Linearised Einstein field equations

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The linearised Einstein field equations (linearised EFE) are an approximation to Einstein's field equations that is valid for a weak gravitational field and is used to simplify many problems in general relativity and to discuss the phenomena of gravitational radiation. It can also be used to derive Newtonian gravity as the weak-field approximation of Einsteinian gravity.

They are obtained by assuming the spacetime metric is only slightly different from some baseline metric (usually a Minkowski metric). Then the difference in the metrics can be considered as a field on the baseline metric, whose behavior is approximated by a set of linear equations.

Starting with the metric for a spacetime in the form

gab = ηab + hab

where \, \eta_{ab} is the Minkowski metric and \, h_{ab} — sometimes written as \epsilon \, \gamma_{ab} — is the deviation of \, g_{ab} from it. h must be negligible compared to η: \left| h_{\mu \nu} \right| \ll 1 (and similarly for all derivatives of h). Then one ignores all products of h (or its derivatives) with h or its derivatives (equivalent to ignoring all terms of higher order than 1 in ε). It is further assumed in this approximation scheme that all indices of h and its derivatives are raised and lowered with η.

The metric h is clearly symmetric, since g and η are. The consistency condition gabgbc = δac shows that

g^{ab} \, = \eta^{ab} - h^{ab}

The Christoffel symbols can be calculated as

2 \Gamma ^a_{bc} = (h^a{}_{b,c}+h^a{}_{c,b}-h_{bc,}{}^a)

where h_{bc,}{}^a \ \stackrel{\mathrm{def}}{=}\ \eta^{ar} h_{bc,r}, and this is used to calculate the Riemann tensor:

2R^a{}_{bcd} = 2(\Gamma^a_{bd,c}-\Gamma^a_{bc,d}) = \eta^{ae} (h_{eb,dc}+h_{ed,bc}-h_{bd,ec} - h_{eb,cd}-h_{ec,bd}+h_{bc,ed}) =
= \eta^{ae} (h_{ed,bc}-h_{bd,ec}-h_{ec,bd}+h_{bc,ed}) = h^a_{d,bc} - h_{bd,}{}^ a{}_c + h_{bc,}{}^a{}_d - h^a{}_{c,bd}

Using Rbd = δcaRabcd gives

2R_{bd}= h^r_{d,br} + h^r_{b,dr} -h_{,bd} - h_{bd, rs} \eta ^{rs}

Then the linearized Einstein equations are

8\pi T_{bd} \, = R_{bd} - R_{ac} \eta^{ac} \eta_{bd} / 2

or

8\pi T_{bd} = (h^r_{d,br} + h^r_{b,dr} -h_{,bd} - h_{bd, r}{}^r - h^r_{s,r}{}^s \eta_{bd})/2 + ( h_{,a}{}^a \eta_{bd} + h_{ac, r}{}^r \eta^{ac} \eta_{bd}) /4

Or, equivalently:

8\pi (T_{bd} - T_{ac} \eta^{ac} \eta_{bd}/2) \, = R_{bd}
16\pi (T_{bd} - T_{ac} \eta^{ac} \eta_{bd}/2) \, = h^r_{d,br} + h^r_{b,dr} -h_{,bd} - h_{bd, rs} \eta ^{rs}

The linearised EFE are used primarily in the theory of gravitational radiation, where the gravitational field far from the source is approximated by these equations.


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