I’ve been reading a book that goes into a lot of detail and I am wondering about a thought sequence and its validity (and doing some text searching on my Kindle has been unsuccessful in determining if he eventually addresses/mentions this).
So. A black hole with a mass of X has a Schwarszchild radius of Y. Another mass gets too close, falls into the black hole, the mass of the black hole increases and the Schwarszchild radius increases accordingly.
Except that while this is what happens from the point of view of the planetary body it isn’t what happens from the point of view of an outside observer. To that observer, the other mass would appear to experience time dilation to a point of essential infinity as it reaches the event horizon and would never actually be seen crossing it and therefore the size of the black hole would never increase (though the total gravity pf the system would since from the outside observers point of view it is all just mass over there).
So, am I correct in this understanding and that it therefore means that once a black hole comes into existing at mass X radius Y, it can never, within the life of the universe be observe to exist with a larger mass/radius?
If all of that is correct then a second question about evaporation (which I am even less certain about).
For a black hole to evaporate to nothing in an observable timeframe, what mass would have to evaporate? The mass/energy observed to be in it by the outside world, or the mass observed to be in it by the mass/energy that have fallen into it?
Feelings won’t be hurt by responses of “oof, you’ve got this all so painfully wrong. Go back to Trixie Belden books.”
AIUI objects only experience time dilation when they have mass and approach the speed of light. Objects consumed by black holes aren’t necessarily travelling at relativistic speeds.
Anyway; gravity and acceleration are equivalent in relativity, and both produce time dilation. Even the gravity of Earth is enough to produce measurable time dilation with modern instruments. GPS satellites for example have to take time dilation into account due to being farther away from Earth’s mass.
If I’m not mistaken, the infalling mass’s gravity alters the observed event horizon, so yes after a finite time the event horizon increases in size.
Strictly speaking, what’s being observed is an event horizon which serves as a “sieve” for particle-pair production. So technically you’re observing mass/energy appearing out of the vacuum of space with the black hole swallowing the energy deficit, and gradually growing smaller as a result.
An essential problem describing this situation is that a Schwarzschild black hole has a very high degree symmetry: it’s spherically symmetric and it’s stationary (the spatial coordinates can be made independent of the time coordinate), whereas the situation of a mass falling into a black hole is not spherically symmetric and it’s not time-independent. So describing what happens exactly is very difficult and in fact there aren’t any known exact solutions for this situation.
However it is known that Kerr black holes (of which a Schwarzschild black hole is the zero angular momentum case) are stable: a small perturbation of a Kerr black hole will evolve towards a Kerr black hole. As the solution is asymptotically flat, it also known energy is conserved. We can therefore say that sometime not too long after the black hole absorbs the mass it will settle down into something that looks very, very similar to a Kerr black hole with the mass of the original black hole+whatever proportion of the mass it absorbed from the in-falling object.
That is good enough for most people as physics is about making sensible approximations, but in many ways it skips around your question. One thing to realize is that for a true BH event horizon any event inside it must be in the future of any event outside it. So in some ways it is sensible to say nothing can cross the event horizon from the point of view of a faraway observer because any such event must always lie in the future from their pov.
The argument for black hole evaporation is that 1) the Hawking effect in semiclassical gravity has been demonstrated to create a negative energy flux across the BH event horizon 2) this negative energy flux is actually greater the smaller the black hole 3) there is a meaningful definition of energy conservation for asymptotically flat spacetime. Therefore to conserve energy the mass of the black hole decreases at an increasing rate until it reaches zero in some finite time. But there is no actual definitive model for the process though that I’m aware of where the evaporation isn’t put in by hand.
As we would assume the black hole is settled into a state which is well-approximated by all the matter that has fallen into it (including matter that we might argue is actually hovering just above the event horizon), we would assume that all the matter must be taken into account when calculating the evaporation time though.
It’s actually two sensible questions, which aren’t entirely non-controversial.
It’s worth noting that all meaningful results for black holes have been derived in a classical, or at best semiclassical, context.* This is only an approximation, and we know that that approximation breaks down at some point. If we knew of anything better to use, we’d use it, but we don’t.
*“Classical” means that we’re ignoring quantum mechanics. “Semiclassical”, in this context, means that we’re treating the stuff falling in (or radiating out) as being quantum mechanical, but that we’re still treating spacetime itself as classical.