What if the usual predictability of the laws of physics fail in general relativity? In Newtonian mechanics the future is predictable, If we know the initial state of a system exactly, then the laws of physics determine its state arbitrarily far into the future.
What if the usual predictability of the laws of physics fail in general relativity?
 A: Predictability does indeed fail in general relativity, though as far as we know we can never observe such failures because they always occur behind an event horizon. This principle is known as the strong cosmic censorship conjecture. Certain types of black holes contain a surface known as a Cauchy horizon. Provided we don't cross the Cauchy horizon the time evolution is deterministic and therefore completely predictable. However beyond the horizon this is no longer the case.
Until recently we were fairly sure that the Cauchy horizon could only exist inside an event horizon, and therefore was forever hidden from us. So as far as we were concerned general relativity is deterministic in all parts of the universe we could ever observe. This changed with a recent paper suggesting this might not be the case, though I'm not sure to what extent the results of the paper have been accepted by the GR community.
There is a reasonably accessible article on this on the Quanta web site here. The paper is on the Arxiv here, but it will be incomprehensible to most of us so I would stick to the Quanta article unless you are a determined GR head.
A: 
In Newtonian mechanics the future is predictable, If we know the initial state of a system exactly, then the laws of physics determine its state arbitrarily far into the future.

One has to separate the mathematical predictability, i.e. the ability to give a mathematical formula that predicts a situation, and validity i.e. is this a valid prediction and what nature will do in the future?
The answer to the second is that Newtonian mechanics predictions are already invalidated in the earth-satellite system. For GPS to work one has to use both special and General realtivity, even for the very near future.

What if the usual predictability of the laws of physics fail in general relativity

Physicists will have to define new models to  fit and predict in this new regime , if necessary.
A: A short answer to your question is that they don't. Once you specify initial conditions in General relativity (more precisely, once you specify boundary conditions on a Cauchy surface),  the classical phase space functions evolve deterministically. To put it more simply, once the initial conditions are specified (on a Cauchy surface), one can determine the entire past and future of the system in general relativity.
https://en.m.wikipedia.org/wiki/Cauchy_surface
A: General relativity is a pure classic theory (in the sense that is not quantum). Predictability of the dynamics is ensured by imposing global properties of the spacetime. In particular there exist the notion of global hyperbolicity.
Said it short: a universe is globally hyperbolic if it accepts a hypersurface of initial conditions such that no two point of it are causal and satisfying that every point of the spacetime is causally connected with it (a Cauchy surface) 
This question  treats the topic as well, and the reference in it could be helpful. 
