Contracting Tensors: g^{\mu\nu}g_{\mu\nu}=4 & T

[pleasehelpmeno]

Am i right in thinking:

g^{\mu\nu}g_{\mu\nu}=4 \mbox{ and } g^{\mu\nu}T_{\mu\nu}=T ?

 

[WannabeNewton]

The first one is certainly true in 4 dimensions. The second is correct yes assuming by T you mean the trace.

 

[pleasehelpmeno]

sorry I mean T is the scalar stress energy tensor (maybe the same thing). Yeah sorry in 4-D.

 

[Fredrik]

g^{\mu\nu}T_{\mu\nu}=T ?

The left-hand side is a scalar. The right-hand side is not.

 

[pleasehelpmeno]

sorry shouldn't it be T^{\mu \nu}_{\mu \nu} = T?

 

[Fredrik]

sorry shouldn't it be T^{\mu \nu}_{\mu \nu} = T?

Components of the stress-energy tensor have two indices, not four. You could however define T by $T=T^\mu{}_\mu$. The right-hand side is defined by $T^\mu{}_\mu =T^{\mu\nu}g_{\mu\nu}$.
$$g^{\mu\nu}T_{\mu\nu} =g^{\mu\nu} g_{\mu\rho} T^{\rho\sigma} g_{\sigma\nu} =\delta^\nu_\rho T^{\rho\sigma} g_{\sigma\nu} = T^{\nu\sigma} g_{\sigma\nu} = T^\nu{}_\nu=T.$$

 

[pleasehelpmeno]

thank you