Einstein-Hilbert action

From Free net encyclopedia

Template:Expert

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. It has been said that Hilbert independently found the equations a few days before Einstein (discussion). These equations were based on the Einstein-Hilbert action.

In general relativity, the action is assumed to be a functional only of the metric, i.e. the connection is given by the Levi-Civita connection. Some extensions of general relativity assume the metric and connection to be independent, however, and vary with respect to both independently.

The action <math>S[g]</math> which gives rise to the vacuum Einstein equations is given by the following integral of the Lagrangian density (sometimes just called the Lagrangian)

<math>S[g]=k \int R \sqrt{-g} \, d^4x </math>

where <math>g = \, \det \, (g_{ab})</math> is the determinant of the Lorentz metric, R is the Ricci scalar, k is a constant which depends on the units chosen (see below), the Lagrangian being <math>R\sqrt{-g}</math>, and the integral is taken over a region of spacetime. In the rival field theory of gravitation called Brans-Dicke theory, k is replaced by a scalar field.

Contents 1 Derivation of Einstein's field equations 1.1 Variation of the Ricci scalar 1.2 Variation of the determinant 1.3 Equation of motion 2 Cosmological constant 3 See also 4 References

Derivation of Einstein's field equations

To derive the full field equations, it is natural to assume that an extra term - a matter Lagrangian (density) L_M be added:

<math>S = \int \left[ k \, R + L_\mathrm{M} \right] \sqrt{-g} \, d^4x </math>

The variation with respect to the metric yields

<math>\delta S = \int d^4x \sqrt{-g} \left[ k \left( \delta R + R \frac{1}{\sqrt{-g}} \delta \sqrt{-g} \right) + \frac{1}{\sqrt{-g}} \frac{\delta \sqrt{-g} L_\mathrm{M}}{\delta g^{mn}} \right] \delta g^{mn} </math>

the last term of which is defined as the stress-energy tensor T_mn.

<math>-\frac{1}{2} T_{mn} := \frac{1}{\sqrt{-g}} \frac{\delta}{\delta g^{mn}} \sqrt{-g} L_\mathrm{M} </math>

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

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

Variation of the Ricci scalar

The variation of the Riemann curvature tensor with respect to the metric is

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

where δΓ is the variation of the Levi-Civita connection (which is not written down explicitly as it is not required subsequently).

Due to R=g^mnR_mn and R_mn=R^r_{mr n} the variation of the scalar curvature is

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

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 δg^mn is supposed to vanish at infinity.

Variation of the determinant

The following property of a determinant is used

<math>\,\! g = \exp \mathrm{tr} \ln \, (g_{mn})</math>

is used to determine the variation:

<math>\,\! \delta g=g (g^{mn} \delta g_{mn})</math>

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

where δ (g^mng_{mn)=0 has been used.}

Equation of motion

From

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

we read off

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

which is Einstein's field equation and

<math>k = \frac{c^4}{16 \pi G}</math>

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

The stress-energy tensor may be written as

<math> T_{mn} = g_{mn} L_\mathrm{M} - 2 \frac{\delta L_\mathrm{M}}{\delta g^{mn}}</math>

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).

Cosmological constant

Sometimes, a cosmological constant is added to the Lagrangian density so that the new action

<math>S = \int \left[ k\, R + \Lambda + L_\mathrm{M} \right] \sqrt{-g} \, d^4x </math>

yields the field equations:

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

See also

• Belinfante-Rosenfeld tensor
• Einstein-Cartan theory
• Gibbons-Hawking-York boundary term
• Palatini action
• Teleparallelism
• Variational methods in general relativity

References

• Carroll, Sean M. (Dec, 1997). Lecture Notes on General Relativity, NSF-ITP-97-147, 231pp, arXiv:gr-qc/9712019fr:Action d'Einstein-Hilbert

Template:Relativity-stub