Vacuum Solutions: Are They Always Divergence Free?

[quasar_4]

Are vacuum solutions to the Einstein field equations always divergence free? How would one test this assumption?

 

[quasar_4]

actually, let me rephrase this question (it doesn't make much sense). If I understand correctly, the stress-energy tensor for the vacuum case is always the zero tensor. Since the Einstein equation is also divergence free, how does one verify the validity of vacuum solutions? It seems that for dust solutions, there's the option to test whether the divergence of the stress-energy tensor is zero. I am wondering if there's anything analogous in the vacuum case.

 

[Mentz114]

I've wondered the same thing. If the Einstein tensor is identically divergenceless then every space-time that allows the tensor to be calculated is a candidate for a valid solution. If the result is not zero, then what's to stop me from calling it the SET and claiming I have a solution ?

There must be other conditions to be satisfied, as you suggest. This surely is covered in standard texts but I don't remember seeing it.

 

[lbrits]

What Mentz114 said is true. It's not hard to find solutions to Einstein's equations. But then you have some random stress energy tensor. What's much harder is to solve it for a specific matter configuration, with specific initial/boundary conditions and specific symmetries.

Similarly, I can write down almost any $$A_\mu$$ and claim that I have a solution to Maxwell's equations. I would then have to infer where the sources are.

 

[nrqed]

actually, let me rephrase this question (it doesn't make much sense). If I understand correctly, the stress-energy tensor for the vacuum case is always the zero tensor. Since the Einstein equation is also divergence free, how does one verify the validity of vacuum solutions? It seems that for dust solutions, there's the option to test whether the divergence of the stress-energy tensor is zero. I am wondering if there's anything analogous in the vacuum case.

You simply have to check if Einstein's tensor is zero, no?

 

[quasar_4]

it sounds right to me (what on Earth else could there be?). But I was afraid it was too good to be true... lol...

I suppose once you begin talking about initial and boundary conditions then things get much harder much faster. And I know some vacuum solutions only work with specific side conditions as well.