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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 …

Let us call the spacetime $M$ with a metric $g_{ab}$. There is a unit spacelike vector field $\eta^a$ orthogonal to a hypersurface. So that we can define the so-called gravitational electric and magne …

Consider a particular conformal transformation $x^\mu\rightarrow x'^\mu$, and the metric of a flat space transforms in the following way, $$\eta_{\mu\nu}\rightarrow g'_{\mu\nu}=\Lambda^2(x)\eta_{\mu\ …

Einstein-aether theory is a theory of gravity with the local Lorentz violation. In addition to the metric tensor, it contains a unit timelike vector field, called aether $u^a$. Because of the constrai …

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 lost one term, which is the one containing $\nabla^t(\partial_t)^r=g^{tt}\Gamma^r_{tt}=-M/r^2$, so this term is $$ -\frac{1}{8\pi}\int r^2\sin\theta \nabla^t(\partial_t)^r d\theta d\phi=\frac{M}{8\ …

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 …

The tensor product combines two lower rank tensors into a higher rank one. For example, you can put two vectors $v^a$ and $w^b$ together to create a rank-2 tensor $v^aw^b$, which can be thought as a m …

In Wald's GR, Komar integral is Eq. (11.2.9): $$M=-\frac{1}{8\pi}\int_S\epsilon_{abcd}\nabla^c\xi^d$$ $S$ can be chosen as a 2-sphere, the boundary of a spacelike hypersurface $\Sigma$ such that the …