Vacuum solutions in general relativity

[exponent137]

If we look Einstein equation
$G_{\mu\nu}=0$, which nontrivial aspects it can gives?
One phenomenon is gravitational wave.
I suppose that we are also free at initial conditions for this equation, thus we can begin with curved space time? What else?

 

[PeterDonis]

There are many different families of known vacuum solutions to the Einstein Field Equation. A brief summary of some of them is here:

https://en.wikipedia.org/wiki/Vacuum_solution_(general_relativity)

[Note: I have changed the thread title to be more descriptive of the question.]

 

[exponent137]

I thought this quotation:
"In fact, however, the equations of general relativity are perfectly consistent with spacetimes that contain no matter at all. Flat (Minkowski) spacetime is a trivial example, but empty spacetime can also be curved, as demonstrated by Willem de Sitter in 1916".
https://einstein.stanford.edu/SPACETIME/spacetime2.html

I suppose that curved spacetime mainly can be curved as initial condition, if gravitational waves are ignored.
I think that solutions of this empty spacetime are rarer than your wikipedia link gives?

 

[PeterDonis]

I suppose that curved spacetime mainly can be curved as initial condition

You are confusing curvature of spacetime as a whole with curvature of an initial spacelike hypersurface. "Initial condition" is talking about an initial spacelike hypersurface. I'm talking about the curvature of spacetime as a whole; the Wikipedia link I gave lists a number of classes of entire spacetimes--4D manifolds--that are solutions of the vacuum Einstein Field Equation. Initial conditions are irrelevant to that list; for any solution of the EFE that is globally hyperbolic (which all of the ones listed in the Wikipedia article are), you can pick any Cauchy surface (roughly any 3D spacelike hypersurface that covers all of "space" at an instant of time) and evolve it both forwards and backwards in time to obtain the full 4D manifold. So there is no one set of "initial conditions" for the full solution, and every Cauchy surface in such solutions is empty (vacuum). (In most of the solutions, every Cauchy surface is curved as well, considered as a 3D manifold; but I'm not sure if that's true of all of them.)

I think that solutions of this empty spacetime are rarer than your wikipedia link gives?

No. You appear to be unduly pessimistic about the variety of properties an "empty spacetime" can have. See above.

 

[exponent137]

Yes, this is true, although I do not have absolute mental image about this.

But, let us assume, that the universe is without matter all the time. I think that Schwarzschild solution does not exist in such universe, thus the variety of properties is reduced. I think that such universe has many forms of gravitational waves. As second, empty spacetime can be curved, as it is written above. It is a consequence of the number of free parameters is larger than it is given by $G_{\mu\nu}=0$. This is what I thought as initial conditions. Maybe, edge conditions is a better word?

 

[martinbn]

I am not sure what you are after, but it seems to me (and I might be wrong of course) that you are confused about the initial value problem. The initial data is not a space-time, and the equations don't determine its evolution. The space-time is the final solution. The initial data consists of what you can call space (with its Riemannian metric and second fundamental form) and the equations determine its evolution through what one could call time. The whole history of that space in time is the space-time that satisfies Einstein's equation, in the case of interest the vacuum equation $G_{\mu\nu}=0$.

You also seem to be very confident about some properties of the vacuum solutions and how rare some of them are. Do you have any reason for that or is it just intuition?

 

[PeterDonis]

let us assume, that the universe is without matter all the time. I think that Schwarzschild solution does not exist in such universe

You are incorrect. The Schwarzschild solution is a vacuum solution, so it is a possible solution for a "universe without matter all the time". Any vacuum solution is.

If you were to say that the Schwarzschild solution is not a physically realistic solution for a universe without matter all the time, I would agree with that; the "white hole" singularity in the full Schwarzschild solution is not, IMO, physically realistic. But to say the solution "does not exist" is not correct; it exists, it just isn't (IMO) physically realistic. Lots of solutions to the EFE exist that aren't physically realistic.

I think that such universe has many forms of gravitational waves.

There are also vacuum solutions to the EFE that contain gravitational waves, yes. But they are not the only vacuum solutions. So you can't just assert that a universe without matter must be one of the gravitational wave solutions. That isn't correct.

I would not even say the vacuum everywhere gravitational wave solutions are more physically realistic than the Schwarzschild solution, because if the universe is vacuum everywhere, there are no sources to produce gravitational waves, so a physically realistic solution should not contain them.

It is a consequence of the number of free parameters is larger than it is given by $G_{\mu\nu}=0$.

I'm not sure what you mean here. If you just mean that the condition $G_{\mu \nu} = 0$ does not pick out a unique solution, of course it doesn't.

This is what I thought as initial conditions. Maybe, edge conditions is a better word?

Not really. What differentiates the various types of possible vacuum solutions is something more like symmetry assumptions--for example, spherical symmetry is the key assumption that leads to the Schwarzschild solution. Such assumptions are not really initial conditions or boundary conditions.

 

[exponent137]

You also seem to be very confident about some properties of the vacuum solutions and how rare some of them are. Do you have any reason for that or is it just intuition?

$G_{\mu\nu} = 0$ means one parameter less according to general Einstein equation. In infinite points this means infinite less parameters ...

 

[PeterDonis]

$G_{\mu\nu} = 0$ means one parameter less according to general Einstein equation. In infinite points this means infinite less parameters ...

I don't know what you mean by this, but it doesn't mean there is a unique vacuum solution to the EFE. Multiple families of vacuum solutions have been given to you. There does not seem to be any point in further discussion, so this thread is closed.