Editing Einstein field equations (section)

=== Equivalent formulations ===
Taking the [[Scalar curvature#Definition|trace with respect to the metric]] of both sides of the EFE one gets
<math display="block">R - \frac{D}{2} R + D \Lambda = \kappa T ,</math>
where {{mvar|D}} is the spacetime dimension. Solving for {{math|''R''}} and substituting this in the original EFE, one gets the following equivalent "trace-reversed" form:
<math display="block">R_{\mu \nu} - \frac{2}{D-2} \Lambda g_{\mu \nu} = \kappa \left(T_{\mu \nu} - \frac{1}{D-2}Tg_{\mu \nu}\right) .</math>

In {{math|1=''D'' = 4}} dimensions this reduces to
<math display="block">R_{\mu \nu} - \Lambda g_{\mu \nu} = \kappa \left(T_{\mu \nu} - \frac{1}{2}T\,g_{\mu \nu}\right) .</math>

Reversing the trace again would restore the original EFE. The trace-reversed form may be more convenient in some cases (for example, when one is interested in weak-field limit and can replace {{math|''g''{{sub|''μν''}}}} in the expression on the right with the [[Minkowski metric]] without significant loss of accuracy).