Integration by parts in general relativity - boundary terms and Stokes theorem

Question

I am trying to prove that the stress-energy-momentum Hilbert tensor satisfies a conservation law if one assumes diffeomorphism invariance of general relativity. I have taken the definition of the SEM Hilbert tensor field to be: $$T^{\mu \nu}=\frac{-2}{\sqrt{-det(g)}}\frac{\delta(\mathcal{L}_{M}\sqrt{-det(g)})}{\delta g_{\mu \nu}}$$ $\delta_{X}S_{M}$ is the variation originating from some difffeomorphism acting on the manifold, which corresponds to some vector field $X$ in $\Gamma(TM)$. Following the calculation performed by Uldreth in this comment: https://physics.stackexchange.com/a/394031/202198

I come upon the following expression:

$$ 0=\delta_{X}S_{M} = -\int_{M} T^{\mu \nu} (\nabla_{\partial\mu} X)_{\nu} \sqrt{-det(g)}d^{4}x $$ Now, integrating it by parts, at which I fail miserably, ought to yield something resembling (I think!):

$$ 0 = \delta_{X}S_{M} = \int_{M}\text{Div(...)} + \int_{M} (\nabla_{\partial\mu} T)^{\mu \nu} X_{\nu} \sqrt{-det(g)}d^{4}x $$

The rough idea I have is to try to use some covariant Stokes-Gauss law which I cannot really recognise in my situation - and then cross the boundary terms out due to the variation vanishing there. I am not even sure what should be in the first integral - the divergence of some tensor? Perhaps multiplied by $\sqrt{-detg}$?

Could someone help me work out in detail the steps leading to the final expression? Additionally, but not necessarily, how could one formulate all of the above in coordinate-free notation, where the Stokes theorem is much easier to operate with for me?

Answer 1

Your question is essentially about integral theorems. So let us see. You have the expression (up to a sign) $$ I=\int T^{\mu\nu}\nabla_\mu X_\nu\ \sqrt{-\det g}d^nx. $$

Utilizing the derivation property of $\nabla$, we can write this as $$ I=\int\nabla_\mu(T^{\mu\nu}X_\nu)\sqrt{-\det g}d^nx-\int \nabla_\mu T^{\mu\nu}X_\nu\ \sqrt{-\det g}d^nx. $$

So in the divergence term, you have the divergence of a vector field, which is $Y^\mu=T^{\mu\nu}X_\nu$.

The covariant formulation of Gauss' theorem is as follows. Let $\mathcal D$ be a regular domain in the manifold, with boundary $\partial\mathcal D$. Let us denote the coordinates in $M$ with $x$ and the coordinates in $\partial \mathcal D$ by $\xi$. The induced metric is $h_{ij}=(\phi^\ast g)_{ij}$ where $\phi:\partial\mathcal D\rightarrow M$ is the inclusion map.

The Gauss' theorem is $$ \int_{\mathcal D}\nabla_\mu X^\mu\sqrt{-\det g}d^nx=\int_{\partial\mathcal D}n_\mu X^\mu \sqrt{|\det h|}d^{n-1}\xi. $$

Here $n$ is the (usually taken to be) outward-pointing unit normal to the boundary. The boundary's orientation is defined such that if $\{n,e_2,...,e_n\}$ is an positively oriented orthonormal basis of $M$ (along $\partial\mathcal D$), then $e_2,...,e_n$ is considered to be a positively oriented orthonormal basis of $\partial\mathcal D$.

Despite the index notation, this is essentially already coordinate free. Let $đ\mu_g$ be the $n$-form on $M$ whose local expression is given by $$ đ\mu_g=\sqrt{-\det g}dx^1\wedge...\wedge dx^n $$ and let $đ\mu_h$ be the $n-1$-form on $\partial \mathcal D$, whose local expression is given by $$ đ\mu_h=\sqrt{|\det h|}d\xi^2\wedge...\wedge d\xi^n. $$

We define for any vector field $X$ the divergence as either $\text{div}X=\text{Tr}\nabla X$ or as $\text{div}(X)đ\mu_g=\mathcal L_Xđ\mu_g$, two expressions, whose equivalence can be readily verified. Then we have $$ \int_\mathcal D \text{div}Xđ\mu_g=\int_{\partial \mathcal D}\langle n,X\rangle đ\mu_h. $$

This result actually follows from the generalized Stokes theorem for differential forms, which is $$ \int_{\mathcal \partial D}\omega=\int_\mathcal D d\omega $$ (where $\omega$ is an $n-1$-form). To see that this does follow from the generalized Stokes theorem, consult Introduction to Smooth Manifolds by John Lee or General Relativity by Straumann. The omitted proofs are also found in these books.