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By definition, the induced metric $h_{ab}$ is given by $$h_{ab}|_pu^av^b=g_{ab}|_pu^av^b,$$ where $u^a,v^b$ are arbitrary tangent vector fields to the hypersurface, and this expression is valuated a …

Penrose gave a very brief proof to this question. Since the spacetime is paracompact, there exists a positive definite metric called $h_{ab}$. Then, the nowhere vanishing time-like vector field $V^a$ …

Let $S$ be a smooth, compact, 2-dimensional manifold with a positive-definite Riemannian metric $g_{ab}$ with a compatible covariant derivative $\nabla_a$. I want to show that there exists a unique t …

The proof to Geroch's claim uses the fact that the manifold is 2-dimensional. Thanks to @JamalS. In this case, any antisymmetric tensor, such as $\nabla_{[a}\xi_{b]}$, is proportional to the volume el …

I contacted R. Wald. He told me that the variation operation $\delta$ does not act on $\xi^a$. So $$\delta\boldsymbol Q[\xi]\ne\mathscr L_\eta (\boldsymbol Q[\xi]).$$ But they wanted to use Lie deri …

In this work, Wald & Zoupas developed a framework to define the "conserved quantities" in a diffeomorphism-invariant theory using the covariant phase space formalism. Let $\xi^a$ be a vector field o …

The equation of Hughston & Tod is correct. One way to check this is to use the conformal transformation. In fact, $V^a$ is a conformal Killing vector. This conformal Killing vector induces a diffeomor …