Conservation laws in a curve spacetime

[Magister]

Homework Statement

Given the energy-momentum tensor for a perfect fluid, what is the conservation laws that I can compute from

$$\nabla_b T^{ab}=0$$

in a curve space-time.

Homework Equations

$$T^{ab}=(\rho + p) u^a u^b - p g^{ab}$$

where p is the pressure and $\rho$ is the density.

The Attempt at a Solution

I have already compute the geodesic equation, the equivalent to the equation of motion in a flat space-time.
I would like to know if there is any other conservation law to get. I supose that might be another one equivalent to the Navie-Strokes in flat space-time...

Thanks in advance

 

[Dick]

Just write $$\nabla_b T^{ab}=0$$ explicitly in components (in terms of density, pressure and scale factor). You will get a conservation type eqn.

 

[Magister]

Just write $$\nabla_b T^{ab}=0$$ explicitly in components (in terms of density, pressure and scale factor). You will get a conservation type eqn.

In terms of scale factor? The problem is that they don't give me a metric. I don't know what to with the $g^{ab}$ term...

 

[Dick]

I guess I was thinking of the problem in FRW background. Can you simplify anything w/o a metric? Not sure...

 

[Magister]

Well. I have already compute the equation of motion, in a curve space, from $\nabla_b T^{ab}=0$. So I suppose that there may be another one...

 

[Barnak]

In general, there isn't any other conservation law, besides the covariant divergence of the energy-momentum tensor. However, if you have a space-time with a continuous symetry, there's a Killing field defined on the space-time, associated to that symettry. It can then serves to define another conservation law, in a similar way as a Noether current.

 

[Magister]

What about this:

$$<br /> \nabla_b T^{ab}=\nabla_b((\rho + p) u^a u^b) - g^{ab} \nabla_b p = 0<br />$$

where I have used $\nabla_b g^{ab} = 0$

$$u^a \nabla_b((\rho + p) u^b) + (\rho + p) u^b \nabla_b u^a -g^{ab} \nabla_b p = 0$$

$$u_a u^a \nabla_b((\rho + p) u^b) + (\rho + p) u^b u_a \nabla_bu^a -g^{ab} u_a \nabla_b p = 0$$

using the fact that $\nabla_b (u_a u^a) = \nabla_b 1 \Leftrightarrow 2 u_a \nabla_b u^a = 0$, I get

$$\nabla_b((\rho + p) u^b) = g^{ab} u_a \nabla_b p$$

using this in the second equation I get

$$(\rho + p) u^b \nabla_b u^a = 0$$

or

$$\nabla_u u^a = 0$$

So I can saw that the equation of motion (a conservation law) can be derived from $\nabla_b T^{ab}=0$.
Am I wrong?

In general, there isn't any other conservation law, besides the covariant divergence of the energy-momentum tensor. However, if you have a space-time with a continuous symetry, there's a Killing field defined on the space-time, associated to that symettry. It can then serves to define another conservation law, in a similar way as a Noether current.

Just has I said above, they don't give me any metric, so I can't compute the Killing vectors.

 

[Barnak]

From equation

$$<br /> \nabla_b T^{ab}=\nabla_b((\rho + p) u^a u^b) - g^{ab} \nabla_b p = 0,<br />$$

you get the following :

(1) $$\nabla_a ((\rho + p) u^a) = u^a \nabla_a p,$$

AND :

(2) $$(\rho + p) u^a \nabla_a ( u^b) = (g^{ab} - u^a u^b) \nabla_a p.$$

In the case of a pressureless gaz, p = 0, and you get the conservation of "matter" AND the geodesics equation :

(3) $$\nabla_a (\rho u^a) = 0,$$

AND :

(4) $$u^a \nabla_a ( u^b) = 0.$$There's no need of any particular metric to define conserved quantites with the Killing fields.