Can the future affect the past in general relativity? Hypothetically, let us suppose that a black hole were to suddenly appear at time $t=1$ at position $x$. Can the effect of the black hole be felt at time $t = 1-\epsilon$ near $x$ due to the bending of space-time by the black hole?
 A: Technically: in general relativity, if there there exist a Cauchy hypersurface (that is, a subset of space-time which is intersected by every inextensible, non-spacelike (i.e. causal) curve exactly once) then the spacetime is globally hyperbolic. That means there is a homeomorphism from the spacetime manifold $M$ to $S \times \mathbb{R}$ where $S$ is the 3D Cauchy hypersurface (Geroch 1970). 
Less technically: if there are Cauchy hypersurfaces - a generalisation of "an instant of time" that can be extended everywhere - then spacetime behaves as the product of the shape it has on the surface and some time coordinate. 
At the very least this rules out making closed causal curves since there is a well-defined time direction. Sudden appearances of things that change the topology of spacetime are also banned. 
Global hyperbolicity ensures predictability. A system is "predictable" if its state on a Cauchy surface uniquely determines its state at any future point. A physical theory is "prognostic" if all systems described by the theory are predictable. Theories such as relativistic mechanics and electrodynamics are prognostic theories on globally hyperbolic manifolds. If predictability does not hold then the extra information needed to determine the state at a point needs to either "come out of nowhere" somehow, or arrive along spacelike curves implying superluminal information transfer. Conversely, prognostic theories do not allow superluminal signals. For globally hyperbolic spacetimes and predictable matter fields general relativity is prognostic: the metric of spacetime is uniquely determined by the field equations and knowing it on a Cauchy surface (Hawking 1973,Krasnikov 2006).
A: Nope. The bending of spacetime itself propagates at the speed of light so a nearby observer would not experience the sudden appearance of gravity sooner than (s)he could detect the black hole's appearance with a telescope.
That said, note that per Ben Crowell's comment below, it isn't possible in the first place to make a black hole or any other chunk of mass appear suddenly from out of nowhere. 
