi> </msub> </semantics></math>. Additionally, it can be observed that as <span class='html-italic'>M</span> approaches infinity, both the position and width converge to a fixed limiting value, a result already demonstrated in [<a href="#B37-universe-10-00393" class="html-bibr">37</a>].</div></section><section id='sec3dot3-universe-10-00393' type=''><h4 class='html-italic' data-nested='2'> 3.3. Light Intensity Distribution on Accretion Disk</h4><div class='html-p'>We used the Novikov–Thorne model to analyze the luminous images of accretion disks. As described in [<a href="#B40-universe-10-00393" class="html-bibr">40</a>], the radiation flux of photons emitted by particles on the accretion disk at a radius <span class='html-italic'>r</span> is given by <div class='html-disp-formula-info' id='FD13-universe-10-00393'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>F</mi> <mrow> <mi>e</mi> <mi>m</mi> </mrow> </msub> <mfenced open="(" close=")"> <mi>r</mi> </mfenced> <mo>=</mo> <mo>−</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi mathvariant="script">M</mi> <msup> <mo>Ω</mo> <mo>′</mo> </msup> </mrow> <mrow> <mn>4</mn> <mi>π</mi> <msqrt> <mrow> <mo>−</mo> <mi>g</mi> </mrow> </msqrt> <msup> <mfenced separators="" open="(" close=")"> <mi>E</mi> <mo>−</mo> <mo>Ω</mo> <mi>L</mi> </mfenced> <mn>2</mn> </msup> </mrow> </mfrac> </mstyle> <msubsup> <mo>∫</mo> <mrow> <msub> <mi>r</mi> <mrow> <mi>i</mi> <mi>s</mi> <mi>c</mi> <mi>o</mi> </mrow> </msub> </mrow> <mi>r</mi> </msubsup> <mfenced separators="" open="(" close=")"> <mi>E</mi> <mo>−</mo> <mo>Ω</mo> <mi>L</mi> </mfenced> <msup> <mi>L</mi> <mo>′</mo> </msup> <mi>d</mi> <mi>r</mi> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(13)</label> </div> </div> where <math display='inline'><semantics> <mi mathvariant="script">M</mi> </semantics></math> denotes the black hole’s accretion rate, <span class='html-italic'>g</span> represents the determinant of the metric, and <span class='html-italic'>E</span>, <span class='html-italic'>L</span>, and <math display='inline'><semantics> <mo>Ω</mo> </semantics></math> stand for the energy, angular momentum, and angular velocity of the particles in the accretion disk, respectively.</div><div class='html-p'>Due to the differences in gravitational fields between the disk and the observer, as well as their relative motion, a frequency shift occurs. Consequently, the radiation flux observed is [<a href="#B41-universe-10-00393" class="html-bibr">41</a>,<a href="#B42-universe-10-00393" class="html-bibr">42</a>,<a href="#B43-universe-10-00393" class="html-bibr">43</a>] <div class='html-disp-formula-info' id='FD14-universe-10-00393'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>F</mi> <mrow> <mi>o</mi> <mi>b</mi> <mi>s</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mi>F</mi> <mrow> <mi>e</mi> <mi>m</mi> </mrow> </msub> <mfenced open="(" close=")"> <mi>r</mi> </mfenced> </mrow> <msup> <mfenced separators="" open="(" close=")"> <mn>1</mn> <mo>+</mo> <mi>z</mi> </mfenced> <mn>4</mn> </msup> </mfrac> </mstyle> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(14)</label> </div> </div> where <span class='html-italic'>z</span> is redshift factor. From [<a href="#B44-universe-10-00393" class="html-bibr">44</a>], we obtain <div class='html-disp-formula-info' id='FD15-universe-10-00393'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>F</mi> <mrow> <mi>o</mi> <mi>b</mi> <mi>s</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mo>−</mo> <mfrac> <mrow> <mi mathvariant="script">M</mi> <msup> <mo>Ω</mo> <mo>′</mo> </msup> </mrow> <mrow> <mn>4</mn> <mi>π</mi> <msqrt> <mrow> <mo>−</mo> <mi>g</mi> </mrow> </msqrt> <msup> <mfenced separators="" open="(" close=")"> <mi>E</mi> <mo> </mo> <mo>−</mo> <mo> </mo> <mo>Ω</mo> <mi>L</mi> </mfenced> <mn>2</mn> </msup> </mrow> </mfrac> <msubsup> <mo>∫</mo> <mrow> <msub> <mi>r</mi> <mrow> <mi>i</mi> <mi>s</mi> <mi>c</mi> <mi>o</mi> </mrow> </msub> </mrow> <mi>r</mi> </msubsup> <mfenced separators="" open="(" close=")"> <mi>E</mi> <mo>−</mo> <mo>Ω</mo> <mi>L</mi> </mfenced> <msup> <mi>L</mi> <mo>′</mo> </msup> <mi>d</mi> <mi>r</mi> </mrow> <msup> <mfenced separators="" open="(" close=")"> <mfrac> <mrow> <mn>1</mn> <mo> </mo> <mo>+</mo> <mo> </mo> <mo>Ω</mo> <mi>b</mi> <mo form="prefix">sin</mo> <mi>θ</mi> <mo form="prefix">cos</mo> <mi>α</mi> </mrow> <msqrt> <mrow> <mo>−</mo> <msub> <mi>g</mi> <mrow> <mi>t</mi> <mi>t</mi> </mrow> </msub> <mo> </mo> <mo>−</mo> <mo> </mo> <msub> <mi>g</mi> <mrow> <mi>ϕ</mi> <mi>ϕ</mi> </mrow> </msub> <msup> <mo>Ω</mo> <mn>2</mn> </msup> </mrow> </msqrt> </mfrac> </mfenced> <mn>4</mn> </msup> </mfrac> </mstyle> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(15)</label> </div> </div> where <div class='html-disp-formula-info' id='FD16-universe-10-00393'> <div class='f'> <math display='block'><semantics> <mrow> <mo>Ω</mo> <mo>=</mo> <mo>±</mo> <msqrt> <mrow> <mo>−</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <msubsup> <mi>g</mi> <mrow> <mi>t</mi> <mi>t</mi> </mrow> <mo>′</mo> </msubsup> <msubsup> <mi>g</mi> <mrow> <mi>ϕ</mi> <mi>ϕ</mi> </mrow> <mo>′</mo> </msubsup> </mfrac> </mstyle> </mrow> </msqrt> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(16)</label> </div> </div><div class='html-disp-formula-info' id='FD17-universe-10-00393'> <div class='f'> <math display='block'><semantics> <mrow> <mi>E</mi> <mo>=</mo> <mo>−</mo> <msub> <mi>g</mi> <mrow> <mi>t</mi> <mi>μ</mi> </mrow> </msub> <msup> <mi>U</mi> <mi>μ</mi> </msup> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mo>−</mo> <msub> <mi>g</mi> <mrow> <mi>t</mi> <mi>t</mi> </mrow> </msub> </mrow> <msqrt> <mrow> <mo>−</mo> <msub> <mi>g</mi> <mrow> <mi>t</mi> <mi>t</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>g</mi> <mrow> <mi>ϕ</mi> <mi>ϕ</mi> </mrow> </msub> <msup> <mo>Ω</mo> <mn>2</mn> </msup> </mrow> </msqrt> </mfrac> </mstyle> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(17)</label> </div> </div><div class='html-disp-formula-info' id='FD18-universe-10-00393'> <div class='f'> <math display='block'><semantics> <mrow> <mi>L</mi> <mo>=</mo> <msub> <mi>g</mi> <mrow> <mi>ϕ</mi> <mi>μ</mi> </mrow> </msub> <msup> <mi>U</mi> <mi>μ</mi> </msup> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mi>g</mi> <mrow> <mi>ϕ</mi> <mi>ϕ</mi> </mrow> </msub> <mo>Ω</mo> </mrow> <msqrt> <mrow> <mo>−</mo> <msub> <mi>g</mi> <mrow> <mi>t</mi> <mi>t</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>g</mi> <mrow> <mi>ϕ</mi> <mi>ϕ</mi> </mrow> </msub> <msup> <mo>Ω</mo> <mn>2</mn> </msup> </mrow> </msqrt> </mfrac> </mstyle> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(18)</label> </div> </div></div><div class='html-p'>Another crucial physical property of the accretion disk is its radiation efficiency, which quantifies the energy released as radiation during matter accretion onto a black hole. Assuming all photons emitted by the particles can escape to infinity, the radiation efficiency is expressed as <div class='html-disp-formula-info' id='FD19-universe-10-00393'> <div class='f'> <math display='block'><semantics> <mrow> <mi>ϵ</mi> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mi>E</mi> <mo>∞</mo> </msub> <mo>−</mo> <msub> <mi>E</mi> <mrow> <mi>I</mi> <mi>S</mi> <mi>C</mi> <mi>O</mi> </mrow> </msub> </mrow> <msub> <mi>E</mi> <mo>∞</mo> </msub> </mfrac> </mstyle> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(19)</label> </div> </div> where <math display='inline'><semantics> <msub> <mi>E</mi> <mo>∞</mo> </msub> </semantics></math> is the energy of the particle at infinity, and <math display='inline'><semantics> <msub> <mi>E</mi> <mrow> <mi>I</mi> <mi>S</mi> <mi>C</mi> <mi>O</mi> </mrow> </msub> </semantics></math> is the energy at the ISCO. From Equation (<a href="#FD17-universe-10-00393" class="html-disp-formula">17</a>), since <math display='inline'><semantics> <mrow> <msub> <mi>E</mi> <mo>∞</mo> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, we have <div class='html-disp-formula-info' id='FD20-universe-10-00393'> <div class='f'> <math display='block'><semantics> <mrow> <mi>ϵ</mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>E</mi> <mrow> <mi>I</mi> <mi>S</mi> <mi>C</mi> <mi>O</mi> </mrow> </msub> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(20)</label> </div> </div></div><div class='html-p'><a href="#universe-10-00393-f013" class="html-fig">Figure 13</a> illustrates the variation of <math display='inline'><semantics> <mi>ϵ</mi> </semantics></math> with <math display='inline'><semantics> <mi>γ</mi> </semantics></math> for different values of <span class='html-italic'>M</span>. When <math display='inline'><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>ϵ</mi> <mo>=</mo> <mn>0.0572</mn> </mrow> </semantics></math>, it corresponds to the radiation efficiency of the Schwarzschild black hole. As <math display='inline'><semantics> <mi>γ</mi> </semantics></math> increases, <math display='inline'><semantics> <mi>ϵ</mi> </semantics></math> rises, while as <span class='html-italic'>M</span> increases, <math display='inline'><semantics> <mi>ϵ</mi> </semantics></math> decreases. This suggests that LQG corrections convert more particle energy into radiation, enhancing the brightness of the accretion disk, as shown in <a href="#universe-10-00393-f014" class="html-fig">Figure 14</a>.</div><div class='html-p'><a href="#universe-10-00393-f014" class="html-fig">Figure 14</a> illustrates the variation of <math display='inline'><semantics> <mrow> <msub> <mi>F</mi> <mrow> <mi>e</mi> <mi>m</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <msub> <mi>F</mi> <mrow> <mi>o</mi> <mi>b</mi> <mi>s</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math>, with respect to <span class='html-italic'>r</span> (left) and the variation of <math display='inline'><semantics> <msub> <mi>F</mi> <mrow> <mi>o</mi> <mi>b</mi> <mi>s</mi> </mrow> </msub> </semantics></math> with <span class='html-italic'>b</span> (right) when <math display='inline'><semantics> <mrow> <msub> <mi>θ</mi> <mn>0</mn> </msub> <mo>=</mo> <msup> <mn>0</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>. Due to redshift, <math display='inline'><semantics> <mrow> <msub> <mi>F</mi> <mrow> <mi>o</mi> <mi>b</mi> <mi>s</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math> is significantly smaller than <math display='inline'><semantics> <mrow> <msub> <mi>F</mi> <mrow> <mi>e</mi> <mi>m</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math>, while LQG corrections enhance the brightness of the accretion disk. Additionally, <math display='inline'><semantics> <mrow> <msub> <mi>F</mi> <mrow> <mi>o</mi> <mi>b</mi> <mi>s</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math> is compressed within each order’s image, with each order containing the full information of <math display='inline'><semantics> <mrow> <msub> <mi>F</mi> <mrow> <mi>o</mi> <mi>b</mi> <mi>s</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math>, but at varying levels of compression.</div><div class='html-p'><a href="#universe-10-00393-f015" class="html-fig">Figure 15</a> presents the luminous images of accretion disks. Similar to traditional accretion disk images, the light intensity distribution is asymmetrical about the <span class='html-italic'>y</span>-axis due to gravitational redshift and Doppler effects. As <math display='inline'><semantics> <msub> <mi>θ</mi> <mn>0</mn> </msub> </semantics></math> increases, the left side of the image progressively brightens, while the right side darkens. Unlike traditional accretion disk images, in which the inner region is illuminated by particles on orbits with smaller radii, here, the inner region of each order’s image is illuminated by particles on orbits with larger radii, while the outer region is illuminated by particles on orbits with smaller radii. Consequently, a broader dark area forms on the inner side and a narrower dark area on the outer side, as <math display='inline'><semantics> <mrow> <msub> <mi>F</mi> <mrow> <mi>o</mi> <mi>b</mi> <mi>s</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math> in <a href="#universe-10-00393-f014" class="html-fig">Figure 14</a> flattens more at <math display='inline'><semantics> <mrow> <mi>r</mi> <mo><</mo> <msub> <mi>r</mi> <mi>m</mi> </msub> </mrow> </semantics></math> than at <math display='inline'><semantics> <mrow> <mi>r</mi> <mo>></mo> <msub> <mi>r</mi> <mi>m</mi> </msub> </mrow> </semantics></math>, where <math display='inline'><semantics> <msub> <mi>r</mi> <mi>m</mi> </msub> </semantics></math> is the radius at which <math display='inline'><semantics> <mrow> <msub> <mi>F</mi> <mrow> <mi>o</mi> <mi>b</mi> <mi>s</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math> reaches its maximum. A close examination of <a href="#universe-10-00393-f015" class="html-fig">Figure 15</a> reveals this phenomenon, which becomes even more apparent in Figure 3 of [<a href="#B37-universe-10-00393" class="html-bibr">37</a>], as the emission model in [<a href="#B37-universe-10-00393" class="html-bibr">37</a>] exhibits a discontinuity at <math display='inline'><semantics> <msub> <mi>r</mi> <mi>m</mi> </msub> </semantics></math>. In that figure, the traditional black hole image shows a darker outer region, while the image produced by a white hole displays a darker inner region.</div></section></section><section id='sec4-universe-10-00393' type=''><h2 data-nested='1'> 4. Summary</h2><div class='html-p'>In this paper, we obtained accretion disk images produced by a white hole from various observational angles and analyzed their properties. In <a href="#sec2-universe-10-00393" class="html-sec">Section 2</a>, we studied the geodesics of photons with <math display='inline'><semantics> <mrow> <mi>b</mi> <mo><</mo> <msub> <mi>b</mi> <mi>c</mi> </msub> </mrow> </semantics></math> in a quantum-corrected spacetime. The results indicate that photons do not vanish at the black hole’s event horizon; rather, they continue their motion, sequentially crossing the outer and inner horizons, reaching periastron, and eventually re-emerging from a white hole. In <a href="#sec3-universe-10-00393" class="html-sec">Section 3</a>, we established an observation system and generated accretion disk images produced by the white hole, analyzing their light intensity distribution using the Novikov–Thorne model. The results indicate that most properties of accretion disk images produced by the white hole are opposite to those of traditional accretion disk images. The images produced by the white hole are confined within a circle of radius <math display='inline'><semantics> <msub> <mi>b</mi> <mi>c</mi> </msub> </semantics></math>. As the order <span class='html-italic'>n</span> increases, these images gradually transform into circles with no width, clustering near <math display='inline'><semantics> <msub> <mi>b</mi> <mi>c</mi> </msub> </semantics></math>. The images corresponding to orbits with larger radii are located on the outer edge of each order’s accretion disk image, while those with smaller radii appear on the inner side. This leads to a broader dark region on the inner side of the bright accretion disk image compared to the outer side. Compared to the Schwarzschild black hole, LQG corrections reduce the ISCO radius, enhancing the radiation efficiency of the accretion disk and increasing its brightness. As the observation angle <math display='inline'><semantics> <msub> <mi>θ</mi> <mn>0</mn> </msub> </semantics></math> increases from 0, the accretion disk image produced by the white hole transforms from a ring to a shell-like shape, with the upper half expanding and the lower half contracting. Additionally, the 2nd-order accretion disk image has the greatest width, while the 0th-order image has the narrowest, though both are much narrower than the first two orders of traditional accretion disk images. Similar to traditional accretion disk images, these images are all symmetric about the <span class='html-italic'>y</span>-axis.</div><div class='html-p'>Our work complements and extends the research of Zhang et al. [<a href="#B37-universe-10-00393" class="html-bibr">37</a>]. While this idea has many limitations, it qualitatively represents a bold and intriguing attempt. Quantum gravity theories hold significant promise in resolving singularity issues, although the observable effects are often subtle. However, the additional accretion disk images produced by white holes are notably distinct. These images could aid in validating quantum gravity theories and distinguishing between different quantum gravity models. </div></section> </div> <div class="html-back"> <section class='html-notes'><h2 >Author Contributions</h2><div class='html-p'>Conceptualization, L.Y. and Y.-H.F.; methodology, L.Y.; software, L.Y.; validation, L.Y.; investigation, L.Y.; writing—original draft preparation, L.Y. and R.-B.W.; writing—review and editing, J.-B.D. and X.-R.H. All authors have read and agreed to the published version of the manuscript.</div></section><section class='html-notes'><h2>Funding</h2><div class='html-p'>This research was funded by the National Natural Science Foundation of China (Grant No. 11571342).</div></section><section class='html-notes'><h2 >Data Availability Statement</h2><div class='html-p'>No new data were created or analyzed in this study. Data sharing is not applicable to this article.</div></section><section id='html-ack' class='html-ack'><h2 >Acknowledgments</h2><div class='html-p'>We are grateful to Jie-Shi Ma and Yu-Cheng Tang for their useful suggestions. We also thank the National Natural Science Foundation of China (Grant No. 11571342) for supporting us on this work.</div></section><section class='html-notes'><h2 >Conflicts of Interest</h2><div class='html-p'>The authors declare no conflicts of interest.</div></section><section id='html-references_list'><h2>References</h2><ol class='html-xx'><li id='B1-universe-10-00393' class='html-x' data-content='1.'>Penrose, R. Gravitational collapse and space-time singularities. <span class='html-italic'>Phys. Rev. Lett.</span> <b>1965</b>, <span class='html-italic'>14</span>, 57. 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The green dotted line represents the orbit of an accretion disk with a radius <span class='html-italic'>r</span>. The red and blue lines represent the trajectories of photons emitted from point <math display='inline'><semantics> <msub> <mi>Q</mi> <mi>BH</mi> </msub> </semantics></math> at location <math display='inline'><semantics> <mi mathvariant="normal">A</mi> </semantics></math> and point <math display='inline'><semantics> <msub> <mi>Q</mi> <msup> <mrow> <mi>BH</mi> </mrow> <mo>′</mo> </msup> </msub> </semantics></math> at location <math display='inline'><semantics> <msup> <mrow> <mi mathvariant="normal">A</mi> </mrow> <mo>′</mo> </msup> </semantics></math>, respectively, reaching an observer at infinity, denoted as point <span class='html-italic'>q</span>.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_universe-10-00393-f001"> <div class="html-caption"> <b>Figure 1.</b> The Penrose diagram of the quantum extended black hole spacetime. The green dotted line represents the orbit of an accretion disk with a radius <span class='html-italic'>r</span>. The red and blue lines represent the trajectories of photons emitted from point <math display='inline'><semantics> <msub> <mi>Q</mi> <mi>BH</mi> </msub> </semantics></math> at location <math display='inline'><semantics> <mi mathvariant="normal">A</mi> </semantics></math> and point <math display='inline'><semantics> <msub> <mi>Q</mi> <msup> <mrow> <mi>BH</mi> </mrow> <mo>′</mo> </msup> </msub> </semantics></math> at location <math display='inline'><semantics> <msup> <mrow> <mi mathvariant="normal">A</mi> </mrow> <mo>′</mo> </msup> </semantics></math>, respectively, reaching an observer at infinity, denoted as point <span class='html-italic'>q</span>.</div> <div class="html-img"><img data-large="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g001.png" data-original="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g001.png" alt="Universe 10 00393 g001" data-lsrc="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g001.png" /></div> </div> <div class="html-fig-wrap" id="universe-10-00393-f002"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/10/393/display" href="#fig_body_display_universe-10-00393-f002"> <img data-large="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g002.png" data-original="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g002.png" alt="Universe 10 00393 g002" data-lsrc="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g002-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/10/393/display" href="#fig_body_display_universe-10-00393-f002"></a> </div> </div> <div class="html-fig_description"> <b>Figure 2.</b> The variation of <math display='inline'><semantics> <msub> <mi>r</mi> <mo>±</mo> </msub> </semantics></math> with <math display='inline'><semantics> <mi>γ</mi> </semantics></math> for different values of <span class='html-italic'>M</span>. The solid line represents <math display='inline'><semantics> <msub> <mi>r</mi> <mo>−</mo> </msub> </semantics></math>, and the dashed line represents <math display='inline'><semantics> <msub> <mi>r</mi> <mo>+</mo> </msub> </semantics></math>.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_universe-10-00393-f002"> <div class="html-caption"> <b>Figure 2.</b> The variation of <math display='inline'><semantics> <msub> <mi>r</mi> <mo>±</mo> </msub> </semantics></math> with <math display='inline'><semantics> <mi>γ</mi> </semantics></math> for different values of <span class='html-italic'>M</span>. The solid line represents <math display='inline'><semantics> <msub> <mi>r</mi> <mo>−</mo> </msub> </semantics></math>, and the dashed line represents <math display='inline'><semantics> <msub> <mi>r</mi> <mo>+</mo> </msub> </semantics></math>.</div> <div class="html-img"><img data-large="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g002.png" data-original="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g002.png" alt="Universe 10 00393 g002" data-lsrc="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g002.png" /></div> </div> <div class="html-fig-wrap" id="universe-10-00393-f003"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/10/393/display" href="#fig_body_display_universe-10-00393-f003"> <img data-large="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g003.png" data-original="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g003.png" alt="Universe 10 00393 g003" data-lsrc="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g003-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/10/393/display" href="#fig_body_display_universe-10-00393-f003"></a> </div> </div> <div class="html-fig_description"> <b>Figure 3.</b> The variation of <math display='inline'><semantics> <msub> <mi>r</mi> <mrow> <mi>p</mi> <mi>e</mi> </mrow> </msub> </semantics></math> with <span class='html-italic'>b</span> is depicted, with the inset illustrating the trajectory connecting the periastron of all photons with <math display='inline'><semantics> <mrow> <mi>b</mi> <mo><</mo> <msub> <mi>b</mi> <mi>c</mi> </msub> </mrow> </semantics></math> in spacetime.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_universe-10-00393-f003"> <div class="html-caption"> <b>Figure 3.</b> The variation of <math display='inline'><semantics> <msub> <mi>r</mi> <mrow> <mi>p</mi> <mi>e</mi> </mrow> </msub> </semantics></math> with <span class='html-italic'>b</span> is depicted, with the inset illustrating the trajectory connecting the periastron of all photons with <math display='inline'><semantics> <mrow> <mi>b</mi> <mo><</mo> <msub> <mi>b</mi> <mi>c</mi> </msub> </mrow> </semantics></math> in spacetime.</div> <div class="html-img"><img data-large="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g003.png" data-original="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g003.png" alt="Universe 10 00393 g003" data-lsrc="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g003.png" /></div> </div> <div class="html-fig-wrap" id="universe-10-00393-f004"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/10/393/display" href="#fig_body_display_universe-10-00393-f004"> <img data-large="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g004.png" data-original="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g004.png" alt="Universe 10 00393 g004" data-lsrc="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g004-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/10/393/display" href="#fig_body_display_universe-10-00393-f004"></a> </div> </div> <div class="html-fig_description"> <b>Figure 4.</b> The <math display='inline'><semantics> <mrow> <mi>G</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> function in the spacetimes of QBH (<b>left</b>) and Schwarzschild black hole (<b>right</b>). Where <math display='inline'><semantics> <mrow> <msub> <mi>u</mi> <mn>0</mn> </msub> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <mo>∞</mo> </mfrac> </mstyle> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mi>u</mi> <mrow> <mi>p</mi> <mi>h</mi> </mrow> </msub> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <msub> <mi>r</mi> <mrow> <mi>p</mi> <mi>h</mi> </mrow> </msub> </mfrac> </mstyle> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mi>u</mi> <mo>±</mo> </msub> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <msub> <mi>r</mi> <mo>±</mo> </msub> </mfrac> </mstyle> </mrow> </semantics></math>, and <math display='inline'><semantics> <mrow> <msub> <mi>u</mi> <mrow> <mi>p</mi> <mi>e</mi> </mrow> </msub> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <msub> <mi>r</mi> <mrow> <mi>p</mi> <mi>e</mi> </mrow> </msub> </mfrac> </mstyle> </mrow> </semantics></math>.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_universe-10-00393-f004"> <div class="html-caption"> <b>Figure 4.</b> The <math display='inline'><semantics> <mrow> <mi>G</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> function in the spacetimes of QBH (<b>left</b>) and Schwarzschild black hole (<b>right</b>). Where <math display='inline'><semantics> <mrow> <msub> <mi>u</mi> <mn>0</mn> </msub> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <mo>∞</mo> </mfrac> </mstyle> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mi>u</mi> <mrow> <mi>p</mi> <mi>h</mi> </mrow> </msub> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <msub> <mi>r</mi> <mrow> <mi>p</mi> <mi>h</mi> </mrow> </msub> </mfrac> </mstyle> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mi>u</mi> <mo>±</mo> </msub> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <msub> <mi>r</mi> <mo>±</mo> </msub> </mfrac> </mstyle> </mrow> </semantics></math>, and <math display='inline'><semantics> <mrow> <msub> <mi>u</mi> <mrow> <mi>p</mi> <mi>e</mi> </mrow> </msub> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <msub> <mi>r</mi> <mrow> <mi>p</mi> <mi>e</mi> </mrow> </msub> </mfrac> </mstyle> </mrow> </semantics></math>.</div> <div class="html-img"><img data-large="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g004.png" data-original="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g004.png" alt="Universe 10 00393 g004" data-lsrc="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g004.png" /></div> </div> <div class="html-fig-wrap" id="universe-10-00393-f005"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/10/393/display" href="#fig_body_display_universe-10-00393-f005"> <img data-large="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g005.png" data-original="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g005.png" alt="Universe 10 00393 g005" data-lsrc="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g005-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/10/393/display" href="#fig_body_display_universe-10-00393-f005"></a> </div> </div> <div class="html-fig_description"> <b>Figure 5.</b> The left panel illustrates the photon entering the white hole in spacetime <span class='html-italic'>A</span>, while the right panel depicts the photon exiting the black hole in spacetime <span class='html-italic'>B</span> (considering the reverse process for clarity). The orange dashed line indicates the trajectory of the photon with <math display='inline'><semantics> <mrow> <mi>b</mi> <mo>=</mo> <msub> <mi>b</mi> <mi>c</mi> </msub> </mrow> </semantics></math>.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_universe-10-00393-f005"> <div class="html-caption"> <b>Figure 5.</b> The left panel illustrates the photon entering the white hole in spacetime <span class='html-italic'>A</span>, while the right panel depicts the photon exiting the black hole in spacetime <span class='html-italic'>B</span> (considering the reverse process for clarity). The orange dashed line indicates the trajectory of the photon with <math display='inline'><semantics> <mrow> <mi>b</mi> <mo>=</mo> <msub> <mi>b</mi> <mi>c</mi> </msub> </mrow> </semantics></math>.</div> <div class="html-img"><img data-large="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g005.png" data-original="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g005.png" alt="Universe 10 00393 g005" data-lsrc="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g005.png" /></div> </div> <div class="html-fig-wrap" id="universe-10-00393-f006"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/10/393/display" href="#fig_body_display_universe-10-00393-f006"> <img data-large="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g006.png" data-original="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g006.png" alt="Universe 10 00393 g006" data-lsrc="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g006-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/10/393/display" href="#fig_body_display_universe-10-00393-f006"></a> </div> </div> <div class="html-fig_description"> <b>Figure 6.</b> The full trajectory of certain photons with <math display='inline'><semantics> <mrow> <mi>b</mi> <mo><</mo> <msub> <mi>b</mi> <mi>c</mi> </msub> </mrow> </semantics></math> in quantum-corrected spacetime.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_universe-10-00393-f006"> <div class="html-caption"> <b>Figure 6.</b> The full trajectory of certain photons with <math display='inline'><semantics> <mrow> <mi>b</mi> <mo><</mo> <msub> <mi>b</mi> <mi>c</mi> </msub> </mrow> </semantics></math> in quantum-corrected spacetime.</div> <div class="html-img"><img data-large="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g006.png" data-original="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g006.png" alt="Universe 10 00393 g006" data-lsrc="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g006.png" /></div> </div> <div class="html-fig-wrap" id="universe-10-00393-f007"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/10/393/display" href="#fig_body_display_universe-10-00393-f007"> <img data-large="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g007.png" data-original="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g007.png" alt="Universe 10 00393 g007" data-lsrc="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g007-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/10/393/display" href="#fig_body_display_universe-10-00393-f007"></a> </div> </div> <div class="html-fig_description"> <b>Figure 7.</b> Observational schematic diagram.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_universe-10-00393-f007"> <div class="html-caption"> <b>Figure 7.</b> Observational schematic diagram.</div> <div class="html-img"><img data-large="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g007.png" data-original="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g007.png" alt="Universe 10 00393 g007" data-lsrc="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g007.png" /></div> </div> <div class="html-fig-wrap" id="universe-10-00393-f008"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/10/393/display" href="#fig_body_display_universe-10-00393-f008"> <img data-large="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g008.png" data-original="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g008.png" alt="Universe 10 00393 g008" data-lsrc="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g008-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/10/393/display" href="#fig_body_display_universe-10-00393-f008"></a> </div> </div> <div class="html-fig_description"> <b>Figure 8.</b> The variation of <math display='inline'><semantics> <msub> <mi>r</mi> <mrow> <mi>I</mi> <mi>S</mi> <mi>C</mi> <mi>O</mi> </mrow> </msub> </semantics></math> with <math display='inline'><semantics> <mi>γ</mi> </semantics></math> for different values of <span class='html-italic'>M</span>.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_universe-10-00393-f008"> <div class="html-caption"> <b>Figure 8.</b> The variation of <math display='inline'><semantics> <msub> <mi>r</mi> <mrow> <mi>I</mi> <mi>S</mi> <mi>C</mi> <mi>O</mi> </mrow> </msub> </semantics></math> with <math display='inline'><semantics> <mi>γ</mi> </semantics></math> for different values of <span class='html-italic'>M</span>.</div> <div class="html-img"><img data-large="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g008.png" data-original="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g008.png" alt="Universe 10 00393 g008" data-lsrc="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g008.png" /></div> </div> <div class="html-fig-wrap" id="universe-10-00393-f009"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/10/393/display" href="#fig_body_display_universe-10-00393-f009"> <img data-large="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g009.png" data-original="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g009.png" alt="Universe 10 00393 g009" data-lsrc="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g009-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/10/393/display" href="#fig_body_display_universe-10-00393-f009"></a> </div> </div> <div class="html-fig_description"> <b>Figure 9.</b> The variation of <span class='html-italic'>b</span> with <math display='inline'><semantics> <mi>ϕ</mi> </semantics></math> for various values of <span class='html-italic'>r</span>.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_universe-10-00393-f009"> <div class="html-caption"> <b>Figure 9.</b> The variation of <span class='html-italic'>b</span> with <math display='inline'><semantics> <mi>ϕ</mi> </semantics></math> for various values of <span class='html-italic'>r</span>.</div> <div class="html-img"><img data-large="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g009.png" data-original="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g009.png" alt="Universe 10 00393 g009" data-lsrc="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g009.png" /></div> </div> <div class="html-fig-wrap" id="universe-10-00393-f010"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/10/393/display" href="#fig_body_display_universe-10-00393-f010"> <img data-large="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g010.png" data-original="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g010.png" alt="Universe 10 00393 g010" data-lsrc="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g010-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/10/393/display" href="#fig_body_display_universe-10-00393-f010"></a> </div> </div> <div class="html-fig_description"> <b>Figure 10.</b> The accretion disk image produced by the white hole. The red, orange, green, blue, and purple lines represent the 0th-, 1st-, 2nd-, 3rd-, and 4th-order images, respectively. We set <math display='inline'><semantics> <mrow> <mi>r</mi> <mo>=</mo> <msub> <mi>r</mi> <mi>ISCO</mi> </msub> </mrow> </semantics></math> and <math display='inline'><semantics> <msub> <mi>r</mi> <mo>∞</mo> </msub> </semantics></math>.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_universe-10-00393-f010"> <div class="html-caption"> <b>Figure 10.</b> The accretion disk image produced by the white hole. The red, orange, green, blue, and purple lines represent the 0th-, 1st-, 2nd-, 3rd-, and 4th-order images, respectively. We set <math display='inline'><semantics> <mrow> <mi>r</mi> <mo>=</mo> <msub> <mi>r</mi> <mi>ISCO</mi> </msub> </mrow> </semantics></math> and <math display='inline'><semantics> <msub> <mi>r</mi> <mo>∞</mo> </msub> </semantics></math>.</div> <div class="html-img"><img data-large="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g010.png" data-original="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g010.png" alt="Universe 10 00393 g010" data-lsrc="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g010.png" /></div> </div> <div class="html-fig-wrap" id="universe-10-00393-f011"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/10/393/display" href="#fig_body_display_universe-10-00393-f011"> <img data-large="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g011.png" data-original="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g011.png" alt="Universe 10 00393 g011" data-lsrc="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g011-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/10/393/display" href="#fig_body_display_universe-10-00393-f011"></a> </div> </div> <div class="html-fig_description"> <b>Figure 11.</b> A composite image displaying the accretion disk image produced by the white hole (blue) and the accretion disk image generated by the black hole (red). For the black hole’s accretion disk image, we adjust <span class='html-italic'>r</span> from <math display='inline'><semantics> <msub> <mi>r</mi> <mo>∞</mo> </msub> </semantics></math> to <math display='inline'><semantics> <mrow> <mn>15</mn> <mi>M</mi> </mrow> </semantics></math>.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_universe-10-00393-f011"> <div class="html-caption"> <b>Figure 11.</b> A composite image displaying the accretion disk image produced by the white hole (blue) and the accretion disk image generated by the black hole (red). For the black hole’s accretion disk image, we adjust <span class='html-italic'>r</span> from <math display='inline'><semantics> <msub> <mi>r</mi> <mo>∞</mo> </msub> </semantics></math> to <math display='inline'><semantics> <mrow> <mn>15</mn> <mi>M</mi> </mrow> </semantics></math>.</div> <div class="html-img"><img data-large="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g011.png" data-original="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g011.png" alt="Universe 10 00393 g011" data-lsrc="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g011.png" /></div> </div> <div class="html-fig-wrap" id="universe-10-00393-f012"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/10/393/display" href="#fig_body_display_universe-10-00393-f012"> <img data-large="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g012.png" data-original="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g012.png" alt="Universe 10 00393 g012" data-lsrc="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g012-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/10/393/display" href="#fig_body_display_universe-10-00393-f012"></a> </div> </div> <div class="html-fig_description"> <b>Figure 12.</b> With <math display='inline'><semantics> <mrow> <msub> <mi>θ</mi> <mn>0</mn> </msub> <mo>=</mo> <msup> <mn>0</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, the left panel illustrates the variation of <math display='inline'><semantics> <mrow> <msub> <mi>b</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> </msub> <mo>/</mo> <mi>M</mi> </mrow> </semantics></math> (solid line) and <math display='inline'><semantics> <mrow> <msub> <mi>b</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> <mo>/</mo> <mi>M</mi> </mrow> </semantics></math> (dashed line) with <span class='html-italic'>M</span>, while the right panel depicts the variation of <math display='inline'><semantics> <mrow> <mo>Δ</mo> <mi>b</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>b</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>/</mo> <mi>M</mi> </mrow> </semantics></math> with <span class='html-italic'>M</span>. The red, orange, green, blue, and purple lines represent the 0th-, 1st-, 2nd-, 3rd-, and 4th-order images, respectively.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_universe-10-00393-f012"> <div class="html-caption"> <b>Figure 12.</b> With <math display='inline'><semantics> <mrow> <msub> <mi>θ</mi> <mn>0</mn> </msub> <mo>=</mo> <msup> <mn>0</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, the left panel illustrates the variation of <math display='inline'><semantics> <mrow> <msub> <mi>b</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> </msub> <mo>/</mo> <mi>M</mi> </mrow> </semantics></math> (solid line) and <math display='inline'><semantics> <mrow> <msub> <mi>b</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> <mo>/</mo> <mi>M</mi> </mrow> </semantics></math> (dashed line) with <span class='html-italic'>M</span>, while the right panel depicts the variation of <math display='inline'><semantics> <mrow> <mo>Δ</mo> <mi>b</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>b</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>/</mo> <mi>M</mi> </mrow> </semantics></math> with <span class='html-italic'>M</span>. The red, orange, green, blue, and purple lines represent the 0th-, 1st-, 2nd-, 3rd-, and 4th-order images, respectively.</div> <div class="html-img"><img data-large="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g012.png" data-original="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g012.png" alt="Universe 10 00393 g012" data-lsrc="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g012.png" /></div> </div> <div class="html-fig-wrap" id="universe-10-00393-f013"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/10/393/display" href="#fig_body_display_universe-10-00393-f013"> <img data-large="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g013.png" data-original="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g013.png" alt="Universe 10 00393 g013" data-lsrc="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g013-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/10/393/display" href="#fig_body_display_universe-10-00393-f013"></a> </div> </div> <div class="html-fig_description"> <b>Figure 13.</b> The variation of <math display='inline'><semantics> <mi>ϵ</mi> </semantics></math> with <math display='inline'><semantics> <mi>γ</mi> </semantics></math> for different values of <span class='html-italic'>M</span>.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_universe-10-00393-f013"> <div class="html-caption"> <b>Figure 13.</b> The variation of <math display='inline'><semantics> <mi>ϵ</mi> </semantics></math> with <math display='inline'><semantics> <mi>γ</mi> </semantics></math> for different values of <span class='html-italic'>M</span>.</div> <div class="html-img"><img data-large="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g013.png" data-original="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g013.png" alt="Universe 10 00393 g013" data-lsrc="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g013.png" /></div> </div> <div class="html-fig-wrap" id="universe-10-00393-f014"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/10/393/display" href="#fig_body_display_universe-10-00393-f014"> <img data-large="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g014.png" data-original="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g014.png" alt="Universe 10 00393 g014" data-lsrc="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g014-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/10/393/display" href="#fig_body_display_universe-10-00393-f014"></a> </div> </div> <div class="html-fig_description"> <b>Figure 14.</b> With <math display='inline'><semantics> <mrow> <msub> <mi>θ</mi> <mn>0</mn> </msub> <mo>=</mo> <msup> <mn>0</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, the left panel shows the variation of <math display='inline'><semantics> <msub> <mi>F</mi> <mrow> <mi>e</mi> <mi>m</mi> </mrow> </msub> </semantics></math> and <math display='inline'><semantics> <msub> <mi>F</mi> <mrow> <mi>o</mi> <mi>b</mi> <mi>s</mi> </mrow> </msub> </semantics></math> with <span class='html-italic'>r</span> for <math display='inline'><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> (Schwarzschild black hole) and <math display='inline'><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>0.2375</mn> </mrow> </semantics></math>, while the right panel shows the variation of <math display='inline'><semantics> <msub> <mi>F</mi> <mrow> <mi>o</mi> <mi>b</mi> <mi>s</mi> </mrow> </msub> </semantics></math> with <span class='html-italic'>b</span> for <math display='inline'><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>0.2375</mn> </mrow> </semantics></math>. The red, orange, green, blue, and purple lines represent the 0th-, 1st-, 2nd-, 3rd-, and 4th-order images, respectively.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_universe-10-00393-f014"> <div class="html-caption"> <b>Figure 14.</b> With <math display='inline'><semantics> <mrow> <msub> <mi>θ</mi> <mn>0</mn> </msub> <mo>=</mo> <msup> <mn>0</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, the left panel shows the variation of <math display='inline'><semantics> <msub> <mi>F</mi> <mrow> <mi>e</mi> <mi>m</mi> </mrow> </msub> </semantics></math> and <math display='inline'><semantics> <msub> <mi>F</mi> <mrow> <mi>o</mi> <mi>b</mi> <mi>s</mi> </mrow> </msub> </semantics></math> with <span class='html-italic'>r</span> for <math display='inline'><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> (Schwarzschild black hole) and <math display='inline'><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>0.2375</mn> </mrow> </semantics></math>, while the right panel shows the variation of <math display='inline'><semantics> <msub> <mi>F</mi> <mrow> <mi>o</mi> <mi>b</mi> <mi>s</mi> </mrow> </msub> </semantics></math> with <span class='html-italic'>b</span> for <math display='inline'><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>0.2375</mn> </mrow> </semantics></math>. The red, orange, green, blue, and purple lines represent the 0th-, 1st-, 2nd-, 3rd-, and 4th-order images, respectively.</div> <div class="html-img"><img data-large="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g014.png" data-original="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g014.png" alt="Universe 10 00393 g014" data-lsrc="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g014.png" /></div> </div> <div class="html-fig-wrap" id="universe-10-00393-f015"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/10/393/display" href="#fig_body_display_universe-10-00393-f015"> <img data-large="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g015.png" data-original="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g015.png" alt="Universe 10 00393 g015" data-lsrc="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g015-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/10/393/display" href="#fig_body_display_universe-10-00393-f015"></a> </div> </div> <div class="html-fig_description"> <b>Figure 15.</b> The accretion disk images with brightness produced by the white hole, with the panels from left to right corresponding to <math display='inline'><semantics> <mrow> <msub> <mi>θ</mi> <mn>0</mn> </msub> <mo>=</mo> <msup> <mn>0</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msup> <mn>40</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msup> <mn>80</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, and <math display='inline'><semantics> <mrow> <msup> <mn>88</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, respectively.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_universe-10-00393-f015"> <div class="html-caption"> <b>Figure 15.</b> The accretion disk images with brightness produced by the white hole, with the panels from left to right corresponding to <math display='inline'><semantics> <mrow> <msub> <mi>θ</mi> <mn>0</mn> </msub> <mo>=</mo> <msup> <mn>0</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msup> <mn>40</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msup> <mn>80</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, and <math display='inline'><semantics> <mrow> <msup> <mn>88</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, respectively.</div> <div class="html-img"><img data-large="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g015.png" data-original="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g015.png" alt="Universe 10 00393 g015" data-lsrc="/universe/universe-10-00393/article_deploy/html/images/universe-10-00393-g015.png" /></div> </div> </section><section class='html-fn_group'><table><tr id=''><td></td><td><div class='html-p'><b>Disclaimer/Publisher’s Note:</b> The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). 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