Meaning of r in Schwarzchild coordinates

[pervect]

I was doing some more reading about the general case, and I suspect that Frobenius' theroem may be involved in the more general solution.

But it's all terribly abstract, I'm not getting much of a "physical" feel.

"Hypersurface orthogonality" is mentioned as a special case of Frobenius' theorem (see pg 434, for instance).

 

[PWiz]

I was doing some more reading about the general case, and I suspect that Frobenius' theroem may be involved in the more general solution.

But it's all terribly abstract, I'm not getting much of a "physical" feel.

"Hypersurface orthogonality" is mentioned as a special case of Frobenius' theorem (see pg 434, for instance).

I don't know much about this, but after some reading on the net, it would appear that the theorem only applies to homogenous, linear 1st order PDEs. The EFEs are coupled, non-linear 2nd order PDEs. Can the theorem be applied to GR?

 

[PeterDonis]

Can the theorem be applied to GR?

Yes. See, for example, Wald, Appendix B.3. Google Books has it here:

https://books.google.com/books?id=9...=frobenius theorem general relativity&f=false

 

[PeterDonis]

it would appear that the theorem only applies to homogenous, linear 1st order PDEs.

Can you give a reference? The discussions I've seen (such as the one in Wald) don't talk about differential equations at all; they talk about manifolds and vector fields or differential forms. (Also, in GR the theorem isn't applied to the EFE; it is applied after you've already solved the EFE, to the manifold with metric that you obtain as the solution.)

 

[PWiz]

...the theorem addresses the problem of finding a maximal set of independent solutions of a regular system of first-order linear homogeneouspartial differential equations.

I read this. I'll emphasize again - I know nothing about this theorem. I've yet to go through a completely formal treatment in GR and topology.

 

[PeterDonis]

I read this.

Yes; the differential equations referred to are not Einstein's Field Equations, so the fact that the EFE is second-order and nonlinear is irrelevant. The equations referred to are equations describing integral curves of vector fields in a spacetime manifold that is a solution of Einstein's Field Equations. AFAIK those equations are first order and linear. (The Wiki page also describes a geometric interpretation of what is going on, which is much closer to how the theorem is applied in GR.)

 

[aleazk]

Can you give a reference? The discussions I've seen (such as the one in Wald) don't talk about differential equations at all; they talk about manifolds and vector fields or differential forms. (Also, in GR the theorem isn't applied to the EFE; it is applied after you've already solved the EFE, to the manifold with metric that you obtain as the solution.)

Check "Introduction to Smooth Manifolds" by J.Lee. From page 361: "...explicitly finding integral manifolds boils down to solving a system of PDEs, we can interpret the Frobenius theorem as an existence and uniqueness result for such equations." And then it goes to show this connection.