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How do we know that the direction of increasing time inside the event horizon a Schwarzchild black hole is that of decreasing $r$, instead of increasing $r$? Both directions would be timelike, but how do we know which is 'future-pointing'?

In other words, is there an easy way to see that gravitational collapse produces a black hole instead of a white hole?

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In the Schwarzschild solution, the time orientation beyond the horizon at $r=2M$ can be found by considering a particle moving through it. Since nothing physical is happening to the particle at $r=2M$ (as can be seen by using some other coordinate system, for example Penrose's), its future and past lightcones must point to the same spacetime directions before and after crossing. If the particle was entering the horizon from outside, his future lightcone pointed inwards just before entrance, and therefore must point inwards just after entering. This continuity argument shows that there is no white hole inside the horizon.

One may ask another question: whether there is another, different solution (not Schwarzschild) which looks like Schwarzchild outside the horizon, but is different inside. The answer to that is No - due to Birkhoff's theorem.

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The Schwarzschild spacetime is time-orientable, hence it is possible to define a time orientation inside the horizon that matches what you have outside of it.

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  • $\begingroup$ Do you mean that a black hole would always match and a white hole would never match the orientation outside? $\endgroup$ – Andrew Aug 2 '17 at 11:02

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