# Poisson, Einstein, Weak Energy Condition

Hello

In newtonian theory Poisson's equation holds: ## \nabla ^{2} U = 4 \pi G \rho ##. So: given a density ##\rho ##, it is possible to find a potential U. On the other hand, I can choose a random function U and give it a gravitational significance if it gives, by Poisson's eq., a density which is always positive.

In General Relativity I must use Einstein's equation: ## G_{\mu \nu} = 8 \pi G T_{\mu \nu} ##.
Thus, given a certain tensor field ##T_{\mu \nu}## i solve the equations and find the right metric ##g_{\mu \nu}##. But I could choose an arbitrary metric, put it in ## G_{\mu \nu}## and see which stress energy tensor describes the matter that bends the space as the given metric tensor says. The problem is exactly the same as in Newtonian theory. But now how can I tell if this stress energy tensor has a physical significance?

• bcrowell

## Answers and Replies

bcrowell
Staff Emeritus
Science Advisor
Gold Member
The bare minimum is that it should satisfy reasonable energy conditions: https://en.wikipedia.org/wiki/Energy_condition . These are requirements that we think all forms of matter (other than dark energy) obey, such as the fact that gravity is attractive, and sound waves can't travel faster than ##c##.

If you have a certain type of matter in mind, e.g., electromagnetic radiation, or nonrelativistic matter ("dust"), then there are more specific requirements you can impose.

If your matter has certain types of quantum-mechanical behavior, then it may disobey all the energy conditions. See Twilight for the energy conditions?, Barcelo and Visser, http://arxiv.org/abs/gr-qc/0205066 .