The Einstein equation should be divided by two

Abstract

We present three reasons for rewriting the Einstein equation. The new version is physically equivalent but geometrically more clear. 1. We write $4 \pi$ instead of $8 \pi$ at the r.h.s, and we explain how this factor enters as surface area of the unit 2--sphere. 2. We define the Riemann curvature tensor and its contractions (including the Einstein tensor at the l.h.s.) with one half of its usual value. This compensates not only for the change made at the r.h.s., but it gives the result that the curvature scalar of the unit 2--sphere equals one, i.e., in two dimensions, now the Gaussian curvature and the Ricci scalar coincide. 3. For the commutator $[u,v]$ of the vector fields $u$ and $v$ we prefer to write (because of the analogy with the antisymmetrization of tensors) $$[u,v]\ = \ \frac{1}{2} \, ( \, u\, v \ - \ v \, u \,)$$ which is one half of the usual value. Then, the curvature operator defined by $$ \nabla_{[u} \ \nabla_{v]} \quad - \quad \nabla_{[u,\,v]}$$ (where $\nabla $ denotes the covariant derivative) is consistent with point 2, i.e., it equals one half of the usual value.