# Could micro black holes obey the Eddington limit?

A stellar-mass black hole has recently been discovered in the Andromeda galaxy. One interesting part of the release is that this black hole shines close to its Eddington limit.

Quasars are supermassive black holes shining at or near the Eddington luminosity, and microquasars are likely stellar-mass black holes whose accretion is similarly close to the theoretical maximum.

Wikipedia derives the equation for the Eddington limit (although I'm not sure about the applicability of its assumptions to small black holes) to be proportional to $M$. The Hawking radiation from a black hole goes as $1/M^2$. This implies that a cross-over point where both formulas calculate the same power. I can calculate this to be about $4 \times 10^{10} kg$.

What would be an accurate picture of a micro black hole, given sufficient matter around to feed it? I think that the Eddington limit is a balance between the gravitational pull and the radiation pressure, but in the case where Hawking radiation is present, would the picture be significantly different?

-

## 1 Answer

This is really a comment since I haven't attempted any calculation (it got too long for the comment box), but assuming there's plenty of accretable matter the repulsive force on the infalling matter will simply be the sum of radiation from accretion and the Hawking radiation. So as the black hole gets smaller the rate of accretion will slow and eventually stop.

You can't shrink a black hole and watch what happens, because if you start with a large black hole the accretion will always exceed the mass loss due to Hawking radiation and the black hole will grow not shrink. You could take isolated black holes and immerse them (thought experiment :-) in a cloud of accretable matter. The Hawking radiation from a very small black hole would blow away accreting matter so there wouldn't be any accretion and the BH would shrink. I guess somewhere in the middle there would be a size where the energy lost due to Hawking radiation is exactly balanced by the rate of accretion.

-
i would be very curious if Kerr black holes emit Hawking radiation at the same temperature in the equatorial bulges and in their polar regions. I've been looking some reference for this for a couple of months now but i haven't been successful yet – lurscher Dec 14 '12 at 16:17
the point of that previous comment is that, could possibly a (very small) Kerr black hole be able to accept infalling matter if it falls along the rotational axis? Possibly, since 4D Kerr black holes angular momentum are bounded by extremality condition, the possible difference in temperature between polar and equatorial regions will not be wide enough to be interesting (possibly not bigger than an order of magnitude), but it would be good to know, you know, to be sure – lurscher Dec 14 '12 at 16:20
John, updgraded comments to a question: physics.stackexchange.com/q/46862/955 – lurscher Dec 14 '12 at 18:21