Why did Einstein choose the form R_ab - 1/2 g_ab R for the field equations?

[Phymath]

Why did Einstein chose R_ab - 1/2 g_ab R = k T_ab the Right side is basically saying the mass/energy causes the curvature and must be a conserved quantity, but why is the curvature expressed like it is on the LHS i know it is conserved through differentiation but why that and not any other variant of Riemann curvature?

 

[pervect]

The short answer is that this is the only second rank tensor that gives us an automatic concept of the conservation and energy.

This happens because $T_{uv} = 8 \pi G_{uv}$, so that $\nabla^u T_{uv}$ is equal to zero, since $\nabla^u G_{uv}$=0.

A longer answer involves many subtle points, unfortunately.

 

[Phymath]

found it in MTW gravitation thank you thou