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Is there a limit to the amount of matter that a black hole can accrete per second and if so could a certain sized black hole bounce off a dense enough surface?

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Let's answer the first part of your question first. In general, astrophysical objects are limited in the amount they can accrete by the Eddington limit.

What happens is as matter is accreted onto the object (black hole in our case), it heats up due to conversion of gravitational potential energy into kinetic thermal energy. This hot matter emits photons whose total energy per time is called the luminosity of the object. If this luminosity is high enough (and you should get a higher luminosity for more accreted matter) then the outward pressure of the photons on the matter actually can overcome the gravitational inward pull. This point (rate of matter accretion) is called the Eddington limit.

To partially answer the second part of your question, if the black hole had non-zero total net electric charge, then it could certainly bounce off another material of sufficient density. However if the charge on the black hole was identically zero, there would be no repulsive force to create a bounce.

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In more detail, it is possible for two merging black holes to get a 'kick' from purely gravitational interactions. This paper discusses black hole "superkicks," which are results from simulations where the two merging spinning black holes are oriented in a specific way designed to get the largest 'kick' possible. – jeffdk Feb 26 at 23:32
Thankyou, the Edington limit was what I had in mind but could not recall. I'll have a go at calculating some masses and velocities now and see what I end up with. Would I get better bounces with smaller or larger BHs I wonder;-) – Jitter Feb 27 at 3:17
If you are taking the charged BH scenario, then the important quantity would be the ratio of charge, $q$ to BH mass, $M$. The larger $q/M$ the better the bounce. In fact you could probably calculate a minimum $q$ necessary to get a good bounce by requiring that the electrostatic repulsion is sufficiently great as to force the BH to turn around before it could get close enough to accrete any matter from what it is bouncing off of. – jeffdk Feb 27 at 10:49
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Look, there's no limit to the accretion because the blackhole simply starts growing as it accretes more mass. I don't think it can "bounce off" any surface.

Yes, if the gravitational forces are comparable, for example, if it encounters another black-hole, they can get in equilibrium, bounce off each other or simply merge.

This happens in the universe.

Every galaxy is found to have a supermassive black-hole at the center. We have evidence for galaxies merging along with their black-holes. Sometimes the galaxies just tear each other apart from tidal influence and the black-holes simply don't interact.

Although, I have never heard of anything like bouncing or merging happening on garden-variety black-holes. The truth is, we haven't detected any significant sum of black-holes. All we have and are sure of are the black-holes at center of galaxies.

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LIGO's main purpose is to look for mergers of stellar-mass black holes. It's an expected event, it just hasn't been observed optically. – Jerry Schirmer Feb 26 at 23:02
@JerrySchirmer Exactly! – Cheeku Feb 26 at 23:03
but you can't say that stellar mass black holes are unobserverd. You have plenty of examples of those, such as things like Cygnus X1. – Jerry Schirmer Feb 26 at 23:07
@JerrySchirmer Yes, I didn't actually mean unobserved. I didn't use 'unobserved'. We have like millions of galaxies to run simulations and do data analysis, but not such a "significant sum(number)" of black-holes. – Cheeku Feb 26 at 23:11

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