Recent content by ilp89

Yes, it is. Alright - everything works out perfectly now. Thanks again.

Just to check: I also need to use \square \phi = g^{ab} \nabla_a \nabla_b \phi - correct?

PeterDonis - Thank you very much! You figured it out. Indeed \nabla_a \phi = \partial_a \phi since \phi is a scalar field, but the second covariant derivative must incorporate the connection coefficients and the \partial_a \phi's. I feel silly for overlooking this - my mind was stuck for...

You are right - I have accidentally been using \phi' and \varphi' interchangeably, since they differ only by a multiplicative factor. You can see this from equation (2.4) since \phi_0 is an arbitrary constant. But I still don't understand why \varphi' should appear anywhere in any equation of...

Does anyone have any ideas? I might need to reproduce this calculation in front of my professor tomorrow evening, so I will be extremely grateful for any help soon. Chris

Maybe I should clarify my question, in order not to scare people away. What I am asking should not take too much time for someone familiar with general relativity. I am only asking that you look at equation (2.1a), mentally insert (2.4) into it, and let me know how a phi-prime appeared in all...

I have a question about the paper: C. G. . Callan, R. C. Myers and M. J. Perry, “Black Holes In String Theory,” Nucl. Phys. B 311, 673 (1989). I have attached the relevant section. I am having trouble using equations (2.1) and (2.4) to derive (2.5) and (2.6). When I do the calculation...

The problem statement: A particle travels along the x-axis with uniformly accelerated motion. At times t and s its position is x and y, respectively. Show that its acceleration is a = 2(yt-xs)/ts(s-t). The attempt at a solution: I could be wrong, but it seems to me this problem is...

Thanks a lot.

But I think we've strayed from the initial question... Does anyone know if there's something similar to Sylvester's criterion for a countably infinite square matrix?

I guess I haven't really told you about the specific problem I'm working on. My infinite matrix is mapping a Banach space into itself, so the matrix operator itself lives in a Banach space, which has a topology.

I guess if all principal minors are positive, then the infinite matrix is the limit of positive definite matrices and is thus positive semi-definite.

Yes, but isn't that just the natural way to word the condition in the finite-dimensional case? Does the proof actually use the fact that the entire matrix's determinant is positive or that the determinant is finite-dimensional?

Well, Sylvester's criterion only requires us to find determinants of the principal minors, which are all finite square matrices. There are just an infinite number of them.

Does Sylvester's Criterion hold for infinite-dimensional matrices? Thanks!