Compact support of matter fields

[haushofer]

Hi,

I have a question concerning the asymptotic boundary conditions on matter fields and the Riemann tensor. What is the precise relation between saying that "the matter fields go to zero at spatial infinity" and "the matter fields have compact support"? And how natural is it to state that the Riemann tensor has "compact support" on a certain spacetime? I would say that if the matter fields have compact support, the Riemann tensor also has, right?

Thanks!

[haushofer]

It seems to me that people often abuse the concept of "having compact support" whenever they actually mean a certain asymptotic behaviour. Can anyone comment on this?

[Stingray]

To say that the matter fields have compact support means that their intersection with a Cauchy surface has compact support in the mathematical sense. This is usually used loosely to mean that all matter fields (exactly) vanish outside of some finite spatial region. Saying that the matter fields fall off at some rate near infinity is therefore a weaker condition.

Einstein's equation implies that the Ricci tensor will have compact support if the matter fields do. The same is not true of the Riemann tensor. It will usually fall off at some rate as one moves away from the matter, but there is no radius beyond which it will vanish.

[haushofer]

Ok, thanks! But in three dimensions compactly supported matter fields then do imply that the Riemann tensor has compact support, right?

[Stingray]

Yes, that's right.