# trace of einstein equation - general relativity

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 ?

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

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