Editing Einstein tensor (section)

== Trace ==
The [[trace (linear algebra)|trace]] of the Einstein tensor can be computed by [[Tensor contraction|contract]]ing the equation in the [[#Definition|definition]] with the [[metric tensor]] {{tmath|1= g^{\mu\nu} }}. In <math>n</math> dimensions (of arbitrary signature):
<math display="block">\begin{align}
g^{\mu\nu}G_{\mu\nu} &= g^{\mu\nu}R_{\mu\nu} - {1\over2} g^{\mu\nu}g_{\mu\nu}R \\
G &= R - {1\over2} (nR) = {{2-n}\over2}R
\end{align}</math>

Therefore, in the special case of {{tmath|1= n = 4 }} dimensions, {{tmath|1= G = -R }}. That is, the trace of the Einstein tensor is the negative of the [[Ricci tensor]]'s trace. Thus, another name for the Einstein tensor is the ''trace-reversed Ricci tensor''. This <math>n=4</math> case is especially relevant in the [[theory of general relativity]].