General Tensor contraction: Trace of Energy-Momentum Tensor (Einstein metric)

[tetris11]

Okay so I have:

Eqn1) T_ij=\rhou_iu_j-ph_ij = \rhou_iu_j-p(g_ij-u_iu_j)

Where T_ij is the energy-momentum tensor, being approximated as a fluid with \rho as the energy density and p as the pressure in the medium.


My problem:
Eqn2) Trace(T) = Tⁱ_i = g^ijT_ij = \rho-3p

My attempt:

Tr(T) = Tⁱ_i = g^ij[\rhou_iu_j-p(g_ij-u_iu_j)]
= [\rhog^iju_iu_j-pg^ijg_ij+pg^iju_iu_j)]
= \rhou - p + pu

which doesn't equal rho-3p (eqn2) as required, so I've done something wrong.
I think I've contracted incorrectly but I don't know why... please help?

 

[George Jones]

What are the following?

g^{i j} g_{i j} = ?

g^{i j} u_i u_j = ?

 

[tetris11]

Well,
g^{i j} u_i u_j = 1

g^{i j} g_{i j} = ??
uh... g? or 0?

Might need to help me out here, maths isn't my first language...

 

[Bill_K]

g^ij g_ij = δⁱ_i = 4.

 

[tetris11]

Cheers man, that actually makes complete sense - but just for the record:

gij gij = δii = n, where n is number of dimensions?

I'm just wondering how you knew it was four without knowing how dimensions it was.
Tensors aren't all 4-d, right?

 

[TobyC]

Cheers man, that actually makes complete sense - but just for the record:

gij gij = δii = n, where n is number of dimensions?

I'm just wondering how you knew it was four without knowing how dimensions it was.
Tensors aren't all 4-d, right?

Well this is the 'Special & General Relativity' board so it was probably just a good guess?