Sgr A* Shadow Study with KTN Space Time and Investigation of NUT Charge Existence

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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?1732286508"> <meta name="title" content="Sgr A* Shadow Study with KTN Space Time and Investigation of NUT Charge Existence"> <meta name="description" content="In this paper, I investigate the existence of the NUT charge through the KTN spacetime using shadow observations of Sgr A*. I report that the range of my constraint for the NUT charge is between −0.5 and 0.5 for Schwarzschild-like and very slowly rotating KTN black holes. This range extends to 1.5 for spins up to −2 and −1.5 for spins up to 2 based on Keck observations for both 40° and 10° viewing angles. For VLTI observations, Schwarzschild-like and very slowly rotating KTN black holes are excluded for a 40° viewing angle, and the NUT charge is constrained to a very narrow range for a 10° viewing angle. I report that the possibility of having KTN naked singularities in Sgr A* is small, considering the uncertainties in the shadow size." > <link rel="image_src" href="https://pub.mdpi-res.com/img/journals/universe-logo.png?8600e93ff98dbf14" > <meta name="dc.title" content="Sgr A* Shadow Study with KTN Space Time and Investigation of NUT Charge Existence"> <meta name="dc.creator" content="Masoumeh Ghasemi-Nodehi"> <meta name="dc.type" content="Communication"> <meta name="dc.source" content="Universe 2024, Vol. 10, Page 378"> <meta name="dc.date" content="2024-09-23"> <meta name ="dc.identifier" content="10.3390/universe10090378"> <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="In this paper, I investigate the existence of the NUT charge through the KTN spacetime using shadow observations of Sgr A*. I report that the range of my constraint for the NUT charge is between −0.5 and 0.5 for Schwarzschild-like and very slowly rotating KTN black holes. This range extends to 1.5 for spins up to −2 and −1.5 for spins up to 2 based on Keck observations for both 40° and 10° viewing angles. For VLTI observations, Schwarzschild-like and very slowly rotating KTN black holes are excluded for a 40° viewing angle, and the NUT charge is constrained to a very narrow range for a 10° viewing angle. I report that the possibility of having KTN naked singularities in Sgr A* is small, considering the uncertainties in the shadow size." > <meta name="dc.subject" content="general relativity" > <meta name="dc.subject" content="black hole shadow" > <meta name="dc.subject" content="test of Kerr paradigm" > <meta name ="prism.issn" content="2218-1997"> <meta name ="prism.publicationName" content="Universe"> <meta name ="prism.publicationDate" content="2024-09-23"> <meta name ="prism.volume" content="10"> <meta name ="prism.number" content="9"> <meta name ="prism.section" content="Communication" > <meta name ="prism.startingPage" content="378" > <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="Sgr A* Shadow Study with KTN Space Time and Investigation of NUT Charge Existence"> <meta name="citation_publication_date" content="2024/9"> <meta name="citation_online_date" content="2024/09/23"> <meta name="citation_volume" content="10"> <meta name="citation_issue" content="9"> <meta name="citation_firstpage" content="378"> <meta name="citation_author" content="Ghasemi-Nodehi, Masoumeh"> <meta name="citation_doi" content="10.3390/universe10090378"> <meta name="citation_id" content="mdpi-universe10090378"> <meta name="citation_abstract_html_url" content="https://www.mdpi.com/2218-1997/10/9/378"> <meta name="citation_pdf_url" content="https://www.mdpi.com/2218-1997/10/9/378/pdf?version=1727246214"> <link rel="alternate" type="application/pdf" title="PDF Full-Text" href="https://www.mdpi.com/2218-1997/10/9/378/pdf?version=1727246214"> <meta name="fulltext_pdf" content="https://www.mdpi.com/2218-1997/10/9/378/pdf?version=1727246214"> <meta name="citation_fulltext_html_url" content="https://www.mdpi.com/2218-1997/10/9/378/htm"> <link rel="alternate" type="text/html" title="HTML Full-Text" href="https://www.mdpi.com/2218-1997/10/9/378/htm"> <meta name="fulltext_html" content="https://www.mdpi.com/2218-1997/10/9/378/htm"> <link rel="alternate" type="text/xml" title="XML Full-Text" href="https://www.mdpi.com/2218-1997/10/9/378/xml"> <meta name="fulltext_xml" content="https://www.mdpi.com/2218-1997/10/9/378/xml"> <meta name="citation_xml_url" content="https://www.mdpi.com/2218-1997/10/9/378/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/2218-1997/10/9/378" /> <meta property="og:title" content="Sgr A* Shadow Study with KTN Space Time and Investigation of NUT Charge Existence" /> <meta property="og:description" content="In this paper, I investigate the existence of the NUT charge through the KTN spacetime using shadow observations of Sgr A*. I report that the range of my constraint for the NUT charge is between −0.5 and 0.5 for Schwarzschild-like and very slowly rotating KTN black holes. This range extends to 1.5 for spins up to −2 and −1.5 for spins up to 2 based on Keck observations for both 40° and 10° viewing angles. For VLTI observations, Schwarzschild-like and very slowly rotating KTN black holes are excluded for a 40° viewing angle, and the NUT charge is constrained to a very narrow range for a 10° viewing angle. I report that the possibility of having KTN naked singularities in Sgr A* is small, considering the uncertainties in the shadow size." /> <meta property="og:image" content="https://pub.mdpi-res.com/universe/universe-10-00378/article_deploy/html/images/universe-10-00378-g001-550.jpg?1727246433" /> <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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<div class='row'> <div class='html-font-bg large-6 medium-6 small-12 columns end'> <div class='row'> <div class='html-font-label large-4 medium-4 small-12 columns'>Background:</div> <div class='large-8 medium-8 small-12 columns'> <div class="html-article-menu-option html-nav-bg html-nav-bright" data-bg="bright"> <i class="fa fa-file-text"></i> </div> <div class="html-article-menu-option html-nav-bg html-nav-dark" data-bg="dark"> <i class="fa fa-file-text-o"></i> </div> <div class="html-article-menu-option html-nav-bg html-nav-creme" data-bg="creme"> <i class="fa fa-file-text"></i> </div> </div> </div> </div> </div> </div> </div> <article ><div class='html-article-content'> <span itemprop="publisher" content="Multidisciplinary Digital Publishing Institute"></span><span itemprop="url" content="https://www.mdpi.com/2218-1997/10/9/378"></span> <div class="article-icons"><span class="label openaccess" data-dropdown="drop-article-label-openaccess" aria-expanded="false">Open Access</span><span class="label articletype">Communication</span></div> <h1 class="title hypothesis_container" itemprop="name"> Sgr A* Shadow Study with KTN Space Time and Investigation of NUT Charge Existence </h1> <div class="art-authors hypothesis_container"> by <span class="inlineblock "><div class='profile-card-drop' data-dropdown='profile-card-drop13182962' data-options='is_hover:true, hover_timeout:5000'> Masoumeh Ghasemi-Nodehi</div><div id="profile-card-drop13182962" 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">Masoumeh Ghasemi-Nodehi</span></div></div></div><div class="profile-card__buttons" style="margin-bottom: 10px;"><a href="https://sciprofiles.com/profile/3726831?utm_source=mdpi.com&utm_medium=website&utm_campaign=avatar_name" class="button button--color-inversed" target="_blank"> SciProfiles </a><a href="https://scilit.net/scholars?q=Masoumeh%20Ghasemi-Nodehi" class="button button--color-inversed" target="_blank"> Scilit </a><a href="https://www.preprints.org/search?search1=Masoumeh%20Ghasemi-Nodehi&field1=authors" class="button button--color-inversed" target="_blank"> Preprints.org </a><a href="https://scholar.google.com/scholar?q=Masoumeh%20Ghasemi-Nodehi" class="button button--color-inversed" target="_blank" rels="noopener noreferrer"> Google Scholar </a></div><div class="profile-card__biography"><div class="js-profile-card__biography-item js-default-open"> The author is currently an Associate Research Professor at Xinjiang Astronomical Observatory, CAS, a [...] </div><div class="js-profile-card__biography-item js-default-closed" style="display: none;"> The author is currently an Associate Research Professor at Xinjiang Astronomical Observatory, CAS, Urumqi, China. She received her PhD from Fudan University, Shanghai, China, graduating in 2017. She was awarded a CSC scholarship for her PhD studies. After completing her PhD, she joined the National Astronomical Observatories of China (NAOC), CAS, Beijing, China as a postdoctoral researcher, supported by a fellowship from the China Postdoctoral Research Foundation. She then joined the Institute for Research in Fundamental Sciences (IPM), Tehran, Iran with a fellowship and research grant. Before pursuing her PhD, she was a postgraduate pre-PhD diploma student at The International Center for Theoretical Physics (ICTP), Trieste, Italy. She is interested in simulations concerning tests of General Relativity, as well as simulations for SKA telescope data in preparation for the arrival of real data. In addition to the China CSC scholarship, Italy ICTP scholarship, China Postdoctoral Research Foundation fellowship, and Iran fellowship and research grant, she has also been awarded the research grant of the Xinjiang Tianchi Talent program and the CAS Talent program. </div><div><a href="#" class='js-profile-card__biography-item js-default-open'>Read more</a><a href="#" class='js-profile-card__biography-item js-default-closed' style="display: none;">Read less</a></div></div></div><sup></sup><span style="display: inline; margin-left: 5px;"></span><a class="toEncode emailCaptcha visibility-hidden" data-author-id="13182962" href="/cdn-cgi/l/email-protection#b897dbd6dc95dbdfd197d497ddd5d9d1d495c8cad7ccdddbccd1d7d69b888888d98edc888d88db89dd88808888888c888b8adc898d88db888a8c8b88db88dd8c8b88dd888b"><sup><i class="fa fa-envelope-o"></i></sup></a><a href="https://orcid.org/0000-0001-6113-0317" target="_blank" rel="noopener noreferrer"><img src="https://pub.mdpi-res.com/img/design/orcid.png?0465bc3812adeb52?1732286508" 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 ">Xinjiang Astronomical Observatory, CAS, 150 Science1-Street, Urumqi 830011, China</div> </div> </div> </div> <div class="bib-identity" style="margin-bottom: 10px;"> <em>Universe</em> <b>2024</b>, <em>10</em>(9), 378; <a href="https://doi.org/10.3390/universe10090378">https://doi.org/10.3390/universe10090378</a> </div> <div class="pubhistory" style="font-weight: bold; padding-bottom: 10px;"> <span style="display: inline-block">Submission received: 26 July 2024</span> / <span style="display: inline-block">Revised: 2 September 2024</span> / <span style="display: inline-block">Accepted: 6 September 2024</span> / <span style="display: inline-block">Published: 23 September 2024</span> </div> <div class="belongsTo" style="margin-bottom: 10px;"> (This article belongs to the Section <a href="/journal/universe/sections/gravitation">Gravitation</a>)<br/> </div> <div class="highlight-box1"> <div class="download"> <a class="button button--color-inversed button--drop-down" data-dropdown="drop-download-1483553" aria-controls="drop-supplementary-1483553" aria-expanded="false"> Download <i class="material-icons">keyboard_arrow_down</i> </a> <div id="drop-download-1483553" class="f-dropdown label__btn__dropdown label__btn__dropdown--button" data-dropdown-content aria-hidden="true" tabindex="-1"> <a class="UD_ArticlePDF" href="/2218-1997/10/9/378/pdf?version=1727246214" data-name="Sgr A* Shadow Study with KTN Space Time and Investigation of NUT Charge Existence" data-journal="universe">Download PDF</a> <br/> <a id="js-pdf-with-cover-access-captcha" href="#" data-target="/2218-1997/10/9/378/pdf-with-cover" class="accessCaptcha">Download PDF with Cover</a> <br/> <a id="js-xml-access-captcha" href="#" data-target="/2218-1997/10/9/378/xml" class="accessCaptcha">Download XML</a> <br/> <a href="/2218-1997/10/9/378/epub" id="epub_link">Download Epub</a> <br/> </div> <div class="js-browse-figures" 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/2218-1997/10/9/378/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-10-00378/article_deploy/html/images/universe-10-00378-g001.png?1727246432" title=" <strong>Figure 1</strong><br/> <p>I first measure center of shadow as discussed in the text. <math display="inline"><semantics> <mrow> <mi>ϕ</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> is from shorter segment of symmetry axis of shadow which is <math display="inline"><semantics> <msup> <mi>x</mi> <mo>′</mo> </msup> </semantics></math>. Starting from <math display="inline"><semantics> <mrow> <mi>ϕ</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>R</mi> <mfenced open="(" close=")"> <mi>ϕ</mi> </mfenced> </mrow> </semantics></math> defines each point of boundary of shadow from center of shadow.</p> "> </a> <a href="https://pub.mdpi-res.com/universe/universe-10-00378/article_deploy/html/images/universe-10-00378-g002.png?1727246433" title=" <strong>Figure 2</strong><br/> <p>The colored regions show deviation from circular shadow of Schwarzschild black hole that is <math display="inline"><semantics> <msub> <mi>δ</mi> <mi>metric</mi> </msub> </semantics></math> in in Equation (<a href="#FD12-universe-10-00378" class="html-disp-formula">12</a>). The range is according to EHT observation of Keck, (<b>a</b>), and VLTI, (<b>b</b>), as Equation (<a href="#FD13-universe-10-00378" class="html-disp-formula">13</a>). The viewing angle is 40° in these plots. The central region within the two dashed green lines is KTN black holes and the outer part is KTN naked singularities.</p> "> </a> <a href="https://pub.mdpi-res.com/universe/universe-10-00378/article_deploy/html/images/universe-10-00378-g003.png?1727246434" title=" <strong>Figure 3</strong><br/> <p>The colored regions show deviation from circular shadow of Schwarzschild black hole that is <math display="inline"><semantics> <msub> <mi>δ</mi> <mi>metric</mi> </msub> </semantics></math> in Equation (<a href="#FD12-universe-10-00378" class="html-disp-formula">12</a>). The range is according to EHT observation of Keck, (<b>a</b>), and VLTI, (<b>b</b>), as in Equation (<a href="#FD13-universe-10-00378" class="html-disp-formula">13</a>). The viewing angle is 10° in these plots. The central region within the two dashed green lines is KTN black holes and the outer part is KTN naked singularities.</p> "> </a> <a href="https://pub.mdpi-res.com/universe/universe-10-00378/article_deploy/html/images/universe-10-00378-g004.png?1727246435" title=" <strong>Figure 4</strong><br/> <p>Plot for fixed value of spin, 0.2, and viewing angle 40°. The red dashed line is observational limits from EHT collaborations.</p> "> </a> <a href="https://pub.mdpi-res.com/universe/universe-10-00378/article_deploy/html/images/universe-10-00378-g005.png?1727246436" title=" <strong>Figure 5</strong><br/> <p>Plot for fixed value of spin, 0.2, and viewing angle 10°. The red dashed line is observational limits from EHT collaborations.</p> "> </a> <a href="https://pub.mdpi-res.com/universe/universe-10-00378/article_deploy/html/images/universe-10-00378-g006.png?1727246437" title=" <strong>Figure 6</strong><br/> <p>Plot for fixed value of spin, 0.8, and viewing angle 40°. The red dashed line is observational limits from EHT collaborations.</p> "> </a> <a href="https://pub.mdpi-res.com/universe/universe-10-00378/article_deploy/html/images/universe-10-00378-g007.png?1727246438" title=" <strong>Figure 7</strong><br/> <p>Plot for fixed value of spin, 0.8, and viewing angle 10°. The red dashed line is observational limits from EHT collaborations.</p> "> </a> </div> <a class="button button--color-inversed" href="/2218-1997/10/9/378/review_report">Review Reports</a> <a class="button button--color-inversed" href="/2218-1997/10/9/378/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">In this paper, I investigate the existence of the NUT charge through the KTN spacetime using shadow observations of Sgr A*. I report that the range of my constraint for the NUT charge is between −0.5 and 0.5 for Schwarzschild-like and very slowly rotating KTN black holes. This range extends to 1.5 for spins up to −2 and −1.5 for spins up to 2 based on Keck observations for both 40° and 10° viewing angles. For VLTI observations, Schwarzschild-like and very slowly rotating KTN black holes are excluded for a 40° viewing angle, and the NUT charge is constrained to a very narrow range for a 10° viewing angle. I report that the possibility of having KTN naked singularities in Sgr A* is small, considering the uncertainties in the shadow size.</div> </section> <div id="html-keywords"> <div class="html-gwd-group"><div id="html-keywords-title">Keywords: </div><a href="/search?q=general+relativity">general relativity</a>; <a href="/search?q=black+hole+shadow">black hole shadow</a>; <a href="/search?q=test+of+Kerr+paradigm">test of Kerr paradigm</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-10-00378' type='intro'><h2 data-nested='1'> 1. Introduction</h2><div class='html-p'>General Relativity predicts that the final product of gravitational collapse is Kerr black holes. These black holes are characterized by mass and spin [<a href="#B1-universe-10-00378" class="html-bibr">1</a>,<a href="#B2-universe-10-00378" class="html-bibr">2</a>]. The Kerr paradigm has been successfully tested in weak regimes. However, precise measurements in strong gravity regimes are still lacking.</div><div class='html-p'>A test of General Relativity in the strong gravity regime can be executed by different methods such as Gravitational Waves [<a href="#B3-universe-10-00378" class="html-bibr">3</a>,<a href="#B4-universe-10-00378" class="html-bibr">4</a>], X-ray reflection spectroscopy [<a href="#B5-universe-10-00378" class="html-bibr">5</a>], the Continuum fitting method [<a href="#B6-universe-10-00378" class="html-bibr">6</a>] and the black hole shadow [<a href="#B7-universe-10-00378" class="html-bibr">7</a>,<a href="#B8-universe-10-00378" class="html-bibr">8</a>,<a href="#B9-universe-10-00378" class="html-bibr">9</a>]. Methods such as X-ray reflection spectroscopy and the continuum fitting method are parametrically degenerate in measuring spin and deviation from General Relativity. All deformation parameters should vanish to verify the Kerr black holes scenario.</div><div class='html-p'>One way to test the Kerr black holes paradigm is to consider deviations from Kerr spacetime by introducing extra parameters and trying to constrain the parameters. In this work, I consider the shadow of black holes.</div><div class='html-p'>The shadow of a black hole is a dark area in the observer plane. This dark area appears over a bright background region. The boundary of the shadow depends on the background metric and the observer viewing angle in the case of a geometrically thick and optically thin accretion disk structure. So far, there are two observations from the Event Horizon Telescope group of the shadow of a compact object. One is the shadow of M87* [<a href="#B7-universe-10-00378" class="html-bibr">7</a>,<a href="#B8-universe-10-00378" class="html-bibr">8</a>] and the other is the shadow of Sgr A* [<a href="#B9-universe-10-00378" class="html-bibr">9</a>].</div><div class='html-p'>In paper [<a href="#B10-universe-10-00378" class="html-bibr">10</a>], my collaborators and I studied the shadow of M87* for the Kerr-Taub-NUT (KTN) spacetime [<a href="#B11-universe-10-00378" class="html-bibr">11</a>]. KTN spacetime has a Newman-Unti-Tamburino (NUT) [<a href="#B12-universe-10-00378" class="html-bibr">12</a>] charge in addition to Kerr parameters. In the case of a vanishing NUT charge, the metric reduces to the Kerr metric, and in the case of a vanishing spin parameter, the metric is the Taub-NUT spacetime with mass and NUT charge. There are reports in [<a href="#B13-universe-10-00378" class="html-bibr">13</a>] and also [<a href="#B14-universe-10-00378" class="html-bibr">14</a>,<a href="#B15-universe-10-00378" class="html-bibr">15</a>,<a href="#B16-universe-10-00378" class="html-bibr">16</a>] about the existence of the effect of a gravitomagnetic monopole, also known as NUT charge, in the spectra of quasars, supernovae, or active galactic nuclei. Based on X-ray observations of GRO J1655-40, it has been reported that this is more consistent with KTN spacetime rather than Kerr spacetime [<a href="#B17-universe-10-00378" class="html-bibr">17</a>].</div><div class='html-p'>I investigate the possibility of the existence of a gravitomagnetic monopole, that is, the NUT charge, in the shadow of M87*. In this paper, I consider the KTN spacetime again, but my simulation is for the case of the shadow of Sgr A*. In the M87* case, I found that a non-zero NUT charge is still compatible with the EHT result of M87*. I show that in prograde rotation (<math display='inline'><semantics> <mrow> <msub> <mi>a</mi> <mo>*</mo> </msub> <mo>></mo> <mn>0</mn> </mrow> </semantics></math>), <math display='inline'><semantics> <msub> <mi>n</mi> <mo>*</mo> </msub> </semantics></math> cannot be greater than 1.1, and in retrograde rotation (<math display='inline'><semantics> <mrow> <msub> <mi>a</mi> <mo>*</mo> </msub> <mo><</mo> <mn>0</mn> </mrow> </semantics></math>), <math display='inline'><semantics> <msub> <mi>n</mi> <mo>*</mo> </msub> </semantics></math> cannot be less than −1.1 for the shadow of M87*. Here, <math display='inline'><semantics> <mrow> <msub> <mi>a</mi> <mo>*</mo> </msub> <mo>=</mo> <mi>a</mi> <mo>/</mo> <mi>M</mi> </mrow> </semantics></math> is the spin parameter, and <math display='inline'><semantics> <mrow> <msub> <mi>n</mi> <mo>*</mo> </msub> <mo>=</mo> <mi>n</mi> <mo>/</mo> <mi>M</mi> </mrow> </semantics></math> is the NUT charge with mass <span class='html-italic'>M</span>. In addition, the results show that there can be a shadow for both cases of the KTN black hole and the KTN naked singularity. In this paper on the Sgr A* shadow, the allowed range for the KTN naked singularity is narrow, so I conclude that the possibility of having KTN naked singularities in Sgr A* is small, considering the uncertainties in the shadow size. The range for the KTN black hole is between −0.5 and 0.5 for spin zero and up to 1.5 for spin −2 and −1.5 for spin two.</div><div class='html-p'>As for the shadow of naked singularities, due to the presence of a strong gravitational field, naked singularities can still cast a shadow. Depending on the geometry, there is a region where light cannot escape. Also, the light can be deflected and create a shadow-like region. The bending or trapping can occur in a specific way. However, the nature and specifically the shape and size of the shadow can be different from those of a standard black hole. The difference in these distortions or asymmetry effects can lead to potential signatures in observations to distinguish a naked singularity from a standard black hole. Even in the absence of a photon sphere, due to the strong gravitational field resulting in light bending, photons can reach the observer plane. There are a number of references that calculate the shadow of naked singularities. I cite some of them as [<a href="#B18-universe-10-00378" class="html-bibr">18</a>,<a href="#B19-universe-10-00378" class="html-bibr">19</a>,<a href="#B20-universe-10-00378" class="html-bibr">20</a>,<a href="#B21-universe-10-00378" class="html-bibr">21</a>,<a href="#B22-universe-10-00378" class="html-bibr">22</a>,<a href="#B23-universe-10-00378" class="html-bibr">23</a>].</div><div class='html-p'>The significance of this study is in probing deviations from Kerr spacetime, quantifying the NUT charge or gravitomagnetic monopole, testing the cosmic censorship conjecture, linking observations with theory through the mathematically interesting theory of KTN spacetime and introducing new and exotic physics.</div><div class='html-p'>The structure of this paper is as follows. I present the KTN spacetime in <a href="#sec2-universe-10-00378" class="html-sec">Section 2</a>. The shadow description is presented in <a href="#sec3-universe-10-00378" class="html-sec">Section 3</a>. <a href="#sec4-universe-10-00378" class="html-sec">Section 4</a> is devoted to constraining the NUT charge and the results of this paper. The conclusion is written in <a href="#sec5-universe-10-00378" class="html-sec">Section 5</a>.</div></section><section id='sec2-universe-10-00378' type=''><h2 data-nested='1'> 2. Theoretical Framework of Kerr-Taub-NUT Space Time</h2><div class='html-p'>In this section, I describe the KTN space time as follows. The metric of KTN space time can be written as [<a href="#B11-universe-10-00378" class="html-bibr">11</a>] <div class='html-disp-formula-info' id='FD1-universe-10-00378'> <div class='f'> <math display='block'><semantics> <mtable displaystyle="true"> <mtr> <mtd columnalign="right"> <mrow> <mi>d</mi> <msup> <mi>s</mi> <mn>2</mn> </msup> </mrow> </mtd> <mtd> <mo>=</mo> </mtd> <mtd columnalign="left"> <mrow> <mo>−</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mi mathvariant="normal">Δ</mi> <msup> <mi>p</mi> <mn>2</mn> </msup> </mfrac> </mstyle> <msup> <mrow> <mo>(</mo> <mi>d</mi> <mi>t</mi> <mo>−</mo> <mi>A</mi> <mi>d</mi> <mi>ϕ</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <msup> <mi>p</mi> <mn>2</mn> </msup> <mi mathvariant="normal">Δ</mi> </mfrac> </mstyle> <mi>d</mi> <msup> <mi>r</mi> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>p</mi> <mn>2</mn> </msup> <mi>d</mi> <msup> <mi>θ</mi> <mn>2</mn> </msup> </mrow> </mtd> </mtr> <mtr> <mtd/> <mtd> <mo>+</mo> </mtd> <mtd columnalign="left"> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <msup> <mi>p</mi> <mn>2</mn> </msup> </mfrac> </mstyle> <msup> <mo form="prefix">sin</mo> <mn>2</mn> </msup> <mi>θ</mi> <msup> <mrow> <mo>(</mo> <mi>a</mi> <mi>d</mi> <mi>t</mi> <mo>−</mo> <mi>B</mi> <mi>d</mi> <mi>ϕ</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </mtd> </mtr> </mtable> </semantics></math> </div> <div class='l'> <label >(1)</label> </div> </div> with <div class='html-disp-formula-info' id='FD2-universe-10-00378'> <div class='f'> <math display='block'><semantics> <mtable displaystyle="true"> <mtr> <mtd columnalign="right"> <mi mathvariant="normal">Δ</mi> </mtd> <mtd> <mo>=</mo> </mtd> <mtd columnalign="left"> <mrow> <msup> <mi>r</mi> <mn>2</mn> </msup> <mo>−</mo> <mn>2</mn> <mi>M</mi> <mi>r</mi> <mo>+</mo> <msup> <mi>a</mi> <mn>2</mn> </msup> <mo>−</mo> <msup> <mi>n</mi> <mn>2</mn> </msup> <mo>,</mo> <mspace width="0.166667em"/> <mspace width="0.166667em"/> <mspace width="0.166667em"/> <mspace width="0.166667em"/> <mspace width="0.166667em"/> <mspace width="0.166667em"/> <mspace width="0.166667em"/> <mspace width="0.166667em"/> <mspace width="0.166667em"/> <msup> <mi>p</mi> <mn>2</mn> </msup> <mo>=</mo> <msup> <mi>r</mi> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mi>n</mi> <mo>+</mo> <mi>a</mi> <mo form="prefix">cos</mo> <mi>θ</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="right"> <mi>A</mi> </mtd> <mtd> <mo>=</mo> </mtd> <mtd columnalign="left"> <mrow> <mi>a</mi> <msup> <mo form="prefix">sin</mo> <mn>2</mn> </msup> <mi>θ</mi> <mo>−</mo> <mn>2</mn> <mi>n</mi> <mo form="prefix">cos</mo> <mi>θ</mi> <mo>,</mo> <mspace width="0.166667em"/> <mspace width="0.166667em"/> <mspace width="0.166667em"/> <mspace width="0.166667em"/> <mspace width="0.166667em"/> <mspace width="0.166667em"/> <mspace width="0.166667em"/> <mspace width="0.166667em"/> <mi>B</mi> <mo>=</mo> <msup> <mi>r</mi> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>a</mi> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>n</mi> <mn>2</mn> </msup> </mrow> </mtd> </mtr> </mtable> </semantics></math> </div> <div class='l'> <label >(2)</label> </div> </div> where <span class='html-italic'>M</span> is the mass, <math display='inline'><semantics> <mrow> <msub> <mi>a</mi> <mo>*</mo> </msub> <mo>=</mo> <mi>a</mi> <mo>/</mo> <mi>M</mi> </mrow> </semantics></math> is spin parameter and <math display='inline'><semantics> <mrow> <msub> <mi>n</mi> <mo>*</mo> </msub> <mo>=</mo> <mi>n</mi> <mo>/</mo> <mi>M</mi> </mrow> </semantics></math> is NUT charge. The NUT parameter is known as the gravitomagnetic monopole of a compact object as well. The outer horizon is expressed as <div class='html-disp-formula-info' id='FD3-universe-10-00378'> <div class='f'> <math display='block'><semantics> <mtable displaystyle="true"> <mtr> <mtd columnalign="right"> <msub> <mi>r</mi> <mi>h</mi> </msub> </mtd> <mtd> <mo>=</mo> </mtd> <mtd columnalign="left"> <mrow> <mi>M</mi> <mo>(</mo> <mn>1</mn> <mo>+</mo> <msqrt> <mrow> <mn>1</mn> <mo>+</mo> <msubsup> <mi>n</mi> <mo>*</mo> <mn>2</mn> </msubsup> <mo>−</mo> <msubsup> <mi>a</mi> <mo>*</mo> <mn>2</mn> </msubsup> </mrow> </msqrt> <mo>)</mo> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </semantics></math> </div> <div class='l'> <label >(3)</label> </div> </div> There is indication of location of singularity in KTN space time when <math display='inline'><semantics> <msup> <mi>p</mi> <mn>2</mn> </msup> </semantics></math> vanishes [<a href="#B24-universe-10-00378" class="html-bibr">24</a>], <div class='html-disp-formula-info' id='FD4-universe-10-00378'> <div class='f'> <math display='block'><semantics> <mtable displaystyle="true"> <mtr> <mtd columnalign="right"> <mrow> <mi>r</mi> <mo>=</mo> <mn>0</mn> <mspace width="0.166667em"/> <mspace width="0.166667em"/> <mspace width="0.166667em"/> <mspace width="0.166667em"/> <mspace width="0.166667em"/> <mi>and</mi> <mspace width="0.166667em"/> <mspace width="0.166667em"/> <mspace width="0.166667em"/> <mspace width="0.166667em"/> <msub> <mi>θ</mi> <mi>s</mi> </msub> <mo>=</mo> <msup> <mo form="prefix">cos</mo> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <mo>−</mo> <msub> <mi>n</mi> <mo>*</mo> </msub> <mo>/</mo> <msub> <mi>a</mi> <mo>*</mo> </msub> <mo>)</mo> </mrow> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </semantics></math> </div> <div class='l'> <label >(4)</label> </div> </div> Expression (<a href="#FD4-universe-10-00378" class="html-disp-formula">4</a>) shows that singularity does not arise for <math display='inline'><semantics> <mrow> <mrow> <mo>|</mo> </mrow> <msub> <mi>n</mi> <mo>*</mo> </msub> <mrow> <mo>|</mo> <mo>></mo> <mo>|</mo> </mrow> <msub> <mi>a</mi> <mo>*</mo> </msub> <mrow> <mo>|</mo> </mrow> </mrow> </semantics></math>. Singularity is covered by horizon at <math display='inline'><semantics> <mrow> <msub> <mi>θ</mi> <mi>s</mi> </msub> <mo>=</mo> <mi>π</mi> </mrow> </semantics></math>. Depending on the numerical value of <math display='inline'><semantics> <msub> <mi>a</mi> <mo>*</mo> </msub> </semantics></math> and <math display='inline'><semantics> <msub> <mi>n</mi> <mo>*</mo> </msub> </semantics></math> for the KTN black hole or KTN naked singularity, singularity always arises for <math display='inline'><semantics> <mrow> <mrow> <mo>|</mo> </mrow> <msub> <mi>n</mi> <mo>*</mo> </msub> <mrow> <mo>|</mo> <mo>≤</mo> <mo>|</mo> </mrow> <msub> <mi>a</mi> <mo>*</mo> </msub> <mrow> <mo>|</mo> </mrow> </mrow> </semantics></math>. The horizon, Equation (<a href="#FD3-universe-10-00378" class="html-disp-formula">3</a>), vanishes for <math display='inline'><semantics> <mrow> <mrow> <mo>|</mo> </mrow> <msub> <mi>a</mi> <mo>*</mo> </msub> <mrow> <mo>|</mo> <mo>></mo> </mrow> <msqrt> <mrow> <mn>1</mn> <mo>+</mo> <msubsup> <mi>n</mi> <mo>*</mo> <mn>2</mn> </msubsup> </mrow> </msqrt> </mrow> </semantics></math>. This leads to KTN naked singularity. Meanwhile, for <math display='inline'><semantics> <mrow> <mrow> <mo>|</mo> </mrow> <msub> <mi>n</mi> <mo>*</mo> </msub> <mrow> <mo>|</mo> <mo>≤</mo> <mo>|</mo> </mrow> <msub> <mi>a</mi> <mo>*</mo> </msub> <mrow> <mo>|</mo> <mo>≤</mo> </mrow> <msqrt> <mrow> <mn>1</mn> <mo>+</mo> <msubsup> <mi>n</mi> <mo>*</mo> <mn>2</mn> </msubsup> </mrow> </msqrt> </mrow> </semantics></math>, the KTN black hole with (covered) singularity is obtained. One can refer to paper [<a href="#B10-universe-10-00378" class="html-bibr">10</a>] for more information about the metric of space time.</div></section><section id='sec3-universe-10-00378' type='methods'><h2 data-nested='1'> 3. Methods and Description of Shadow Boundary of a Compact Object</h2><div class='html-p'>The image of a black hole, also called the black hole shadow, is a dark area over a brighter region. Capture, scatter to infinity, and the critical curve are the trajectories of photons approaching the black hole from distant radii around the black hole. The critical curve is the separation between capture and scatter to infinity trajectories. In the case of a nearly tangential to circular orbit, the 3-momentum of the light ray is unstable and orbits around the black hole several times, which creates a brighter area around the darker region. The 2D dark region in the observer’s sky plane is the shadow of the black hole. To obtain the shadow for a specific spacetime geometry, one should solve the geodesic equations. In this paper, I solve these equations numerically using the ray tracing technique. Solving the equations is achieved with a class of the Runge–Kutta–Nystrom method with adaptive step size and error control [<a href="#B25-universe-10-00378" class="html-bibr">25</a>]. Ray tracing images the observer’s perception of a distant object or portion of the sky. Each observed light ray is traced back to the object. Ray tracing studies the distortion or appearance of a background distant star by local gravitational fields.</div><div class='html-p'>The Runge–Kutta–Nystrom method is a numerical technique for second-order ordinary differential equations. There is no need to convert them into a system of first-order equations. I start from equations like <div class='html-disp-formula-info' id='FD5-universe-10-00378'> <div class='f'> <math display='block'><semantics> <mtable displaystyle="true"> <mtr> <mtd columnalign="right"> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msup> <mrow> <mi mathvariant="normal">d</mi> </mrow> <mn>2</mn> </msup> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">d</mi> <msup> <mi>τ</mi> <mn>2</mn> </msup> </mrow> </mfrac> </mstyle> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>τ</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi mathvariant="normal">d</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">d</mi> <mi>τ</mi> </mrow> </mfrac> </mstyle> <mo>)</mo> </mrow> <mspace width="0.166667em"/> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </semantics></math> </div> <div class='l'> <label >(5)</label> </div> </div></div><div class='html-p'>As geodesic equations are second-order differential equations, I can use the Runge–Kutta–Nystrom method to solve them numerically. In the case of geodesic equations <div class='html-disp-formula-info' id='FD6-universe-10-00378'> <div class='f'> <math display='block'><semantics> <mtable displaystyle="true"> <mtr> <mtd columnalign="right"> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msup> <mrow> <mi mathvariant="normal">d</mi> </mrow> <mn>2</mn> </msup> <msup> <mi>x</mi> <mi>μ</mi> </msup> </mrow> <mrow> <mi mathvariant="normal">d</mi> <msup> <mi>τ</mi> <mn>2</mn> </msup> </mrow> </mfrac> </mstyle> <mo>+</mo> <msubsup> <mi mathvariant="sans-serif">Γ</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> <mi>μ</mi> </msubsup> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi mathvariant="normal">d</mi> <msup> <mi>x</mi> <mi>α</mi> </msup> </mrow> <mrow> <mi mathvariant="normal">d</mi> <mi>τ</mi> </mrow> </mfrac> </mstyle> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi mathvariant="normal">d</mi> <msup> <mi>x</mi> <mi>β</mi> </msup> </mrow> <mrow> <mi mathvariant="normal">d</mi> <mi>τ</mi> </mrow> </mfrac> </mstyle> <mo>=</mo> <mn>0</mn> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </semantics></math> </div> <div class='l'> <label >(6)</label> </div> </div></div><div class='html-p'>I have <math display='inline'><semantics> <mrow> <mi>y</mi> <mo>=</mo> <msup> <mi>x</mi> <mi>μ</mi> </msup> <mrow> <mo>(</mo> <mi>τ</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>. The Runge–Kutta–Nystrom method updates the position of <math display='inline'><semantics> <msup> <mi>x</mi> <mi>μ</mi> </msup> </semantics></math> and <math display='inline'><semantics> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi mathvariant="normal">d</mi> <msup> <mi>x</mi> <mi>μ</mi> </msup> </mrow> <mrow> <mi mathvariant="normal">d</mi> <mi>τ</mi> </mrow> </mfrac> </mstyle> </semantics></math>. It uses a specific iterative scheme to perform this update. As for adaptive step sizes, I have control over errors in each step, ensuring the solution remains accurate by controlling the errors dynamically and computing with high precision.</div><div class='html-p'>I start from the observer plane. The observer’s sky plane in my setup is located at the Cartesian coordinate <math display='inline'><semantics> <mfenced separators="" open="(" close=")"> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>,</mo> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo>,</mo> <msup> <mi>z</mi> <mo>′</mo> </msup> </mfenced> </semantics></math> and the compact object is at <math display='inline'><semantics> <mfenced separators="" open="(" close=")"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> </mfenced> </semantics></math>. The distance of the observer is <span class='html-italic'>D</span> and the viewing angle is <span class='html-italic'>i</span>. <math display='inline'><semantics> <mrow> <msub> <mi>k</mi> <mn>0</mn> </msub> <mo>=</mo> <mrow> <mo>(</mo> <msubsup> <mi>k</mi> <mn>0</mn> <mi>t</mi> </msubsup> <mo>,</mo> <msubsup> <mi>k</mi> <mn>0</mn> <mi>r</mi> </msubsup> <mo>,</mo> <msubsup> <mi>k</mi> <mn>0</mn> <mi>θ</mi> </msubsup> <mo>,</mo> <msubsup> <mi>k</mi> <mn>0</mn> <mi>ϕ</mi> </msubsup> <mo>)</mo> </mrow> </mrow> </semantics></math> is the photon 4-momentum perpendicular to the observer plane from which I fire the photon. This observer plane is a grid of fired photons and the ray is traced back. The initial conditions are as follows.</div><div class='html-p'>The transformation from image plane <math display='inline'><semantics> <mrow> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>,</mo> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo>,</mo> <msup> <mi>z</mi> <mo>′</mo> </msup> </mrow> </semantics></math> to compact object <math display='inline'><semantics> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> </mrow> </semantics></math> is <div class='html-disp-formula-info' id='FD7-universe-10-00378'> <div class='f'> <math display='block'><semantics> <mtable displaystyle="true"> <mtr> <mtd columnalign="right"> <mi>x</mi> </mtd> <mtd> <mo>=</mo> </mtd> <mtd columnalign="left"> <mrow> <mi>D</mi> <mo form="prefix">sin</mo> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> <mo>−</mo> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo form="prefix">cos</mo> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> <mo>+</mo> <msup> <mi>z</mi> <mo>′</mo> </msup> <mo form="prefix">sin</mo> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="right"> <mi>y</mi> </mtd> <mtd> <mo>=</mo> </mtd> <mtd columnalign="left"> <mrow> <msup> <mi>x</mi> <mo>′</mo> </msup> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="right"> <mi>z</mi> </mtd> <mtd> <mo>=</mo> </mtd> <mtd columnalign="left"> <mrow> <mi>D</mi> <mo form="prefix">cos</mo> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> <mo>+</mo> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo form="prefix">sin</mo> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> <mo>+</mo> <msup> <mi>z</mi> <mo>′</mo> </msup> <mo form="prefix">cos</mo> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> <mspace width="0.166667em"/> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </semantics></math> </div> <div class='l'> <label >(7)</label> </div> </div></div><div class='html-p'>The coordinate system reduces to a spherical coordinate far from the compact object with transformation. <div class='html-disp-formula-info' id='FD8-universe-10-00378'> <div class='f'> <math display='block'><semantics> <mtable displaystyle="true"> <mtr> <mtd columnalign="right"> <mi>r</mi> </mtd> <mtd> <mo>=</mo> </mtd> <mtd columnalign="left"> <mrow> <msqrt> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>y</mi> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>z</mi> <mn>2</mn> </msup> </mrow> </msqrt> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="right"> <mi>θ</mi> </mtd> <mtd> <mo>=</mo> </mtd> <mtd columnalign="left"> <mrow> <mo form="prefix">arccos</mo> <mfenced separators="" open="(" close=")"> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mi>z</mi> <mi>r</mi> </mfrac> </mstyle> </mfenced> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="right"> <mi>ϕ</mi> </mtd> <mtd> <mo>=</mo> </mtd> <mtd columnalign="left"> <mrow> <mo form="prefix">arctan</mo> <mfenced separators="" open="(" close=")"> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mi>y</mi> <mi>x</mi> </mfrac> </mstyle> </mfenced> <mspace width="0.166667em"/> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </semantics></math> </div> <div class='l'> <label >(8)</label> </div> </div></div><div class='html-p'>Considering photon initial position at <math display='inline'><semantics> <mfenced separators="" open="(" close=")"> <msubsup> <mi>x</mi> <mn>0</mn> <mo>′</mo> </msubsup> <mo>,</mo> <msubsup> <mi>y</mi> <mn>0</mn> <mo>′</mo> </msubsup> <mo>,</mo> <mn>0</mn> </mfenced> </semantics></math> with 3-momentum <math display='inline'><semantics> <mrow> <msub> <mi>k</mi> <mn>0</mn> </msub> <mo>=</mo> <mo>−</mo> <msub> <mi>k</mi> <mn>0</mn> </msub> <msup> <mi>z</mi> <mo>′</mo> </msup> </mrow> </semantics></math> perpendicular to the image plane, the photon initial conditions are obtained by substituting Equation (<a href="#FD7-universe-10-00378" class="html-disp-formula">7</a>) into Equation (<a href="#FD8-universe-10-00378" class="html-disp-formula">8</a>) at <math display='inline'><semantics> <mfenced separators="" open="(" close=")"> <msubsup> <mi>x</mi> <mn>0</mn> <mo>′</mo> </msubsup> <mo>,</mo> <msubsup> <mi>y</mi> <mn>0</mn> <mo>′</mo> </msubsup> <mo>,</mo> <mn>0</mn> </mfenced> </semantics></math>, <div class='html-disp-formula-info' id='FD9-universe-10-00378'> <div class='f'> <math display='block'><semantics> <mtable displaystyle="true"> <mtr> <mtd columnalign="right"> <msub> <mi>t</mi> <mn>0</mn> </msub> </mtd> <mtd> <mo>=</mo> </mtd> <mtd columnalign="left"> <mrow> <mn>0</mn> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="right"> <msub> <mi>r</mi> <mn>0</mn> </msub> </mtd> <mtd> <mo>=</mo> </mtd> <mtd columnalign="left"> <mrow> <msqrt> <mrow> <msubsup> <mi>x</mi> <mn>0</mn> <mrow> <mo>′</mo> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>y</mi> <mn>0</mn> <mrow> <mo>′</mo> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msup> <mi>D</mi> <mn>2</mn> </msup> </mrow> </msqrt> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="right"> <msub> <mi>θ</mi> <mn>0</mn> </msub> </mtd> <mtd> <mo>=</mo> </mtd> <mtd columnalign="left"> <mrow> <mo form="prefix">arccos</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msubsup> <mi>y</mi> <mn>0</mn> <mo>′</mo> </msubsup> <mo form="prefix">sin</mo> <mi>i</mi> <mo>+</mo> <mi>D</mi> <mo form="prefix">cos</mo> <mi>i</mi> </mrow> <msqrt> <mrow> <msubsup> <mi>x</mi> <mn>0</mn> <mrow> <mo>′</mo> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>y</mi> <mn>0</mn> <mrow> <mo>′</mo> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msup> <mi>D</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mfrac> </mstyle> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="right"> <msub> <mi>ϕ</mi> <mn>0</mn> </msub> </mtd> <mtd> <mo>=</mo> </mtd> <mtd columnalign="left"> <mrow> <mo form="prefix">arctan</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <msubsup> <mi>x</mi> <mn>0</mn> <mo>′</mo> </msubsup> <mrow> <mi>D</mi> <mo form="prefix">sin</mo> <mi>i</mi> <mo>−</mo> <msubsup> <mi>y</mi> <mn>0</mn> <mo>′</mo> </msubsup> <mo form="prefix">cos</mo> <mi>i</mi> </mrow> </mfrac> </mstyle> <mspace width="0.166667em"/> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </semantics></math> </div> <div class='l'> <label >(9)</label> </div> </div></div><div class='html-p'>The initial condition of the 4-momentum is <div class='html-disp-formula-info' id='FD10-universe-10-00378'> <div class='f'> <math display='block'><semantics> <mtable displaystyle="true"> <mtr> <mtd columnalign="right"> <msubsup> <mi>k</mi> <mn>0</mn> <mi>r</mi> </msubsup> </mtd> <mtd> <mo>=</mo> </mtd> <mtd columnalign="left"> <mrow> <mo>−</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mi>D</mi> <msqrt> <mrow> <msubsup> <mi>x</mi> <mn>0</mn> <mrow> <mo>′</mo> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>y</mi> <mn>0</mn> <mrow> <mo>′</mo> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msup> <mi>D</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mfrac> </mstyle> <mrow> <mo>|</mo> <msub> <mi>k</mi> <mn>0</mn> </msub> <mo>|</mo> </mrow> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="right"> <msubsup> <mi>k</mi> <mn>0</mn> <mi>θ</mi> </msubsup> </mtd> <mtd> <mo>=</mo> </mtd> <mtd columnalign="left"> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mo form="prefix">cos</mo> <mi>i</mi> <mo>−</mo> <mi>D</mi> <mfrac> <mrow> <msubsup> <mi>y</mi> <mn>0</mn> <mo>′</mo> </msubsup> <mo form="prefix">sin</mo> <mi>i</mi> <mo>+</mo> <mi>D</mi> <mo form="prefix">cos</mo> <mi>i</mi> </mrow> <mrow> <msubsup> <mi>x</mi> <mn>0</mn> <mrow> <mo>′</mo> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>y</mi> <mn>0</mn> <mrow> <mo>′</mo> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msup> <mi>D</mi> <mn>2</mn> </msup> </mrow> </mfrac> </mrow> <msqrt> <mrow> <msubsup> <mi>x</mi> <mn>0</mn> <mrow> <mo>′</mo> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mi>D</mi> <mo form="prefix">sin</mo> <mi>i</mi> <mo>−</mo> <msubsup> <mi>y</mi> <mn>0</mn> <mo>′</mo> </msubsup> <mo form="prefix">cos</mo> <mi>i</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </msqrt> </mfrac> </mstyle> <mrow> <mo>|</mo> <msub> <mi>k</mi> <mn>0</mn> </msub> <mo>|</mo> </mrow> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="right"> <msubsup> <mi>k</mi> <mn>0</mn> <mi>ϕ</mi> </msubsup> </mtd> <mtd> <mo>=</mo> </mtd> <mtd columnalign="left"> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msubsup> <mi>x</mi> <mn>0</mn> <mo>′</mo> </msubsup> <mo form="prefix">sin</mo> <mi>i</mi> </mrow> <mrow> <msubsup> <mi>x</mi> <mn>0</mn> <mrow> <mo>′</mo> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mi>D</mi> <mo form="prefix">sin</mo> <mi>i</mi> <mo>−</mo> <msubsup> <mi>y</mi> <mn>0</mn> <mo>′</mo> </msubsup> <mo form="prefix">cos</mo> <mi>i</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </mfrac> </mstyle> <mrow> <mo>|</mo> <msub> <mi>k</mi> <mn>0</mn> </msub> <mo>|</mo> </mrow> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="right"> <msubsup> <mi>k</mi> <mn>0</mn> <mi>t</mi> </msubsup> </mtd> <mtd> <mo>=</mo> </mtd> <mtd columnalign="left"> <mrow> <msqrt> <mrow> <msup> <mfenced separators="" open="(" close=")"> <msubsup> <mi>k</mi> <mn>0</mn> <mi>r</mi> </msubsup> </mfenced> <mn>2</mn> </msup> <mo>+</mo> <msubsup> <mi>r</mi> <mn>0</mn> <mn>2</mn> </msubsup> <msup> <mfenced separators="" open="(" close=")"> <msubsup> <mi>k</mi> <mn>0</mn> <mi>θ</mi> </msubsup> </mfenced> <mn>2</mn> </msup> <mo>+</mo> <msubsup> <mi>r</mi> <mn>0</mn> <mn>2</mn> </msubsup> <msup> <mo form="prefix">sin</mo> <mn>2</mn> </msup> <msub> <mi>θ</mi> <mn>0</mn> </msub> <msup> <mrow> <mo>(</mo> <msubsup> <mi>k</mi> <mn>0</mn> <mi>ϕ</mi> </msubsup> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </msqrt> <mspace width="0.166667em"/> </mrow> </mtd> </mtr> </mtable> </semantics></math> </div> <div class='l'> <label >(10)</label> </div> </div> where <math display='inline'><semantics> <mrow> <msub> <mi>g</mi> <mrow> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <msup> <mi>k</mi> <mi>μ</mi> </msup> <msup> <mi>k</mi> <mi>ν</mi> </msup> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> is used to calculate <math display='inline'><semantics> <msubsup> <mi>k</mi> <mn>0</mn> <mi>t</mi> </msubsup> </semantics></math>.</div><div class='html-p'>To define the boundary of the shadow, I use the following definition: <math display='inline'><semantics> <mrow> <mi>ρ</mi> <mo>(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>,</mo> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo>)</mo> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> for inside the shadow and <math display='inline'><semantics> <mrow> <mi>ρ</mi> <mo>(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>,</mo> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo>)</mo> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> for outside the shadow. <div class='html-disp-formula-info' id='FD11-universe-10-00378'> <div class='f'> <math display='block'><semantics> <mtable displaystyle="true"> <mtr> <mtd columnalign="right"> <msubsup> <mi>x</mi> <mi>center</mi> <mo>′</mo> </msubsup> </mtd> <mtd> <mo>=</mo> </mtd> <mtd columnalign="left"> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mo>∫</mo> <mo>∫</mo> <mi>ρ</mi> <mrow> <mo>(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>,</mo> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo>)</mo> </mrow> <msup> <mi>x</mi> <mo>′</mo> </msup> <mi>d</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> <mi>d</mi> <msup> <mi>y</mi> <mo>′</mo> </msup> </mrow> <mrow> <mo>∫</mo> <mo>∫</mo> <mi>ρ</mi> <mrow> <mo>(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>,</mo> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo>)</mo> </mrow> <mi>d</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> <mi>d</mi> <msup> <mi>y</mi> <mo>′</mo> </msup> </mrow> </mfrac> </mstyle> <mspace width="0.166667em"/> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="right"> <msubsup> <mi>y</mi> <mi>center</mi> <mo>′</mo> </msubsup> </mtd> <mtd> <mo>=</mo> </mtd> <mtd columnalign="left"> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mo>∫</mo> <mo>∫</mo> <mi>ρ</mi> <mrow> <mo>(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>,</mo> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo>)</mo> </mrow> <msup> <mi>y</mi> <mo>′</mo> </msup> <mi>d</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> <mi>d</mi> <msup> <mi>y</mi> <mo>′</mo> </msup> </mrow> <mrow> <mo>∫</mo> <mo>∫</mo> <mi>ρ</mi> <mrow> <mo>(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>,</mo> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo>)</mo> </mrow> <mi>d</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> <mi>d</mi> <msup> <mi>y</mi> <mo>′</mo> </msup> </mrow> </mfrac> </mstyle> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </semantics></math> </div> <div class='l'> <label >(11)</label> </div> </div></div><div class='html-p'><math display='inline'><semantics> <mrow> <mi>ϕ</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> is the shorter segment of the symmetry axis of the shadow, which is <math display='inline'><semantics> <msup> <mi>x</mi> <mo>′</mo> </msup> </semantics></math>. Starting from <math display='inline'><semantics> <mrow> <mi>ϕ</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>R</mi> <mfenced open="(" close=")"> <mi>ϕ</mi> </mfenced> </mrow> </semantics></math> defines each point of the boundary of the shadow from the center of the shadow. Please see <a href="#universe-10-00378-f001" class="html-fig">Figure 1</a> for shadow boundary definitions. Also refer to [<a href="#B26-universe-10-00378" class="html-bibr">26</a>,<a href="#B27-universe-10-00378" class="html-bibr">27</a>] for shadow description.</div></section><section id='sec4-universe-10-00378' type=''><h2 data-nested='1'> 4. Constraining the NUT Charges in Sgr A*</h2><div class='html-p'>EHT collaborations [<a href="#B9-universe-10-00378" class="html-bibr">9</a>] characterize the deviation from the shadow size of the Schwarzschild black hole of Sgr A* as <div class='html-disp-formula-info' id='FD12-universe-10-00378'> <div class='f'> <math display='block'><semantics> <mtable displaystyle="true"> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi>δ</mi> <mi>metric</mi> </msub> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <msub> <mover accent="true"> <mi>d</mi> <mo stretchy="false">˜</mo> </mover> <mi>metric</mi> </msub> <mrow> <mn>6</mn> <msqrt> <mn>3</mn> </msqrt> </mrow> </mfrac> </mstyle> <mo>−</mo> <mn>1</mn> </mrow> </mtd> </mtr> </mtable> </semantics></math> </div> <div class='l'> <label >(12)</label> </div> </div> where <math display='inline'><semantics> <msub> <mover accent="true"> <mi>d</mi> <mo stretchy="false">˜</mo> </mover> <mi>metric</mi> </msub> </semantics></math> is the shadow size, that is, the diameter between two points of the boundary of the shadow along the <math display='inline'><semantics> <msup> <mi>x</mi> <mo>′</mo> </msup> </semantics></math> axis. The size of the Schwarzschild shadow is <math display='inline'><semantics> <mrow> <mn>6</mn> <msqrt> <mn>3</mn> </msqrt> </mrow> </semantics></math>; it is circular. The authors report that the <math display='inline'><semantics> <mi>δ</mi> </semantics></math> values for Sgr A* are as follows [<a href="#B9-universe-10-00378" class="html-bibr">9</a>]:<div class='html-disp-formula-info' id='FD13-universe-10-00378'> <div class='f'> <math display='block'><semantics> <mtable displaystyle="true"> <mtr> <mtd columnalign="right"> <mrow> <mi>Keck</mi> <mo>:</mo> </mrow> </mtd> <mtd/> <mtd columnalign="left"> <mrow> <mi>δ</mi> <mo>=</mo> <mo>−</mo> <mn>0</mn> <mo>.</mo> <msubsup> <mn>04</mn> <mrow> <mo>−</mo> <mn>0.10</mn> </mrow> <mrow> <mo>+</mo> <mn>0.09</mn> </mrow> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="right"> <mrow> <mi>VLTI</mi> <mo>:</mo> </mrow> </mtd> <mtd/> <mtd columnalign="left"> <mrow> <mi>δ</mi> <mo>=</mo> <mo>−</mo> <mn>0</mn> <mo>.</mo> <msubsup> <mn>08</mn> <mrow> <mo>−</mo> <mn>0.09</mn> </mrow> <mrow> <mo>+</mo> <mn>0.09</mn> </mrow> </msubsup> </mrow> </mtd> </mtr> </mtable> </semantics></math> </div> <div class='l'> <label >(13)</label> </div> </div></div><div class='html-p'>In this paper, I use these values to constrain the NUT charge.</div><div class='html-p'>I calculate <math display='inline'><semantics> <msub> <mi>δ</mi> <mi>metric</mi> </msub> </semantics></math> of Equation (<a href="#FD12-universe-10-00378" class="html-disp-formula">12</a>) for each <math display='inline'><semantics> <msub> <mi>a</mi> <mo>*</mo> </msub> </semantics></math> and <math display='inline'><semantics> <msub> <mi>n</mi> <mo>*</mo> </msub> </semantics></math>. I consider a viewing angle of 40° and 10°. I plot both values of the reported <math display='inline'><semantics> <mi>δ</mi> </semantics></math> of Keck and VLTI observations. <a href="#universe-10-00378-f002" class="html-fig">Figure 2</a> and <a href="#universe-10-00378-f003" class="html-fig">Figure 3</a> show my results. I limit the color region to the value reported in Equation (<a href="#FD13-universe-10-00378" class="html-disp-formula">13</a>) as the EHT collaboration stated [<a href="#B9-universe-10-00378" class="html-bibr">9</a>].</div><div class='html-p'>As can be seen in plots <a href="#universe-10-00378-f002" class="html-fig">Figure 2</a>a and <a href="#universe-10-00378-f003" class="html-fig">Figure 3</a>a for Keck observations, the NUT charge can vary from about −0.5 to about 0.5 for the central region, that is, Schwarzschild and very slowly rotating KTN black holes. The colored region is extended to a NUT charge of 1.5 for a negative spin up to −2 and −1.5 for a positive spin up to 2 for KTN black holes.</div><div class='html-p'>As can be seen in plot <a href="#universe-10-00378-f002" class="html-fig">Figure 2</a>b for VLTI observations, the central region, that is, for Schwarzschild-like and very slowly rotating KTN black holes, is excluded. However, for plot <a href="#universe-10-00378-f003" class="html-fig">Figure 3</a>b for VLTI observations, the middle part is not excluded, but the constrained range for the NUT charge is narrow and close to the zero value. The colored region is extended to a NUT charge of 1.5 for a negative spin up to −2 and −1.5 for a positive spin up to 2 for KTN black holes.</div><div class='html-p'>In <a href="#universe-10-00378-f004" class="html-fig">Figure 4</a>, <a href="#universe-10-00378-f005" class="html-fig">Figure 5</a>, <a href="#universe-10-00378-f006" class="html-fig">Figure 6</a> and <a href="#universe-10-00378-f007" class="html-fig">Figure 7</a>, I fix some spin values as examples in order to see better ranges for the NUT charge. The red dashed lines in all plots show the limits from the EHT observation of Sgr A*, reported in Equation (<a href="#FD13-universe-10-00378" class="html-disp-formula">13</a>) for <math display='inline'><semantics> <mi>δ</mi> </semantics></math>. As discussed above, for slowly rotating KTN black holes, such as the plots in <a href="#universe-10-00378-f004" class="html-fig">Figure 4</a> and <a href="#universe-10-00378-f005" class="html-fig">Figure 5</a> with a spin of 0.2, the constraints are narrower and for VLTI observations are nearly zero. For larger spin values, such as 0.8 in <a href="#universe-10-00378-f006" class="html-fig">Figure 6</a> and <a href="#universe-10-00378-f007" class="html-fig">Figure 7</a>, it spans a larger range as shown in the plots. All plots include a zero value for he NUT charge. A slowly rotating KTN black hole in the sample plots, <a href="#universe-10-00378-f004" class="html-fig">Figure 4</a> and <a href="#universe-10-00378-f005" class="html-fig">Figure 5</a>, does not contain negative values for <math display='inline'><semantics> <mi>δ</mi> </semantics></math>, but a stronger gravitational field with larger spin values in the plots, <a href="#universe-10-00378-f006" class="html-fig">Figure 6</a> and <a href="#universe-10-00378-f007" class="html-fig">Figure 7</a>, introduces negative values for <math display='inline'><semantics> <mi>δ</mi> </semantics></math>, causing the shape to become distorted compared with the shape of Schwarzschild black holes.</div><div class='html-p'>As the colored region for KTN naked singularities is narrow and considering uncertainties, I conclude that the possibility for Sgr A* as a KTN naked singularities is small for both cases of Keck and VLTI for both 40° and 10° viewing angles. Qualitatively speaking, according to <a href="#universe-10-00378-f003" class="html-fig">Figure 3</a>, for a fixed spin value within the naked singularity region, the color distribution is symmetric, with the middle part showing a more bluish or greenish hue, while moving towards higher or lower NUT charges shifts the color towards red. This indicates that the middle part corresponds to more negative <math display='inline'><semantics> <mi>δ</mi> </semantics></math> values, transitioning to positive values as the color becomes redder. This implies that the shadow shape is becoming oblate as it transitions to a reddish color. However, when the viewing angle is increased to 40° in <a href="#universe-10-00378-f002" class="html-fig">Figure 2</a>, the colors become mixed, and no specific trend is observed. Nevertheless, qualitatively speaking, the dominant color in the Keck observation is red, and possibly yellow (positive values), with a smaller blue region. For VLTI, the blue color (negative values) is more prominent. In all cases depicted in <a href="#universe-10-00378-f002" class="html-fig">Figure 2</a> and <a href="#universe-10-00378-f003" class="html-fig">Figure 3</a>, the NUT charge appears constant with increasing spin values (<math display='inline'><semantics> <mrow> <mrow> <mo>|</mo> </mrow> <msub> <mi>a</mi> <mo>*</mo> </msub> <mrow> <mo>|</mo> </mrow> </mrow> </semantics></math>) in the naked singularity region.</div></section><section id='sec5-universe-10-00378' type='conclusions'><h2 data-nested='1'> 5. Conclusions and Discussion</h2><div class='html-p'>The KTN space time is generalization of the Kerr space time as a rotating black hole in General Relativity. The KTN space time introduces additional parameter, the NUT charge. As key features of the KTN space time, it is rotating like the Kerr case with mass and angular momentum. The additional parameter, the NUT charge or a related gravitomagnetic monopole, provides a non-trivial topology and alters the causal structure. Like the Kerr case, the off-diagonal component in the metric leads to a frame-dragging signature, but the KTN case is more complex because of the presence of the NUT charge. As for limitations, the physical interpretation of the NUT charge is not clear in nature. It is kind of a dual mass or a gravitational analog to a magnetic monopole that there is no direct observational evidence for its existence so far. The KTN space time is mathematically interesting, but it physical implications and relevance to astrophysics is an open question.</div><div class='html-p'>In this paper, I investigate the existence of the NUT charge in the shadow of Sgr A* reported by the EHT collaborations. I consider the metric of KTN spacetime in this study. I calculate the shadow of Sgr A* with KTN spacetime, including the NUT charge. Then, I constrain my results with the reported values of deviations from the Schwarzschild shadow size from the EHT collaborations. In a previous paper, we calculated, the shadow of the same spacetime for M87*. We show that M87* can be both a KTN black hole and a KTN naked singularity. We report that the upper limit for the NUT charge is not greater than 1.1 for a positive spin. The lower limit reports that the NUT charge cannot be less than −1.1 for negative spin values. Also, a KTN naked singularity can still be compatible with M87* observations.</div><div class='html-p'>In this work, the ranges for constraining the NUT charge from the Keck observation of Sgr A* can be a maximum of 0.5 for Schwarzschild-like and very slowly rotating KTN black holes, and extended to 1.5 for negative spin values up to −2. The minimum value for the NUT charge for Keck observations can be −0.5 for Schwarzschild-like and very slowly rotating KTN black holes, and extended to −1.5 for positive spin values up to 2. I consider two values of 40° and 10° for viewing angles. For VLTI observations, the Schwarzschild-like and very slowly rotating KTN black holes are excluded for 40° and the range is very narrow in the NUT charge for 10°. The colored region is extended to 1.5 for the NUT charge for negative spin values up to −2 and −1.5 for positive spin values up to 2. For all four plots I report, there is a small possibility of indications of KTN naked singularities considering uncertainties.</div><div class='html-p'>The main significance of the result is testing General Relativity and exploring alternative gravity theories, deviations from the Kerr solution, and exploring the possibility of exotic physics. Constraining the NUT charge, which would be interpreted as the gravitational analog of a magnetic monopole through observations, could lead to certain extensions and modifications of standard General Relativity. Confirming the existence of the NUT charge has implications for the type of objects that can exist, the end state of gravitational collapse and black hole formationand Galactic dynamics.</div><div class='html-p'>In addition to the possible constraining of exotic solutions and the implications of a gravitational monopole, my work supports the cosmic censorship conjecture (singularities should be hidden within the event horizon) by reducing the possibility of the existence of naked singularities in the shadow of Sgr A*. Naked singularities are a theoretical challenge in General Relativity.</div><div class='html-p'>In general, the possibility of singularities with other observations is reduced. As one example, one can study X-ray data through the iron line method to probe strong gravity regimes. X-ray reflection spectroscopy, also known as the iron line method, studies the prominent feature of the iron <math display='inline'><semantics> <mrow> <mi>K</mi> <mi>α</mi> </mrow> </semantics></math> line in X-ray reflection spectra. In the inner region of the accretion disk, close to the compact object in the presence of a strong gravitational field, the line is broadened and asymmetric. Currently, this method is used for measuring black hole spin considering the Kerr spacetime. It is also used to test deviations from the Kerr case. In paper [<a href="#B28-universe-10-00378" class="html-bibr">28</a>], the authors show that the iron line of naked singularities is different from that of a black hole. For instance, the line introduces three peaks instead of two peaks. One can also simulate the KTN naked singularity iron line and see the differences.</div><div class='html-p'>Regarding the test of General Relativity and black holes through X-ray observations, there are some problems as well. These observations suffer from parametric degeneracy problems with current facilities. That means that the same observations can be fitted by considering a spacetime deviated from the Kerr case. Shadow observations and the narrow range of the NUT charge in this study can help to better understand alternatives to the Kerr paradigm.</div><div class='html-p'>Moreover, in another recent manuscript [<a href="#B29-universe-10-00378" class="html-bibr">29</a>], the authors consider some naked singular spacetimes and study them using the shadow of Sgr A* and M87*. They show that a broad class of naked singularities is excluded.</div><div class='html-p'>Regarding the viewing angle, one can consider different angles as there are no strong constraints from observation on the viewing angle. The EHT group only report values below 50°, so I consider 40° and 10° as examples. There is also some reported range from infrared, radio, and other observations; it is mostly between 30° and 60°. One can also consider higher values, but as they are not reported by observations, do not consider them here.</div><div class='html-p'>A crucial point is understanding the impact of plasma on the shadow; for example, blob of plasma can affect it as a Doppler shift on the shadow image due to the plasma’s high velocity and the resulting relativistic Doppler effect. Additionally, it can distort the appearance of the black hole. Observationally, the brightness of the photon ring caused by moving/orbiting plasma causes variability in time for the shadow image. GRMHD simulations can model the plasma and its behavior. The Doppler shift and beaming signature can be studied. The brightness and spectrum of the accretion disk and jet can lead to an asymmetric structure in these studies, and the resulting photon ring appearance would be asymmetric as well.</div></section> </div> <div class="html-back"> <section class='html-notes'><h2>Funding</h2><div class='html-p'>This paper is supported by the CAS Telent Program and the Xinjiang Tianchi Telent Program.</div></section><section class='html-notes'><h2 >Data Availability Statement</h2><div class='html-p'>The raw data supporting the conclusions of this article will be made available by the authors on request.</div></section><section id='html-ack' class='html-ack'><h2 >Acknowledgments</h2><div class='html-p'>I acknowledge the support from the CAS Talent Program and the Xinjiang Tianchi Talent Program. I also thank Youjun Lu and Golshan Ejlali for fruitful discussions. Discussion with Chandrachur Chakraborty at early stage of work is appreciated as well. I would like to express my sincere gratitude to the referees for their valuable comments and suggestions, which have greatly improved the quality of this manuscript.</div></section><section class='html-notes'><h2 >Conflicts of Interest</h2><div class='html-p'>The author declares no conflicts of interest.</div></section><section id='html-glossary'><h2 >Abbreviations</h2><div class='html-p'>The following abbreviations are used in this manuscript: <table class='html-array_table'><tbody ><tr ><td align='left' valign='middle' class='html-align-left' >KTN</td><td align='left' valign='middle' class='html-align-left' >Kerr-Taub-NUT</td></tr><tr ><td align='left' valign='middle' class='html-align-left' >NUT</td><td align='left' valign='middle' class='html-align-left' >Newman-Unti-Tamburino</td></tr><tr ><td align='left' valign='middle' class='html-align-left' >EHT</td><td align='left' valign='middle' class='html-align-left' >Event Horizon Telescope</td></tr></tbody></table></div></section><section id='html-references_list'><h2>References</h2><ol class='html-xx'><li id='B1-universe-10-00378' class='html-x' data-content='1.'>Kerr, R.P. Gravitational field of a spinning mass as an example of algebraically special metrics. <span class='html-italic'>Phys. Rev. Lett.</span> <b>1963</b>, <span class='html-italic'>11</span>, 237. [<a href="https://scholar.google.com/scholar_lookup?title=Gravitational+field+of+a+spinning+mass+as+an+example+of+algebraically+special+metrics&author=Kerr,+R.P.&publication_year=1963&journal=Phys.+Rev.+Lett.&volume=11&pages=237&doi=10.1103/PhysRevLett.11.237" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevLett.11.237" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B2-universe-10-00378' class='html-x' data-content='2.'>Kerr, R.P. Gravitational collapse and rotation. In <span class='html-italic'>Quasistellar Sources and Gravitational Collapse: Including the Proceedings of the First Texas Symposium on Relativistic Astrophysics</span>; Robinson, I., Schild, A., Schuecking, E.L., Eds.; University of Chicago Press: Chicago, IL, USA, 1965; pp. 99–102. [<a href="https://scholar.google.com/scholar_lookup?title=Gravitational+collapse+and+rotation&author=Kerr,+R.P.&publication_year=1965&pages=99%E2%80%93102" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>]</li><li id='B3-universe-10-00378' class='html-x' data-content='3.'>Gair, J.R.; Vallisneri, M.; Larson, S.L.; Baker, J.G. Testing General Rel- ativity with Low-Frequency, Space-Based Gravitational-Wave Detectors. <span class='html-italic'>Living Rev. Relativ.</span> <b>2013</b>, <span class='html-italic'>16</span>, 7. [<a href="https://scholar.google.com/scholar_lookup?title=Testing+General+Rel-+ativity+with+Low-Frequency,+Space-Based+Gravitational-Wave+Detectors&author=Gair,+J.R.&author=Vallisneri,+M.&author=Larson,+S.L.&author=Baker,+J.G.&publication_year=2013&journal=Living+Rev.+Relativ.&volume=16&pages=7&doi=10.12942/lrr-2013-7&pmid=28163624" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.12942/lrr-2013-7" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>] [<a href="https://www.ncbi.nlm.nih.gov/pubmed/28163624" class='cross-ref pub_med' data-typ='pmid' target='_blank' rel='noopener noreferrer'>PubMed</a>]</li><li id='B4-universe-10-00378' class='html-x' data-content='4.'>Yunes, N.; Siemens, X. Gravitational-Wave Tests of General Relativity with Ground-Based Detectors and Pulsar Timing-Arrays. <span class='html-italic'>Living Rev. Relativ.</span> <b>2013</b>, <span class='html-italic'>16</span>, 9. [<a href="https://scholar.google.com/scholar_lookup?title=Gravitational-Wave+Tests+of+General+Relativity+with+Ground-Based+Detectors+and+Pulsar+Timing-Arrays&author=Yunes,+N.&author=Siemens,+X.&publication_year=2013&journal=Living+Rev.+Relativ.&volume=16&pages=9&doi=10.12942/lrr-2013-9" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.12942/lrr-2013-9" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B5-universe-10-00378' class='html-x' data-content='5.'>Guilbert, P.W.; Rees, M.J. ‘Cold’ material in non-thermal sources. <span class='html-italic'>Mon. Not. R. Astron. Soc.</span> <b>1988</b>, <span class='html-italic'>233</span>, 475. [<a href="https://scholar.google.com/scholar_lookup?title=%E2%80%98Cold%E2%80%99+material+in+non-thermal+sources&author=Guilbert,+P.W.&author=Rees,+M.J.&publication_year=1988&journal=Mon.+Not.+R.+Astron.+Soc.&volume=233&pages=475&doi=10.1093/mnras/233.2.475" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1093/mnras/233.2.475" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B6-universe-10-00378' class='html-x' data-content='6.'>Zhang, S.N.; Cui, W.; Chen, W. Black Hole Spin in X-ray Binaries: Observational Consequences. <span class='html-italic'>Astrophys. J.</span> <b>1997</b>, <span class='html-italic'>482</span>, L155. [<a href="https://scholar.google.com/scholar_lookup?title=Black+Hole+Spin+in+X-ray+Binaries:+Observational+Consequences&author=Zhang,+S.N.&author=Cui,+W.&author=Chen,+W.&publication_year=1997&journal=Astrophys.+J.&volume=482&pages=L155&doi=10.1086/310705" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1086/310705" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B7-universe-10-00378' class='html-x' data-content='7.'>The Event Horizon Telescope Collaboration. First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole. <span class='html-italic'>Astrophys. J.</span> <b>2019</b>, <span class='html-italic'>875</span>, L1. [<a href="https://scholar.google.com/scholar_lookup?title=First+M87+Event+Horizon+Telescope+Results.+I.+The+Shadow+of+the+Supermassive+Black+Hole&author=The+Event+Horizon+Telescope+Collaboration&publication_year=2019&journal=Astrophys.+J.&volume=875&pages=L1&doi=10.3847/2041-8213/ab0ec7" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.3847/2041-8213/ab0ec7" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B8-universe-10-00378' class='html-x' data-content='8.'>The Event Horizon Telescope Collaboration. First M87 Event Horizon Telescope Results. V. Physical Origin of the Asymmetric Ring. <span class='html-italic'>Astrophys. J.</span> <b>2019</b>, <span class='html-italic'>875</span>, L5. [<a href="https://scholar.google.com/scholar_lookup?title=First+M87+Event+Horizon+Telescope+Results.+V.+Physical+Origin+of+the+Asymmetric+Ring&author=The+Event+Horizon+Telescope+Collaboration&publication_year=2019&journal=Astrophys.+J.&volume=875&pages=L5&doi=10.3847/2041-8213/ab0f43" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.3847/2041-8213/ab0f43" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B9-universe-10-00378' class='html-x' data-content='9.'>Akiyama, K.; Alberdi, A.; Alef, W.; Algaba, J.C.; Anantua, R.; Asada, K.; Azulay, R.; Bach, U.; Baczko, A.-K.; Ball, D.; et al. [Event Horizon Telescope Collaboration]. First Sagittarius A* Event Horizon Telescope Results. VI. Testing the Black Hole Metric. <span class='html-italic'>Astrophys. J. Lett.</span> <b>2022</b>, <span class='html-italic'>930</span>, L17. [<a href="https://scholar.google.com/scholar_lookup?title=%5BEvent+Horizon+Telescope+Collaboration%5D.+First+Sagittarius+A*+Event+Horizon+Telescope+Results.+VI.+Testing+the+Black+Hole+Metric&author=Akiyama,+K.&author=Alberdi,+A.&author=Alef,+W.&author=Algaba,+J.C.&author=Anantua,+R.&author=Asada,+K.&author=Azulay,+R.&author=Bach,+U.&author=Baczko,+A.-K.&author=Ball,+D.&publication_year=2022&journal=Astrophys.+J.+Lett.&volume=930&pages=L17" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>]</li><li id='B10-universe-10-00378' class='html-xx' data-content='10.'>Ghasemi-Nodehi, M.; Chakraborty, C.; Yu, Q.; Lu, Y. Investigating the existence of gravitomagnetic monopole in M87*. <span class='html-italic'>Eur. Phys. J. C</span> <b>2021</b>, <span class='html-italic'>81</span>, 939. [<a href="https://scholar.google.com/scholar_lookup?title=Investigating+the+existence+of+gravitomagnetic+monopole+in+M87*&author=Ghasemi-Nodehi,+M.&author=Chakraborty,+C.&author=Yu,+Q.&author=Lu,+Y.&publication_year=2021&journal=Eur.+Phys.+J.+C&volume=81&pages=939&doi=10.1140/epjc/s10052-021-09696-3" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1140/epjc/s10052-021-09696-3" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B11-universe-10-00378' class='html-xx' data-content='11.'>Miller, J.G. Global analysis of the Kerr-Taub-NUT metric. <span class='html-italic'>J. Math. Phys.</span> <b>1973</b>, <span class='html-italic'>14</span>, 486. [<a href="https://scholar.google.com/scholar_lookup?title=Global+analysis+of+the+Kerr-Taub-NUT+metric&author=Miller,+J.G.&publication_year=1973&journal=J.+Math.+Phys.&volume=14&pages=486&doi=10.1063/1.1666343" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1063/1.1666343" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B12-universe-10-00378' class='html-xx' data-content='12.'>Newman, E.; Tamburino, L.; Unti, T. Empty-Space Generalization of the Schwarzschild Metric. <span class='html-italic'>J. Math. Phys.</span> <b>1963</b>, <span class='html-italic'>4</span>, 915. [<a href="https://scholar.google.com/scholar_lookup?title=Empty-Space+Generalization+of+the+Schwarzschild+Metric&author=Newman,+E.&author=Tamburino,+L.&author=Unti,+T.&publication_year=1963&journal=J.+Math.+Phys.&volume=4&pages=915&doi=10.1063/1.1704018" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1063/1.1704018" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B13-universe-10-00378' class='html-xx' data-content='13.'>Lynden-Bell, D.; Nouri-Zonoz, M. Classical monopoles: Newton, NUT space, gravomagnetic lensing, and atomic spectra. <span class='html-italic'>Rev. Mod. Phys.</span> <b>1998</b>, <span class='html-italic'>70</span>, 427. [<a href="https://scholar.google.com/scholar_lookup?title=Classical+monopoles:+Newton,+NUT+space,+gravomagnetic+lensing,+and+atomic+spectra&author=Lynden-Bell,+D.&author=Nouri-Zonoz,+M.&publication_year=1998&journal=Rev.+Mod.+Phys.&volume=70&pages=427&doi=10.1103/RevModPhys.70.427" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/RevModPhys.70.427" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B14-universe-10-00378' class='html-xx' data-content='14.'>Kagramanova, V.; Kunz, J.; Hackmann, E.; Lämmerzahl, C. Analytic treatment of complete and incomplete geodesics in Taub-NUT space-times. <span class='html-italic'>Phys. Rev.</span> <b>2010</b>, <span class='html-italic'>D 81</span>, 124044. [<a href="https://scholar.google.com/scholar_lookup?title=Analytic+treatment+of+complete+and+incomplete+geodesics+in+Taub-NUT+space-times&author=Kagramanova,+V.&author=Kunz,+J.&author=Hackmann,+E.&author=L%C3%A4mmerzahl,+C.&publication_year=2010&journal=Phys.+Rev.&volume=D+81&pages=124044&doi=10.1103/PhysRevD.81.124044" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevD.81.124044" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B15-universe-10-00378' class='html-xx' data-content='15.'>Liu, C.; Chen, S.; Ding, C.; Jing, J. Particle acceleration on the background of the Kerr-Taub-NUT spacetime. <span class='html-italic'>Phys. Lett.</span> <b>2011</b>, <span class='html-italic'>B 701</span>, 285. [<a href="https://scholar.google.com/scholar_lookup?title=Particle+acceleration+on+the+background+of+the+Kerr-Taub-NUT+spacetime&author=Liu,+C.&author=Chen,+S.&author=Ding,+C.&author=Jing,+J.&publication_year=2011&journal=Phys.+Lett.&volume=B+701&pages=285&doi=10.1016/j.physletb.2011.05.070" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1016/j.physletb.2011.05.070" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B16-universe-10-00378' class='html-xx' data-content='16.'>Chakraborty, C. Anomalous Lense-Thirring precession in Kerr-Taub-NUT spacetimes. <span class='html-italic'>Eur. Phys. J.</span> <b>2015</b>, <span class='html-italic'>C 75</span>, 572. [<a href="https://scholar.google.com/scholar_lookup?title=Anomalous+Lense-Thirring+precession+in+Kerr-Taub-NUT+spacetimes&author=Chakraborty,+C.&publication_year=2015&journal=Eur.+Phys.+J.&volume=C+75&pages=572&doi=10.1140/epjc/s10052-015-3785-y" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1140/epjc/s10052-015-3785-y" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B17-universe-10-00378' class='html-xx' data-content='17.'>Chakraborty, C.; Bhattacharyya, S. Circular orbits in Kerr-Taub-NUT spacetime and their implications for accreting black holes and naked singularities. <span class='html-italic'>J. Cosmol. Astropart. Phys</span> <b>2019</b>, <span class='html-italic'>5</span>, 34. [<a href="https://scholar.google.com/scholar_lookup?title=Circular+orbits+in+Kerr-Taub-NUT+spacetime+and+their+implications+for+accreting+black+holes+and+naked+singularities&author=Chakraborty,+C.&author=Bhattacharyya,+S.&publication_year=2019&journal=J.+Cosmol.+Astropart.+Phys&volume=5&pages=34&doi=10.1088/1475-7516/2019/05/034" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1088/1475-7516/2019/05/034" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B18-universe-10-00378' class='html-xx' data-content='18.'>Ortiz, N.; Sarbach, O.; Zannias, T. Shadow of a naked singularity. <span class='html-italic'>Phys. Rev. D</span> <b>2015</b>, <span class='html-italic'>92</span>, 044035. [<a href="https://scholar.google.com/scholar_lookup?title=Shadow+of+a+naked+singularity&author=Ortiz,+N.&author=Sarbach,+O.&author=Zannias,+T.&publication_year=2015&journal=Phys.+Rev.+D&volume=92&pages=044035&doi=10.1103/PhysRevD.92.044035" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevD.92.044035" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B19-universe-10-00378' class='html-xx' data-content='19.'>Joshi, A.B.; Dey, D.; Joshi, P.S.; Bambhaniya, P. Shadow of a Naked Singularity without Photon Sphere. <span class='html-italic'>Phys. Rev. D</span> <b>2020</b>, <span class='html-italic'>102</span>, 024022. [<a href="https://scholar.google.com/scholar_lookup?title=Shadow+of+a+Naked+Singularity+without+Photon+Sphere&author=Joshi,+A.B.&author=Dey,+D.&author=Joshi,+P.S.&author=Bambhaniya,+P.&publication_year=2020&journal=Phys.+Rev.+D&volume=102&pages=024022&doi=10.1103/PhysRevD.102.024022" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevD.102.024022" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B20-universe-10-00378' class='html-xx' data-content='20.'>Dey, D.; Shaikh, R.; Joshi, P.S. Shadow of nulllike and timelike naked singularities without photon spheres. <span class='html-italic'>Phys. Rev. D</span> <b>2021</b>, <span class='html-italic'>103</span>, 024015. [<a href="https://scholar.google.com/scholar_lookup?title=Shadow+of+nulllike+and+timelike+naked+singularities+without+photon+spheres&author=Dey,+D.&author=Shaikh,+R.&author=Joshi,+P.S.&publication_year=2021&journal=Phys.+Rev.+D&volume=103&pages=024015&doi=10.1103/PhysRevD.103.024015" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevD.103.024015" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B21-universe-10-00378' class='html-xx' data-content='21.'>Kaur, K.P.; Joshi, P.S.; Dey, D.; Joshi, A.B.; Desai, R.P. Comparing Shadows of Blackhole and Naked Singularity. <span class='html-italic'>arXiv</span> <b>2021</b>, arXiv:2106.13175. [<a href="https://scholar.google.com/scholar_lookup?title=Comparing+Shadows+of+Blackhole+and+Naked+Singularity&author=Kaur,+K.P.&author=Joshi,+P.S.&author=Dey,+D.&author=Joshi,+A.B.&author=Desai,+R.P.&publication_year=2021&journal=arXiv" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>]</li><li id='B22-universe-10-00378' class='html-xx' data-content='22.'>Patel, V.; Tahelyani, D.; Joshi, A.B.; Dey, D.; Joshi, P.S. Light trajectory and shadow shape in the rotating naked singularity. <span class='html-italic'>Eur. Phys. J. C</span> <b>2022</b>, <span class='html-italic'>82</span>, 798. [<a href="https://scholar.google.com/scholar_lookup?title=Light+trajectory+and+shadow+shape+in+the+rotating+naked+singularity&author=Patel,+V.&author=Tahelyani,+D.&author=Joshi,+A.B.&author=Dey,+D.&author=Joshi,+P.S.&publication_year=2022&journal=Eur.+Phys.+J.+C&volume=82&pages=798&doi=10.1140/epjc/s10052-022-10638-w" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1140/epjc/s10052-022-10638-w" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B23-universe-10-00378' class='html-xx' data-content='23.'>Wang, M.; Guo, G.; Yan, P.; Chen, S.; Jing, J. The ring-shaped shadow of rotating naked singularity with a complete photon sphere. <span class='html-italic'>arXiv</span> <b>2024</b>, arXiv:2307.16748. [<a href="https://scholar.google.com/scholar_lookup?title=The+ring-shaped+shadow+of+rotating+naked+singularity+with+a+complete+photon+sphere&author=Wang,+M.&author=Guo,+G.&author=Yan,+P.&author=Chen,+S.&author=Jing,+J.&publication_year=2024&journal=arXiv&doi=10.1088/1674-1137/ad5660" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1088/1674-1137/ad5660" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B24-universe-10-00378' class='html-xx' data-content='24.'>Mukherjee, S.; Chakraborty, S.; Dadhich, N. On some novel features of the Kerr-Newman-NUT spacetime. <span class='html-italic'>Eur. Phys. J.</span> <b>2019</b>, <span class='html-italic'>C 79</span>, 161. [<a href="https://scholar.google.com/scholar_lookup?title=On+some+novel+features+of+the+Kerr-Newman-NUT+spacetime&author=Mukherjee,+S.&author=Chakraborty,+S.&author=Dadhich,+N.&publication_year=2019&journal=Eur.+Phys.+J.&volume=C+79&pages=161&doi=10.1140/epjc/s10052-019-6662-2" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1140/epjc/s10052-019-6662-2" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B25-universe-10-00378' class='html-xx' data-content='25.'>Lund, E.; Bugge, L.; Gavrilenko, I.; Strandlie, A. Track parameter propagation through the application of a new adaptive Runge-Kutta-Nyström method in the ATLAS experiment. <span class='html-italic'>JINST</span> <b>2009</b>, <span class='html-italic'>4</span>, P04001. [<a href="https://scholar.google.com/scholar_lookup?title=Track+parameter+propagation+through+the+application+of+a+new+adaptive+Runge-Kutta-Nystr%C3%B6m+method+in+the+ATLAS+experiment&author=Lund,+E.&author=Bugge,+L.&author=Gavrilenko,+I.&author=Strandlie,+A.&publication_year=2009&journal=JINST&volume=4&pages=P04001&doi=10.1088/1748-0221/4/04/P04001" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1088/1748-0221/4/04/P04001" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B26-universe-10-00378' class='html-xx' data-content='26.'>Ghasemi-Nodehi, M.; Li, Z.; Bambi, C. Shadows of CPR black holes and tests of the Kerr metric. <span class='html-italic'>Eur. Phys. J.</span> <b>2015</b>, <span class='html-italic'>C 75</span>, 315. [<a href="https://scholar.google.com/scholar_lookup?title=Shadows+of+CPR+black+holes+and+tests+of+the+Kerr+metric&author=Ghasemi-Nodehi,+M.&author=Li,+Z.&author=Bambi,+C.&publication_year=2015&journal=Eur.+Phys.+J.&volume=C+75&pages=315&doi=10.1140/epjc/s10052-015-3539-x" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1140/epjc/s10052-015-3539-x" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B27-universe-10-00378' class='html-xx' data-content='27.'>Ghasemi-Nodehi, M.; Bambi, C. Note on a new parametrization for testing the Kerr metric. <span class='html-italic'>Eur. Phys. J.</span> <b>2016</b>, <span class='html-italic'>C 76</span>, 290. [<a href="https://scholar.google.com/scholar_lookup?title=Note+on+a+new+parametrization+for+testing+the+Kerr+metric&author=Ghasemi-Nodehi,+M.&author=Bambi,+C.&publication_year=2016&journal=Eur.+Phys.+J.&volume=C+76&pages=290&doi=10.1140/epjc/s10052-016-4137-2" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1140/epjc/s10052-016-4137-2" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B28-universe-10-00378' class='html-xx' data-content='28.'>Mummery, A.; Ingram, A. Reflecting on naked singularities: Iron line fitting as a probe of the cosmic censorship conjecture. <span class='html-italic'>Mon. Not. Roy. Astron. Soc.</span> <b>2024</b>, <span class='html-italic'>528</span>, 2015–2025. [<a href="https://scholar.google.com/scholar_lookup?title=Reflecting+on+naked+singularities:+Iron+line+fitting+as+a+probe+of+the+cosmic+censorship+conjecture&author=Mummery,+A.&author=Ingram,+A.&publication_year=2024&journal=Mon.+Not.+Roy.+Astron.+Soc.&volume=528&pages=2015%E2%80%932025&doi=10.1093/mnras/stae140" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1093/mnras/stae140" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B29-universe-10-00378' class='html-xx' data-content='29.'>Broderick, A.B.; Salehi, K. Cosmic Censorship in Sgr A* and M87*: Observationally Excluding Naked Singularities. <span class='html-italic'>arXiv</span> <b>2024</b>, arXiv:2406.05181. [<a href="https://scholar.google.com/scholar_lookup?title=Cosmic+Censorship+in+Sgr+A*+and+M87*:+Observationally+Excluding+Naked+Singularities&author=Broderick,+A.B.&author=Salehi,+K.&publication_year=2024&journal=arXiv" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>]</li></ol></section><section id='Figures' type='display-objects'><div class="html-fig-wrap" id="universe-10-00378-f001"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/9/378/display" href="#fig_body_display_universe-10-00378-f001"> <img data-large="/universe/universe-10-00378/article_deploy/html/images/universe-10-00378-g001.png" data-original="/universe/universe-10-00378/article_deploy/html/images/universe-10-00378-g001.png" alt="Universe 10 00378 g001" data-lsrc="/universe/universe-10-00378/article_deploy/html/images/universe-10-00378-g001-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/9/378/display" href="#fig_body_display_universe-10-00378-f001"></a> </div> </div> <div class="html-fig_description"> <b>Figure 1.</b> I first measure center of shadow as discussed in the text. <math display='inline'><semantics> <mrow> <mi>ϕ</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> is from shorter segment of symmetry axis of shadow which is <math display='inline'><semantics> <msup> <mi>x</mi> <mo>′</mo> </msup> </semantics></math>. Starting from <math display='inline'><semantics> <mrow> <mi>ϕ</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>R</mi> <mfenced open="(" close=")"> <mi>ϕ</mi> </mfenced> </mrow> </semantics></math> defines each point of boundary of shadow from center of shadow.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_universe-10-00378-f001"> <div class="html-caption"> <b>Figure 1.</b> I first measure center of shadow as discussed in the text. <math display='inline'><semantics> <mrow> <mi>ϕ</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> is from shorter segment of symmetry axis of shadow which is <math display='inline'><semantics> <msup> <mi>x</mi> <mo>′</mo> </msup> </semantics></math>. Starting from <math display='inline'><semantics> <mrow> <mi>ϕ</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>R</mi> <mfenced open="(" close=")"> <mi>ϕ</mi> </mfenced> </mrow> </semantics></math> defines each point of boundary of shadow from center of shadow.</div> <div class="html-img"><img data-large="/universe/universe-10-00378/article_deploy/html/images/universe-10-00378-g001.png" data-original="/universe/universe-10-00378/article_deploy/html/images/universe-10-00378-g001.png" alt="Universe 10 00378 g001" data-lsrc="/universe/universe-10-00378/article_deploy/html/images/universe-10-00378-g001.png" /></div> </div> <div class="html-fig-wrap" id="universe-10-00378-f002"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/9/378/display" href="#fig_body_display_universe-10-00378-f002"> <img data-large="/universe/universe-10-00378/article_deploy/html/images/universe-10-00378-g002.png" data-original="/universe/universe-10-00378/article_deploy/html/images/universe-10-00378-g002.png" alt="Universe 10 00378 g002" data-lsrc="/universe/universe-10-00378/article_deploy/html/images/universe-10-00378-g002-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/9/378/display" href="#fig_body_display_universe-10-00378-f002"></a> </div> </div> <div class="html-fig_description"> <b>Figure 2.</b> The colored regions show deviation from circular shadow of Schwarzschild black hole that is <math display='inline'><semantics> <msub> <mi>δ</mi> <mi>metric</mi> </msub> </semantics></math> in in Equation (<a href="#FD12-universe-10-00378" class="html-disp-formula">12</a>). The range is according to EHT observation of Keck, (<b>a</b>), and VLTI, (<b>b</b>), as Equation (<a href="#FD13-universe-10-00378" class="html-disp-formula">13</a>). The viewing angle is 40° in these plots. The central region within the two dashed green lines is KTN black holes and the outer part is KTN naked singularities.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_universe-10-00378-f002"> <div class="html-caption"> <b>Figure 2.</b> The colored regions show deviation from circular shadow of Schwarzschild black hole that is <math display='inline'><semantics> <msub> <mi>δ</mi> <mi>metric</mi> </msub> </semantics></math> in in Equation (<a href="#FD12-universe-10-00378" class="html-disp-formula">12</a>). The range is according to EHT observation of Keck, (<b>a</b>), and VLTI, (<b>b</b>), as Equation (<a href="#FD13-universe-10-00378" class="html-disp-formula">13</a>). The viewing angle is 40° in these plots. The central region within the two dashed green lines is KTN black holes and the outer part is KTN naked singularities.</div> <div class="html-img"><img data-large="/universe/universe-10-00378/article_deploy/html/images/universe-10-00378-g002.png" data-original="/universe/universe-10-00378/article_deploy/html/images/universe-10-00378-g002.png" alt="Universe 10 00378 g002" data-lsrc="/universe/universe-10-00378/article_deploy/html/images/universe-10-00378-g002.png" /></div> </div> <div class="html-fig-wrap" id="universe-10-00378-f003"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/9/378/display" href="#fig_body_display_universe-10-00378-f003"> <img data-large="/universe/universe-10-00378/article_deploy/html/images/universe-10-00378-g003.png" data-original="/universe/universe-10-00378/article_deploy/html/images/universe-10-00378-g003.png" alt="Universe 10 00378 g003" data-lsrc="/universe/universe-10-00378/article_deploy/html/images/universe-10-00378-g003-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/9/378/display" href="#fig_body_display_universe-10-00378-f003"></a> </div> </div> <div class="html-fig_description"> <b>Figure 3.</b> The colored regions show deviation from circular shadow of Schwarzschild black hole that is <math display='inline'><semantics> <msub> <mi>δ</mi> <mi>metric</mi> </msub> </semantics></math> in Equation (<a href="#FD12-universe-10-00378" class="html-disp-formula">12</a>). The range is according to EHT observation of Keck, (<b>a</b>), and VLTI, (<b>b</b>), as in Equation (<a href="#FD13-universe-10-00378" class="html-disp-formula">13</a>). The viewing angle is 10° in these plots. The central region within the two dashed green lines is KTN black holes and the outer part is KTN naked singularities.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_universe-10-00378-f003"> <div class="html-caption"> <b>Figure 3.</b> The colored regions show deviation from circular shadow of Schwarzschild black hole that is <math display='inline'><semantics> <msub> <mi>δ</mi> <mi>metric</mi> </msub> </semantics></math> in Equation (<a href="#FD12-universe-10-00378" class="html-disp-formula">12</a>). The range is according to EHT observation of Keck, (<b>a</b>), and VLTI, (<b>b</b>), as in Equation (<a href="#FD13-universe-10-00378" class="html-disp-formula">13</a>). The viewing angle is 10° in these plots. The central region within the two dashed green lines is KTN black holes and the outer part is KTN naked singularities.</div> <div class="html-img"><img data-large="/universe/universe-10-00378/article_deploy/html/images/universe-10-00378-g003.png" data-original="/universe/universe-10-00378/article_deploy/html/images/universe-10-00378-g003.png" alt="Universe 10 00378 g003" data-lsrc="/universe/universe-10-00378/article_deploy/html/images/universe-10-00378-g003.png" /></div> </div> <div class="html-fig-wrap" id="universe-10-00378-f004"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/9/378/display" href="#fig_body_display_universe-10-00378-f004"> <img data-large="/universe/universe-10-00378/article_deploy/html/images/universe-10-00378-g004.png" data-original="/universe/universe-10-00378/article_deploy/html/images/universe-10-00378-g004.png" alt="Universe 10 00378 g004" data-lsrc="/universe/universe-10-00378/article_deploy/html/images/universe-10-00378-g004-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/9/378/display" href="#fig_body_display_universe-10-00378-f004"></a> </div> </div> <div class="html-fig_description"> <b>Figure 4.</b> Plot for fixed value of spin, 0.2, and viewing angle 40°. The red dashed line is observational limits from EHT collaborations.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_universe-10-00378-f004"> <div class="html-caption"> <b>Figure 4.</b> Plot for fixed value of spin, 0.2, and viewing angle 40°. The red dashed line is observational limits from EHT collaborations.</div> <div class="html-img"><img data-large="/universe/universe-10-00378/article_deploy/html/images/universe-10-00378-g004.png" data-original="/universe/universe-10-00378/article_deploy/html/images/universe-10-00378-g004.png" alt="Universe 10 00378 g004" data-lsrc="/universe/universe-10-00378/article_deploy/html/images/universe-10-00378-g004.png" /></div> </div> <div class="html-fig-wrap" id="universe-10-00378-f005"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/9/378/display" href="#fig_body_display_universe-10-00378-f005"> <img data-large="/universe/universe-10-00378/article_deploy/html/images/universe-10-00378-g005.png" data-original="/universe/universe-10-00378/article_deploy/html/images/universe-10-00378-g005.png" alt="Universe 10 00378 g005" data-lsrc="/universe/universe-10-00378/article_deploy/html/images/universe-10-00378-g005-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/9/378/display" href="#fig_body_display_universe-10-00378-f005"></a> </div> </div> <div class="html-fig_description"> <b>Figure 5.</b> Plot for fixed value of spin, 0.2, and viewing angle 10°. The red dashed line is observational limits from EHT collaborations.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_universe-10-00378-f005"> <div class="html-caption"> <b>Figure 5.</b> Plot for fixed value of spin, 0.2, and viewing angle 10°. The red dashed line is observational limits from EHT collaborations.</div> <div class="html-img"><img data-large="/universe/universe-10-00378/article_deploy/html/images/universe-10-00378-g005.png" data-original="/universe/universe-10-00378/article_deploy/html/images/universe-10-00378-g005.png" alt="Universe 10 00378 g005" data-lsrc="/universe/universe-10-00378/article_deploy/html/images/universe-10-00378-g005.png" /></div> </div> <div class="html-fig-wrap" id="universe-10-00378-f006"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/9/378/display" href="#fig_body_display_universe-10-00378-f006"> <img data-large="/universe/universe-10-00378/article_deploy/html/images/universe-10-00378-g006.png" data-original="/universe/universe-10-00378/article_deploy/html/images/universe-10-00378-g006.png" alt="Universe 10 00378 g006" data-lsrc="/universe/universe-10-00378/article_deploy/html/images/universe-10-00378-g006-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/9/378/display" href="#fig_body_display_universe-10-00378-f006"></a> </div> </div> <div class="html-fig_description"> <b>Figure 6.</b> Plot for fixed value of spin, 0.8, and viewing angle 40°. The red dashed line is observational limits from EHT collaborations.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_universe-10-00378-f006"> <div class="html-caption"> <b>Figure 6.</b> Plot for fixed value of spin, 0.8, and viewing angle 40°. The red dashed line is observational limits from EHT collaborations.</div> <div class="html-img"><img data-large="/universe/universe-10-00378/article_deploy/html/images/universe-10-00378-g006.png" data-original="/universe/universe-10-00378/article_deploy/html/images/universe-10-00378-g006.png" alt="Universe 10 00378 g006" data-lsrc="/universe/universe-10-00378/article_deploy/html/images/universe-10-00378-g006.png" /></div> </div> <div class="html-fig-wrap" id="universe-10-00378-f007"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/9/378/display" href="#fig_body_display_universe-10-00378-f007"> <img data-large="/universe/universe-10-00378/article_deploy/html/images/universe-10-00378-g007.png" data-original="/universe/universe-10-00378/article_deploy/html/images/universe-10-00378-g007.png" alt="Universe 10 00378 g007" data-lsrc="/universe/universe-10-00378/article_deploy/html/images/universe-10-00378-g007-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/9/378/display" href="#fig_body_display_universe-10-00378-f007"></a> </div> </div> <div class="html-fig_description"> <b>Figure 7.</b> Plot for fixed value of spin, 0.8, and viewing angle 10°. The red dashed line is observational limits from EHT collaborations.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_universe-10-00378-f007"> <div class="html-caption"> <b>Figure 7.</b> Plot for fixed value of spin, 0.8, and viewing angle 10°. The red dashed line is observational limits from EHT collaborations.</div> <div class="html-img"><img data-large="/universe/universe-10-00378/article_deploy/html/images/universe-10-00378-g007.png" data-original="/universe/universe-10-00378/article_deploy/html/images/universe-10-00378-g007.png" alt="Universe 10 00378 g007" data-lsrc="/universe/universe-10-00378/article_deploy/html/images/universe-10-00378-g007.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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