Einstein-Hilbert action

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The Einstein-Hilbert action is a mathematical object (an action) that is used to derive Einstein's field equations of general relativity.

In 1913, Albert Einstein was working on the equivalence of gravity and acceleration. He realized the geometry would work if spacetime was curved, and not flat as had always been the classical assumption. Einstein proposed that mass and energy would warp spacetime and that gravitational force was merely an expression of the spacetime curvature. Einstein initially found an equation which was correct for vacuum spacetime (but in the presence of matter was wrong as it omitted the term −Rgmn/2) and finally found the correct equations in 1915 by trial and error. Hilbert, who had been studying Einstein's work, also found the missing term (in Hilbert (1915)) at about the same time as Einstein, by deriving Einstein's equation as the Euler-Lagrange equations for the Einstein-Hilbert action with respect to variation of the metric.

In general relativity, the action is usually assumed to be a functional of the metric (and matter fields), and the connection is given by the Levi-Civita connection. The 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-integral spin.

The action S[g] which gives rise to the vacuum Einstein equations is given by the following integral of the Lagrangian

S[g]= \int kR \sqrt{-g} \, \mathrm{d}^4x

where g = \, \det \, (g_{ab}) is the determinant of a spacetime Lorentz metric, R is the Ricci scalar, k is the constant c4/16πG, the Lagrangian being R\sqrt{-g}, and the integral is taken over a region of spacetime. The Einstein equations in the presence of matter are given by adding the Lagrangian for the matter into the integral. (In this article "Lagrangian" means "Lagrangian density"; other authors occasionally use it to mean the integral of the Lagrangian density over space or over spacetime.)

Note that \sqrt{-g} \, \mathrm{d}^4x is an invariant 4-volume element, so the action can be also written in the following (somewhat more elegant) fashion:

S[g]= k \int R \, \mathrm{dV}

Contents

[edit] Derivation of Einstein's field equations

To derive the full field equations, it is natural to assume that an extra term - a matter Lagrangian \mathcal{L}_\mathrm{M} be added:

S = \int \left[ k \, R + \mathcal{L}_\mathrm{M} \right] \sqrt{-g} \, \mathrm{d}^4x

The variation with respect to the reciprocal of the metric yields

0 = \delta S = \int k \delta (R \sqrt{-g}) + \delta (\sqrt{-g} \mathcal{L}_\mathrm{M}) \mathrm{d}^4x
= \int \left[ k \left( \frac{\delta R}{\delta g^{mn}} + R \frac{\delta \sqrt{-g}}{\sqrt{-g}\delta g^{mn} } \right) + \frac{1}{\sqrt{-g}} \frac{\delta (\sqrt{-g} \mathcal{L}_\mathrm{M})}{\delta g^{mn}} \right] (\delta g^{mn}) \sqrt{-g}\, \mathrm{d}^4x

The following are standard text book calculations which have in part been taken from Carroll (see References).

[edit] Variation of the Ricci scalar

Recall that the Riemann curvature tensor is

{R^\rho}_{\sigma\mu\nu} = \partial_\mu\Gamma^\rho_{\nu\sigma} - \partial_\nu\Gamma^\rho_{\mu\sigma} + \Gamma^\rho_{\mu\lambda}\Gamma^\lambda_{\nu\sigma} - \Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\sigma}

If we calculate its variation, the first two terms on the right side become

\partial_\mu\delta\Gamma^\rho_{\nu\sigma} - \partial_\nu\delta\Gamma^\rho_{\mu\sigma}

Notice that \delta\Gamma^\rho_{\nu\sigma} is a tensor. So we can transform to a coordinate system where \Gamma^\rho_{\nu\sigma} is zero. This causes the terms without partial derivatives to vanish. Also, in the terms with partial derivatives, they become covariant derivatives. So we get

\delta R^r{}_{mln} = \nabla_l (\delta \Gamma^r_{nm}) - \nabla_n (\delta \Gamma^r_{lm})

where δΓ is the variation of the Levi-Civita connection (which is not expanded in terms of δg because those details are not required subsequently).

Due to R=gmnRmn and Rmn=Rrmr n the variation of the scalar curvature is

\delta R = R_{mn} \delta g^{mn} + \nabla_s ( g^{mn} \delta\Gamma^s_{nm} - g^{ms}\delta\Gamma^r_{rm} )

where the second term yields a surface term by Stokes' theorem as long as k is a constant and does not contribute when the variation δgmn is supposed to vanish at infinity. So we get

\frac{\delta R}{\delta g^{mn}} = R_{mn}

[edit] Variation of the determinant

The rule for differentiating a determinant (see Determinant#Derivative) gives:

\,\! \delta g=g (g^{mn} \delta g_{mn})

or one could transform to a coordinate system where g_{mn}\! is diagonal and then apply the product rule to differentiate the product of factors on the main diagonal.

Using this we get

\delta \sqrt{-g} = \frac{1}{2\sqrt{-g}}\cdot -\delta g = \frac{1}{2} \sqrt{-g} (g^{mn} \delta g_{mn})

Now use 0=\delta(4)=\delta(g^{mn}g_{mn})=(\delta g^{mn})g_{mn}+g^{mn}\delta g_{mn}\! to infer

\delta \sqrt{-g} = \frac{1}{2} \sqrt{-g} (-g_{mn} \delta g^{mn})

So we conclude

\frac{\delta \sqrt{-g}}{\sqrt{-g}\delta g^{mn} } = -\frac{1}{2}\, g_{mn}

[edit] Variation of matter term

The matter term is used to define the stress-energy tensor Tmn.

-\frac{1}{2}\, T_{mn} := \frac{1}{\sqrt{-g}} \frac{\delta (\sqrt{-g} \mathcal{L}_\mathrm{M})}{\delta g^{mn}} = -\frac{1}{2}g_{mn}\mathcal{L}_\mathrm{M} + \frac{\delta \mathcal{L}_\mathrm{M}}{\delta g^{mn}}

Multiplying by minus two, we get that the stress-energy tensor may be written as

T_{mn} = g_{mn} \mathcal{L}_\mathrm{M} - 2 \frac{\delta \mathcal{L}_\mathrm{M}}{\delta g^{mn}}

where the functional derivative can be replaced by a partial derivative, if the matter Lagrangian does not depend on derivatives of the metric (as is common in general relativity). Also the derivative of the matter Lagrangian must be made symmetric in its indices m and n, if it is not already so, because the variation of g is symmetric since g itself is constrained to be symmetric.

Note that this is the conventional definition in general relativity, although there are several inequivalent definitions, in particular the canonical stress-energy tensor.

[edit] Equation of motion

Substituting the results into the variational equation, we get

0 = \delta S = \int \left[ k ( R_{mn} - \frac{1}{2} g_{mn} R) - \frac{1}{2} T_{mn} \right] (\delta g^{mn}) \sqrt{-g} \, \mathrm{d}^4x

from which we can see that the expression in brackets must be zero. So we can read off

R_{mn} - \frac{1}{2} g_{mn} R = \frac{8 \pi G}{c^4} T_{mn}

which is Einstein's field equation and

k = \frac{c^4}{16 \pi G}

has been chosen such that the non-relativistic limit yields the usual form of Newton's gravity law, where G is the gravitational constant.

[edit] Cosmological constant

Sometimes, a cosmological constant Λ is included in the Lagrangian so that the new action

S = \int \left[ k\left(R-2\Lambda\right) + \mathcal{L}_\mathrm{M} \right] \sqrt{-g} \, \mathrm{d}^4x

yields the field equations:

R_{mn} - \frac{1}{2} g_{mn} R + \Lambda g_{mn}= \frac{8 \pi G}{c^4} T_{mn}

[edit] See also

[edit] References

  • Carroll, Sean M. (Dec, 1997). Lecture Notes on General Relativity, NSF-ITP-97-147, 231pp, arXiv:gr-qc/9712019
  • Hilbert, D. (1915) Die Grundlagen der Physik, Konigl. Gesell. d. Wiss. Gottingen, Nachr. Math.-Phys. Kl. 395-407

D.D. Sokolov, "Cosmological constant" SpringerLink Encyclopaedia of Mathematics (2001)

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