According to an answer to this question, it is possible for two events to be connected by both spacelike and timelike curves even in a globally hyperbolic spacetime. The example given is the "canonical Lorentzian cylinder" which has a cylindrical topology ($R\times S$) and a Lorentzian metric $ds^2=dt^2-d\theta^2$. Now as far as I understand the Schwarzschild spacetime also has a "hyper-cylindrical" topology $R^2\times S^2$ (See What are the topological properties of a Schwarzschild black hole, and of its horizon and singularity? and references therein), so this made me wonder: does the above property exist also in the Schwarzschild spacetime? Namely, can we find there two events that are connected by both a spacelike and a timelike curve?
1 Answers
In general, of course, for the same reason you can do it it Minkowski spacetime. In flat spacetime, the events $A=(0,0,0,0)$ and $B=(1,0,0,0)$ are connected by both the obvious timelike curves (like $C^\mu(\tau)=(\tau,0,0,0)$) as well as spacelike curves that start at $A$, do a spacelike loop, and then come back to the neighborhood of $B$.
In a black hole spacetime, two events $A,B$ both outside the event horizon can be connected by both a timelike curve (whatever timelike curve works) and a spacelike curve (a curve that starts at $A$, goes into the event horizon, then comes out and hits $B$).
There will be greatly constrained results if you consider only geodesics connecting two events. I can’t immediately think of a case where two events can be connected by two different geodesics of different likeness, but it is possible I am mistaken in this regard.
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