What does it mean that Einstein's equations are hyperbolic-elliptical?

The key point of this distinction is the type of initial conditions you have to give for an equation.

The canonical example of a hyperbolic set of equations is the wave equation, where the characteristic polynomical that you get when you do a Fourier transform in all of the variables gives you a graph of a hyperbola in configuration space. For this type of equation, you specify the initial value for your wave on a spacelike slice, and then you evolve that in time in a way predicted by the equation of motion you started with.

The basic example of an elliptical set of equations would be Poisson's equation--here, you don't evolve in time, so you don't specify an initial time value to evolve the equation. Rather, you specify the value of the function on some boundary, and then you integrate out/in from that boundary using the equation to find the values elsewhere. In the most basic case, think about when you chose a surface where $V=0$ when you were solving electrostatics problems. This often arises when solving for constraints in dynamical system.

Both of these are relevant in the case of numerical relativity, since, in an ADM formulation, the evolution equations of the metric and the extrinsic curvature are hyperbolic equations, while the two constraint equations, the Hamiltonian and Momentum constraints, are elliptical.

And obviously, you can also have parabolic equations--which is the edge case of the two, but acts in a way most like hyperbolic equations, with initial time values that evolve. The most famous of these is the diffusion equation. This will show up when you do a double-null decomposition of Einstein's equation and attempt to evolve it.

v. Good answer Jerry. May I add an examples from another field. The electromagnetic field is described by a wave equation which is hyperbolic. An elliptic field allows for closed paths which is not allowed in a hyperbolic field. So if we wanted to create mass(a trapped energy or a trapped momentum), we need an elliptic field augmentation. An example is the Dirac operator. see this ref and quote; https://en.wikipedia.org/wiki/Elliptic_operator ''On the other hand, a weakly elliptic first-order operator, such as the Dirac operator can square to become a strongly elliptic operator, such as the Laplacian. The composition of weakly elliptic operators is weakly elliptic ''.

Then we note that a wave equation like; fxx=c^2 ftt for example, can be turned into an elliptic equation with the help of imaginary numbers. So if c is imaginary or t=time is imaginary or x- is imaginary, the equation becomes fxx+c^2 ftt=0 since i^2=-1, which is elliptic. Thus if we wanted to create mass(elliptic fields) within a bigger massless Hyperbolic field, we need to introduce a complex factor. This is what the complex Higgs mechanism is doing to give mass to massless particles.