The Geometric Tool That Solved Einstein’s Relativity Problem

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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/explainers/">explainers</a> </div> <h1 class='post__title__title mv025 noe theme__text' >The Geometric Tool That Solved Einstein’s Relativity Problem</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/josephhowlett/'> <span class='byline__author uppercase kern light small'>Joseph Howlett</span> </a> </div> <p class="h6 mv1 pv025 o6"><em>August 12, 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 ' > Tensors are used all over math and science to reveal hidden geometric truths. What are they? </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='139724 https://www.quantamagazine.org/?p=139724'></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='139724' data-active-class='' aria-expanded='false'> <svg 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src="https://www.quantamagazine.org/wp-content/uploads/2024/08/Tensors-crMicheleSclafani-Lede-v2-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/Tensors-crMicheleSclafani-Lede-v2-scaled.webp 2560w, https://www.quantamagazine.org/wp-content/uploads/2024/08/Tensors-crMicheleSclafani-Lede-v2-1720x968.webp 1720w, https://www.quantamagazine.org/wp-content/uploads/2024/08/Tensors-crMicheleSclafani-Lede-v2-520x293.webp 520w, https://www.quantamagazine.org/wp-content/uploads/2024/08/Tensors-crMicheleSclafani-Lede-v2-768x432.webp 768w, https://www.quantamagazine.org/wp-content/uploads/2024/08/Tensors-crMicheleSclafani-Lede-v2-1536x864.webp 1536w, https://www.quantamagazine.org/wp-content/uploads/2024/08/Tensors-crMicheleSclafani-Lede-v2-2048x1152.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>Michele Sclafani for <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/josephhowlett/'> <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="1000" height="1000" src="https://www.quantamagazine.org/wp-content/uploads/2024/08/joseph-howlett-edited2.jpg" class="absolute fit-x fill-h fill-v mxa" alt="" decoding="async" srcset="https://www.quantamagazine.org/wp-content/uploads/2024/08/joseph-howlett-edited2.jpg 1000w, https://www.quantamagazine.org/wp-content/uploads/2024/08/joseph-howlett-edited2-520x520.jpg 520w, https://www.quantamagazine.org/wp-content/uploads/2024/08/joseph-howlett-edited2-768x768.jpg 768w, https://www.quantamagazine.org/wp-content/uploads/2024/08/joseph-howlett-edited2-160x160.jpg 160w" sizes="(max-width: 1000px) 100vw, 1000px" /> </div> </div> </div> <div class='h3t mv05'> <span class='screen-reader-text'>By </span>Joseph Howlett </div> <p class="o8 theme__text h5 mt05 mb1"><em>Staff Writer</em></p> </a> </div> <hr class='mb075 o1'> <p class="h6 mv1 pv025 o6"><em>August 12, 2024</em></p> </div> <form action='https://www.quantamagazine.org/the-geometric-tool-that-solved-einsteins-relativity-problem-20240812' 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/computer-science'> <span class='absolute fit-x theme__text-background o1'></span> <span class='theme__text theme__accent-hover transition--color relative'>computer science</span></a> <a class='sidebar__tag h6 pangram medium inline-block mb0 relative float-left' href='/tag/explainers'> <span class='absolute fit-x theme__text-background o1'></span> <span class='theme__text theme__accent-hover transition--color relative'>explainers</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/geometry'> <span class='absolute fit-x theme__text-background o1'></span> <span class='theme__text theme__accent-hover transition--color relative'>geometry</span></a> <a class='sidebar__tag h6 pangram medium inline-block mb0 relative float-left' href='/tag/linear-algebra'> <span class='absolute fit-x theme__text-background o1'></span> <span class='theme__text theme__accent-hover transition--color relative'>linear algebra</span></a> <a class='sidebar__tag h6 pangram medium inline-block mb0 relative float-left' href='/tag/mathematics'> <span class='absolute fit-x theme__text-background o1'></span> <span class='theme__text theme__accent-hover transition--color relative'>mathematics</span></a> <a class='sidebar__tag h6 pangram medium inline-block mb0 relative float-left' href='/tag/physics'> <span class='absolute fit-x theme__text-background o1'></span> <span class='theme__text theme__accent-hover transition--color relative'>physics</span></a> <a class='sidebar__tag h6 pangram medium inline-block mb0 relative theme__accent w-full' href='/topics'> <span class='absolute fit-x theme__text-background o1'></span> <span class='theme__text-hover transition--color relative'>All topics</span> <svg class='icon ml05' viewbox='0 0 50 50' enable-background='new 0 0 50 50'><path fill='currentColor' d='M50 25l-17.4-8.7v6.5H0v4.4h32.6v6.5'></path></svg> </a> </div> </div> <div class='sidebar__poster'> <a href='https://www.quantabooks.org/' title='' target='_blank' class=''> <img width="640" height="1100" src="https://www.quantamagazine.org/wp-content/uploads/2025/01/D-Article-Qbooks-Jan2025-1.png" class="fill-h" alt="" decoding="async" srcset="https://www.quantamagazine.org/wp-content/uploads/2025/01/D-Article-Qbooks-Jan2025-1.png 640w, https://www.quantamagazine.org/wp-content/uploads/2025/01/D-Article-Qbooks-Jan2025-1-303x520.png 303w" sizes="(max-width: 640px) 100vw, 640px" /></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>After Albert Einstein published his special theory of relativity in 1905, he spent the next decade trying to come up with a theory of gravity. But for years, he kept running up against a problem.</p> <p>He wanted to show that gravity is really a warping of the geometry of space-time caused by the presence of matter. But he also knew that time and distance are counterintuitively relative: They change depending on your frame of reference. Moving quickly makes distances shrink and time slow down. How, then, might you describe gravity objectively, regardless of whether you’re stationary or moving?</p> <p>Einstein found the solution in a new geometric theory published a few years earlier by the Italian mathematicians Gregorio Ricci-Curbastro and Tullio Levi-Civita. In this theory lay the mathematical foundation for what would later be dubbed a “tensor.”</p> <p>Since then, tensors have become instrumental not just in Einstein’s general theory of relativity, but also in machine learning, quantum mechanics and even biology. “Tensors are the most efficient packaging device we have to organize our equations,” said <a href="https://www.kcl.ac.uk/people/dionysios-anninos">Dionysios Anninos</a>, a theoretical physicist at King’s College London. “They’re the natural language for geometric objects.”</p> <p>They’re also tough to define. Talk to a computer scientist, and they might tell you that a tensor is an array of numbers that stores important data. A single number is a “rank 0” tensor. A list of numbers, called a vector, is a rank 1 tensor. A grid of numbers, or matrix, is a rank 2 tensor. And so on.</p> <p>But talk to a physicist or mathematician, and they’ll find this definition wanting. To them, though tensors can be represented by such arrays of numbers, they have a deeper geometric meaning.</p> <p>To understand the geometric notion of a tensor, start with vectors. You can think of a vector as an arrow floating in space — it has a length and a direction. (This arrow does not need to be anchored to a particular point: If you move it around in space, it remains the same vector.) A vector might represent the velocity of a particle, for example, with the length denoting its speed and the direction its bearing.</p> <p>This information gets packaged in a list of numbers. For instance, a vector in two-dimensional space is defined by a pair of numbers. The first tells you how many units the arrow stretches to the right or left, and the second tells you how far it stretches up or down.</p> <p>But these numbers depend on how you’ve defined your coordinate system. Say you change your coordinate system:</p> <figure class='mb2 mt1 image--shortcode s:mt-0'> <div class='relative image mx0'> <img src="https://www.quantamagazine.org/wp-content/uploads/2024/08/Tensors-Fig1-crMarkBelan-Desktop.svg" class="block fit-x fill-h fill-v is-loaded mxa large-print-img s:hidden m:hidden" alt="" decoding="async" /><img src="https://www.quantamagazine.org/wp-content/uploads/2024/08/Tensors-Fig1-crMarkBelan-Mobile.svg" class="block fit-x fill-h fill-v is-loaded mxa large-print-img l:hidden" alt="" decoding="async" /> </div> <figcaption class='image__meta mt1'> <div class='attribution theme__anchors--solid wysiwyg pangram h6 mb1 fill-h post__aside__attribution'> <p>All graphics: Mark Belan for <i data-stringify-type="italic">Quanta Magazine</i></p> </div> </figcaption> </figure> <p>You now express the vector in terms of how far it stretches in each direction of the new coordinate system. This gives you a different pair of numbers. But the vector itself has not changed: Its length and orientation remain the same, no matter what coordinate system you’re in. Moreover, if you know how to move from one coordinate system to another, you’ll also automatically know how your list of numbers should change.</p> <p>Tensors generalize these ideas. A vector is a rank 1 tensor; tensors of higher rank contain more complicated geometric information.</p> <p>For example, imagine you have a block of steel, and you want to describe all the forces that can be exerted on it. A rank 2 tensor — written as a matrix — can do this. Each of the faces of the block feels forces in three different directions. (For instance, the right face of the block can experience forces in the up-down direction, the left-right direction, and the forward-back direction.)</p> <figure class='mb2 mt1 image--shortcode s:mt-0'> <div class='relative image mx0'> <img src="https://www.quantamagazine.org/wp-content/uploads/2024/08/Tensors-Fig2-crMarkBelan-Desktop.svg" class="block fit-x fill-h fill-v is-loaded mxa large-print-img s:hidden m:hidden" alt="" decoding="async" /><img src="https://www.quantamagazine.org/wp-content/uploads/2024/08/Tensors-Fig2-crMarkBelan-Mobile.svg" class="block fit-x fill-h fill-v is-loaded mxa large-print-img l:hidden" alt="" decoding="async" /> </div> </figure> <p>The tensor that encapsulates all these forces can therefore be represented by a matrix of nine numbers — one number for each direction for each of three faces. (Opposite faces, in this example, are considered redundant.)</p> <p>Mathematicians often conceive of tensors as functions that take one or more vectors as inputs and produce another vector, or a number, as an output. This output doesn’t depend on the choice of coordinate system. (That constraint is what makes tensors distinct from functions more generally.) A tensor might, for instance, take in two vectors that form the edges of a rectangle, and output the rectangle’s area. If you rotate the rectangle, its length along the <em>x</em>-axis and height along the <em>y</em>-axis will change. 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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-medium justify-center s:mb-4'> </div> </section> </figcaption> </figure> <div class='acf-content scale1 mt2'> <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>In Einstein’s theory of relativity, distance and time — once thought to be absolute — turn out to change for different observers. But just as length and height can be combined to calculate area, distance and time can be combined to define other fixed properties, or invariants. Tensors enabled Einstein to efficiently manipulate these invariants, and to describe the relationship between mass and space-time. He could write a single equation describing how matter curves space-time, condensing what otherwise would have been 16 separate, interconnected equations.</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/the-ai-researcher-giving-her-field-its-bitter-medicine-20220830/"> <span class="' . $span_class . 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'">Strangely Curved Shapes Break 50-Year-Old Geometry Conjecture</span> </a> </h3> </li> </ol> </aside> </div> <p>Since the publication of this equation in 1915, tensors have become ubiquitous. Physicists use them to characterize the motion of electrons around atomic nuclei, or to describe the state of an entangled quantum system. Computer scientists use them to store the parameters of machine learning models. Biologists use them to trace traits back through a lineage. And mathematicians multiply them together to build even more complicated tensors, and then study the new spaces those tensors live in. Tensors can help mathematicians explore complicated symmetries, analyze properties of special shapes called manifolds, and probe the relationships between different functions, among other things.</p> <p>Einstein once begged a friend to help him understand tensors, fearing he was going mad. But understand them he did — and they’ve been key to scientists’ ability to describe our world ever since.</p> </div> </div> </div> </section> </div> <div class='pv2'></div> <div class='post__footer'> <div class='mha container--m'> <div class='flex flex-wrap flex-items-stretch'> <div class='post__footer__col fill-h pv2 flex flex-items-center flex-justify-center'> <section class='outer'> <aside class='post__sidebar hide flex flex-justify-center post__sidebar--footer'> <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/josephhowlett/'> <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="1000" height="1000" src="https://www.quantamagazine.org/wp-content/uploads/2024/08/joseph-howlett-edited2.jpg" 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The Geometric Tool That Solved Einstein’s Relativity Problem | Quanta Magazine