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<div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div 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"><p><b>Scalar theories of gravitation</b> are field theories of <a href="/wiki/Gravitation" class="mw-redirect" title="Gravitation">gravitation</a> in which the gravitational field is described using a <a href="/wiki/Scalar_field_theory" title="Scalar field theory">scalar field</a>, which is required to satisfy some field equation. </p><p><i>Note:</i> This article focuses on <a href="/wiki/Theory_of_relativity" title="Theory of relativity">relativistic</a> <a href="/wiki/Classical_field_theory" title="Classical field theory">classical field theories</a> of gravitation. The best known relativistic classical field theory of gravitation, <a href="/wiki/General_relativity" title="General relativity">general relativity</a>, is a tensor theory, in which the gravitational interaction is described using a <a href="/wiki/Tensor" title="Tensor">tensor</a> field. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Newtonian_gravity">Newtonian gravity</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Scalar_theories_of_gravitation&action=edit&section=1" title="Edit section: Newtonian gravity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The prototypical scalar theory of gravitation is <a href="/wiki/Newtonian_gravity" class="mw-redirect" title="Newtonian gravity">Newtonian gravitation</a>. In this theory, the gravitational interaction is completely described by the <a href="/wiki/Potential" title="Potential">potential</a> <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 \Phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aed80a2011a3912b028ba32a52dfa57165455f24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Phi }"></span>, which is required to satisfy the <a href="/wiki/Poisson_equation" class="mw-redirect" title="Poisson equation">Poisson equation</a> (with the mass density acting as the source of the field). To wit: </p><p><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 \Delta \Phi =4\pi G\rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi mathvariant="normal">Φ</mi> <mo>=</mo> <mn>4</mn> <mi>π</mi> <mi>G</mi> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta \Phi =4\pi G\rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6f0ad867710dbbd1b54433908e60b9ae90e13fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.236ex; height:2.676ex;" alt="{\displaystyle \Delta \Phi =4\pi G\rho }"></span>, where </p> <ul><li><i>G</i> is the gravitational constant and</li> <li><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 \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"></span> is the mass density.</li></ul> <p>This field theory formulation leads directly to the familiar law of universal gravitation, <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 F=m_{1}m_{2}G/r^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F=m_{1}m_{2}G/r^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe86a7f33c6258d6dacbd5e0a218481d1bf886b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.121ex; height:3.176ex;" alt="{\displaystyle F=m_{1}m_{2}G/r^{2}}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Nordström's_theories_of_gravitation"><span id="Nordstr.C3.B6m.27s_theories_of_gravitation"></span>Nordström's theories of gravitation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Scalar_theories_of_gravitation&action=edit&section=2" title="Edit section: Nordström's theories of gravitation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The first attempts to present a relativistic (classical) field theory of gravitation were also scalar theories. <a href="/wiki/Gunnar_Nordstr%C3%B6m" title="Gunnar Nordström">Gunnar Nordström</a> created two such theories.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Nordström's first idea (1912) was to simply replace the divergence operator in the field equation of Newtonian gravity with the <a href="/wiki/D%27Alembert_operator" title="D'Alembert operator">d'Alembertian</a> operator <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 \square =\partial _{t}^{2}-\nabla ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>◻</mi> <mo>=</mo> <msubsup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \square =\partial _{t}^{2}-\nabla ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/260a380f9823aed5f7aba4f6469b241b631447a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.135ex; height:3.343ex;" alt="{\displaystyle \square =\partial _{t}^{2}-\nabla ^{2}}"></span>. This gives the field equation </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 \square \Phi =-4\pi G\rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>◻</mi> <mi mathvariant="normal">Φ</mi> <mo>=</mo> <mo>−</mo> <mn>4</mn> <mi>π</mi> <mi>G</mi> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \square \Phi =-4\pi G\rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/810a6958f527a5018cd6a2f1a1c47caade8d0779" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.916ex; height:2.676ex;" alt="{\displaystyle \square \Phi =-4\pi G\rho }"></span>.</dd></dl> <p>However, several theoretical difficulties with this theory quickly arose, and Nordström dropped it. </p><p>A year later, Nordström tried again, presenting the field equation </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 \Phi \square \Phi =-4\pi GT}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> <mi>◻</mi> <mi mathvariant="normal">Φ</mi> <mo>=</mo> <mo>−</mo> <mn>4</mn> <mi>π</mi> <mi>G</mi> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi \square \Phi =-4\pi GT}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fafae6344e8014a896711a732ca9953b4685480" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.028ex; height:2.343ex;" alt="{\displaystyle \Phi \square \Phi =-4\pi GT}"></span>,</dd></dl> <p>where <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 T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> is the trace of the <a href="/wiki/Stress%E2%80%93energy_tensor" title="Stress–energy tensor">stress–energy tensor</a>. </p><p>Solutions of Nordström's second theory are <a href="/wiki/Conformally_flat" class="mw-redirect" title="Conformally flat">conformally flat</a> Lorentzian spacetimes. That is, the metric tensor can be written as <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 g_{\mu \nu }=A\eta _{\mu \nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>=</mo> <mi>A</mi> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{\mu \nu }=A\eta _{\mu \nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae523e90b50e6e5ed5ee92713563e7e51578e6e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.295ex; height:2.843ex;" alt="{\displaystyle g_{\mu \nu }=A\eta _{\mu \nu }}"></span>, where </p> <ul><li>η<sub>μν</sub> is the <a href="/wiki/Minkowski_metric" class="mw-redirect" title="Minkowski metric">Minkowski metric</a>, and</li> <li><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 A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> is a scalar which is a function of position.</li></ul> <p>This suggestion signifies that the inertial mass should depend on the scalar field. </p><p>Nordström's second theory satisfies the weak <a href="/wiki/Equivalence_principle" title="Equivalence principle">equivalence principle</a>. However: </p> <ul><li>The theory fails to predict any deflection of light passing near a massive body (contrary to observation)</li> <li>The theory predicts an anomalous <a href="/wiki/Perihelion" class="mw-redirect" title="Perihelion">perihelion</a> <a href="/wiki/Precession" title="Precession">precession</a> of <a href="/wiki/Mercury_(planet)" title="Mercury (planet)">Mercury</a>, but this disagrees in both sign and magnitude with the observed anomalous precession (the part which cannot be explained using Newtonian gravitation).</li></ul> <p>Despite these disappointing results, Einstein's critiques of Nordström's second theory played an important role in his development of general relativity. </p> <div class="mw-heading mw-heading2"><h2 id="Einstein's_scalar_theory"><span id="Einstein.27s_scalar_theory"></span>Einstein's scalar theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Scalar_theories_of_gravitation&action=edit&section=3" title="Edit section: Einstein's scalar theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In 1913, Einstein (erroneously) concluded from his <a href="/wiki/Hole_argument" title="Hole argument">hole argument</a> that general covariance was not viable.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Inspired by Nordström's work, he proposed his own scalar theory.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> This theory employs a massless scalar field coupled to the stress–energy tensor, which is the sum of two terms. The first, </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 T_{g}^{\mu \nu }={\frac {1}{4\pi G}}\left[\partial ^{\mu }\phi \,\partial ^{\nu }\phi \,-{\frac {1}{2}}\eta ^{\mu \nu }\partial _{\lambda }\phi \,\partial ^{\lambda }\phi \right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>π</mi> <mi>G</mi> </mrow> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mi>ϕ</mi> <mspace width="thinmathspace" /> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> <mi>ϕ</mi> <mspace width="thinmathspace" /> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msub> <mi>ϕ</mi> <mspace width="thinmathspace" /> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msup> <mi>ϕ</mi> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{g}^{\mu \nu }={\frac {1}{4\pi G}}\left[\partial ^{\mu }\phi \,\partial ^{\nu }\phi \,-{\frac {1}{2}}\eta ^{\mu \nu }\partial _{\lambda }\phi \,\partial ^{\lambda }\phi \right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d80ac96f3361df0db1a23dad103202c393cac63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:39.695ex; height:6.176ex;" alt="{\displaystyle T_{g}^{\mu \nu }={\frac {1}{4\pi G}}\left[\partial ^{\mu }\phi \,\partial ^{\nu }\phi \,-{\frac {1}{2}}\eta ^{\mu \nu }\partial _{\lambda }\phi \,\partial ^{\lambda }\phi \right]}"></span></dd></dl> <p>represents the stress–momentum–energy of the scalar field itself. The second represents the stress-momentum-energy of any matter which may be present: </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 T_{m}^{\mu \nu }=\rho \phi u^{\mu }u^{\nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msubsup> <mo>=</mo> <mi>ρ</mi> <mi>ϕ</mi> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{m}^{\mu \nu }=\rho \phi u^{\mu }u^{\nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/757de6948d249a80366a0916c8462bd71640adc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.487ex; height:3.009ex;" alt="{\displaystyle T_{m}^{\mu \nu }=\rho \phi u^{\mu }u^{\nu }}"></span></dd></dl> <p>where <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^{\mu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u^{\mu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/598b309f281119e95cd083756f650469e1db1de6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.553ex; height:2.343ex;" alt="{\displaystyle u^{\mu }}"></span> is the <a href="/wiki/Four-velocity" title="Four-velocity">velocity</a> vector of an observer, or <a href="/wiki/Tangent_vector" title="Tangent vector">tangent vector</a> to the world line of the observer. (Einstein made no attempt, in this theory, to take account of possible gravitational effects of the field energy of the <a href="/wiki/Electromagnetic_field" title="Electromagnetic field">electromagnetic field</a>.) </p><p>Unfortunately, this theory is not <a href="/wiki/Diffeomorphism" title="Diffeomorphism">diffeomorphism</a> covariant. This is an important consistency condition, so Einstein dropped this theory in late 1914.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Associating the scalar field with the metric leads to Einstein's later conclusions that the theory of gravitation he sought could not be a scalar theory. Indeed, the theory he finally arrived at in 1915, <a href="/wiki/General_relativity" title="General relativity">general relativity</a>, is a tensor theory, not a scalar theory, with a 2-tensor, the metric, as the potential. Unlike his 1913 scalar theory, it is <a href="/wiki/Generally_covariant" class="mw-redirect" title="Generally covariant">generally covariant</a>, and it does take into account the field energy–momentum–stress of the electromagnetic field (or any other nongravitational field). </p> <div class="mw-heading mw-heading2"><h2 id="Additional_variations">Additional variations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Scalar_theories_of_gravitation&action=edit&section=4" title="Edit section: Additional variations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Kaluza%E2%80%93Klein_theory" title="Kaluza–Klein theory">Kaluza–Klein theory</a> involves the use of a scalar gravitational field in addition to the <a href="/wiki/Electromagnetic_field" title="Electromagnetic field">electromagnetic field</a> potential <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 A^{\mu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{\mu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1da55ba40017c25d210f7e269efb2d6d539a1b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.967ex; height:2.343ex;" alt="{\displaystyle A^{\mu }}"></span> in an attempt to create a five-dimensional unification of gravity and electromagnetism. Its generalization with a 5th variable component of the metric that leads to a variable gravitational constant was first given by <a href="/wiki/Pascual_Jordan" title="Pascual Jordan">Pascual Jordan</a>.<sup id="cite_ref-Brans2005_5-0" class="reference"><a href="#cite_note-Brans2005-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Brans%E2%80%93Dicke_theory" title="Brans–Dicke theory">Brans–Dicke theory</a> is a scalar-tensor theory, not a scalar theory, meaning that it represents the gravitational interaction using both a scalar field and a tensor field. We mention it here because one of the field equations of this theory involves only the scalar field and the trace of the stress–energy tensor, as in Nordström's theory. Moreover, the Brans–Dicke theory is equal to the independently derived theory of Jordan (hence it is often referred to as the Jordan-Brans–Dicke or JBD theory). The Brans–Dicke theory couples a scalar field with the curvature of space-time and is self-consistent and, assuming appropriate values for a tunable constant, this theory has not been ruled out by observation. The Brans–Dicke theory is generally regarded as a leading competitor of general relativity, which is a pure tensor theory. However, the Brans–Dicke theory seems to need too high a parameter, which favours general relativity).<sup id="cite_ref-Brans2005_5-1" class="reference"><a href="#cite_note-Brans2005-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></li> <li>Zee combined the idea of the BD theory with the Higgs-Mechanism of Symmetry Breakdown for mass generation, which led to a scalar-tensor theory with Higgs field as scalar field, in which the scalar field is massive (short-ranged). An example of this theory was proposed by H. Dehnen and H. Frommert 1991, parting from the nature of Higgs field interacting gravitational- and Yukawa (long-ranged)-like with the particles that get mass through it.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup></li> <li>The <a href="/w/index.php?title=Watt%E2%80%93Misner_theory_of_gravitation&action=edit&redlink=1" class="new" title="Watt–Misner theory of gravitation (page does not exist)">Watt–Misner theory</a> (1999) is a recent example of a scalar theory of gravitation. It is not intended as a viable theory of gravitation (since, as Watt and Misner point out, it is not consistent with observation), but as a toy theory which can be useful in testing numerical relativity schemes. It also has pedagogical value.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup></li></ul> <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=Scalar_theories_of_gravitation&action=edit&section=5" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Nordstr%C3%B6m%27s_theory_of_gravitation" title="Nordström's theory of gravitation">Nordström's theory of gravitation</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=Scalar_theories_of_gravitation&action=edit&section=6" 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-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</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="CITEREFNorton1992" class="citation journal cs1">Norton, John D. 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M., "On the History of Unified Field Theories"; <i>Living Rev. Relativ. <b>7</b>(2)</i>, 2004, <a rel="nofollow" class="external text" href="http://www.livingreviews.org/lrr-2004-2">lrr-2004-2</a>. Retrieved August 10, 2005.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRavndal,_Finn2004" class="citation arxiv cs1"><a href="/wiki/Finn_Ravndal" title="Finn Ravndal">Ravndal, Finn</a> (2004). "Scalar Gravitation and Extra Dimensions". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/gr-qc/0405030">gr-qc/0405030</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Scalar+Gravitation+and+Extra+Dimensions&rft.date=2004&rft_id=info%3Aarxiv%2Fgr-qc%2F0405030&rft.au=Ravndal%2C+Finn&rfr_id=info%3Asid%2Fen.wikipedia.org%3AScalar+theories+of+gravitation" class="Z3988"></span></li> <li>P. 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gravitation</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Standard</th><td class="navbox-list-with-group navbox-list navbox-odd" 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:1%;font-weight:normal;">Newtonian gravity (NG)</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Newton%27s_law_of_universal_gravitation" title="Newton's law of universal gravitation">Newton's law of universal gravitation</a></li> <li><a href="/wiki/Gauss%27s_law_for_gravity" title="Gauss's law for gravity">Gauss's law for gravity</a></li> <li><a href="/wiki/Poisson%27s_equation#Newtonian_gravity" title="Poisson's equation">Poisson's equation for gravity</a></li> <li><a href="/wiki/History_of_gravitational_theory" title="History of gravitational theory">History of gravitational theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/General_relativity" title="General relativity">General relativity (GR)</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;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/History_of_general_relativity" title="History of general relativity">History</a></li> <li><a href="/wiki/Mathematics_of_general_relativity" title="Mathematics of general relativity">Mathematics</a></li> <li><a href="/wiki/Exact_solutions_in_general_relativity" title="Exact solutions in general relativity">Exact solutions</a></li> <li><a href="/wiki/General_relativity#Further_reading" title="General relativity">Resources</a></li> <li><a href="/wiki/Tests_of_general_relativity" title="Tests of general relativity">Tests</a></li> <li><a href="/wiki/Parameterized_post-Newtonian_formalism" title="Parameterized post-Newtonian formalism">Post-Newtonian formalism</a></li> <li><a href="/wiki/Linearized_gravity" title="Linearized gravity">Linearized gravity</a></li> <li><a href="/wiki/ADM_formalism" title="ADM formalism">ADM formalism</a></li> <li><a href="/wiki/Gibbons%E2%80%93Hawking%E2%80%93York_boundary_term" title="Gibbons–Hawking–York boundary term">Gibbons–Hawking–York boundary term</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;line-height:1.2em;"><a href="/wiki/Alternatives_to_general_relativity" title="Alternatives to general relativity">Alternatives to<br />general relativity</a></th><td class="navbox-list-with-group navbox-list navbox-odd" 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:1%;font-weight:normal;">Paradigms</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alternatives_to_general_relativity" title="Alternatives to general relativity">Classical theories of gravitation</a></li> <li><a href="/wiki/Quantum_gravity" title="Quantum gravity">Quantum gravity</a></li> <li><a href="/wiki/Theory_of_everything" title="Theory of everything">Theory of everything</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Classical</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/w/index.php?title=Poincar%C3%A9_gauge_theory&action=edit&redlink=1" class="new" title="Poincaré gauge theory (page does not exist)">Poincaré gauge theory</a> <ul><li><span style="font-size:85%;"><a href="/wiki/Einstein%E2%80%93Cartan_theory" title="Einstein–Cartan theory">Einstein–Cartan</a></span></li> <li><span style="font-size:85%;"><a href="/wiki/Teleparallelism" title="Teleparallelism">Teleparallelism</a></span></li></ul></li> <li><a href="/wiki/Bimetric_gravity" title="Bimetric gravity">Bimetric theories</a></li> <li><a href="/wiki/Gauge_theory_gravity" title="Gauge theory gravity">Gauge theory gravity</a></li> <li><a href="/wiki/Composite_gravity" title="Composite gravity">Composite gravity</a></li> <li><a href="/wiki/F(R)_gravity" title="F(R) gravity"><i>f</i>(<i>R</i>) gravity</a></li> <li><a href="/wiki/Infinite_derivative_gravity" title="Infinite derivative gravity">Infinite derivative gravity</a></li> <li><a href="/wiki/Massive_gravity" title="Massive gravity">Massive gravity</a></li> <li><a href="/wiki/Modified_Newtonian_dynamics" title="Modified Newtonian dynamics">Modified Newtonian dynamics, MOND</a> <ul><li><span style="font-size:85%;"><a href="/wiki/AQUAL" title="AQUAL">AQUAL</a></span></li> <li><span style="font-size:85%;"><a href="/wiki/Tensor%E2%80%93vector%E2%80%93scalar_gravity" title="Tensor–vector–scalar gravity">Tensor–vector–scalar</a></span></li></ul></li> <li><a href="/wiki/Nonsymmetric_gravitational_theory" title="Nonsymmetric gravitational theory">Nonsymmetric gravitation</a></li> <li><a href="/wiki/Scalar%E2%80%93tensor_theory" title="Scalar–tensor theory">Scalar–tensor theories</a> <ul><li><span style="font-size:85%;"><a href="/wiki/Brans%E2%80%93Dicke_theory" title="Brans–Dicke theory">Brans–Dicke</a></span></li></ul></li> <li><a href="/wiki/Scalar%E2%80%93tensor%E2%80%93vector_gravity" title="Scalar–tensor–vector gravity">Scalar–tensor–vector</a></li> <li><a href="/wiki/Conformal_gravity" title="Conformal gravity">Conformal gravity</a></li> <li><a class="mw-selflink selflink">Scalar theories</a> <ul><li><span style="font-size:85%;"><a href="/wiki/Nordstr%C3%B6m%27s_theory_of_gravitation" title="Nordström's theory of gravitation">Nordström</a></span></li></ul></li> <li><a href="/wiki/Whitehead%27s_theory_of_gravitation" title="Whitehead's theory of gravitation">Whitehead</a></li> <li><a href="/wiki/Geometrodynamics" title="Geometrodynamics">Geometrodynamics</a></li> <li><a href="/wiki/Induced_gravity" title="Induced gravity">Induced gravity</a></li> <li><a href="/wiki/Degenerate_Higher-Order_Scalar-Tensor_theories" title="Degenerate Higher-Order Scalar-Tensor theories">Degenerate Higher-Order Scalar-Tensor theories</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Quantum-mechanical</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Euclidean_quantum_gravity" title="Euclidean quantum gravity">Euclidean quantum gravity</a></li> <li><a href="/wiki/Canonical_quantum_gravity" title="Canonical quantum gravity">Canonical quantum gravity</a> <ul><li><span style="font-size:85%;"><a href="/wiki/Wheeler%E2%80%93DeWitt_equation" title="Wheeler–DeWitt equation">Wheeler–DeWitt equation</a></span></li> <li><span style="font-size:85%;"><a href="/wiki/Loop_quantum_gravity" title="Loop quantum gravity">Loop quantum gravity</a></span></li> <li><span style="font-size:85%;"><a href="/wiki/Spin_foam" title="Spin foam">Spin foam</a></span></li></ul></li> <li><a href="/wiki/Causal_dynamical_triangulation" title="Causal dynamical triangulation">Causal dynamical triangulation</a></li> <li><a href="/wiki/Asymptotic_safety_in_quantum_gravity" title="Asymptotic safety in quantum gravity">Asymptotic safety in quantum gravity</a></li> <li><a href="/wiki/Causal_sets" title="Causal sets">Causal sets</a></li> <li><a href="/wiki/DGP_model" title="DGP model">DGP model</a></li> <li><a href="/wiki/Rainbow_gravity_theory" title="Rainbow gravity theory">Rainbow gravity theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Unified-field-theoric</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Kaluza%E2%80%93Klein_theory" title="Kaluza–Klein theory">Kaluza–Klein theory</a></li> <li><a href="/wiki/Supergravity" title="Supergravity">Supergravity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Unified-field-theoric and <br /> quantum-mechanical</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Noncommutative_quantum_field_theory" title="Noncommutative quantum field theory">Noncommutative geometry</a></li> <li><a href="/wiki/Semiclassical_gravity" title="Semiclassical gravity">Semiclassical gravity</a></li> <li><a href="/wiki/Superfluid_vacuum_theory" title="Superfluid vacuum theory">Superfluid vacuum theory</a> <ul><li><span style="font-size:85%;"><a href="/wiki/Superfluid_vacuum_theory#Logarithmic_BEC_vacuum_theory" title="Superfluid vacuum theory">Logarithmic BEC vacuum</a></span></li></ul></li> <li><a href="/wiki/String_theory" title="String theory">String theory</a> <ul><li><span style="font-size:85%;"><a href="/wiki/M-theory" title="M-theory">M-theory</a></span></li> <li><span style="font-size:85%;"><a href="/wiki/F-theory" title="F-theory">F-theory</a></span></li> <li><span style="font-size:85%;"><a href="/wiki/Heterotic_string_theory" title="Heterotic string theory">Heterotic string theory</a></span></li> <li><span style="font-size:85%;"><a href="/wiki/Type_I_string_theory" title="Type I string theory">Type I string theory</a></span></li> <li><span style="font-size:85%;"><a href="/wiki/Type_0_string_theory" title="Type 0 string theory">Type 0 string theory</a></span></li> <li><span style="font-size:85%;"><a href="/wiki/Bosonic_string_theory" title="Bosonic string theory">Bosonic string theory</a></span></li> <li><span style="font-size:85%;"><a href="/wiki/Type_II_string_theory" title="Type II string theory">Type II string theory</a></span></li> <li><span style="font-size:85%;"><a href="/wiki/Little_string_theory" title="Little string theory">Little string theory</a></span></li></ul></li> <li><a href="/wiki/Twistor_theory" title="Twistor theory">Twistor theory</a> <ul><li><span style="font-size:85%;"><a href="/wiki/Twistor_string_theory" title="Twistor string theory">Twistor string theory</a></span></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Generalisations / <br /> extensions of GR</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Liouville_gravity" class="mw-redirect" title="Liouville gravity">Liouville gravity</a></li> <li><a href="/wiki/Lovelock_theory_of_gravity" title="Lovelock theory of gravity">Lovelock theory</a></li> <li><a href="/wiki/(2%2B1)-dimensional_topological_gravity" title="(2+1)-dimensional topological gravity">(2+1)-dimensional topological gravity</a></li> <li><a href="/wiki/Gauss%E2%80%93Bonnet_gravity" title="Gauss–Bonnet gravity">Gauss–Bonnet gravity</a></li> <li><a href="/wiki/Jackiw%E2%80%93Teitelboim_gravity" title="Jackiw–Teitelboim gravity">Jackiw–Teitelboim gravity</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Pre-Newtonian<br />theories and<br /><a href="/wiki/Toy_model" title="Toy model">toy models</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Aristotelian_physics" title="Aristotelian physics">Aristotelian physics</a></li> <li><a href="/wiki/CGHS_model" title="CGHS model">CGHS model</a></li> <li><a href="/wiki/RST_model" title="RST model">RST model</a></li> <li><a href="/wiki/Mechanical_explanations_of_gravitation" title="Mechanical explanations of gravitation">Mechanical explanations</a> <ul><li><span style="font-size:85%;"><a href="/wiki/Le_Sage%27s_theory_of_gravitation" title="Le Sage's theory of gravitation">Fatio–Le Sage</a></span></li> <li><span style="font-size:85%;"><a href="/wiki/Entropic_gravity" title="Entropic gravity">Entropic gravity</a></span></li></ul></li> <li><a href="/wiki/Gravitational_interaction_of_antimatter" title="Gravitational interaction of antimatter">Gravitational interaction of antimatter</a></li> <li><a href="/wiki/Physics_in_the_medieval_Islamic_world" title="Physics in the medieval Islamic world">Physics in the medieval Islamic world</a></li> <li><a href="/wiki/Theory_of_impetus" title="Theory of impetus">Theory of impetus</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related topics</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Gravitational_wave" title="Gravitational wave">Gravitational wave</a></li> <li><a href="/wiki/Graviton" title="Graviton">Graviton</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; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Scalar_theories_of_gravitation&oldid=1230363165">https://en.wikipedia.org/w/index.php?title=Scalar_theories_of_gravitation&oldid=1230363165</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Category</a>: <ul><li><a href="/wiki/Category:Theories_of_gravity" title="Category:Theories of 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