A question from Einstein's original paper on general relativity [closed]

Question

I'm working on Einstein's original paper on general relativity (1916). I have a problem on its derivation.

I can't understand the process from (52) to (53), how is it derived? Is there something that I didn't get?

Answer 1

You can use the matter-free equation first. You can choose two tensors since it's just a tensor equation. \begin{equation} \frac{\partial \Gamma^{\alpha}_{\mu\nu}}{\partial x^{\alpha}} + \Gamma^{\alpha}_{\mu\beta}\Gamma^{\beta}_{\nu\alpha} = A_{\mu\nu}+B{\mu\nu} \end{equation}

Multiplying by $g^{\nu\sigma}$ on both sides, you get: \begin{equation} g^{\nu\sigma}\left(\frac{\partial \Gamma^{\alpha}_{\mu\nu}}{\partial x^{\alpha}}+\Gamma^{\alpha}_{\mu\beta}\Gamma^{\beta}_{\nu\alpha}\right) = g^{\nu\sigma}(A_{\mu\nu}+B{\mu\nu}) \end{equation}

Now, the trick is to identify the terms on the right-hand side such as $A_{\mu\nu} = -\chi T^{\mu\nu}$ and $B_{\mu\nu}=(1/2)\chi g_{\mu\nu}T$ in order to get (53).

We can do that since on the LHS, you have using (51) : \begin{equation} \frac{\partial}{\partial x^{\alpha}}(g^{\nu\sigma}\Gamma^{\alpha}_{\mu\nu}) +\chi(t^{\sigma}_{\mu}-(1/2)\delta^{\sigma}_{\mu}) = g^{\nu\sigma}(A_{\mu\nu}+B{\mu\nu}) \end{equation}

Thus, to recover (52), you must have $g^{\nu\sigma}A_{\mu\nu} = -\chi T^{\sigma}_{\mu}$ and $g^{\nu\sigma}B_{\mu\nu}=(1/2)\chi\delta^{\sigma}_{\mu}T$.