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When a black hole forms and subsequently accretes matter, what happens to the material that is consumed?

There's no need for anything to 'freeze' anywhere. Defined locally time stops for noone. It's actually a very Ptolemaic assumption to use only one clock and ruler to define a universe from. To do so should also invalidate relativity.

the clock is following a shorter and shorter path to the future. In fact, the path gets shorter at such a rate that it actually reaches (our) future infinity in finite proper time

Quotethe clock is following a shorter and shorter path to the future. In fact, the path gets shorter at such a rate that it actually reaches (our) future infinity in finite proper timeI think that perhaps this description has reversed the clocks, and has (wrongly) produced "our future infinity" time when it should have produced a "near-zero time" for the infalling clock.

My reasoning is as follows: Assume we have:- 1. an observer well away from the black hole (who has some understanding of relativity)2. an observer falling directly towards a block hole (ie not orbiting around it)3. Both observers have a local clock4. Both observers are linked by a laser beam which continually transmits the reading of his local clock. Measuring the frequency of this beam allows red shift to be measured.Observer (1) will see observer (2) falling towards the black hole at a speed which rapidly increases to a fair fraction of the speed of light.

Observer (2) extrapolates the speed of observer (1) to predict that observer (1) will hit the event horizon in (say) 10 minutes by the clock of observer (2).

Observer (1) will never actually see observer (2) cross the event horizon, but will see observer (1) reach a point where his laser beam is red-shifted into undetectability.

On the other hand, observer (2) will measure this "same" time as well under 10 minutes, due to gravitational time dilation; the most extreme time dilation ocurring in the last minute and the last second.

So rather than observer (2) seeing the infinite future of observer (1), I expect that he will see only 10 minutes of the future of observer (1); the last minute of this period taking considerably less than a minute by the clock of observer (2).

Predicting the clock of observer (2) after he crosses the event horizon is the source of a variety of theories; whatever it is, we know of no way that observer (2) can inform observer (1) about it, so it must remain a topic of speculation.

What I can't work out is whether observer (2) will see the laser from observer (1) will be red-shifted, blue shifted or unchanged as observer (2) approaches the event horizon; I guess it all depends on the orbit of observer (1) compared to the falling trajectory of observer (2).]

When a black hole forms and subsequently accretes matter, what happens to the material that is consumed? Is the fate of a black hole to relentlessly and indefinitely grow, assuming there is an endless supply of material with which to feed it?

So what happens to the energy in the photons that are captured by a black hole? Is this converted into mass and hence contributes to the black holes gravitational field?

The light entering has energy, which will be added to the black hole. So perhaps it imparts some momentum and makes the black hole move a little bit?