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As far as I understand it black holes radiate Hawking radiation and probably gravity waves, which over time causes them to lose mass and eventually evaporate after almost unfathomable amounts of time. I've also read that black holes eventually have difficulty gaining size after a certain point because gravitational forces working in a black hole's accreditation disk can eventually begin ejecting in-falling matter before it can be absorbed.

While I know black holes can become quite massive, that leads me to wonder if Black holes have a maximum possible size where they will begin to immediately radiate any additional mass that's added to it. Or can a black hole effectively expand forever as long as the supply of matter is greater than the effects of radiation and ejection?

Is there effectively a size limit to black holes?

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    $\begingroup$ Hawking radiation is theoretical (i.e., not yet observed), and is inversely proportional to the square of mass. The latter means that if it exists, it is a non-factor for anything but small primordial black holes. $\endgroup$ – David Hammen Dec 27 '17 at 4:33
  • $\begingroup$ @DavidHammen - So I guess Hawking radiation isn't really a limiter, what about accretion ejection increase? Doesn't that increase with size to the point where its hard to insert anything? $\endgroup$ – Mark Rogers Dec 27 '17 at 4:44
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As David Hammen commented, the power emitted through Hawking radiation is proportional to $M^{-2}$. Thus the evaporation timescale for a black hole is proportional to $M^3$. This means that a more massive black hole is much more stable against evaporation than a lower mass black hole.

The other issue you mention is the limited rate that you can "feed" a black hole. There is inevitably a feedback; as gas is compressed towards the event horizon it gets hot and emits radiation. The pressure of this radiation can eventually balance the inward gravitational infall. For spherically symmetric accretion this leads to the Eddington limit, which sets the maximum spherical accretion rate, where $\dot{M}_{\rm max}\propto M$. That is, the maximum accretion rate is proportional to the black hole mass.

If accretion proceeds at the Eddington limit then the black hole mass grows exponentially with time and with a characteristic doubling timescale of around 50 million years (independent of the original mass - see this Physics SE page for some mathematical details).

If black holes were limited to this accretion rate (though there is some evidence from the presence of very luminous quasars at high redshift that they may exceed it), then the maximum black hole mass will depend on the age of the universe and the size of the initial "seed" black holes. If we assume an initial mass of 100 solar masses, a doubling timescale of 50 million years and that the seed black holes formed 400 million years after the big bang (all plausible, but contestable), then there have been 266 doubling timescales since and the black hole could have grown by a factor of $10^{80}$ !

Clearly there are no black holes with anywhere near this mass in the observable universe - the largest appear to be of order $10^{10}$ solar masses. Their growth is limited by their food supply. Supermassive black holes are found at the centres of galaxies. There is a poorly understood relationship between the black hole mass and the mass of the bulge of the galaxy it is in. The ratio peaks at about 1 percent for the most massive bulges (e.g. see Hu 2009; McConnell & Ma 2013). Since the most massive massive elliptical galaxies are of order $10^{12}$ solar masses, this appears to set the maximum mass of a black hole in the present day universe.

The future is speculation. If the cosmic expansion rate continues to accelerate, then galaxy mergers will become increasingly uncommon and the opportunities for further black hole growth will be limited.

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Is there a maximum size for a black hole?

No.

As far as I understand it black holes radiate Hawking radiation

That's what people say, but we have no actual evidence for Hawking radiation. However even if Hawking was 100% correct, as Rob said, Hawking radiation has less and less effect as the black hole gets bigger and bigger.

and probably gravity waves

A black hole on its own won't emit any.

which over time causes them to lose mass and eventually evaporate after almost unfathomable amounts of time.

There's no scientific evidence that a black hole will disappear. However there is scientific evidence that black holes exist. There's definitely something very small and very massive at the center of our galaxy:

enter image description here

I've also read that black holes eventually have difficulty gaining size after a certain point because gravitational forces working in a black holes accreditation disk can eventually begin ejecting in-falling matter before it can be absorbed.

Yes, black holes are said to "choke" if they try to eat to much at once. See the physicsworld article Supermassive black hole struggles to swallow Milky Way. There are other issues to do with gamma ray bursters meaning black holes are messy eaters, but they still "eat", as it were.

While I know black holes can become quite massive, that leads me to wonder if Black holes have a maximum possible size where they will begin to immediately radiate any additional mass that's added to it. Or can a black hole effectively expand forever as long as the supply of matter is greater than the effects of radiation and ejection?

It's the latter. Imagine you're near a supermassive black hole. It's so massive that any Hawking radiation is negligible. It has no accretion disk because it's eaten everything or blown it away. What happens next? You fall in. So that black hole gets bigger.

Is there effectively a size limit to black holes?

No. If there was, and the black hole in the above scenario had reached it, you wouldn't fall in.

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There's no maximum theoretical size short of the size of the universe. But there's no way in nature that a Black Hole that big could form.

More interesting, perhaps, are the BHs which formed naturally. The largest natural BHs we know of the the supermassive BHs in galactic centers. There is a very interesting relationship that we have observed: The mass of Galactic BHs is never more than about 0.001 of the mass of their home galaxy's central bulge.

We don't understand -- yet -- in any detail why this is so, but it is strongly suspected that a galactic bulge an its central BH grow at the same time and stop growing together, also. Modelling work has been done which makes this plausible, but as far as I know no one has a fully satisfactory theory explaining their origin and growth.

Supermassive BHs sometimes merge after their host galaxies merge, though in most cases there probably hasn't been enough time for that to happen. (We do know of galaxies with multiple SMBHs in them.)

Putting this all together, the largest BHs found in nature will be a few times more than 0.001 of the mass of the largest galactic bulges, and will be formed when several especially large galaxies merge and their SMBHs eventually also merge.

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