Yesterday, I met Einstein constraint equations in a thesis? I failed to understand them. Do they have physical meaning? And what do they "constrain"?
[Physics] What are Einstein constraint equations
general-relativity
Related Solutions
As asked in the comments, here is one answer :
One formalism where it is somewhat common to expand the Einstein equations into a full set of equations is the Newman-Penrose formalism. Not quite common as it uses both spinors instead of tensors and the coordinates are weird complex null-vectors, but it should give an idea of the whole thing.
https://en.wikipedia.org/wiki/Newman%E2%80%93Penrose_formalism#NP_field_equations
First, we notice that the paths traced by particles through spacetime under the influence of a gravitational field seem to depend only on their positions and velocities, i.e. they are independent of any identifiable charge or composition of the particles. It is almost as if the particles were moving along tracks carved into some curved surface.
Although even Galileo noticed this, we happen to be passingly familiar with differential geometry to the point where we can actually make something of it. We realize that, if this observation really held, it would be possible to describe the curvature mathematically by assigning to spacetime a nonzero Riemann curvature tensor $R^a_{bcd}$. This object describes the rotation of a vector when dragged along a closed loop through the spacetime.
What should we couple $R^a_{bcd}$ to? Well, gravity is obviously sourced by mass somehow, but we know because we invented special relativity that mass and energy are intimately connected. We also know we can uniquely describe the energies and pressures of classical matter fields in a nicely coordinate-independent way by their stress-energy tensor $T_{ab}$. So, making the leap that perhaps pressure can source gravity as well, we write down
$$R^a_{bcd} = -\kappa T_{ab}.$$
Unfortunately this equation makes no sense because the RHS and LHS have a different number of indices. No matter - we can just contract some of the indices on the left hand side. This gives us the Ricci tensor $$R_{ab} \equiv R^j_{ajb}$$ so we write $$R_{ab} = -\kappa T_{ab}.$$
This equation predicts the perihelion precession of Mercury, so we are very happy. Unfortunately it has a rather serious flaw: $T_{ab}$ had better have identically zero divergence if energy is at least locally conserved - i.e. if energy does not simply appear out of nowhere in small regions of spacetime. But $R_{ab}$ unfortunately can have nonzero divergence.
Again, no problem - we can simply subtract the divergence: $$R_{ab} - \frac{1}{2}g_{ab}R = -\kappa T_{ab}$$ where $$R\equiv R^a_a$$ is the so-called Ricci scalar. Actually this is a bit restrictive: the divergence is determined only up to an arbitrary constant $\Lambda$. Thus: $$R_{ab} + \left(\Lambda - \frac{1}{2}R\right)g_{ab} = -\kappa T_{ab}.$$
What about the free parameter $\kappa$? This we can't fix with the theory, so it will have to be measured, and then set to enforce agreement with experiment. In particular, in the limit where the curvature is very weak, we should be able to reproduce Newtonian gravity. It turns out to do this we have to set $\kappa = 8\pi$, so that we end up with
$$R_{ab} + \left(\Lambda - \frac{1}{2}R\right)g_{ab} = -8\pi T_{ab}.$$
Best Answer
Constraint equations, in the sense of the "Einstein constraint equations", arise when you try to write out an initial value formulation of GR. The idea behind an initial value formulation, in general, is to show that if I hand you a set of data about the fields everywhere in space at some initial moment in time, that you can always find a solution to the equations of motion whose values matched the initial data at that initial moment. This is similar in spirit to providing initial conditions for a plain old Newtonian point mass: if I tell you the initial position and the initial velocity of a particle moving in a potential, then there's a unique solution to Newton's Second Law for the rest of time which satisfies these initial conditions.
So what are constraints? Well, it turns out that in some (many?) field theories, you can't just pick any set of initial conditions you want. Before we go to GR, let's deal with an easier example of this: electrodynamics. If we write out the equations for the scalar & vector potential in Lorenz gauge, in the absence of charge, we have \begin{align*} \Box^2 V &= 0 & \Box^2 \vec{A} &= 0 & \frac{\partial V}{\partial t} + \nabla \cdot \vec{A} &= 0. \end{align*} Now, the first two of these equations are just the wave equation; and we know that if you tell me the initial amplitude of a wave and its initial rate of change at every point in space (i.e., $\psi(\vec{r}, t_0)$ and $\dot{\psi}(\vec{r}, t_0)$), then I can find a solution for $\psi(t)$ that satisfies these initial conditions. So you might think that if I hand you $V(\vec{r}, t_0)$, $\dot{V}(\vec{r}, t_0)$, $\vec{A}(\vec{r}, t_0)$, and $\dot{\vec{A}}(\vec{r}, t_0)$, then you can always find a solution to the above equations. Right?
Wrong, of course. Because we can't freely specify our initial data; they must obey an initial-value constraint. In this case, the Lorenz gauge condition demands that $\dot{V}(\vec{r}, t_0)$ is exactly equal to $-(\nabla \cdot \vec{A})(\vec{r}, t_0)$. However, so long as you hand me a set of initial data that satisfies this constraint initially, then the solutions of the wave equation that you find will satisfy this constraint for all time.
So what does this all have to do with GR? Well, when we try to do the same thing for GR, we first have to split up our spacetime into a foliation (i.e., a set of 3D spacelike hypersurfaces that are parametrized by a time coordinate $t$.) Einstein's equations can then be entirely rewritten in terms of $h_{ab}(t)$, the induced metric on the hypersurfaces; and $K_{ab}(t)$, the extrinsic curvature of the spacelike hypersurfaces. (Morally speaking, $K_{ab}$ can be thought of as the time derivative of $h_{ab}$.) But, as in the case of electrodynamics, you can't just write down an arbitrary $h_{ab}(t_0)$ and $K_{ab}(t_0)$ on your initial hypersurface; they are constrained relative to each other, by the equations $$ D_b K^b {}_a - D_a K^b {}_b = 0, \qquad {}^{(3)}R + (K^a {}_a)^2 - K_{ab} K^{ab} = 0, $$ where $D_a$ is the covariant derivative operator on the initial hypersurface and ${}^{(3)}R$ is its associated Ricci scalar. This is analogous to how $\dot{V}$ and $\vec{A}$ couldn't be freely specified at $t_0$, but rather were related to each other.
I'm glossing over a lot of detail here, but that's the basic gist of it. The best reference I know of for this subject is Wald's General Relativity, in Chapter 10; the notions of constraints in classical field theory are also covered quite nicely in the first section of Dirac's Lectures on Quantum Mechanics (not to be confused with his Principles of Quantum Mechanics.)