Everything is surely ok if it has a compact support. But in this case it works fine, as it leaves "zero trace" near infinity.
Say I have a vector field defined on a space such as Actually, it is AdS space im working on but I don't think that matters for what I want to ask. I assume that I can write the coordinates as (time,radial, p-angles)
Now if I take the integral of the divergence of the field
then by the divergence theorem this gives zero if K dies of sufficiently quickly at spatial (and temporal)infinity i.e. (r --->infinity)
Everything works as I want it to if I take it to die of as 1/r or quicker. Unless I have made a mistake, then any other possibility doesn't work like I think it should.
Can any one reassure me about this.
lol, sorry but I don't have a clue what you just said.
I made my 'assumption' to make the answer do what I hoped it would and am relying on the words 'sufficiently quickly' But I still have to justify why I have assumed that a 'fall off' of 1/r is enough to ensure the divergence integrates to zero over spacetime.
Ofcourse it doesn't. Just saw you have time+radial+space=2+p dimensions.
I mean the divergence theorem applies when your vector field is zero near infinity (="compactly supported") over space+time. Are you interested in radial dimensions only? If yes, remember that you must have a finite integral over time also.
Now the divergence cannot go to zero slower than . Can you see why?
My expression for K at large r is
Where depends only on t and Omega , and X is a vector, possibly with some t and omega dependance I think but no r dependance. can be safely assumed to die at temporal infinity sufficiently quickly. Now I want to arrange p and so that . The second and third fourth terms have the highest power of r. These are what I am thinking I should arrange to be 1/r, and then it does what I want it to.
I should also say, that the factor of sqrt(g) where g is the metric determinant is included in K, so there need not be any sqrt(g) in the integral, and it is not the covariant divergence that I'm interested in as K is a tensor density, not a tensor.
Oh, and by the way. X is a killing vector, so its covariant derivative is zero by killing's equation, and I am hoping that where depends p and m and this condition will impose the correct restrictions on m.
I am sort of managing to convince myself that I am right
where is the p-sphere of radius r. And the order of integration cannot be changed.
Now, there's something else I am worried about...
lol, im not, I told you that here.
Not to worry, there's a lot of details that would take forever to write down. Thanks anyway.