Identification of Spacelike hypersurfaces as diffeomorphisms

While developing the Hamiltonian formulation of General Relativity, we break up spacetime into space and time. This is done by foliating the spacetime with spacelike hypersurfaces and defining lapse function and shift vector.

But in the book by Wald, it is mentioned that the hypersurfaces are identified with the integral curves of a vector field $t^a$.

In order to answer this question, we must view general relativity as describing the time evolution of some quantity. Let $(M, g_{ab})$ be a globally hyperbolic spacetime. (We consider only globally hyperbolic spacetimes since the initial value formulation should be relevant only in this case). As proven in theorem 8.3.14, we can foliate $(M, g_{ab})$ by Cauchy surfaces, $\Sigma_t$, parametrized by a global time function, $t$. Let $n^a$ be the unit normal vector field to the hypersurface $\Sigma_t$. The spacetime metric, $g_{ab}$, induces a spatial metric (i.e., a three-dimensional Riemannian metric) $h_{ab}$ on each $\Sigma_t$ by the formula

$$h_{ab} = g_{ab} + n_a n_b \tag{10.2.10}$$

Let $t^a$ be a vector field on $M$ satisfying $t^a \nabla_a t = 1$. We decompose $t^a$ into its parts normal and tangential to $\Sigma_t$ by defining the lapse function, $N$, and the shift vector, $N^a$, with respect to $t^a$ by

\begin{eqnarray} N &=& -t^a n_a = (n^a \nabla_a t)^{-1}, \tag{10.2.11}\\ N^a &=& h_{ab} t^b \tag{10.2.12} \end{eqnarray}

(see Fig. 10.2).

We may interpret the vector field $t^a$ as representing the "flow of time" throughout spacetime. As we "move forward in time" by parameter time $t$ starting from the $t = 0$ surface $\Sigma_0$, we go to the surface $\Sigma_t$. If we identify the hypersurfaces $\Sigma_0$, $\Sigma_t$ by the diffeomorphism resulting from following integral curves of $t^a$, we may view the effect of "moving forward in time" as that of changing the spatial metric on an abstract three-dimensional manifold $\Sigma$ from $h_{ab}(0)$ to $h_{ab}(t)$. Thus, we may view a globally hyperbolic spacetime $(M, g_{ab})$ as representing the time development of a Riemannian metric on a fixed three-dimensional manifold.

How is this identification made? How can hypersurfaces be diffeomorphisms resulting from the integral curves of a vector field?

• In the cited text, hypersurfaces are not identified to diffeomorphisms, they are identified by using diffeomorphisms. Jun 11, 2018 at 8:44
• Then what does that mean? Jun 11, 2018 at 9:16

If we identify hypersurfaces $\Sigma_0,\Sigma_t$ by the diffeomorphism resulting from following the integral curves of $t^a$, we may view the effect of "moving forward in time" as that of changing the spatial metric on an abstract three-dimensional manifold $\Sigma$ from $h_{ab}(0)$ to $h_{ab}(t)$.

I believe the answer is written there quite explicitly. So let us break it down to definitions. A diffeomorphism is a map from a manifold to another manifold which is infinitely-differentiable1 and invertible, with an inverse which is also infinitely-differentiable1. While an integral curve of a field $X$ is a curve $\gamma$, such that at each point $p$ in the curve, its tangent vector coincides with the vector field i.e. $X_p = \dot{\gamma(\lambda)}$ where $\gamma(\lambda) = p$. If you consider the full set of integral curves you obtain the flow of $X$, that is you have to parameters, one which specifies the integral curve you are in and the second one specifies where in that integral curve you are.

So if you consider the submanifold $\Sigma_0$ (3-surface), that is the Cauchy surface specified by the normal vector $n^a$ at time $t=0$, what the flow (or integral curves) of $t^a$ does to $\Sigma_0$ is to move the points around smoothly and returns $\Sigma_t$ to you. So in that sense it is a diffeomorphism (all vector field flows can be identified with local diffeos). In the book it is not stated

that hypersurfaces are identified with integral curves of a vector field $t^a$.

that statement is not what the book meant.

Footnotes

[1]: It could also be fixed to a certain order but this is not relevant here.