Value of energy-momentum tensor in weak field approximations

In summary: The assumption that the Sun is filled with dust is valid because the pressure inside the Sun is too low to be felt by us.
  • #1
peter46464
37
0
My first question, so sorry if it's in the wrong forum.

I'm trying to understand the Newtonian weak field approximations to general relativity. I can't see why, if the Schwarzschild metric (which can describe the gravitational field around the Sun) is a vacuum solution ([itex]T_{\mu\nu}=0[/itex] ) , do textbooks state that [itex]T=\rho c^{2}[/itex] when approximating Poisson's Equation (which also describes the gravitational field around the Sun) from the Einstein Field Equations?

Thank you .
 
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  • #2
In Newtonian gravitation the gravitational potential inside the sun obeys Poisson's equation, ∇2 = -4π Gρ where the mass density ρ ≠ 0. Outside the sun ρ = 0 and we have Laplace's equation ∇2 = 0. General Relativity works the same way except the source is the stress energy tensor Tμν, or in the case of the sun just T00 = ρc2.
 
  • #3
Sorry, I still don't see why if [itex]T^{00}=\rho c[/itex] there can be a Schwarzschild solution for [itex]T^{\mu\nu}=0[/itex] . Why aren't these two contradictory statements?
 
  • #4
I'm not positive I understand your difficulty, but I'll take a guess. Said guess being that you're cofused by how you can get gravity if T_uv = 0 everywhere.

If you look at a Newtonian point charge, the charge density is zero everywhere except for the point, where the charge density is infinite.

So except for one singular point, the charge density is 0.

The Schwarzschild solution is vaguely similar in that you have an infinite singular stress-energy tensor at the center (the sigularity), and it's zero everywhere else, i.e. a vacuuum.
 
  • #5
Take a look at Bill's post again. In order to understand the answer to your question you have to understand the two different situations Possion, and Laplace's equations hold. Laplace's equation talks about a region of space that is void of sources, Possion's equation describes a region that has sources in that region.

Take the Schwarzschild metric, it is a solution of
[tex]
R_{\alpha\beta}=0
[/tex]
Now take the weak field metric, and without any assumptions for now about the sources, to order in [itex]1/c^2[/itex], what does the Einstein tensor reduce to? Now what are the conditions we should put on [itex]T_{\alpha\beta}[/itex] to make our Newtonian limit make sense?

Do you see how Laplace's equation is a special case of Possion's equation?
 
  • #6
So when we approximate Poisson's equation

[itex]\nabla\cdot\nabla\phi=4\pi G\rho [/itex]

by assuming [itex]T=\rho c^{2} [/itex]

we are doing it for a point inside a gravitational source (eg the Sun).

However, the Schwarzschild solution is for points outside a gravitational source, where [itex]T=0 [/itex]

I think the fog may be clearing a little.

But one new confusion. When I read my textbook I see that the derivation of the Newtonian limit is based on the assumption that we are dealing with a region filled with dust (ie no internal pressure). So we are assuming (still using the Sun as our example) that the Sun is “dust”. How is that a valid assumption? When I think of the Sun I think of this great seething mass of gas, with lots of pressure. So it's OK to assume that, even if for the Sun that
[itex]pressure\ll density [/itex]

?

Thanks for persevering.
 

1. What is the energy-momentum tensor in weak field approximations?

The energy-momentum tensor is a mathematical object used in general relativity to describe the distribution of energy and momentum in spacetime. In weak field approximations, it represents the energy and momentum of a gravitational field in a region where the gravitational effects are small.

2. How is the energy-momentum tensor calculated?

The energy-momentum tensor is calculated using the Einstein field equations, which describe how matter and energy affect the curvature of spacetime. It involves solving a set of differential equations that take into account the distribution of mass and energy in a given region of spacetime.

3. Why is the value of energy-momentum tensor important in weak field approximations?

The energy-momentum tensor is important in weak field approximations because it allows us to make predictions about how gravitational fields will behave in regions where the effects of gravity are small. This is especially useful for understanding the dynamics of objects in our solar system, such as planets and satellites.

4. How does the energy-momentum tensor relate to the conservation of energy and momentum?

In general relativity, the energy-momentum tensor is derived from the principle of conservation of energy and momentum. This means that the total energy and momentum of a system are conserved, even in the presence of gravitational fields. The energy-momentum tensor provides a mathematical representation of this principle.

5. Can the energy-momentum tensor be used to study strong gravitational fields?

The energy-momentum tensor is not as useful in studying strong gravitational fields, as it is only accurate in regions where the gravitational effects are small. In these cases, we need to use more advanced mathematical tools, such as the full equations of general relativity, to accurately describe the behavior of space and time in the presence of strong gravitational fields.

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