In a Physics paper the author states without proof something that seemed quite strange to me. The paper is on General Relativity, so he assumes a Lorentzian manifold $(M,g)$ is given.
His hypothesis are:
Suppose $f$ is a continuous scalar function on $M$ whose support $W$ extends to past and future infinity but which is bounded in spacelike directions, and take $L$, parametrized as $\gamma(s)$ by the proper time $s$ along it, to be a timelike worldline representing an observer. We shall now take $\Sigma(s)$ to be an arbitrary spacelike hypersurface through $\gamma(s)$ which depends continuously on $s$. Then if $w^\alpha$ is a vector field such that displacement of every point by $w^\alpha ds$ maps $\Sigma(s)$ into $\Sigma(s+ds)$ for each $s$, we have $$\langle f,\phi\rangle=\int ds \int_{\Sigma(s)} f\phi \sqrt{-g}w^\alpha n_\alpha d\Sigma,$$ for all compactly supported $\phi\in C^\infty_0(M)$ being $n^\alpha$ the normal to $\Sigma(s)$.
I believe that it all boils down to this: given a one-parameter family of spacelike hypersurfaces $\Sigma(s)$, and a vector field $W$ such that its flow satisfies $\Phi^X_\delta(\Sigma(s))=\Sigma(s+\delta)$, we have
$$\int_M f(x) d^nx=\int ds \int_{\Sigma(s)} f \sqrt{-g} W^\alpha n_\alpha d\Sigma.$$
I might be wrong though, and it might only work with $\phi$ and with the hypothesis on the support of $f$.
The thing is: how this is proven? Actually, if $\phi : (-\epsilon,\epsilon)\times M\to M$ is the flow of $W$ so that $\phi_t : M\to M$ is the diffeomorphism moving points a parameter value $t$ on the integral curves, we can consider the set
$$A = \{\phi(s,p) \in M :s\in (-\epsilon,\epsilon),p\in \Sigma(s)\}$$
Then if $\varphi : U\subset \mathbb{R}^{n-1}\to \Sigma(s)$ is a parametrization of the hypersurface, we shall have a parametrization $\psi : (-\epsilon,\epsilon)\times U\to A$ given by
$$\psi(s,q)=\phi(s,\varphi(q)).$$
Let $(y^\alpha)$ a coordinate system on $(-\epsilon,\epsilon)\times U\subset \mathbb{R}^n$ and $(x^\mu)$ a coordinate system on $M$
Now pick $f\in C^\infty_0(M)$, the $n$-form $f\epsilon$ has compact support, where $\epsilon$ is the Lorentzian volume form. But if $h : M\to \mathbb{R}$ is $h = \sqrt{|g|}$ what we want is to integrate $\omega = fh dx^0\wedge\cdots \wedge dx^{n-1}$. But by a theorem on Spivak DG Vol. 1 (7.7) we have
$$\psi^\ast \omega=(fh\circ\psi) \det\left(\dfrac{\partial(x^\mu \circ \psi)}{\partial y^\alpha}\right) dy^0\wedge \cdots dy^{n-1}$$
The issue now is to compute that determinant. I concluded (not much rigorously really) that
$$\dfrac{\partial(x^\mu \circ \psi)}{\partial y^0}=W^\mu\circ \psi$$
While for the other I computed not rigorously also that
$$\dfrac{\partial(x^\mu \circ \psi)}{\partial y^i}=\dfrac{\partial(x^\mu\circ \phi)}{\partial x^\nu}\dfrac{\partial (x^\nu\circ \varphi)}{\partial y^i}$$
The first seems to be the Jacobian of the flow of $W$, the second seems to be the components of $\varphi_\ast e_i$ so the basis of the tangent spaces of $\Sigma$ induced by the basis of $\mathbb{R}^{n-1}$. If I'm not mistaken this is the same as
$$\dfrac{\partial(x^\mu \circ \psi)}{\partial y^i}=\phi_{\ast}(\varphi_\ast e_i)=(\phi\circ\varphi)_\ast e_i$$
So the first column of the matrix are the components of $W$ and the $i$-th column are the components of the $i$-th basis vector of $\mathbb{R}^n$ pushed to $M$ throguh $(\phi\circ\varphi)$ - the parametrization followed by the flow.
Now I think that the normal covector $n$ is
$$n = \star (\phi\circ\varphi)_\ast e_1\wedge \cdots \wedge (\phi\circ\varphi)_\ast e_{n-1}$$
The matrix above has $W$ in one column and these vectors on the others. Computing the determinant with the Levi-Civita symbol $\varepsilon_{\mu_1\dots\mu_n}$ gives
$$\det(J)=\varepsilon_{\mu_1\dots \mu_n} W^{\mu_1} (\phi\circ\varphi)_\ast e_1^{\mu_2}\cdots e_{n-1}^{\mu_n}$$
But then combining with the factor $h\circ \psi$ we factored will turn the Levi Civita symbol to $\epsilon_{\mu_1\cdots\mu_n}$ which in turn gives the Hodge dual
$$\psi^\ast \epsilon = (f \epsilon_{\mu_1\cdots \mu_n}\circ \psi W^{\mu_1})\circ \psi(\phi\circ\varphi)_\ast e_1^{\mu_2}\cdots(\phi\circ\varphi)_\ast e_{n-1}^{\mu_n} dy^0\wedge \cdots \wedge dy^{n-1}= (fW^\mu n_\mu)\circ \psi dy^0\wedge\cdots \wedge dy^{n-1} $$
I just don't know if this is right. Is my approach correct?