Einstein-Hilbert action Details, Meaning Einstein-Hilbert action Article and Explanation Guide

Einstein-Hilbert action Guide, Meaning , Facts, Information and Description

In general relativity, Einstein's field equations can be derived from an action principle starting from the Einstein-Hilbert action:

where g is the (pseudo)Riemannian metric, R is the Ricci scalar, n is the number of spacetime dimensions and k is a constant which depends on the units chosen (see below). In Brans-Dicke theory, k is replaced by a scalar field.

In general relativity, the action is assumed to be a functional of the metric only, 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 Einstein-Hilbert action is said to have been written down first by the german mathematician David Hilbert.

Table of 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 See also
3 References

Derivation of Einstein's field equations

The Einstein-Hilbert action as stated above will actually yield the vacuum Einstein equations. So as starting point a matter Lagrangean L_M should be added:

The variation with respect to the metric yields

the last term of which is by definition called the stress-energy tensor T_mn.

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

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_mrn the variation of the scalar curvature is

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

We use the following property of a determinant

to determine the variation

where δ(g^mng_{mn_{)=0 has been used.}}

Equation of motion

From

we read off

which is Einstein's field equation and

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

The stress-energy tensor may be written as

where the functional derivative can be replaced by a partial derivative if the matter Lagrangean does not depend on derivatives of the metric as is common in general relativity.

References

Carroll, Sean M. (Dec, 1997). Lecture Notes on General Relativity, NSF-ITP-97-147, 231pp,

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