The following passage came from a 1973 article authored by Robert Georch and titled "ENERGY EXTRACTION". The author tries to prove the positive-energy theorem. I would like to ask two questions:
What does the author mean by nested topological two-spheres? After searching on the internet, I know a topological sphere would probably mean a topological space that is homeomorphic to a sphere. But what are nested topological spheres? Also, does the surface $t=\text{const}$ refer to a level set of the mapping $t:S\to\mathbb{R}$? As an example, is $t=0$ symbolic of the level set $\phi^{-1}(0)$?
What's the meaning of the expression $\phi\xi^a\nabla_a t=1$? If we adopt Einstein's summation convention, how do we interpret $\xi^a\nabla_a t$? I never saw it in my collection of references on geometry and relativity.
Thanks for help.
Edit. Sorry, my bad. I just noticed the author had declared the $\nabla_a$ in (Q2) to be the covariant derivative on $S$. But still, I don't know why he includes the index $a$ to write $\nabla_a$. It doesn't look like pretty much the covariant derivative I saw in my geometry books. And how does this operator act on $t$?
