- #1

fluidistic

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## Homework Statement

Hi guys,

I would like to show that if ##t^\mu## is a temporal vector then ##t^\mu t^\nu T_{\mu\nu}## is the density of energy of the EM field measured by an observer with velocity ##t^\mu##. And that it is greater or equal to 0.

Density of energy is proportional to ##E^2+B^2## if my memory doesn't fail me.

## Homework Equations

Mostly tensor "operations"?

Def. of the S-E tensor: ##T^{\mu \nu }=\frac{1}{4\pi} \left ( F^{\mu \sigma } F^{\nu \rho} \eta _{\sigma \rho}-\frac{1}{4} \eta ^{\mu \nu } F^{\sigma \rho} F_{\sigma \rho} \right )##

## The Attempt at a Solution

I'm trying to grasp the basics of tensors through self study. I would like some feedback on this: ##T_{\mu \nu}=\frac{1}{4\pi} \left ( F_{\mu\sigma}F_{\nu\rho}\eta^{\sigma\rho}-\frac{1}{4} \eta_{\mu\nu}F_{\sigma\rho}F^{\sigma\rho} \right )##. Is this expression correct? I just used the definition given and lowered upper indices and rised lower indices.

My idea is to perform/analyse first ##t^\nu T_{\mu\nu}## and then contract it with the velocity vector ##t^\mu##.

So I wrote out the terms of ##t^\nu T_{\mu\nu}##, and since ##t^\nu## is temporal I chose it as (1,0,0,0) in my mind. So the only non zero "terms" are ##t^0T_{00}+t^0T_{10}+t^0T_{20}+t^0T_{30}=\sum _{\mu=0}^3 T_{\mu0}##.

I am not really sure what this sum is. It is a sum of what exactly? Of scalars??? Can't be right... because then ##t^\mu## multiplied by a scalar is another vector and the density of energy is supposed to be a scalar....

More thoughts: Then ##t^\mu t^\nu T_{\mu\nu}=t^\mu T_{\mu 0}=t^0 T_{00}+t^1 T_{10}+t^2 T_{20}+t^3 T_{30}=c\underbrace{t}_{\text {time}}T_{00}+t^1 T_{10}+t^2 T_{20}+t^3 T_{30}##.

Where I considered ##t^\mu=(ct,t^1,t^2,t^3)##, the 4-velocity.

Now I guess I must evaluate ##T_{\mu 0}## and replace back into the expression I got for ##t^\mu t^\nu T_{\mu\nu}##.

Is this correct so far?