Local Existence & Uniqueness of Vacuum EFE Solutions

[ShayanJ]

When I was taking a look at this page, I noticed that she is "known for proving the local existence and uniqueness of solutions to the vacuum Einstein Equations". But this doesn't make sense to me(the uniqueness part). Just consider the Minkowski and Schwarzschild solutions. They're both vacuum solutions but are not equivalent. So is it a mistake in the above page or am I misunderstanding something?
Thanks

 

[martinbn]

The Minkowski and Schwarzschild solutions don't have the same initial data. It is not enough to look at the Einstein's equations alone. You need to do the 3+1 split and formulate the question as an initial value problem. Then prove the existence and uniqueness.

 

[ShayanJ]

The Minkowski and Schwarzschild solutions don't have the same initial data. It is not enough to look at the Einstein's equations alone. You need to do the 3+1 split and formulate the question as an initial value problem. Then prove the existence and uniqueness.

How are the initial data different between those two?
Is it correct to say that the only difference in their foliations is that in the Schwarzschild case the hypersurfaces are $S^3$ while in the Minkowski case they're 3D Euclidean spaces?

 

[martinbn]

I cannot give you the details out of the top of my head, you'll need to look it up, I believe most texts will have the examples. In the initial data the parameter M of the Schwarzschild solution appears(it comes when you solve the constrained equations) and if not zero you don't get Minkowski.

ps The slices are not compact.

 

[PeterDonis]

How are the initial data different between those two?

The Cauchy surfaces for these two spacetimes have different topology and different intrinsic and extrinsic curvatures. (Note that "initial data" is something of a misnomer; data on any Cauchy surface will do, even if it is not the "first" such surface in the spacetime--that notion makes no sense for either of the spacetimes under discussion.)

Is it correct to say that the only difference in their foliations is that in the Schwarzschild case the hypersurfaces are $S^3$ while in the Minkowski case they're 3D Euclidean spaces?

No, because Cauchy surfaces (spacelike hypersurfaces that intersect every causal curve exactly once) in (maximally extended) Schwarzschild spacetime are not 3-spheres. They have topology $S^2 \times R$, and their geometry depends on how and where in the maximally extended spacetime you "cut" them.

 

[ShayanJ]

I searched the internet and also the tables of contents of several numerical relativity books, but I found no numerical derivation of the Schwarzschild spacetime. Is there any website that I can find such code samples in? Preferably in C++ or Python!
Or at least a guide to the calculations needed to write such codes, specifically for the Schwarzschild case.

 

[Ben Niehoff]

When I was taking a look at this page, I noticed that she is "known for proving the local existence and uniqueness of solutions to the vacuum Einstein Equations". But this doesn't make sense to me(the uniqueness part). Just consider the Minkowski and Schwarzschild solutions. They're both vacuum solutions but are not equivalent. So is it a mistake in the above page or am I misunderstanding something?
Thanks

The question is not really "Do solutions to the Einstein equation exist and are they unique?", but rather "When do solutions to the Einstein equation exist, and what data needs to be provided to fix a unique solution?" (Also, existence can be further refined to questions like "What conditions must the initial data satisfy in order that a solution exist?" and "Given initial data satisfying these conditions, for how long in the future and/or past will the solution exist?").

By "vacuum Einstein equation", one means the equation with zero sources and zero cosmological constant. The cosmological constant can make the answer quite a bit messier.

For example, with a negative cosmological constant, initial data is not sufficient to guarantee uniqueness; one also has to specify data along the conformal boundary all the way from past infinity to future infinity. The intuitive reason for this is that signals can propagate from the interior to infinity (and back!) in finite time; thus anti-de-Sitter spacetime is like a universe "in a box". One needs to specify not only the initial conditions within the box, but also boundary conditions that detail all the radiation that might enter or leave the box throughout all of its history. Often this means a complete lack of predictability (i.e., the Cauchy problem is not well-posed).