Hamiltonian Formulation of GR, 3+1 decomposition I was reading Wald´s book and in the Hamiltonian formulation they mention that we must do a $3+1$ decomposition of space and time. I´ve seen that the Legendre transformation of the Einstein-Hilbert Lagrangian density is singular, which would lead to constraints. My question is why would the $3+1$ decomposition be a correct thing to do? Why is it required that we separate space and time?
 A: Whether making a 3+1 split is the "right thing to do" depends on the problem you want to solve.  There are a wide range of interesting astrophysical problems where this makes sense as a starting point because you know or think you can model something about "now" and you want to predict how it will evolve into the "future" - deliberately being loose with the sense of now and later for the moment.
An example might be a stellar collapse scenario where you think you can give reasonably good "initial" condition early in the process when relativistic effects are less important but you want to model the system deep into the collapse process where full relativity is needed.  You'd like to turn this into some sort of initial-value problem, but solving this in the fully covariant form of the Einstein equations is not easy because time and space are mixed by the equation in that form.
The original work by ADM was to show how to unravel space and time in the Einstein equations, which allows for formulating exactly this type of initial value problem.  Where I was loose with the sense of now and later above, the ADM formulation (and various subsequent derivative formulations) shows how to be formal about it within the context of relativity.  They also show how to formulate an initial value problem:  Solve the constraint equations on your initial time slice and then use the remaining equations to evolve the solution forward in time.
This is not the only application but it's a reasonably active one given the interest in gravitational wave sources and detection from astrophysical sources.
A: (Following on from my previous comments, another way of saying this is:)
Assume our spacetime manifold $\mathcal{M}$ is globally hyperbolic, i.e. the topology $\mathbb{R} \times \mathcal{S}$ with $\mathcal{S}$ an orientable 3-manifold. This  is equivalent to the manifold being foliated by hypersurfaces $\mathcal{S}_{t}$ where $t$ is a scalar $t \in \mathbb{R}$, and we can write
$$ \mathcal{M} = \bigcup_{t} \mathcal{S}_t \ ,$$
where $\mathcal{S}_t$ are known as the leaves, which we identify with $\{t\} \times \mathcal{S}$. We interpret the scalar $t$ as the global time function and define the following one-form,
$$ \omega_{a} = \nabla_{a} t \ , $$
which is normal to $\mathcal{S}_t$. You can then go on to define the unit normal $n_a$, which is obviously proportional to $\omega_a$ (normalised using the Lapse function, but not important here). The vector field $\omega^a = g^{ab} \omega_b$ is timelike and can be seen as specifying the 'flow of time'. This is in agreement with your most recent comment, but put slightly differently.
