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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.
The Einstein-Hilbert action as stated above will actually yield the vacuum Einstein equations. So as starting point a matter Lagrangean LM should be added:
The variation with respect to the metric yields
the last term of which is by definition called the stress-energy tensor Tmn.
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).
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=gmnRmn and Rmn=Rrmrn the variation of the scalar curvature is
where the second term yields a surface term by Stokes' theoremStokes' theorem in differential geometry is a statement about the integration of differential forms which generalizes several theorems from vector calculus. It is named after Sir George Gabriel Stokes ( 1819- 1903). Let M be an oriented piecewise smooth m as long as k is a constant and does not contribute when the variation δgmn is supposed to vanish at infinity.
We use the following property of a determinantIn linear algebra, the determinant is a function that associates a scalar det A to every square matrix A''. The fundamental geometric meaning of the determinant is as the scale factor for volume when A is regarded as a linear transformation. Determinants
to determine the variation
where δ(gmngmn)=0 has been used.
From
we read off
which is Einstein's field equationGeneral relativity In physics, the Einstein field equation or the Einstein equation is an equation in the theory of gravitation, called general relativity, that describes how matter creates gravity and, conversely, how gravity affects matter. The Einstein and
has been chosen such that the non-relativistic limit yields the usual form of Newtons gravity lawThis article covers the physics of gravitation. See also gravity (disambiguation). Gravitation is the tendency of masses to move toward each other. The first mathematical formulation of the theory of gravitation was made by Sir Isaac Newton and proved ast, where G is the gravitational constantAccording to the law of universal gravitation, the attractive force between two bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them. The constant of proportionality is called , the gr.
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.
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