Conservation laws in a curve spacetime

In fact, given a Killing field, you define the conserved quantity as :(5) I = u^a K_a,where K_a is the Killing field. In this case, the conserved quantity is the momentum, as I said in the previous post.In summary, the conservation laws that can be computed from \nabla_b T^{ab}=0 in a curved space-time include the covariant divergence of the energy-momentum tensor and, if there is a continuous symmetry in the space-time, a conserved quantity defined by a Killing field. In the case of a pressureless gas, this results in the conservation of matter and the geodesic equation. No specific metric is needed to define conserved
  • #1
Magister
83
0

Homework Statement



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

[tex]
\nabla_b T^{ab}=0
[/tex]

in a curve space-time.

Homework Equations



[tex]
T^{ab}=(\rho + p) u^a u^b - p g^{ab}
[/tex]

where p is the pressure and [itex]\rho[/itex] 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
 
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  • #2
Just write [tex]\nabla_b T^{ab}=0[/tex] explicitly in components (in terms of density, pressure and scale factor). You will get a conservation type eqn.
 
  • #3
Dick said:
Just write [tex]\nabla_b T^{ab}=0[/tex] 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 [itex]g^{ab}[/itex] term...
 
  • #4
I guess I was thinking of the problem in FRW background. Can you simplify anything w/o a metric? Not sure...
 
  • #5
Well. I have already compute the equation of motion, in a curve space, from [itex]\nabla_b T^{ab}=0[/itex]. So I suppose that there may be another one...
 
  • #6
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.
 
  • #7
What about this:

[tex]

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

[/tex]

where I have used [itex]\nabla_b g^{ab} = 0[/itex]

[tex]
u^a \nabla_b((\rho + p) u^b) + (\rho + p) u^b \nabla_b u^a -g^{ab} \nabla_b p = 0
[/tex]

[tex]
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
[/tex]

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

[tex]
\nabla_b((\rho + p) u^b) = g^{ab} u_a \nabla_b p
[/tex]

using this in the second equation I get

[tex]
(\rho + p) u^b \nabla_b u^a = 0
[/tex]

or

[tex]
\nabla_u u^a = 0
[/tex]

So I can saw that the equation of motion (a conservation law) can be derived from [itex]\nabla_b T^{ab}=0[/itex].
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.
 
Last edited:
  • #8
From equation

[tex]

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

[/tex]

you get the following :

(1) [tex]
\nabla_a ((\rho + p) u^a) = u^a \nabla_a p,
[/tex]

AND :

(2) [tex]
(\rho + p) u^a \nabla_a ( u^b) = (g^{ab} - u^a u^b) \nabla_a p.
[/tex]

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

(3) [tex]
\nabla_a (\rho u^a) = 0,
[/tex]

AND :

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

1. What are conservation laws in a curve spacetime?

Conservation laws in a curve spacetime refer to the principles that govern the conservation of physical quantities, such as energy, momentum, and angular momentum, in a curved space. These laws are derived from the fundamental principles of general relativity and have been extensively studied in the field of theoretical physics.

2. How do conservation laws apply in a curved spacetime?

In a curved spacetime, conservation laws are still valid but must be modified to account for the effects of gravity. This is because the presence of matter and energy in a curved space affects the curvature of that space, which in turn affects the behavior of physical quantities. Therefore, the conservation laws must be expressed in terms of the curvature of spacetime.

3. How do conservation laws differ in a curved spacetime compared to a flat spacetime?

In a flat spacetime, conservation laws are expressed in terms of the conservation of energy and momentum. However, in a curved spacetime, these laws must also account for the curvature of spacetime and its effects on physical quantities. This means that the conservation laws are more complex and must be expressed in terms of the curvature of spacetime.

4. Are conservation laws always valid in a curved spacetime?

Yes, conservation laws are always valid in a curved spacetime. However, the equations that govern these laws may be more complex and must take into account the effect of gravity on physical quantities. Additionally, in extreme conditions such as near black holes, the laws of conservation may break down due to the extreme curvature of spacetime.

5. How do conservation laws in a curve spacetime impact our understanding of the universe?

Conservation laws in a curve spacetime play a crucial role in our understanding of the universe. They help us understand how matter and energy behave in the presence of gravity and how the curvature of spacetime affects the behavior of physical quantities. The study of conservation laws in a curved spacetime has also led to important discoveries, such as the existence of black holes and the expansion of the universe.

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