- #1

Mmmm

- 63

- 0

## Homework Statement

For a perfect fluid verify that the spatial components of [itex]T^{\mu \nu};_{\nu} = 0[/itex] in the Newtonian limit reduce to

[tex](\rho v^{i}),_{t} + (\rho v^{i} v^{j}),_{j} + P,_{i} + \rho \phi ,_{i}[/tex]

## Homework Equations

Metric

[tex]ds^{2} = -(1+2 \phi )dt^{2} + (1-2 \phi) (dx^{2} + dy^{2} + dz^{2})[/tex]

Christoffel Symbols

[tex] \Gamma^{t}_{t \alpha} = \phi ,_{\alpha}[/tex]

[tex] \Gamma^{t}_{ij} = - \phi ,_{t} \delta _{ij}[/tex]

[tex] \Gamma^{i}_{tt} = \phi ,_{i}[/tex]

[tex] \Gamma^{i}_{tj} = - \phi ,_{t} \delta _{ij}[/tex]

[tex] \Gamma^{i}_{jk} = \phi ,_{i} \delta _{jk} - \phi ,_{k} \delta _{ij} - \phi ,_{j} \delta _{ik}[/tex]

Stress-energy tensor for a perfect fluid

[tex] T^{\alpha \beta} = (\rho + P)U^{\alpha}U^{\beta} + Pg^{\alpha \beta}[/tex]

In Newtonian limit,

[tex]T^{tt} = \rho[/tex]

[tex]T^{ti} = \rho v^{i}[/tex]

[tex]T^{ij} = \rho v^{i}v^{j} + P\delta ^{ij}[/tex]

## The Attempt at a Solution

Spacial components are

[tex]T^{\i \nu};_{\nu} [/tex]

[tex] = T^{it};_{t} + T^{ij};_{j}[/tex]

[tex] = T^{it},_{t} + T^{\alpha t} \Gamma ^{i}_{\alpha t} +T^{i \alpha}\Gamma ^{t}_{\alpha t} + T^{ij},_{j} + T^{\alpha j} \Gamma ^{i}_{\alpha j} + T^{i \alpha} \Gamma ^{j}_{\alpha j}[/tex]

[tex] = T^{it},_{t} + T^{tt} \Gamma ^{i}_{tt} + T^{jt} \Gamma ^{i}_{jt} + T^{it}\Gamma ^{t}_{tt} + T^{ij}\Gamma ^{t}_{jt} + T^{ij},_{j} + T^{tj} \Gamma ^{i}_{tj} + T^{kj} \Gamma ^{i}_{kj} + T^{it} \Gamma ^{j}_{tj} + T^{ik} \Gamma ^{j}_{kj}[/tex]

Then substituting in the above values and simplifying I get

[tex](\rho v^{i}),_{t} + (\rho v^{i} v^{j}),_{j} + P,_{i} + \rho \phi ,_{i} + P\phi,_{i} + \rho v^{k}v^{j} \delta _{kj} \phi ,_{i} - 4 \rho v^{i} \phi ,_{t} - 2 \rho v^{i} v^{j} \phi ,_{j}[/tex]

and as [itex]T^{\i \nu};_{\nu} = 0[/itex], this expression is also = 0.

As you can see the first 4 terms are what I'm looking for, but I somehow have a load of terms on the end which should vanish but I just don't know how...

(if it's helpful I'll post my full working with the simplification etc)

Last edited: