Energy & Momentum of Spacetime w/o Mass

[sqljunkey]

So the gravitational field has energy and momentum, and so you can never have a spacetime with an energy stress tensor set to 0. Is this like a constant then? that you can calculate for a portion of spacetime with no "stress energy tensor"?

 

[PeterDonis]

the gravitational field has energy and momentum

You have to be very careful interpreting this ordinary language statement properly. There is a sense in which it is true, but that sense does not mean what you think it does. See below.

you can never have a spacetime with an energy stress tensor set to 0.

This is not correct. The "energy and momentum in the gravitational field" does not appear in the stress-energy tensor; only non-gravitational stress-energy does. So it is perfectly possible to have a vacuum solution to the Einstein Field Equation, i.e., a solution where the stress-energy tensor is zero everywhere. A very commonly used example (other than the trivial one of flat Minkowski spacetime) is Schwarzschild spacetime, which, as a solution with vacuum everywhere, describes a black hole.

 

[sqljunkey]

wouldn't the black hole itself have a mass in the schwarzchild spacetime? or because of the singularity it's discarded? I haven't gotten there yet though...

 

[PAllen]

A gravitational wave traversing space without matter has stress energy tensor identically zero.

If you want some technical terms, the stress energy tensor is related to Ricci curvature. Weyl curvature describes vacuum gravitational effects. A gravitational wave is progagating Weyl curvature. An idealized black hole is all Weyl curvature, with zero stress energy tensor (effectively equal Ricci curvature) everywhere. An 'old' black hole has mass but no matter, with the mass being a property of the Weyl curvature.

 

[sqljunkey]

ok thanks

 

[PeroK]

wouldn't the black hole itself have a mass in the schwarzchild spacetime? or because of the singularity it's discarded? I haven't gotten there yet though...

Although the Schwarzschild spacetime is called a black hole, this doesn't mean what you think it means! The spacetime around any isolated spherical mass, e.g. the Sun to a good approximation, is Schwarzschild. But, in this case the Schwarzschild spacetime only extends as far as the surface of the star. With a black hole, the Schwarzschild spacetime extends all the way to $r = 0$.

The point about the vacuum solution is that you have no additional stress-energy in the spacetime you are looking at. This is really no different from the vacuum solution for the Newtonian gravitational field, where the field is caused by a single central mass and varies across the vacuum outside this mass.

A non-vacuum solution, on the other hand, would be a solution taking into account the mass or stress-energy distributed across the spacetime you are looking at. E.g. the cosmological models of the universe.

 

[sweet springs]

Hi, sqliunky.
As Peter Denis said in #2 Minkowsky spacetime in SR, whose metic is $$ds^2=c^2dt^2 - dx^2 -dy^2 -dz^2$$, belongs to spacetime with no mass you said.
Spacetime with no mass is not necessary Minkowsky spacetime, shown Rotating System as an example. We need a boundary condition to identify the proper spacetime satisfying no mass or everywhere zero energy momentum tensor more precisely.

 

[PeterDonis]

wouldn't the black hole itself have a mass in the schwarzchild spacetime?

The spacetime that describes a black hole that forms from the collapse of a massive object like a star will have nonzero stress-energy somewhere in it, yes. However, that doesn't change the fact that the hole itself is vacuum, and the solution to the Einstein Field Equation that describes it is a vacuum solution, but it is still a solution (and is curved).