Mathematicians Prove Hawking Wrong About ‘Extremal’ Black Holes

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class='theme__background-background theme__text'> <section class='post__title__wrapper relative '> <section class='outer fill-h relative outer--content '> <div class='mha container--s'> <div class='header-spacer'></div> <div class='post__title pv1 scale1 mha'> <div class='scale0'> <div class='mb1 pb025 mt0 h6t post__title__kicker '> <a class="kicker theme__accent theme__text-hover uppercase" href="/tag/mathematical-physics/">mathematical physics</a> </div> <h1 class='post__title__title mv025 noe theme__text' >Mathematicians Prove Hawking Wrong About the Most Extreme Black Holes</h1> <div class='post__title__author-date h5 mt1'> <div class='byline relative flex flex-items-start merriweather mv025 mr1 gray3 h6t theme__text-hover'> <em class='byline__by gray4 mr075'>By </em> <a href='https://www.quantamagazine.org/authors/nadis_steven/'> <span class='byline__author uppercase kern light small'>Steve Nadis</span> </a> </div> <p class="h6 mv1 pv025 o6"><em>August 21, 2024</em></p> </div> <div class='post__title__meta flex flex-wrap flex-items-start'> <div class='post__title__excerpt wysiwyg p italic mb1 mt025 pr2 o4 theme__text ' > For decades, extremal black holes were considered mathematically impossible. A new proof reveals otherwise. </div> <div class='post__title__actions h5 mt025 flex-auto flex flex-items-center'> <div class='comments-button mr05 flex flex-items-center theme__accent-hover h5 theme__accent' data-name='comment-bubble' > <a href='#comments' class='flex flex-items-center pangram'> <svg viewbox='0 0 50 50' enable-background='new 0 0 50 50' xmlns='http://www.w3.org/2000/svg' class=''><title>Comment</title><path fill='currentColor' d='M9.4 4.2h31.2c8.6 0 9.4 7 9.4 15.6s-.7 15.6-9.4 15.6h-2.2l-.9 9.4-18.8-9.4H9.4c-8.6 0-9.4-7-9.4-15.6S.7 4.2 9.4 4.2z'></path></svg> <div> <span class='comments-button__count ml075 small mbold disqus-comment-count' data-disqus-identifier='139952 https://www.quantamagazine.org/?p=139952'></span> </div> </a> </div> <span class='o-divider mh075 relative'></span> <div class='relative'> <button class='bookmark-button mh05 flex flex-items-center theme__accent-hover ' data-toggle-hover='tooltip' data-id='139952' data-active-class='' 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src="https://www.quantamagazine.org/wp-content/uploads/2024/08/BlackHoleMath-crKristinaArmitage-Lede-scaled.webp" class="mb-6 w-full s:mb-4" alt="" decoding="async" fetchpriority="high" srcset="https://www.quantamagazine.org/wp-content/uploads/2024/08/BlackHoleMath-crKristinaArmitage-Lede-scaled.webp 2560w, https://www.quantamagazine.org/wp-content/uploads/2024/08/BlackHoleMath-crKristinaArmitage-Lede-1720x968.webp 1720w, https://www.quantamagazine.org/wp-content/uploads/2024/08/BlackHoleMath-crKristinaArmitage-Lede-520x293.webp 520w, https://www.quantamagazine.org/wp-content/uploads/2024/08/BlackHoleMath-crKristinaArmitage-Lede-768x432.webp 768w, https://www.quantamagazine.org/wp-content/uploads/2024/08/BlackHoleMath-crKristinaArmitage-Lede-1536x864.webp 1536w, https://www.quantamagazine.org/wp-content/uploads/2024/08/BlackHoleMath-crKristinaArmitage-Lede-2048x1152.webp 2048w" sizes="(max-width: 2560px) 100vw, 2560px" /> </div> </div> <figcaption class='block my-0 mx-auto w-full'> <section class='py-0 px-12 s:p-0 m:px-16'> <div class='w-full my-0 mx-auto flex flex-col s:mx-5 s:w-full-minus-25 s:max-w-mod-inline m:max-w-mod-inline l:flex-row l:max-w-mod-large justify-end s:mt-4 s:mb-4'> <div class='attribution theme__anchors--solid wysiwyg pangram h6 mb1 fill-h text-left w-auto before:block l:before:hidden l:w-half-minus-17 l:text-right l:pl-24'> <p>Kristina Armitage/<em>Quanta Magazine</em></p> </div> </div> </section> </figcaption> </figure> <div class='acf-content scale1 mt2'> <h2 class='screen-reader-text'>Introduction</h2> <div class='post__wrapper scale0 show-dropcap'> <div class='mha container--m'> <div class='post__content relative flex flex-items-start flex-justify-between'> <aside class='post__sidebar hide flex flex-justify-center '> <div class='post__sidebar__content'> <div class='post__sidebar__content__inner'> <div class='align-c mb075'> <div class='sidebar__author'> <a class='theme__accent-hover transition--color ' href='https://www.quantamagazine.org/authors/nadis_steven/'> <div class='sidebar__author__avatar mha mb1'> <div class='image mx0 relative image--circle '> <div class='image__inner absolute fit-x is-loaded'> <img width="1654" height="1720" src="https://www.quantamagazine.org/wp-content/uploads/2023/01/nadis-1654x1720.jpg" class="absolute fit-x fill-h fill-v mxa" alt="" decoding="async" srcset="https://www.quantamagazine.org/wp-content/uploads/2023/01/nadis-1654x1720.jpg 1654w, https://www.quantamagazine.org/wp-content/uploads/2023/01/nadis-500x520.jpg 500w, https://www.quantamagazine.org/wp-content/uploads/2023/01/nadis-768x798.jpg 768w, https://www.quantamagazine.org/wp-content/uploads/2023/01/nadis-1477x1536.jpg 1477w, https://www.quantamagazine.org/wp-content/uploads/2023/01/nadis-1970x2048.jpg 1970w" sizes="(max-width: 1654px) 100vw, 1654px" /> </div> </div> </div> <div class='h3t mv05'> <span class='screen-reader-text'>By </span>Steve Nadis </div> <p class="o8 theme__text h5 mt05 mb1"><em>Contributing Writer</em></p> </a> </div> <hr class='mb075 o1'> <p class="h6 mv1 pv025 o6"><em>August 21, 2024</em></p> </div> <form action='https://www.quantamagazine.org/mathematicians-prove-hawking-wrong-about-extremal-black-holes-20240821' method='get' class='sidebar__actions'> <hr class='mt075 mb1 o1'> <input type='hidden' id='print' name='print' value='1' /> <button type="submit" class='h6 pangram uppercase mv05 pb1 bold kern flex flex-items-center flex-justify-between sidebar__print transition--color theme__accent-hover'> <small>View PDF/Print Mode</small> <svg class="icon-l theme__accent ml05" viewbox="0 0 50 50" enable-background="new 0 0 50 50"><path fill="currentColor" d="M39.9,27.5h4.9v22.4H0.1V5.1h22.4V10H5v35h35V27.5z M49.8,0.1h-2.4h-1H33.8V5h7.6L20.7,25.8l3.4,3.4L45,8.4v7.7h4.9V2.6L49.8,0.1z"></path></svg> </button> </form> <div class='mt1 pt05 sidebar__tag-wrap'> <a class='sidebar__tag h6 pangram medium inline-block mb0 relative float-left' href='/tag/black-holes'> <span class='absolute fit-x theme__text-background o1'></span> <span class='theme__text theme__accent-hover transition--color relative'>black holes</span></a> <a class='sidebar__tag h6 pangram medium inline-block mb0 relative float-left' href='/tag/general-relativity'> <span class='absolute fit-x theme__text-background o1'></span> <span class='theme__text theme__accent-hover transition--color relative'>general relativity</span></a> <a class='sidebar__tag h6 pangram medium inline-block mb0 relative float-left' href='/tag/gravity'> <span class='absolute fit-x theme__text-background o1'></span> <span class='theme__text theme__accent-hover transition--color relative'>gravity</span></a> <a class='sidebar__tag h6 pangram medium inline-block mb0 relative float-left' href='/tag/mathematical-physics'> <span class='absolute fit-x theme__text-background o1'></span> <span class='theme__text theme__accent-hover transition--color relative'>mathematical physics</span></a> <a 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href='https://quantamagazine.us1.list-manage.com/subscribe?u=0d6ddf7dc1a0b7297c8e06618&id=f0cb61321c' title='' target='_blank' class=''> <img width="320" height="550" src="https://www.quantamagazine.org/wp-content/uploads/2024/02/FundamentalsNewsletter-Ads-Article.gif" class="fill-h" alt="Introducing Fundamentals, a new newsletter from Quanta Magazine." decoding="async" /></a> </div> </div> </aside> <section class='outer mha js-router-anchors outer--content'> <div class='flex-auto mha container--xs '> <div class='post__content__section wysiwyg p theme__anchors--underline' data-role='selectable'> <div class='post__content wysiwyg p theme__anchors--underline '> <p>To understand the universe, scientists look to its outliers. “You always want to know about the extreme cases — the special cases that lie at the edge,” said <a href="https://www.southampton.ac.uk/people/5wy8rn/professor-carsten-gundlach">Carsten Gundlach</a>, a mathematical physicist at the University of Southampton.</p> <p>Black holes are the enigmatic extremes of the cosmos. Within them, matter is packed so tightly that, according to Einstein’s general theory of relativity, nothing can escape. For decades, physicists and mathematicians have used them to probe the limits of their ideas about gravity, space and time.</p> <p>But even black holes have edge cases — and those cases have their own insights to give. Black holes rotate in space. As matter falls into them, they start to spin faster; if that matter has charge, they also become electrically charged. In principle, a black hole can reach a point where it has as much charge or spin as it possibly can, given its mass. Such a black hole is called “extremal” — the extreme of the extremes.</p> <p>These black holes have some bizarre properties. In particular, the so-called surface gravity at the boundary, or event horizon, of such a black hole is zero. “It is a black hole whose surface doesn’t attract things anymore,” Gundlach said. But if you were to nudge a particle slightly toward the black hole’s center, it would be unable to escape.</p> <p>In 1973, the prominent physicists Stephen Hawking, James Bardeen and Brandon Carter asserted that extremal black holes can’t exist in the real world — that there is simply no plausible way that they can form. Nevertheless, for the past 50 years, extremal black holes have served as useful models in theoretical physics. “They have nice symmetries that make it easier to calculate things,” said <a href="https://web.uri.edu/physics/meet/gaurav-khanna/">Gaurav Khanna</a> of the University of Rhode Island, and this allows physicists to test theories about the mysterious relationship between quantum mechanics and gravity.</p> <figure class='mb2 mt1 image--shortcode s:mt-0'> <div class='relative image mx0'> <img width="2560" height="1713" src="https://www.quantamagazine.org/wp-content/uploads/2024/08/StephenHawking-crSantiVisalli_GettyImages-scaled.webp" class="block fit-x fill-h fill-v is-loaded mxa" alt="" decoding="async" srcset="https://www.quantamagazine.org/wp-content/uploads/2024/08/StephenHawking-crSantiVisalli_GettyImages-scaled.webp 2560w, https://www.quantamagazine.org/wp-content/uploads/2024/08/StephenHawking-crSantiVisalli_GettyImages-1720x1151.webp 1720w, https://www.quantamagazine.org/wp-content/uploads/2024/08/StephenHawking-crSantiVisalli_GettyImages-520x348.webp 520w, https://www.quantamagazine.org/wp-content/uploads/2024/08/StephenHawking-crSantiVisalli_GettyImages-768x514.webp 768w, https://www.quantamagazine.org/wp-content/uploads/2024/08/StephenHawking-crSantiVisalli_GettyImages-1536x1028.webp 1536w, https://www.quantamagazine.org/wp-content/uploads/2024/08/StephenHawking-crSantiVisalli_GettyImages-2048x1370.webp 2048w" sizes="(max-width: 2560px) 100vw, 2560px" /> </div> <figcaption class='image__meta mt1'> <div class='caption wysiwyg h5 theme__anchors--solid fill-h post__aside__caption post__aside__caption--shortcode'> <p>In 1973, Stephen Hawking and two other prominent physicists hypothesized that extremal black holes could never form.</p> </div> <div class='attribution theme__anchors--solid wysiwyg pangram h6 mb1 fill-h post__aside__attribution'> <p>Santi Visalli/Getty Images</p> </div> </figcaption> </figure> <p>Now two mathematicians have proved Hawking and his colleagues wrong. The new work — contained in <a class="c-link c-link--focus-visible" href="https://arxiv.org/abs/2211.15742" target="_blank" rel="noopener noreferrer" data-stringify-link="https://arxiv.org/abs/2211.15742" data-sk="tooltip_parent">a pair of</a> <a class="c-link c-link--focus-visible" href="https://arxiv.org/abs/2402.10190" target="_blank" rel="noopener noreferrer" data-stringify-link="https://arxiv.org/abs/2402.10190" data-sk="tooltip_parent">recent papers</a> by <a class="c-link c-link--focus-visible" href="https://math.mit.edu/~kehle/" target="_blank" rel="noopener noreferrer" data-stringify-link="https://math.mit.edu/~kehle/" data-sk="tooltip_parent">Christoph Kehle</a> of the Massachusetts Institute of Technology and <a class="c-link c-link--focus-visible" href="https://rcunger.github.io/" target="_blank" rel="noopener noreferrer" data-stringify-link="https://rcunger.github.io/" data-sk="tooltip_parent">Ryan Unger</a> of Stanford University and the University of California, Berkeley — demonstrates that there is nothing in our known laws of physics to prevent the formation of an extremal black hole.</p> <p>Their mathematical proof is “beautiful, technically innovative and physically surprising,” said <a href="https://web.math.princeton.edu/~dafermos/">Mihalis Dafermos</a>, a mathematician at Princeton University (and Kehle’s and Unger’s doctoral adviser). It hints at a potentially richer and more varied universe in which “extremal black holes could be out there astrophysically,” he added.</p> <p>That doesn’t mean they are. “Just because a mathematical solution exists that has nice properties doesn’t necessarily mean that nature will make use of it,” Khanna said. “But if we somehow find one, that would really [make] us think about what we are missing.” Such a discovery, he noted, has the potential to raise “some pretty radical kinds of questions.”</p> <h2><strong>The Law of Impossibility</strong></h2> <p>Before Kehle and Unger’s proof, there was good reason to believe that extremal black holes couldn’t exist.</p> <p>In 1973, Bardeen, Carter and Hawking introduced four laws about the behavior of black holes. They resembled the four long-established laws of thermodynamics — a set of sacrosanct principles that state, for instance, that the universe becomes more disordered over time, and that energy cannot be created or destroyed.</p> <aside class="post__aside mb2 relative post__aside--right"><figure class='mb2 mt1 image--shortcode s:mt-0'> <div class='relative image mx0'> <img width="1707" height="2560" src="https://www.quantamagazine.org/wp-content/uploads/2024/08/ChristophKehle-crDanKomoda_InstituteforAdvancedStudy-scaled.webp" class="block fit-x fill-h fill-v is-loaded mxa vertical" alt="The mathematician Christoph Kehle." decoding="async" srcset="https://www.quantamagazine.org/wp-content/uploads/2024/08/ChristophKehle-crDanKomoda_InstituteforAdvancedStudy-scaled.webp 1707w, https://www.quantamagazine.org/wp-content/uploads/2024/08/ChristophKehle-crDanKomoda_InstituteforAdvancedStudy-1147x1720.webp 1147w, https://www.quantamagazine.org/wp-content/uploads/2024/08/ChristophKehle-crDanKomoda_InstituteforAdvancedStudy-1920x2880.webp 1920w, https://www.quantamagazine.org/wp-content/uploads/2024/08/ChristophKehle-crDanKomoda_InstituteforAdvancedStudy-347x520.webp 347w, https://www.quantamagazine.org/wp-content/uploads/2024/08/ChristophKehle-crDanKomoda_InstituteforAdvancedStudy-768x1152.webp 768w, https://www.quantamagazine.org/wp-content/uploads/2024/08/ChristophKehle-crDanKomoda_InstituteforAdvancedStudy-1024x1536.webp 1024w, https://www.quantamagazine.org/wp-content/uploads/2024/08/ChristophKehle-crDanKomoda_InstituteforAdvancedStudy-1365x2048.webp 1365w" sizes="(max-width: 1707px) 100vw, 1707px" /> </div> <figcaption class='image__meta mt1'> <div class='caption wysiwyg h5 theme__anchors--solid fill-h post__aside__caption post__aside__caption--shortcode'> <p>Christoph Kehle, a mathematician at the Massachusetts Institute of Technology, recently disproved a 1973 conjecture about extremal black holes.</p> </div> <div class='attribution theme__anchors--solid wysiwyg pangram h6 mb1 fill-h post__aside__attribution'> <p>Dan Komoda/Institute for Advanced Study</p> </div> </figcaption> </figure> </aside> <p>In their paper, the physicists proved their first three laws of black hole thermodynamics: the zeroth, first and second. By extension, they assumed that the third law (like its standard thermodynamics counterpart) would also be true, even though they were not yet able to prove it.</p> <p>That law stated that the surface gravity of a black hole cannot decrease to zero in a finite amount of time — in other words, that there is no way to create an extremal black hole. To support their claim, the trio argued that any process that would allow a black hole’s charge or spin to reach the extremal limit could also potentially result in its event horizon disappearing altogether. It is widely believed that black holes without an event horizon, called naked singularities, cannot exist. Moreover, because a black hole’s temperature is known to be proportional to its surface gravity, a black hole with no surface gravity would also have no temperature. Such a black hole would not emit thermal radiation — something that Hawking later proposed black holes had to do.</p> <p>In 1986, a physicist named Werner Israel seemed to put the issue to rest when he <a href="https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.57.397">published a proof</a> of the third law. Say you want to create an extremal black hole from a regular one. You might try to do so by making it spin faster or by adding more charged particles. Israel’s proof seemed to demonstrate that doing so could not force a black hole’s surface gravity to drop to zero in a finite amount of time.</p> <p>As Kehle and Unger would ultimately discover, Israel’s argument concealed a flaw.</p> <h2><strong>Death of the Third Law</strong></h2> <p>Kehle and Unger did not set out to find extremal black holes. They stumbled on them entirely by accident.</p> <p>They were studying the formation of electrically charged black holes. “We realized that we could do it” — make a black hole — “for all charge-to-mass ratios,” Kehle said. That included the case where the charge is as high as possible, a hallmark of an extremal black hole.</p> <aside class="post__aside mb2 relative post__aside--right"><figure class='mb2 mt1 image--shortcode s:mt-0'> <div class='relative image mx0'> <img width="1106" height="1629" src="https://www.quantamagazine.org/wp-content/uploads/2024/08/RyanUnger-coRyanUnger.webp" class="block fit-x fill-h fill-v is-loaded mxa vertical" alt="" decoding="async" srcset="https://www.quantamagazine.org/wp-content/uploads/2024/08/RyanUnger-coRyanUnger.webp 1106w, https://www.quantamagazine.org/wp-content/uploads/2024/08/RyanUnger-coRyanUnger-353x520.webp 353w, https://www.quantamagazine.org/wp-content/uploads/2024/08/RyanUnger-coRyanUnger-768x1131.webp 768w, https://www.quantamagazine.org/wp-content/uploads/2024/08/RyanUnger-coRyanUnger-1043x1536.webp 1043w" sizes="(max-width: 1106px) 100vw, 1106px" /> </div> <figcaption class='image__meta mt1'> <div class='caption wysiwyg h5 theme__anchors--solid fill-h post__aside__caption post__aside__caption--shortcode'> <p>After proving that highly charged extremal black holes are mathematically possible, Ryan Unger of Stanford University is now trying to show that fast-spinning ones are, too. But it’s a much harder problem.</p> </div> <div class='attribution theme__anchors--solid wysiwyg pangram h6 mb1 fill-h post__aside__attribution'> <p>Dimitris Fetsios</p> </div> </figcaption> </figure> </aside> <p>Dafermos recognized that his former students had uncovered a counterexample to Bardeen, Carter and Hawking’s third law: They’d shown that they could indeed change a typical black hole into an extremal one within a finite stretch of time.</p> <p>Kehle and Unger started with a black hole that doesn’t rotate and has no charge, and modeled what might happen if it was placed in a simplified environment called a scalar field, which assumes a background of uniformly charged particles. They then buffeted the black hole with pulses from the field to add charge to it.</p> <p>These pulses also contributed electromagnetic energy to the black hole, which added to its mass. By sending diffuse, low-frequency pulses, the mathematicians realized that they could increase the black hole’s charge faster than its mass — precisely what they needed to complete their proof.</p> <p>After discussing their result with Dafermos, they pored over Israel’s 1986 proof and identified his error. They also constructed two other solutions to Einstein’s equations of general relativity that involved different ways of adding charge to a black hole. Having disproved Bardeen, Carter and Hawking’s hypothesis in three different contexts, the work should leave no doubt, Unger said: “The third law is dead.”</p> <p>The pair also showed that the formation of an extremal black hole would not open the door to a naked singularity, as physicists had feared. Instead, extremal black holes seem to lie on a critical threshold: Add the right amount of charge to a dense cloud of charged matter, and it will collapse to form an extremal black hole. Add more than that, and rather than collapse into a naked singularity, the cloud will disperse. No black hole will form at all. Kehle and Unger are just as excited by this result as they are by their proof that extremal black holes can exist.</p> <p>“This is a beautiful example of math giving back to physics,” said <a href="https://www.math.columbia.edu/~egiorgi/">Elena Giorgi</a>, a mathematician at Columbia University.</p> <h2><strong>The Impossible Made Visible</strong></h2> <p>While Kehle and Unger proved that it’s theoretically possible for extremal black holes to exist in nature, there’s no guarantee that they do.</p> <p>For one thing, the theoretical examples possess maximal charge. But black holes with a discernible charge have never been observed. It’s far more likely to see a black hole that’s quickly rotating. Kehle and Unger want to construct an example that reaches the extremal threshold for spin, rather than charge.</p> <div class="related-list"> <aside class="post__aside mb2 relative hide-on-print post__aside--right"> <h2 class="mv0 h5 bold uppercase kern">Related:</h2> <hr class="mt1 o2"> <ol class="post__aside__list theme__anchors--none pt05"> <li class="mv075 flex"> <h3 class="h5t mt0 mb025 medium"> <a class="theme__accent theme__text-hover" href="https://www.quantamagazine.org/black-hole-paradoxes-reveal-a-fundamental-link-between-energy-and-order-20200528/"> <span class="' . $span_class . '">Black Hole Paradoxes Reveal a Fundamental Link Between Energy and Order</span> </a> </h3> </li> <li class="mv075 flex"> <h3 class="h5t mt0 mb025 medium"> <a class="theme__accent theme__text-hover" href="https://www.quantamagazine.org/a-century-later-new-math-smooths-out-general-relativity-20231130/"> <span class="' . $span_class . '">A Century Later, New Math Smooths Out General Relativity</span> </a> </h3> </li> <li class="mv075 flex"> <h3 class="h5t mt0 mb025 medium"> <a class="theme__accent theme__text-hover" href="https://www.quantamagazine.org/in-violation-of-einstein-black-holes-might-have-hair-20210211/"> <span class="' . $span_class . '">In Violation of Einstein, Black Holes Might Have ‘Hair’</span> </a> </h3> </li> </ol> </aside> </div> <p>But working with spin is much more mathematically challenging. “You need a lot of new math, and new ideas, to do that,” Unger said. He and Kehle are just starting to <a href="https://arxiv.org/abs/2304.08455">investigate the problem</a>.</p> <p>In the meantime, a better understanding of extremal black holes can provide further insights into near-extremal black holes, which are thought to be plentiful in the universe. “Einstein didn’t think that black holes could be real [because] they’re just too weird,” Khanna said. “But now we know the universe is teeming with black holes.”</p> <p>For similar reasons, he added, “we shouldn’t give up on extremal black holes. I just don’t want to put limits on nature’s creativity.”</p> <p><b data-stringify-type="bold"><i data-stringify-type="italic">Correction: </i></b><i data-stringify-type="italic">August 25, 2024<br /> This article originally stated that the physicist John Bardeen hypothesized that extremal black holes cannot exist. 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