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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 ?

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    $\begingroup$ "How can I prove that the trace of Einstein equation in general relativity is zero?" you cant because in general it is not. $\endgroup$ – AccidentalFourierTransform Apr 16 '16 at 14:53
  • $\begingroup$ if i have only two dimension ? can you write the components of the matrix ?? $\endgroup$ – Lucas G Leite F Pollito Apr 18 '16 at 1:11
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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

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