View source for Einstein–Hilbert action

{{Short description|Concept in general relativity}}
{{General relativity sidebar}}
The '''Einstein–Hilbert action''' in [[general relativity]] is the [[action (physics)|action]] that yields the [[Einstein field equations]] through the [[stationary-action principle]]. With the [[Sign convention#Relativity|{{nowrap|(− + + +)}} metric signature]], the gravitational part of the action is given as<ref>{{cite book |first=Richard P. |last=Feynman |title=Feynman Lectures on Gravitation |url=https://archive.org/details/feynmanlectureso0000feyn_g4q1 |url-access=registration |publisher=Addison-Wesley |year=1995 |isbn=0-201-62734-5 |at=p. 136, eq. (10.1.2) }}</ref>

:<math>S = {1 \over 2\kappa} \int R \sqrt{-g} \, \mathrm{d}^4x,</math>

where <math>g=\det(g_{\mu\nu})</math> is the determinant of the [[metric tensor]] matrix, <math>R</math> is the [[Ricci scalar]], and <math>\kappa = 8\pi Gc^{-4}</math> is the [[Einstein gravitational constant]] (<math>G</math> is the [[gravitational constant]] and <math>c</math> is the [[speed of light]] in vacuum). If it converges, the integral is taken over the whole [[spacetime]]. If it does not converge, <math>S</math> is no longer well-defined, but a modified definition where one integrates over arbitrarily large, relatively compact domains, still yields the Einstein equation as the [[Euler–Lagrange equation]] of the Einstein–Hilbert action. The action was proposed<ref>{{Citation
|author-first=David
|author-last=Hilbert
|author-link =David Hilbert
|title = Die Grundlagen der Physik
|trans-title= Foundations of Physics
|journal = Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen – Mathematisch-Physikalische Klasse
|volume =3
|issue=
|pages =395–407
|year =1915
|language =German
|url =
|doi =
|jfm = 
}}</ref> by [[David Hilbert]] in 1915 as part of his application of the [[stationary action principle|variational principle]] to a combination of gravity and electromagnetism.<ref>{{Cite book |last=Mehra |first=Jagdish |chapter=Einstein, Hilbert, and the Theory of Gravitation |editor-last=Mehra |editor-first=Jagdish |title=The physicist's conception of nature |date=1987 |publisher=Reidel |isbn=978-90-277-2536-3 |edition=Reprint |location=Dordrecht}}</ref>{{rp|119}}

== Discussion ==
Deriving equations of motion from an action has several advantages. First, it allows for easy unification of general relativity with other classical field theories (such as [[Maxwell theory]]), which are also formulated in terms of an action. In the process, the derivation identifies a natural candidate for the source term coupling the metric to matter fields. Moreover, symmetries of the action allow for easy identification of conserved quantities through [[Noether's theorem]].

In general relativity, the action is usually assumed to be a [[functional (mathematics)|functional]] of the metric (and matter fields), and the [[connection (mathematics)|connection]] is given by the [[Levi-Civita connection]]. The [[Palatini action|Palatini formulation]] of general relativity assumes the metric and connection to be independent, and varies with respect to both independently, which makes it possible to include fermionic matter fields with non-integer spin.

The Einstein equations in the presence of matter are given by adding the matter action to the Einstein–Hilbert action.

==Derivation of Einstein field equations==
Suppose that the full action of the theory is given by the Einstein–Hilbert term plus a term <math>\mathcal{L}_\mathrm{M}</math> describing any matter fields appearing in the theory.

{{NumBlk|:|<math>S = \int \left[ \frac{1}{2\kappa} R + \mathcal{L}_\mathrm{M} \right] \sqrt{-g} \, \mathrm{d}^4 x </math>.|{{EquationRef|1}}}}

The [[stationary-action principle]] then tells us that to recover a physical law, we must demand that the variation of this [[Action (physics)|action]] with respect to the inverse metric be zero, yielding

:<math>\begin{align}
0 &= \delta S \\
  &= \int \left[ \frac{1}{2\kappa} \frac{\delta \left(\sqrt{-g}R\right)}{\delta g^{\mu\nu}} + \frac{\delta \left(\sqrt{-g} \mathcal{L}_\mathrm{M}\right)}{\delta g^{\mu\nu}}
 \right] \delta g^{\mu\nu} \, \mathrm{d}^4x \\
  &= \int \left[ \frac{1}{2\kappa} \left( \frac{\delta R}{\delta g^{\mu\nu}} + \frac{R}{\sqrt{-g}} \frac{\delta \sqrt{-g}}{\delta g^{\mu\nu}}
 \right) + \frac{1}{\sqrt{-g}} \frac{\delta \left(\sqrt{-g} \mathcal{L}_\mathrm{M}\right)}{\delta g^{\mu\nu}} \right] \delta g^{\mu\nu} \sqrt{-g}\, \mathrm{d}^4x
\end{align}</math>.

Since this equation should hold for any variation <math>\delta g^{\mu\nu}</math>, it implies that

{{NumBlk|:|<math>\frac{\delta R}{\delta g^{\mu\nu}} + \frac{R}{\sqrt{-g}} \frac{\delta \sqrt{-g}}{\delta g^{\mu\nu}} = -2\kappa \frac{1}{\sqrt{-g}} \frac{\delta (\sqrt{-g} \mathcal{L}_\mathrm{M})}{\delta g^{\mu\nu}}</math>|{{EquationRef|2}}}}

is the [[equation of motion]] for the metric field. The right hand side of this equation is (by definition) proportional to the [[stress–energy tensor]],<ref>{{Citation
 | last = Blau
 | first = Matthias
 | title = Lecture Notes on General Relativity
 | journal = 
 | volume = 
 | url = http://www.blau.itp.unibe.ch/newlecturesGR.pdf
 | page = 207 
 | date = August 27, 2024
}}</ref>

:<math>T_{\mu\nu} := \frac{-2}{\sqrt{-g}}\frac{\delta (\sqrt{-g} \mathcal{L}_\mathrm{M})}{\delta g^{\mu\nu}} = -2 \frac{\delta \mathcal{L}_\mathrm{M}}{\delta g^{\mu\nu}} + g_{\mu\nu} \mathcal{L}_\mathrm{M}</math>.

To calculate the left hand side of the equation we need the variations of the Ricci scalar <math>R</math> and the determinant of the metric. These can be obtained by standard textbook calculations such as the one given below, which is strongly based on the one given in Carroll (2004).<ref>{{Citation|author=Carroll, Sean M. |author-link=Sean M. Carroll |title=Spacetime and Geometry: An Introduction to General Relativity |location=San Francisco |publisher=Addison-Wesley |date=2004 |isbn=978-0-8053-8732-2}}</ref>

===Variation of the Ricci scalar===
The variation of the [[Ricci scalar]] follows from varying the [[Riemann curvature tensor]], and then the [[Ricci curvature tensor]].

The first step is captured by the [[Palatini identity]]

:<math>
  \delta R_{\sigma\nu} \equiv \delta {R^\rho}_{\sigma\rho\nu} =
  \nabla_\rho \left( \delta \Gamma^\rho_{\nu\sigma} \right) - \nabla_\nu \left( \delta \Gamma^\rho_{\rho\sigma} \right)</math>.

Using the product rule, the variation of the Ricci scalar <math>R = g^{\sigma\nu} R_{\sigma\nu}</math> then becomes

:<math>\begin{align}
\delta R &= R_{\sigma\nu} \delta g^{\sigma\nu} + g^{\sigma\nu} \delta R_{\sigma\nu}\\
         &= R_{\sigma\nu} \delta g^{\sigma\nu} + \nabla_\rho \left( g^{\sigma\nu} \delta\Gamma^\rho_{\nu\sigma} - g^{\sigma\rho} \delta \Gamma^\mu_{\mu\sigma} \right),
\end{align}</math>

where we also used the [[Metric connection#Riemannian connection|metric compatibility]] <math>\nabla_\sigma g^{\mu\nu} = 0</math>, and renamed the summation indices <math>(\rho,\nu) \rightarrow (\mu,\rho)</math> in the last term.

When multiplied by <math>\sqrt{-g}</math>, the term <math>\nabla_\rho \left( g^{\sigma\nu} \delta\Gamma^\rho_{\nu\sigma} - g^{\sigma\rho}\delta\Gamma^\mu_{\mu\sigma} \right)</math> becomes a [[total derivative]], since for any [[Ricci calculus|vector]] <math>A^\lambda</math> and any [[tensor density]] <math>\sqrt{-g}\,A^\lambda</math>, we have 
:<math>
        \sqrt{-g} \, A^\lambda_{;\lambda} =
  \left(\sqrt{-g} \, A^\lambda\right)_{;\lambda} =
  \left(\sqrt{-g} \, A^\lambda\right)_{,\lambda}
</math> or <math>
  \sqrt{-g} \, \nabla_\mu A^\mu =
    \nabla_\mu\left(\sqrt{-g} \, A^\mu\right) =
  \partial_\mu\left(\sqrt{-g} \, A^\mu\right)
</math>.

By [[Stokes' theorem]], this only yields a boundary term when integrated. The boundary term is in general non-zero, because the integrand depends not only on <math>\delta g^{\mu\nu},</math> but also on its partial derivatives <math>\partial_\lambda\, \delta g^{\mu\nu} \equiv \delta\, \partial_\lambda g^{\mu\nu}</math>; see the article [[Gibbons–Hawking–York boundary term]] for details. However, when the variation of the metric <math>\delta g^{\mu\nu}</math> vanishes in a neighbourhood of the boundary or when there is no boundary, this term does not contribute to the variation of the action. Thus, we can forget about this term and simply obtain

{{NumBlk|:|<math>\frac{\delta R}{\delta g^{\mu\nu}} = R_{\mu\nu}</math>.|{{EquationRef|3}}}}

at [[event (relativity)|events]] not in the [[closure (topology)|closure]] of the boundary.

===Variation of the determinant===
[[Jacobi's formula]], the rule for differentiating a [[determinant#Derivative|determinant]], gives:

:<math>\delta g = \delta \det(g_{\mu\nu}) = g g^{\mu\nu} \delta g_{\mu\nu}</math>,

or one could transform to a coordinate system where <math>g_{\mu\nu}</math> is diagonal and then apply the product rule to differentiate the product of factors on the main diagonal. Using this we get

:<math>\delta \sqrt{-g} = -\frac{1}{2\sqrt{-g}}\delta g = \frac{1}{2} \sqrt{-g} \left( g^{\mu\nu} \delta g_{\mu\nu} \right) = -\frac{1}{2} \sqrt{-g} \left( g_{\mu\nu} \delta g^{\mu\nu} \right)</math>

In the last equality we used the fact that

:<math>g_{\mu\nu}\delta g^{\mu\nu} = -g^{\mu\nu} \delta g_{\mu\nu}</math>

which follows from the rule for differentiating the inverse of a matrix

:<math>\delta g^{\mu\nu} = - g^{\mu\alpha} \left( \delta g_{\alpha\beta} \right) g^{\beta\nu}</math>.

Thus we conclude that

{{NumBlk|:|<math>\frac{1}{\sqrt{-g}} \frac{\delta \sqrt{-g}}{\delta g^{\mu\nu} } = -\frac{1}{2} g_{\mu\nu}</math>.|{{EquationRef|4}}}}

===Equation of motion===
Now that we have all the necessary variations at our disposal, we can insert ({{EquationNote|3}}) and ({{EquationNote|4}}) into the equation of motion ({{EquationNote|2}}) for the metric field to obtain

{{NumBlk|:|<math>R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R = \frac{8 \pi G}{c^4} T_{\mu\nu}</math>,|{{EquationRef|5}}}}

which is the [[Einstein field equations]], and

:<math>\kappa = \frac{8\pi G}{c^4}</math>

has been chosen such that the non-relativistic limit yields [[Newton's law of universal gravitation|the usual form of Newton's gravity law]], where <math>G</math> is the [[gravitational constant]] (see [[Einstein field equations#The correspondence principle|here]] for details).

== Cosmological constant ==
When a [[cosmological constant]] Λ is included in the [[Lagrangian (field theory)|Lagrangian]], the action:

:<math>S = \int \left[ \frac{1}{2\kappa} (R-2 \Lambda ) + \mathcal{L}_\mathrm{M} \right] \sqrt{-g} \, \mathrm{d}^4 x </math>

Taking variations with respect to the inverse metric:

:<math>\begin{align} 
  \delta S
    &= \int \left[ \frac{\sqrt{-g}}{2\kappa} \frac{\delta R}{\delta g^{\mu \nu}} + \frac{R}{2\kappa} \frac{\delta \sqrt{-g}}{\delta g^{\mu \nu}} - \frac{\Lambda}{\kappa} \frac{\delta \sqrt{-g}}{\delta g^{\mu \nu}} + \sqrt{-g}\frac{\delta \mathcal{L}_\mathrm{M}}{\delta g^{\mu \nu}} + \mathcal{L}_\mathrm{M} \frac{\delta \sqrt{-g}}{\delta g^{\mu \nu}} \right] \delta g^{\mu \nu} \mathrm{d}^4 x \\
    &= \int \left[ \frac{1}{2\kappa} \frac{\delta R}{\delta g^{\mu \nu}} + \frac{R}{2\kappa} \frac{1}{\sqrt{-g}} \frac{\delta \sqrt{-g}}{\delta g^{\mu \nu}} - \frac{\Lambda}{\kappa} \frac{1}{\sqrt{-g}} \frac{\delta \sqrt{-g}}{\delta g^{\mu \nu}} + \frac{\delta \mathcal{L}_\mathrm{M}}{\delta g^{\mu \nu}} + \frac{\mathcal{L}_\mathrm{M}}{\sqrt{-g}} \frac{\delta \sqrt{-g}}{\delta g^{\mu \nu}} \right] \delta g^{\mu \nu} \sqrt{-g} \, \mathrm{d}^4 x 
\end{align}</math>

Using the [[action principle]]: 
:<math>
  0 = \delta S =
  \frac{1}{2\kappa} \frac{\delta R}{\delta g^{\mu \nu}} + \frac{R}{2\kappa} \frac{1}{\sqrt{-g}} \frac{\delta \sqrt{-g}}{\delta g^{\mu \nu}} - 
  \frac{\Lambda}{\kappa} \frac{1}{\sqrt{-g}} \frac{\delta \sqrt{-g}}{\delta g^{\mu \nu}} + \frac{\delta \mathcal{L}_\mathrm{M}}{\delta g^{\mu \nu}} + 
  \frac{\mathcal{L}_\mathrm{M}}{\sqrt{-g}} \frac{\delta \sqrt{-g}}{\delta g^{\mu \nu}}
</math>

Combining this expression with the results obtained before:

:<math>\begin{align}
  \frac{\delta R}{\delta g^{\mu \nu}} &= R_{\mu \nu} \\
  \frac{1}{\sqrt{-g}} \frac{\delta \sqrt{-g}}{\delta g^{\mu \nu}} &= \frac{-g_{\mu \nu}}{2} \\
  T_{\mu \nu} &= \mathcal{L}_\mathrm{M} g_{\mu \nu} - 2 \frac{\delta\mathcal{L}_\mathrm{M}}{\delta g^{\mu \nu}}
\end{align}</math>

We can obtain:

:<math>\begin{align}
  \frac{1}{2\kappa} R_{\mu \nu} + \frac{R}{2\kappa} \frac{-g_{\mu \nu}}{2} - \frac{\Lambda}{\kappa} \frac{-g_{\mu \nu}}{2} +  \left(\frac{\delta \mathcal{L}_\mathrm{M}}{\delta g^{\mu \nu}} + \mathcal{L}_\mathrm{M}\frac{-g_{\mu \nu}}{2} \right) &= 0 \\
  R_{\mu \nu} - \frac{R}{2} g_{\mu \nu} + \Lambda g_{\mu \nu} + \kappa \left(2 \frac{\delta \mathcal{L}_\mathrm{M}}{\delta g^{\mu \nu}} - \mathcal{L}_\mathrm{M}g_{\mu \nu} \right) &= 0 \\
  R_{\mu \nu} - \frac{R}{2} g_{\mu \nu} + \Lambda g_{\mu \nu} - \kappa T_{\mu \nu} &= 0
\end{align} </math>

With <math display="inline">\kappa = \frac{8 \pi G}{c^4} </math>, the expression becomes the field equations with a [[cosmological constant]]:

:<math>R_{\mu \nu} - \frac{1}{2} g_{\mu \nu} R + \Lambda g_{\mu \nu} = \frac{8 \pi G}{c^4} T_{\mu \nu}.</math>

==See also==
*[[Belinfante–Rosenfeld stress–energy tensor|Belinfante–Rosenfeld tensor]]
*[[Brans–Dicke theory]] (in which the constant ''k'' is replaced by a scalar field).
*[[Einstein–Cartan theory]]
*[[f(R) gravity]] (in which the Ricci scalar is replaced by a function of the Ricci curvature)
*[[Gibbons–Hawking–York boundary term]]
*[[Kaluza–Klein theory]]
*[[Komar superpotential]]
*[[Palatini action]]
*[[Teleparallelism]]
*[[Tetradic Palatini action]]
*[[Variational methods in general relativity]]
*[[Vermeil's theorem]]

==Notes==
{{reflist}}

==Bibliography==
* {{Citation|first1=Charles W.|last1=Misner|author-link=Charles W. Misner|first2=Kip. S.|last2=Thorne|author2-link=Kip Thorne|first3=John A.|last3=Wheeler|author3-link=John A. Wheeler|title=Gravitation|publisher= W. H. Freeman|date=1973|isbn=978-0-7167-0344-0|title-link=Gravitation (book)}}
* {{Citation|last=Wald|first=Robert M.|author-link=Robert Wald|title=General Relativity|publisher=University of Chicago Press|date=1984|isbn=978-0-226-87033-5|title-link=General Relativity (book)}}
* {{Citation|author=Carroll, Sean M. |author-link=Sean M. Carroll |title=Spacetime and Geometry: An Introduction to General Relativity |location=San Francisco |publisher=Addison-Wesley |date=2004 |isbn=978-0-8053-8732-2}}
*[[David Hilbert|Hilbert, D.]] (1915) [http://einstein-annalen.mpiwg-berlin.mpg.de/related_texts/relativity_rev/hilbert'' Die Grundlagen der Physik'' (German original for free)] [https://doi.org/10.1007%2F978-1-4020-4000-9_44 (English translation for $25)], Konigl. Gesell. d. Wiss. Göttingen, Nachr. Math.-Phys. Kl. 395–407
*{{springer|id=C/c026670|last=Sokolov |first=D.D. |title=Cosmological constant}}
*{{Citation | last = Feynman | first = Richard P. | year = 1995 | author-link = Richard Feynman | title = Feynman Lectures on Gravitation | publisher = Addison-Wesley | isbn = 0-201-62734-5 | url = https://archive.org/details/feynmanlectureso0000feyn_g4q1 | url-access = registration }}
*Christopher M. Hirata [http://www.tapir.caltech.edu/~chirata/ph236/2011-12/lec33.pdf Lecture 33: Lagrangian formulation of GR] (27 April 2012).

{{DEFAULTSORT:Einstein-Hilbert action}}
[[Category:Variational formalism of general relativity]]
[[Category:General relativity]]
[[Category:Albert Einstein]]
[[Category:Gravity]]
[[Category:David Hilbert]]