- #1
binbagsss
- 1,284
- 11
Homework Statement
This should be pretty simple and I guess I am doing something stupid?
##T_{bv}=(p+\rho)U_bU_v-\rho g_{bv}##
compute ##T^u_v##:
##T^0_0=\rho, T^i_i=-p##
Homework Equations
##U^u=\delta^t_u##
##g_{uv}## is the FRW metric,in particular ##g_{tt}=1##
##g^{bu}T_{bv}=T^u_v##
## g^{ab}g_{ca}=\delta^{(4)b} _c=4## if b=c(where I haven't paid attention to order of the indices when lowering and raising since both metric and energy-momentum tensor are symmetric objects)
The Attempt at a Solution
[/B]
##T^u_v=g^{bu}T_{bv}=g^{bu}(p+\rho)U_bU_v-\rho g_{bv}g^{bu}##
##= (p+\rho)U^bU_v-p\delta^u_v##
similarly ##U^u=\delta^u_t##
Now I get the correct components if ##\delta^u_v=1 ## but ## \delta^u_v=4 ## if u=v because we are in 4d-d space-time I thought?
Many thanks