Contracting tensors

1. Feb 24, 2013

Am i right in thinking:

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

2. Feb 24, 2013

WannabeNewton

Re: contracting tenors

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

3. Feb 25, 2013

Re: contracting tenors

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

4. Feb 25, 2013

Fredrik

Staff Emeritus
Re: contracting tenors

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

5. Feb 25, 2013

Re: contracting tenors

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

6. Feb 25, 2013

Fredrik

Staff Emeritus
Re: contracting tenors

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 $T^\mu{}_\mu =T^{\mu\nu}g_{\mu\nu}$.
$$g^{\mu\nu}T_{\mu\nu} =g^{\mu\nu} g_{\mu\rho} T^{\rho\sigma} g_{\sigma\nu} =\delta^\nu_\rho T^{\rho\sigma} g_{\sigma\nu} = T^{\nu\sigma} g_{\sigma\nu} = T^\nu{}_\nu=T.$$

7. Feb 25, 2013