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Available in 15 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-15" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">15 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Diagrama_de_Penrose-Carter" title="Diagrama de Penrose-Carter – Catalan" lang="ca" hreflang="ca" data-title="Diagrama de Penrose-Carter" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Penrose-Diagramm" title="Penrose-Diagramm – German" lang="de" hreflang="de" data-title="Penrose-Diagramm" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Diagrama_de_Penrose-Carter" title="Diagrama de Penrose-Carter – Spanish" lang="es" hreflang="es" data-title="Diagrama de Penrose-Carter" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Penrose_diagrama" title="Penrose diagrama – Basque" lang="eu" hreflang="eu" data-title="Penrose diagrama" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link 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class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Diagramma_di_Penrose" title="Diagramma di Penrose – Italian" lang="it" hreflang="it" data-title="Diagramma di Penrose" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Penrose-diagram" title="Penrose-diagram – Dutch" lang="nl" hreflang="nl" data-title="Penrose-diagram" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%9A%E3%83%B3%E3%83%AD%E3%83%BC%E3%82%BA%E5%9B%B3" title="ペンローズ図 – Japanese" lang="ja" hreflang="ja" data-title="ペンローズ図" data-language-autonym="日本語" 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id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Two-dimensional diagram capturing the causal relations between different points in spacetime</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">For the tensor diagram notation, see <a href="/wiki/Penrose_graphical_notation" title="Penrose graphical notation">Penrose graphical notation</a>.</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Penrose_diagram.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Penrose_diagram.svg/320px-Penrose_diagram.svg.png" decoding="async" width="320" height="289" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Penrose_diagram.svg/480px-Penrose_diagram.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Penrose_diagram.svg/640px-Penrose_diagram.svg.png 2x" data-file-width="651" data-file-height="588" /></a><figcaption>Penrose diagram of an infinite <a href="/wiki/Hermann_Minkowski" title="Hermann Minkowski">Minkowski</a> universe, horizontal axis <i>u</i>, vertical axis <i>v</i></figcaption></figure> <p>In <a href="/wiki/Theoretical_physics" title="Theoretical physics">theoretical physics</a>, a <b>Penrose diagram</b> (named after mathematical physicist <a href="/wiki/Roger_Penrose" title="Roger Penrose">Roger Penrose</a>) is a <a href="/wiki/Two-dimensional" class="mw-redirect" title="Two-dimensional">two-dimensional</a> diagram capturing the <a href="/wiki/Causal_relation" class="mw-redirect" title="Causal relation">causal relations</a> between different points in <a href="/wiki/Spacetime" title="Spacetime">spacetime</a> through a <a href="/wiki/Conformal_map" title="Conformal map">conformal</a> treatment of infinity. It is an extension (suitable for the curved spacetimes of e.g. <a href="/wiki/General_relativity" title="General relativity">general relativity</a>) of the <a href="/wiki/Minkowski_diagram" class="mw-redirect" title="Minkowski diagram">Minkowski diagram</a> of <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a> where the vertical dimension represents time, and the <a href="/wiki/Horizontal_plane" class="mw-redirect" title="Horizontal plane">horizontal</a> dimension represents a space dimension. Using this design, all light rays take a 45° path <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (c=1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>c</mi> <mo>=</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (c=1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2226a75eeb89c3edd90adbc45eb521622d68337" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.077ex; height:2.843ex;" alt="{\displaystyle (c=1)}"></span>. Locally, the <a href="/wiki/Metric_tensor" title="Metric tensor">metric</a> on a Penrose diagram is <a href="/wiki/Conformal_equivalence" class="mw-redirect" title="Conformal equivalence">conformally equivalent</a> to the metric of the spacetime depicted. The conformal factor is chosen such that the entire infinite spacetime is transformed into a Penrose diagram of finite size, with infinity on the boundary of the diagram. For <a href="/wiki/Spherically_symmetric_spacetime" title="Spherically symmetric spacetime">spherically symmetric spacetimes</a>, every point in the Penrose diagram corresponds to a 2-dimensional sphere <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\theta ,\phi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\theta ,\phi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c907dc1e8b19ea4c73c6bd56ef5d3e9d32e9cb40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.319ex; height:2.843ex;" alt="{\displaystyle (\theta ,\phi )}"></span>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Basic_properties">Basic properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Penrose_diagram&action=edit&section=1" title="Edit section: Basic properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>While Penrose diagrams share the same basic <a href="/wiki/Coordinate_vector" title="Coordinate vector">coordinate vector</a> system of other spacetime diagrams for local <a href="/wiki/Asymptotically_flat_spacetime" title="Asymptotically flat spacetime">asymptotically flat spacetime</a>, it introduces a system of representing distant spacetime by shrinking or "triturando" distances that are further away. Straight lines of constant time and straight lines of constant space coordinates therefore become <a href="/wiki/Hyperbola" title="Hyperbola">hyperbolae</a>, which appear to converge at <a href="/wiki/Point_(geometry)" title="Point (geometry)">points</a> in the corners of the diagram. These points and boundaries represent <b>conformal infinity</b> for spacetime, which was first introduced by Penrose in 1963.<sup id="cite_ref-penrose1963_1-0" class="reference"><a href="#cite_note-penrose1963-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Penrose diagrams are more properly (but less frequently) called <b>Penrose–Carter diagrams</b> (or <b>Carter–Penrose diagrams</b>),<sup id="cite_ref-carroll2004_2-0" class="reference"><a href="#cite_note-carroll2004-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> acknowledging both <a href="/wiki/Brandon_Carter" title="Brandon Carter">Brandon Carter</a> and Roger Penrose, who were the first researchers to employ them. They are also called <b>conformal diagrams</b>, or simply spacetime diagrams (although the latter may refer to <a href="/wiki/Minkowski_diagram" class="mw-redirect" title="Minkowski diagram">Minkowski diagrams</a>). </p><p>Two lines drawn at 45° angles should intersect in the diagram only if the corresponding two light rays intersect in the actual spacetime. So, a Penrose diagram can be used as a concise illustration of spacetime regions that are accessible to observation. The <a href="/wiki/Diagonal" title="Diagonal">diagonal</a> boundary lines of a Penrose diagram correspond to the region called "<a href="/wiki/Null_infinity" title="Null infinity">null infinity</a>", or to singularities where light rays must end. Thus, Penrose diagrams are also useful in the study of asymptotic properties of spacetimes and singularities. An infinite static <a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski universe</a>, coordinates <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/baa3647f3b8798f94f0f2ac249637b0b709f3718" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.013ex; height:2.843ex;" alt="{\displaystyle (x,t)}"></span> is related to Penrose coordinates <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (u,v)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (u,v)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eadf12294edccd7a29c99cfc1765e4a14bf47e58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.301ex; height:2.843ex;" alt="{\displaystyle (u,v)}"></span> by: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan(u\pm v)=x\pm t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo>±</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mo>±</mo> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan(u\pm v)=x\pm t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e19967d1eafafed99e20dc808e298e53e4548591" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.575ex; height:2.843ex;" alt="{\displaystyle \tan(u\pm v)=x\pm t}"></span></dd></dl> <p>The corners of the Penrose diagram, which represent the spacelike and timelike conformal infinities, are <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi /2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi /2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b44e3d874a0b229fded7ffce67a0677dd5b8b67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.657ex; height:2.843ex;" alt="{\displaystyle \pi /2}"></span> from the origin. </p> <div class="mw-heading mw-heading2"><h2 id="Black_holes">Black holes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Penrose_diagram&action=edit&section=2" title="Edit section: Black holes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Penrose diagrams are frequently used to illustrate the causal structure of spacetimes containing <a href="/wiki/Black_holes" class="mw-redirect" title="Black holes">black holes</a>. Singularities in the <a href="/wiki/Schwarzchild_Black_Hole" class="mw-redirect" title="Schwarzchild Black Hole">Schwarzschild solution</a> are denoted by a spacelike boundary, unlike the timelike boundary found on conventional spacetime diagrams. This is due to the interchanging of timelike and spacelike coordinates within the horizon of a black hole (since space is uni-directional within the horizon, just as time is uni-directional outside the horizon). The singularity is represented by a spacelike boundary to make it clear that once an object has passed the horizon it will inevitably hit the singularity even if it attempts to take evasive action. </p><p>Penrose diagrams are often used to illustrate the hypothetical <a href="/wiki/Einstein%E2%80%93Rosen_bridge" class="mw-redirect" title="Einstein–Rosen bridge">Einstein–Rosen bridge</a> connecting two separate universes in the maximally extended <a href="/wiki/Schwarzschild_solution" class="mw-redirect" title="Schwarzschild solution">Schwarzschild black hole solution</a>. The precursors to the Penrose diagrams were <a href="/wiki/Kruskal%E2%80%93Szekeres_coordinates" title="Kruskal–Szekeres coordinates">Kruskal–Szekeres diagrams</a>. (The Penrose diagram adds to Kruskal and Szekeres' diagram the conformal crunching of the regions of flat spacetime far from the hole.) These introduced the method of aligning the <a href="/wiki/Event_horizon" title="Event horizon">event horizon</a> into past and future horizons oriented at 45° angles (since one would need to travel <a href="/wiki/Faster_than_light" class="mw-redirect" title="Faster than light">faster than light</a> to cross from the <a href="/wiki/Schwarzschild_radius" title="Schwarzschild radius">Schwarzschild radius</a> back into flat spacetime); and splitting the <a href="/wiki/Mathematical_singularity" class="mw-redirect" title="Mathematical singularity">singularity</a> into past and future horizontally-oriented lines (since the singularity "cuts off" all paths into the future once one enters the hole). </p><p>The Einstein–Rosen bridge closes off (forming "future" singularities) so rapidly that passage between the two asymptotically flat exterior regions would require faster-than-light velocity, and is therefore impossible. In addition, highly <a href="/wiki/Blue-shift" class="mw-redirect" title="Blue-shift">blue-shifted</a> light rays (called a <b>blue sheet</b>) would make it impossible for anyone to pass through. </p> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Penrose_Diagrams_of_various_black_hole_solutions.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Penrose_Diagrams_of_various_black_hole_solutions.svg/330px-Penrose_Diagrams_of_various_black_hole_solutions.svg.png" decoding="async" width="330" height="268" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Penrose_Diagrams_of_various_black_hole_solutions.svg/495px-Penrose_Diagrams_of_various_black_hole_solutions.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Penrose_Diagrams_of_various_black_hole_solutions.svg/660px-Penrose_Diagrams_of_various_black_hole_solutions.svg.png 2x" data-file-width="875" data-file-height="710" /></a><figcaption>Penrose diagrams of various black hole solutions</figcaption></figure> <p>The maximally extended solution does not describe a typical black hole created from the collapse of a star, as the surface of the collapsed star replaces the sector of the solution containing the past-oriented <i><a href="/wiki/White_hole" title="White hole">white hole</a></i> geometry and other universe. </p><p>While the basic <a href="/wiki/Space-like" class="mw-redirect" title="Space-like">space-like</a> passage of a static black hole cannot be traversed, the Penrose diagrams for solutions representing <a href="/wiki/Kerr_metric" title="Kerr metric">rotating</a> and/or <a href="/wiki/Reissner%E2%80%93Nordstr%C3%B6m_metric" title="Reissner–Nordström metric">electrically charged</a> black holes illustrate these solutions' inner event horizons (lying in the future) and vertically oriented singularities, which open up what is known as a <a href="/wiki/Time-like" class="mw-redirect" title="Time-like">time-like</a> "wormhole" allowing passage into future universes. In the case of the rotating hole, there is also a "negative" universe entered through a ring-shaped singularity (still portrayed as a line in the diagram) that can be passed through if entering the hole close to its <a href="/wiki/Coordinate_axis" class="mw-redirect" title="Coordinate axis">axis</a> of rotation. These features of the solutions are, however, not stable under perturbations and not believed to be a realistic description of the interior regions of such black holes; the <a href="/wiki/Cosmic_censorship_hypothesis" title="Cosmic censorship hypothesis">true character of their interiors is still an open question</a>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Penrose_diagram&action=edit&section=3" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Causality" title="Causality">Causality</a></li> <li><a href="/wiki/Causal_structure" title="Causal structure">Causal structure</a></li> <li><a href="/wiki/Conformal_cyclic_cosmology" title="Conformal cyclic cosmology">Conformal cyclic cosmology</a></li> <li><a href="/wiki/Weyl_transformation" title="Weyl transformation">Weyl transformation</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Penrose_diagram&action=edit&section=4" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-penrose1963-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-penrose1963_1-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFPenrose1963" class="citation journal cs1">Penrose, Roger (15 January 1963). <a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.10.66">"Asymptotic proprierties of fields and space-times"</a>. <i>Physical Review Letters</i>. <b>10</b> (2): 66–68. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1963PhRvL..10...66P">1963PhRvL..10...66P</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.10.66">10.1103/PhysRevLett.10.66</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+Letters&rft.atitle=Asymptotic+proprierties+of+fields+and+space-times&rft.volume=10&rft.issue=2&rft.pages=66-68&rft.date=1963-01-15&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.10.66&rft_id=info%3Abibcode%2F1963PhRvL..10...66P&rft.aulast=Penrose&rft.aufirst=Roger&rft_id=https%3A%2F%2Fdoi.org%2F10.1103%252FPhysRevLett.10.66&rfr_id=info%3Asid%2Fen.wikipedia.org%3APenrose+diagram" class="Z3988"></span></span> </li> <li id="cite_note-carroll2004-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-carroll2004_2-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCarroll2004" class="citation book cs1"><a href="/wiki/Sean_M._Carroll" title="Sean M. Carroll">Carroll, Sean</a> (2004). <i>Spacetime and Geometry – An Introduction to General Relativity</i>. Addison Wesley. p. 471. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8053-8732-3" title="Special:BookSources/0-8053-8732-3"><bdi>0-8053-8732-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Spacetime+and+Geometry+%E2%80%93+An+Introduction+to+General+Relativity&rft.pages=471&rft.pub=Addison+Wesley&rft.date=2004&rft.isbn=0-8053-8732-3&rft.aulast=Carroll&rft.aufirst=Sean&rfr_id=info%3Asid%2Fen.wikipedia.org%3APenrose+diagram" class="Z3988"></span></span> </li> </ol></div></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFD'Inverno1992" class="citation book cs1">D'Inverno, Ray (1992). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/introducingeinst0000dinv"><i>Introducing Einstein's relativity</i></a></span> (Repr ed.). Oxford: Clarendon Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-859686-8" title="Special:BookSources/978-0-19-859686-8"><bdi>978-0-19-859686-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introducing+Einstein%27s+relativity&rft.place=Oxford&rft.edition=Repr&rft.pub=Clarendon+Press&rft.date=1992&rft.isbn=978-0-19-859686-8&rft.aulast=D%27Inverno&rft.aufirst=Ray&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroducingeinst0000dinv&rfr_id=info%3Asid%2Fen.wikipedia.org%3APenrose+diagram" class="Z3988"></span> See <i>Chapter 17</i> (and various succeeding sections) for a very readable introduction to the concept of conformal infinity plus examples.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFrauendiener2004" class="citation journal cs1">Frauendiener, Jörg (2004). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5256109">"Conformal Infinity"</a>. <i>Living Reviews in Relativity</i>. <b>7</b> (1): 1. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2004LRR.....7....1F">2004LRR.....7....1F</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.12942%2Flrr-2004-1">10.12942/lrr-2004-1</a></span>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2367-3613">2367-3613</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5256109">5256109</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/28179863">28179863</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Living+Reviews+in+Relativity&rft.atitle=Conformal+Infinity&rft.volume=7&rft.issue=1&rft.pages=1&rft.date=2004&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC5256109%23id-name%3DPMC&rft_id=info%3Abibcode%2F2004LRR.....7....1F&rft_id=info%3Apmid%2F28179863&rft_id=info%3Adoi%2F10.12942%2Flrr-2004-1&rft.issn=2367-3613&rft.aulast=Frauendiener&rft.aufirst=J%C3%B6rg&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC5256109&rfr_id=info%3Asid%2Fen.wikipedia.org%3APenrose+diagram" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCarter1966" class="citation journal cs1">Carter, Brandon (1966). <a rel="nofollow" class="external text" href="https://link.aps.org/doi/10.1103/PhysRev.141.1242">"Complete Analytic Extension of the Symmetry Axis of Kerr's Solution of Einstein's Equations"</a>. <i>Physical Review</i>. <b>141</b> (4): 1242–1247. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1966PhRv..141.1242C">1966PhRv..141.1242C</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.141.1242">10.1103/PhysRev.141.1242</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0031-899X">0031-899X</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review&rft.atitle=Complete+Analytic+Extension+of+the+Symmetry+Axis+of+Kerr%27s+Solution+of+Einstein%27s+Equations&rft.volume=141&rft.issue=4&rft.pages=1242-1247&rft.date=1966&rft.issn=0031-899X&rft_id=info%3Adoi%2F10.1103%2FPhysRev.141.1242&rft_id=info%3Abibcode%2F1966PhRv..141.1242C&rft.aulast=Carter&rft.aufirst=Brandon&rft_id=https%3A%2F%2Flink.aps.org%2Fdoi%2F10.1103%2FPhysRev.141.1242&rfr_id=info%3Asid%2Fen.wikipedia.org%3APenrose+diagram" class="Z3988"></span> See also <a rel="nofollow" class="external text" href="http://prola.aps.org/abstract/PR/v141/i4/p1242_1">on-line version</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110927004227/http://prola.aps.org/abstract/PR/v141/i4/p1242_1">Archived</a> 2011-09-27 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> (requires a subscription to access)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHawkingEllisHawkingEllis1973" class="citation book cs1">Hawking, Stephen W.; Ellis, George F. R.; Hawking, S. W. & Ellis, G. F. R. (1973). <i>The large scale structure of space-time</i>. Cambridge monographs on mathematical physics (21st ed.). Cambridge: Cambridge Univ. Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-09906-6" title="Special:BookSources/978-0-521-09906-6"><bdi>978-0-521-09906-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+large+scale+structure+of+space-time&rft.place=Cambridge&rft.series=Cambridge+monographs+on+mathematical+physics&rft.edition=21st&rft.pub=Cambridge+Univ.+Press&rft.date=1973&rft.isbn=978-0-521-09906-6&rft.aulast=Hawking&rft.aufirst=Stephen+W.&rft.au=Ellis%2C+George+F.+R.&rft.au=Hawking%2C+S.+W.&rft.au=Ellis%2C+G.+F.+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APenrose+diagram" class="Z3988"></span> See <i>Chapter 5</i> for a very clear discussion of Penrose diagrams (the term used by Hawking & Ellis) with many examples.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKaufmann1977" class="citation book cs1">Kaufmann, William J. III (1977). <i>The Cosmic Frontiers of General Relativity</i>. Little Brown & Co. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-316-48341-4" title="Special:BookSources/978-0-316-48341-4"><bdi>978-0-316-48341-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Cosmic+Frontiers+of+General+Relativity&rft.pub=Little+Brown+%26+Co&rft.date=1977&rft.isbn=978-0-316-48341-4&rft.aulast=Kaufmann&rft.aufirst=William+J.+III&rfr_id=info%3Asid%2Fen.wikipedia.org%3APenrose+diagram" class="Z3988"></span> Really breaks down the transition from simple Minkowski diagrams, to <a href="/wiki/Martin_David_Kruskal" title="Martin David Kruskal">Kruskal</a>-Szekeres diagrams to Penrose diagrams, and goes into much detail the facts and fiction concerning wormholes. Plenty of easy to understand illustrations. A less involved, but still very informative book is his <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKaufmann1979" class="citation book cs1">Kaufmann, William J. (1979). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/blackholeswarped00kauf"><i>Black holes and warped spacetime</i></a></span>. San Francisco: W. H. Freeman. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7167-1152-0" title="Special:BookSources/978-0-7167-1152-0"><bdi>978-0-7167-1152-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Black+holes+and+warped+spacetime&rft.place=San+Francisco&rft.pub=W.+H.+Freeman&rft.date=1979&rft.isbn=978-0-7167-1152-0&rft.aulast=Kaufmann&rft.aufirst=William+J.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fblackholeswarped00kauf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APenrose+diagram" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Penrose_diagram&action=edit&section=5" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://astro-gr.org/conformal-diagrams/">Conformal diagrams</a> – Introduction to conformal diagrams, series of minilectures by Pau Amaro Seoane</li> <li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Commons-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span> Media related to <a href="https://commons.wikimedia.org/wiki/Category:Penrose_diagrams" class="extiw" title="commons:Category:Penrose diagrams">Penrose diagrams</a> at Wikimedia Commons</li></ul> <div class="navbox-styles"><style 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href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Relativity" title="Template:Relativity"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Relativity" title="Template talk:Relativity"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Relativity" title="Special:EditPage/Template:Relativity"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Relativity" style="font-size:114%;margin:0 4em"><a href="/wiki/Theory_of_relativity" title="Theory of relativity">Relativity</a></div></th></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="/wiki/Special_relativity" title="Special relativity">Special<br />relativity</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Background</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Principle_of_relativity" title="Principle of relativity">Principle of relativity</a> (<a href="/wiki/Galilean_invariance" title="Galilean invariance">Galilean relativity</a></li> <li><a href="/wiki/Galilean_transformation" title="Galilean transformation">Galilean transformation</a>)</li> <li><a href="/wiki/Special_relativity" title="Special relativity">Special relativity</a></li> <li><a href="/wiki/Doubly_special_relativity" title="Doubly special relativity">Doubly special relativity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Fundamental<br />concepts</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Frame_of_reference" title="Frame of reference">Frame of reference</a></li> <li><a href="/wiki/Speed_of_light" title="Speed of light">Speed of light</a></li> <li><a href="/wiki/Hyperbolic_orthogonality" title="Hyperbolic orthogonality">Hyperbolic orthogonality</a></li> <li><a href="/wiki/Rapidity" title="Rapidity">Rapidity</a></li> <li><a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell's equations</a></li> <li><a href="/wiki/Proper_length" title="Proper length">Proper length</a></li> <li><a href="/wiki/Proper_time" title="Proper time">Proper time</a></li> <li><a href="/wiki/Proper_acceleration" title="Proper acceleration">Proper acceleration</a></li> <li><a href="/wiki/Mass_in_special_relativity" title="Mass in special relativity">Relativistic mass</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Formulation</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Lorentz_transformation" title="Lorentz transformation">Lorentz transformation</a></li> <li><a href="/wiki/List_of_textbooks_on_relativity" title="List of textbooks on relativity">Textbooks</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Phenomena</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Time_dilation" title="Time dilation">Time dilation</a></li> <li><a href="/wiki/Mass%E2%80%93energy_equivalence" title="Mass–energy equivalence">Mass–energy equivalence (E=mc<sup>2</sup>)</a></li> <li><a href="/wiki/Length_contraction" title="Length contraction">Length contraction</a></li> <li><a href="/wiki/Relativity_of_simultaneity" title="Relativity of simultaneity">Relativity of simultaneity</a></li> <li><a href="/wiki/Relativistic_Doppler_effect" title="Relativistic Doppler effect">Relativistic Doppler effect</a></li> <li><a href="/wiki/Thomas_precession" title="Thomas precession">Thomas precession</a></li> <li><a href="/wiki/Ladder_paradox" title="Ladder paradox">Ladder paradox</a></li> <li><a href="/wiki/Twin_paradox" title="Twin paradox">Twin paradox</a></li> <li><a href="/wiki/Terrell_rotation" title="Terrell rotation">Terrell rotation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;"><a href="/wiki/Spacetime" title="Spacetime">Spacetime</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Light_cone" title="Light cone">Light cone</a></li> <li><a href="/wiki/World_line" title="World line">World line</a></li> <li><a href="/wiki/Minkowski_diagram" class="mw-redirect" title="Minkowski diagram">Minkowski diagram</a></li> <li><a href="/wiki/Biquaternion" title="Biquaternion">Biquaternions</a></li> <li><a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski space</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="/wiki/General_relativity" title="General relativity">General<br />relativity</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Background</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Introduction_to_general_relativity" title="Introduction to general relativity">Introduction</a></li> <li><a href="/wiki/Mathematics_of_general_relativity" title="Mathematics of general relativity">Mathematical formulation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Fundamental<br />concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Equivalence_principle" title="Equivalence principle">Equivalence principle</a></li> <li><a href="/wiki/Riemannian_geometry" title="Riemannian geometry">Riemannian geometry</a></li> <li><a class="mw-selflink selflink">Penrose diagram</a></li> <li><a href="/wiki/Geodesics_in_general_relativity" title="Geodesics in general relativity">Geodesics</a></li> <li><a href="/wiki/Mach%27s_principle" title="Mach's principle">Mach's principle</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Formulation</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/ADM_formalism" title="ADM formalism">ADM formalism</a></li> <li><a href="/wiki/BSSN_formalism" title="BSSN formalism">BSSN formalism</a></li> <li><a href="/wiki/Einstein_field_equations" title="Einstein field equations">Einstein field equations</a></li> <li><a href="/wiki/Linearized_gravity" title="Linearized gravity">Linearized gravity</a></li> <li><a href="/wiki/Parameterized_post-Newtonian_formalism" title="Parameterized post-Newtonian formalism">Post-Newtonian formalism</a></li> <li><a href="/wiki/Raychaudhuri_equation" title="Raychaudhuri equation">Raychaudhuri equation</a></li> <li><a href="/wiki/Hamilton%E2%80%93Jacobi%E2%80%93Einstein_equation" title="Hamilton–Jacobi–Einstein equation">Hamilton–Jacobi–Einstein equation</a></li> <li><a href="/wiki/Ernst_equation" title="Ernst equation">Ernst equation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Phenomena</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Black_hole" title="Black hole">Black hole</a></li> <li><a href="/wiki/Event_horizon" title="Event horizon">Event horizon</a></li> <li><a href="/wiki/Gravitational_singularity" title="Gravitational singularity">Singularity</a></li> <li><a href="/wiki/Two-body_problem_in_general_relativity" title="Two-body problem in general relativity">Two-body problem</a></li></ul> <ul><li><a href="/wiki/Gravitational_wave" title="Gravitational wave">Gravitational waves</a>: <a href="/wiki/Gravitational-wave_astronomy" title="Gravitational-wave astronomy">astronomy</a></li> <li><a href="/wiki/Gravitational-wave_observatory" title="Gravitational-wave observatory">detectors</a> (<a href="/wiki/LIGO" title="LIGO">LIGO</a> and <a href="/wiki/LIGO_Scientific_Collaboration" title="LIGO Scientific Collaboration">collaboration</a></li> <li><a href="/wiki/Virgo_interferometer" title="Virgo interferometer">Virgo</a></li> <li><a href="/wiki/LISA_Pathfinder" title="LISA Pathfinder">LISA Pathfinder</a></li> <li><a href="/wiki/GEO600" title="GEO600">GEO</a>)</li> <li><a href="/wiki/Hulse%E2%80%93Taylor_binary" class="mw-redirect" title="Hulse–Taylor binary">Hulse–Taylor binary</a></li></ul> <ul><li><a href="/wiki/Tests_of_general_relativity" title="Tests of general relativity">Other tests</a>: <a href="/wiki/Apsidal_precession" title="Apsidal precession">precession</a> of Mercury</li> <li><a href="/wiki/Gravitational_lens" title="Gravitational lens">lensing</a> (together with <a href="/wiki/Einstein_cross" class="mw-redirect" title="Einstein cross">Einstein cross</a> and <a href="/wiki/Einstein_rings" class="mw-redirect" title="Einstein rings">Einstein rings</a>)</li> <li><a href="/wiki/Gravitational_redshift" title="Gravitational redshift">redshift</a></li> <li><a href="/wiki/Shapiro_time_delay" title="Shapiro time delay">Shapiro delay</a></li> <li><a href="/wiki/Frame-dragging" title="Frame-dragging">frame-dragging</a> / <a href="/wiki/Geodetic_effect" title="Geodetic effect">geodetic effect</a> (<a href="/wiki/Lense%E2%80%93Thirring_precession" title="Lense–Thirring precession">Lense–Thirring precession</a>)</li> <li><a href="/wiki/Pulsar_timing_array" title="Pulsar timing array">pulsar timing arrays</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Advanced<br />theories</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Brans%E2%80%93Dicke_theory" title="Brans–Dicke theory">Brans–Dicke theory</a></li> <li><a href="/wiki/Kaluza%E2%80%93Klein_theory" title="Kaluza–Klein theory">Kaluza–Klein</a></li> <li><a href="/wiki/Quantum_gravity" title="Quantum gravity">Quantum gravity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;"><a href="/wiki/Exact_solutions_in_general_relativity" title="Exact solutions in general relativity">Solutions</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li>Cosmological: <a href="/wiki/Friedmann%E2%80%93Lema%C3%AEtre%E2%80%93Robertson%E2%80%93Walker_metric" title="Friedmann–Lemaître–Robertson–Walker metric">Friedmann–Lemaître–Robertson–Walker</a> (<a href="/wiki/Friedmann_equations" title="Friedmann equations">Friedmann equations</a>)</li> <li><a href="/wiki/Lema%C3%AEtre%E2%80%93Tolman_metric" title="Lemaître–Tolman metric">Lemaître–Tolman</a></li> <li><a href="/wiki/Kasner_metric" title="Kasner metric">Kasner</a></li> <li><a href="/wiki/BKL_singularity" title="BKL singularity">BKL singularity</a></li> <li><a href="/wiki/G%C3%B6del_metric" title="Gödel metric">Gödel</a></li> <li><a href="/wiki/Milne_model" title="Milne model">Milne</a></li></ul> <ul><li>Spherical: <a href="/wiki/Schwarzschild_metric" title="Schwarzschild metric">Schwarzschild</a> (<a href="/wiki/Interior_Schwarzschild_metric" title="Interior Schwarzschild metric">interior</a></li> <li><a href="/wiki/Tolman%E2%80%93Oppenheimer%E2%80%93Volkoff_equation" title="Tolman–Oppenheimer–Volkoff equation">Tolman–Oppenheimer–Volkoff equation</a>)</li> <li><a href="/wiki/Reissner%E2%80%93Nordstr%C3%B6m_metric" title="Reissner–Nordström metric">Reissner–Nordström</a></li></ul> <ul><li>Axisymmetric: <a href="/wiki/Kerr_metric" title="Kerr metric">Kerr</a> (<a href="/wiki/Kerr%E2%80%93Newman_metric" title="Kerr–Newman metric">Kerr–Newman</a>)</li> <li><a href="/wiki/Weyl%E2%88%92Lewis%E2%88%92Papapetrou_coordinates" class="mw-redirect" title="Weyl−Lewis−Papapetrou coordinates">Weyl−Lewis−Papapetrou</a></li> <li><a href="/wiki/Taub%E2%80%93NUT_space" title="Taub–NUT space">Taub–NUT</a></li> <li><a href="/wiki/Van_Stockum_dust" title="Van Stockum dust">van Stockum dust</a></li> <li><a href="/wiki/Relativistic_disk" title="Relativistic disk">discs</a></li></ul> <ul><li>Others: <a href="/wiki/Pp-wave_spacetime" title="Pp-wave spacetime">pp-wave</a></li> <li><a href="/wiki/Ozsv%C3%A1th%E2%80%93Sch%C3%BCcking_metric" title="Ozsváth–Schücking metric">Ozsváth–Schücking</a></li> <li><a href="/wiki/Alcubierre_drive" title="Alcubierre drive">Alcubierre</a></li></ul> <ul><li>In computational physics: <a href="/wiki/Numerical_relativity" title="Numerical relativity">Numerical relativity</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Scientists</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Poincaré</a></li> <li><a href="/wiki/Hendrik_Lorentz" title="Hendrik Lorentz">Lorentz</a></li> <li><a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein</a></li> <li><a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert</a></li> <li><a href="/wiki/Karl_Schwarzschild" title="Karl Schwarzschild">Schwarzschild</a></li> <li><a href="/wiki/Willem_de_Sitter" title="Willem de Sitter">de Sitter</a></li> <li><a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Weyl</a></li> <li><a href="/wiki/Arthur_Eddington" title="Arthur Eddington">Eddington</a></li> <li><a href="/wiki/Alexander_Friedmann" title="Alexander Friedmann">Friedmann</a></li> <li><a href="/wiki/Georges_Lema%C3%AEtre" title="Georges Lemaître">Lemaître</a></li> <li><a href="/wiki/Edward_Arthur_Milne" title="Edward Arthur Milne">Milne</a></li> <li><a href="/wiki/Howard_P._Robertson" title="Howard P. Robertson">Robertson</a></li> <li><a href="/wiki/Subrahmanyan_Chandrasekhar" title="Subrahmanyan Chandrasekhar">Chandrasekhar</a></li> <li><a href="/wiki/Fritz_Zwicky" title="Fritz Zwicky">Zwicky</a></li> <li><a href="/wiki/John_Archibald_Wheeler" title="John Archibald Wheeler">Wheeler</a></li> <li><a href="/wiki/Yvonne_Choquet-Bruhat" title="Yvonne Choquet-Bruhat">Choquet-Bruhat</a></li> <li><a href="/wiki/Roy_Kerr" title="Roy Kerr">Kerr</a></li> <li><a href="/wiki/Yakov_Zeldovich" title="Yakov Zeldovich">Zel'dovich</a></li> <li><a href="/wiki/Igor_Dmitriyevich_Novikov" title="Igor Dmitriyevich Novikov">Novikov</a></li> <li><a href="/wiki/J%C3%BCrgen_Ehlers" title="Jürgen Ehlers">Ehlers</a></li> <li><a href="/wiki/Robert_Geroch" title="Robert Geroch">Geroch</a></li> <li><a href="/wiki/Roger_Penrose" title="Roger Penrose">Penrose</a></li> <li><a href="/wiki/Stephen_Hawking" title="Stephen Hawking">Hawking</a></li> <li><a href="/wiki/Joseph_Hooton_Taylor_Jr." title="Joseph Hooton Taylor Jr.">Taylor</a></li> <li><a href="/wiki/Russell_Alan_Hulse" title="Russell Alan Hulse">Hulse</a></li> <li><a href="/wiki/Hermann_Bondi" title="Hermann Bondi">Bondi</a></li> <li><a href="/wiki/Charles_W._Misner" title="Charles W. Misner">Misner</a></li> <li><a href="/wiki/Shing-Tung_Yau" title="Shing-Tung Yau">Yau</a></li> <li><a href="/wiki/Kip_Thorne" title="Kip Thorne">Thorne</a></li> <li><a href="/wiki/Rainer_Weiss" title="Rainer Weiss">Weiss</a></li> <li><a href="/wiki/List_of_contributors_to_general_relativity" title="List of contributors to general relativity"><i>others</i></a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2" style="text-align:center;"><div><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Theory_of_relativity" title="Category:Theory of relativity">Category</a></div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Roger_Penrose" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Roger_Penrose" title="Template:Roger Penrose"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Roger_Penrose" title="Template talk:Roger Penrose"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Roger_Penrose" title="Special:EditPage/Template:Roger Penrose"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Roger_Penrose" style="font-size:114%;margin:0 4em"><a href="/wiki/Roger_Penrose" title="Roger Penrose">Roger Penrose</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Books</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><i><a href="/wiki/The_Emperor%27s_New_Mind" title="The Emperor's New Mind">The Emperor's New Mind</a></i> (1989)</li> <li><i><a href="/wiki/Shadows_of_the_Mind" title="Shadows of the Mind">Shadows of the Mind</a></i> (1994)</li> <li><i><a href="/wiki/The_Road_to_Reality" title="The Road to Reality">The Road to Reality</a></i> (2004)</li> <li><i><a href="/wiki/Cycles_of_Time" title="Cycles of Time">Cycles of Time</a></i> (2010)</li> <li><i><a href="/wiki/Fashion,_Faith,_and_Fantasy_in_the_New_Physics_of_the_Universe" title="Fashion, Faith, and Fantasy in the New Physics of the Universe">Fashion, Faith, and Fantasy in the New Physics of the Universe</a></i> (2016)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Coauthored books</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><span class="wrap"> <i><a href="/wiki/The_Nature_of_Space_and_Time" title="The Nature of Space and Time">The Nature of Space and Time</a></i> (with <a href="/wiki/Stephen_Hawking" title="Stephen Hawking">Stephen Hawking</a>) (1996) </span></li> <li><span class="wrap"> <i><a href="/wiki/The_Large,_the_Small_and_the_Human_Mind" title="The Large, the Small and the Human Mind">The Large, the Small and the Human Mind</a></i> (with <a href="/wiki/Abner_Shimony" title="Abner Shimony">Abner Shimony</a>, <a href="/wiki/Nancy_Cartwright_(philosopher)" title="Nancy Cartwright (philosopher)">Nancy Cartwright</a> and Stephen Hawking) (1997) </span></li> <li><span class="wrap"> <i>White Mars or, The Mind Set Free</i> (with <a href="/wiki/Brian_W._Aldiss" class="mw-redirect" title="Brian W. Aldiss">Brian W. Aldiss</a>) (1999) </span></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Concepts</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Twistor_theory" title="Twistor theory">Twistor theory</a></li> <li><a href="/wiki/Spin_network" title="Spin network">Spin network</a></li> <li><a href="/wiki/Abstract_index_notation" title="Abstract index notation">Abstract index notation</a></li> <li><a href="/wiki/Black_hole_bomb" title="Black hole bomb">Black hole bomb</a></li> <li><a href="/wiki/Spacetime" title="Spacetime">Geometry of spacetime</a></li> <li><a href="/wiki/Cosmic_censorship_hypothesis" title="Cosmic censorship hypothesis">Cosmic censorship</a></li> <li><a href="/wiki/Weyl_curvature_hypothesis" title="Weyl curvature hypothesis">Weyl curvature hypothesis</a></li> <li><a href="/wiki/Contributors_to_general_relativity#P" class="mw-redirect" title="Contributors to general relativity">Penrose inequalities</a></li> <li><a href="/wiki/Penrose_interpretation" title="Penrose interpretation">Penrose interpretation of quantum mechanics</a></li> <li><a href="/wiki/Moore%E2%80%93Penrose_inverse" title="Moore–Penrose inverse">Moore–Penrose inverse</a></li> <li><a href="/wiki/Newman%E2%80%93Penrose_formalism" title="Newman–Penrose formalism">Newman–Penrose formalism</a></li> <li><a class="mw-selflink selflink">Penrose diagram</a></li> <li><a href="/wiki/Penrose%E2%80%93Hawking_singularity_theorems" title="Penrose–Hawking singularity theorems">Penrose–Hawking singularity theorems</a></li> <li><a href="/wiki/Riemannian_Penrose_inequality" title="Riemannian Penrose inequality">Riemannian Penrose inequality</a></li> <li><a href="/wiki/Penrose_process" title="Penrose process">Penrose process</a></li> <li><a href="/wiki/Penrose_tiling" title="Penrose tiling">Penrose tiling</a></li> <li><a href="/wiki/Penrose_triangle" title="Penrose triangle">Penrose triangle</a></li> <li><a href="/wiki/Penrose_stairs" title="Penrose stairs">Penrose stairs</a></li> <li><a href="/wiki/Penrose_graphical_notation" title="Penrose graphical notation">Penrose graphical notation</a></li> <li><a href="/wiki/Penrose_transform" title="Penrose transform">Penrose transform</a></li> <li><a href="/wiki/Terrell_rotation" title="Terrell rotation">Penrose–Terrell effect</a></li> <li><a href="/wiki/Orchestrated_objective_reduction" title="Orchestrated objective reduction">Orchestrated objective reduction</a>/<a href="/wiki/Penrose%E2%80%93Lucas_argument" title="Penrose–Lucas argument">Penrose–Lucas argument</a></li> <li><a href="/wiki/Free-orbit_experiment_with_laser_interferometry_X-rays" title="Free-orbit experiment with laser interferometry X-rays">FELIX experiment</a></li> <li><a href="/wiki/Trapped_surface" title="Trapped surface">Trapped surface</a></li> <li><a href="/wiki/Andromeda_paradox" class="mw-redirect" title="Andromeda paradox">Andromeda paradox</a></li> <li><a href="/wiki/Conformal_cyclic_cosmology" title="Conformal cyclic cosmology">Conformal cyclic cosmology</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Lionel_Penrose" title="Lionel Penrose">Lionel Penrose</a> (father)</li> <li><a href="/wiki/Oliver_Penrose" title="Oliver Penrose">Oliver Penrose</a> (brother)</li> <li><a href="/wiki/Jonathan_Penrose" title="Jonathan Penrose">Jonathan Penrose</a> (brother)</li> <li><a href="/wiki/Shirley_Hodgson" title="Shirley Hodgson">Shirley Hodgson</a> (sister)</li> <li><a href="/wiki/John_Beresford_Leathes" title="John Beresford Leathes">John Beresford Leathes</a> (grandfather)</li> <li><a href="/wiki/Illumination_problem" title="Illumination problem">Illumination problem</a></li> <li><a href="/wiki/Quantum_mind" title="Quantum mind">Quantum mind</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; 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