I am little bit dissapointment with action integral in General relativity. The action integral is:
$$ \int Rd^{4}x=\int R_{ij}g^{ij}d^{4}x\tag{1} $$ Where
$$ R_{ij}=\frac{\partial\Gamma^{l}_{ij}}{\partial x^{l}}-\frac{\partial\Gamma^{l}_{li}}{\partial x^{j}}+\Gamma^{l}_{ij}\Gamma^{m}_{lm}-\Gamma^{l}_{im}\Gamma^{m}_{lj}\tag{2} $$ Is the Ricci tensor. The Ricci tensor is in General relativity connected with hamiltonian, and quantity $$ R=R_{ij}g^{ij}\tag{3} $$ is the scalar curvature, which is an invariant quantity, and also total energy of the system. I can't understand, how can I see in Ricci tensor Hamiltonian function.
In every book from General relativity I found something like this:
Action principle...
Lets have an action:
$$ \int\sqrt{-g} R d^4 x=\int\sqrt{-g} R_{ik}g^{ij} d^4 x $$ Where R is the scalar curvature...aaaand, when we do some variation gymnastics like: $$ \delta \sqrt{-g}=-\frac{1}{2}\sqrt{-g}g_{ik}\delta g^{ik} $$ $$ \delta R=R_{ik}\delta g^{ik} $$ and wait when smoke clears, we hopefully arrive to vacuum Einstein field equations: $$ R_{ik}-\frac{1}{2}Rg_{ik}=0 $$ and $$ R_{ik}-\frac{1}{2}Rg_{ik}=\frac{8\pi G}{c^{4}}T_{ik} $$ in the presence of matter. And now we can go further...:)
This is written in every book. But question is: Ok...we arrive to Einstein equations through variation of R. But, why we do variation of R and not some other quantity? I found this identity: $$ R_{ik}=\kappa(T_{ik}-\frac{1}{2}Tg_{ik}) $$ But I can't prove it yet. And its the same, where I started