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I know quite well what the trace of a matrix is; however, I am not quite sure I understand the meaning of the 'trace' concept when applied to tensors. I would be very grateful to you if:

How can i prove that the trace of einstein equation in general relativity is zero? And how can i find the values of the elements of the principal diagonal? There have 4 elements in the diagonal, thats right ?

Lucas G Leite F Pollito
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For a (0,2)-tensor $T_{ab}$ say, its trace is $T = g^{ab}T_{ab}$ which uses the inverse metric components. If instead you write this tensor as a (1,1)-tensor $T_a^b$, then its trace is simply the usual trace of a matrix: $T^a_a$.

When you mention the Einstein field equations, do you mean instead local conservation of energy: $\nabla_a T^{ab}=0$? See e.g. Wikipedia

Colin MacLaurin
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