Recent content by Bishal Banjara

I believe the book "Gravitation: Foundations and Frontiers" by T. Padmanabhan (and his online lectures) is best. And Landau-Lifshitz's classical theory of fields, Feynman's lectures on Gravitation, Gravitation and Cosmology by S. Weinberg are better.

Correction: This would lead to the expression of matter Lagrangian (not density) as

I followed the reverse back derivation of $$T_{\mu\nu}$$ in the equation $$T_{\mu\nu}=\frac{2\delta(\sqrt{-g}\mathcal{L}_m)}{\sqrt{-g}\delta{g^{\mu\nu}}}$$ multiplying by $$\sqrt{-g}/2$$ and reintroducing the intergand. Further, we get variation of matter action as...

That was my mistake that I apparently saw partial differentiation in my own post. It is even mistake at this time also, please make it as variation. I have no edit option.

Yes, I am asking to retrieve the case. But if there is way to explore the variation of matter Lagrangian density with metric tensor resolving the variation, then don't this makes sense to solve the problem?

I know the metric, then what is the mathematical relation between the Ricci scalar and stress energy tensor?

If this is solved, rest is simple.

Basically, the stress energy tensor is given by $$T_{uv}=-2\frac{\partial (L\sqrt{-g})}{\partial g^{uv}}\frac{1}{\sqrt{-g}}.$$ It makes easy to calculate stress energy tensor if the variation of Lagrangian with the metric tensor is known. But it is possible to retrieve matter Lagrangian if the...

yes, @Orodruin had already mentioned. "If that is the best you can do at explaining what your "core quest" is, we might as well close this thread. Can you do any better?" but why?

my core quest is this...

after 1 hr onwards

In the lecture of Leonardo Susskind, he writes the same form I mentioned when deriving the Christoffels from metric tensors. There are other also, Iets not make this as an issue but my core quest is other as I mentioned above.

Particularly, lets take as as what you mentioned, $${\nabla}_{t}T^{tt}=\partial_{t}T^{tt}+\Gamma^t_{t\gamma}T^{t\gamma}+\Gamma^t_{r\gamma}T^{t\gamma}$$. Is this possible or not, if we extend the equation taking all summation where $$\gamma$$ runs from t to $$\phi$$ as...

We could find numerous example following the covariant type as what I have done. My question is not exactly for that. I am expecting the difference in the result that I mentioned and what you mentioned (you could assume as what you took of contravariant type). I have mentioned the same way you...

Why can't I write the covariant derivative of $$T_{tt}$$ components as $${\nabla}_{t}T_{tt}=\partial_{t}T_{tt}-\Gamma^t_{t\gamma}T_{t\gamma}-\Gamma^t_{r\gamma}T_{t\gamma}$$ where, $$\gamma$$ runs from $$t$$ to $$\phi$$?