Ngô Quốc Anh

December 20, 2009

R-G: Einstein vacuum field equations

Filed under: Riemannian geometry — Tags: — Ngô Quốc Anh @ 13:43

This is a short note concerning the so called Einstein vacuum field equations {\rm Eins}(g)_{\alpha\beta}=0 where {\rm Eins}(g) is nothing but the Einstein curvature tensor defined in this topic. I have not discussed either  Einstein field equations or Einstein vacuum field equations yet, however, we can adop the following equation

\displaystyle {\rm Ric}_{\alpha\beta} - {1 \over 2}g_{\alpha\beta} R + g_{\alpha\beta}\Lambda = {8 \pi G \over c^4} T_{\alpha\beta}

to be the Einstein field equations (EFE) where {\rm Ric}_{\alpha\beta} is the Ricci curvature tensor, R the scalar curvature, g_{\alpha\beta} the metric tensor, \Lambda is the cosmological constant, G is the gravitational constant, c the speed of light, and T_{\alpha\beta} the stress-energy tensor.

One can write the EFE in a more compact form by defining the Einstein tensor

\displaystyle {\rm Eins}_{\alpha\beta} = {\rm Ric}_{\alpha\beta} - {1 \over 2}R g_{\alpha\beta},

which is a symmetric second-rank tensor that is a function of the metric. The EFE can then be written as

\displaystyle {\rm Eins}_{\alpha\beta} = {8 \pi G \over c^4} T_{\alpha\beta},

where the cosmological term has been absorbed into the stress-energy tensor as dark energy.

If the energy-momentum tensor T_{\alpha\beta} is zero in the region under consideration, then the field equations are also referred to as the vacuum field equations. By setting T_{\alpha\beta}=0 in the full field equations, the vacuum equations can be written as

\displaystyle {\rm Ric}_{\alpha\beta} = {1 \over 2} R \, g_{\alpha\beta}.

Taking the trace of this (contracting with g_{\alpha\beta}) and using the fact that g^{\alpha\beta} g_{\alpha\beta} = 4, we get

\displaystyle R = {1 \over 2} R4 = 2 R,

and thus

\displaystyle R = 0.

Substituting back, we get an equivalent form of the vacuum field equations

\displaystyle {\rm Ric}_{\alpha\beta} = 0.

The above equation is frequently used in the literature, sometimes, it is called the Einstein vacuum field equations, for example, we refer the reader to introduction part of the following paper due to James Isenberg. In the case of nonzero cosmological constant, the equations are

\displaystyle {\rm Ric}_{\alpha\beta} = {1 \over 2}R g_{\alpha\beta} - \Lambda g_{\alpha\beta}

which gives

\displaystyle R = 4 \Lambda

yielding the equivalent form

\displaystyle {\rm Ric}_{\alpha\beta} = \Lambda g_{\alpha\beta}.

The solutions to the vacuum field equations are called vacuum solutions. Flat Minkowski space is the simplest example of a vacuum solution. Nontrivial examples include the Schwarzschild solution and the Kerr solution.

Manifolds with a vanishing Ricci tensor, {\rm Ric}_{\alpha\beta} = 0, are referred to as Ricci-flat manifolds and manifolds with a Ricci tensor proportional to the metric as Einstein manifolds.

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