# Are there upper or lower mass limits for black holes?

Natarajan & Treister (2008) describes a practical upper limit for black hole masses at $\sim 10^{10} M_\odot$. This is all due to the black hole's interactions with nearby matter.

However, is there a theoretical upper mass limit for black holes in general relativity? More specifically, do any solutions make note of this? Would this depend on whether the black hole described is eternal or time-variant, static or spinning, charged or uncharged, etc.?

Similarly, do any metrics make note of lower mass limits? Would it be possible for a black hole with the mass of an electron to exist (at any point in time, putting aside Hawking radiation)?

• Talking about electron-mass black holes gets into some weird physics and shows we have a long way to go to. Usually, electrons are described as point particles in QM, meaning that technically they're also black holes since their mass is interior to their Schwarzschild Radius. Obviously electrons aren't black holes, which illuminates the difficulty in using these various conflicting theories in extreme realms. I don't know if there's really a good answer to this good question. – zephyr Oct 26 '16 at 14:02
• @zephyr Electrons don't have a size in quantum mechanics. The concept of size isn't really relavant at those scales. – Sir Cumference Oct 26 '16 at 14:38
• That was my whole point though. – zephyr Oct 26 '16 at 14:41
• qzephyr: Electrons have also angular momentum and electrical charge. These two variables influence the formation of a black hole. When you just stubbornly try to compute the classical Schwarzschild radius of an electron, you'll get a negative value (i.e., no black hole at all) – jk - Reinstate Monica Oct 26 '16 at 20:20

In classical General Relativity, black holes can exist at any size (mass) without any problem. The upper limit is given by the available mass of the universe and there is no theoretical lower limit.

As already noted in the question, quantum effects like Hawking radiation set up lower limits on stable black holes; the ones with too low mass will decay quickly into radiation.

The discovery of TON 618 have created a new black hole species (already fingerprinted by M87 or even IC1101 cores): the ultramassive black holes with masses greater than $$10^{10}M_\odot$$. As said in the previous answer, in classical settings, there is no upper limit of the mass of black holes (I am not so sure if you get a theory beyond General Relativity even in classical settings).

Maybe, one day we will learn that quantum gravity says something about that. Interestingly, any supermassive, stellar, intermediate and ultramassive black hole has a mass much much greater than Planck mass, about a microgram. The issue is that we think quantum gravity applies only to VERY MASSIVE TINY (very dense) objects, not to very massive only. Indeed, any person has a mass much greater than Planck mass, but it is not "concentrated". When you have concentrated mass in very tiny regions, we have no idea of how to handle quantum fluctuations and amplitudes excepting with superstring theory. Another related question, is if you can have black holes of any DENSITY. Again, as said, you need to consider quantum processes like Hawking radiation, ... However, there is a subtle point, called the transplanckian problem. In principle, as the black holes evaporates it gets smaller and smaller, such as at certain size the wavelength would be lesser than the Planck length. We have to expect for a definitive theory of quantum gravity before to answer the ultimate fate of black holes and thus, the destiny of both: black holes and the whole universe (even the spacetime could be metastable and provisional/transitional state).

How large can a black hole formed from the collapse of a massive star grow in 1 Gyr? Suppose the black hole can grow as fast as it can. Suppose, by the moment, it satisfies the Eddington limit. Then, an exponential law follows up: $$\dot{M}=kM=M/\tau$$ where $$k=4\cdot 10^{-16}s^{-1}$$ for a ten solar masses inicial mass function accordingtly to the Eddington limit. Then, as

$$M=M_0\exp(kt)$$

Plugin into this formula $$M_0=10M_\odot$$ and the value of k, you get that the maximum mass it yields is in the range of ultramassive BH, i.e., $$M_f\sim 10^{10}M_\odot$$ for a timescale about 1 Gyr (be aware, the numbers are tricky). Of course, transEddington limit is tricky, but there are some reasons to believe black holes bigger than $$10^{10}M_\odot$$ are unstable and eject material. Of course, in the absence of any other argument, the above argument does NOT provide an upper limit in principle. Only other considerations relativo to quasars and jets seem to apply. But the issue is a hot topic of debate in astrophysics. By the other hand, the minimal (or tiniest) black hole mass is also a mystery. In macroscale, we have NOT found black holes tinier than 3-5 solar masses (stellar black holes). However, primordial black holes or microblack holes could made some bits of the dark matter hidden in clusters and other parts of the galaxies. Again, the only hint are inflationary ideas, astronomical measures and experimental bounds (recently, it has been analyzed the probability of dark matter being totally black holes, but some evidence seems to say that that is not the case: black holes can not be all the dark matter).

• This doesn't really address the question. – Peter Erwin Oct 15 '18 at 7:10
• Rewritten arguments... – riemannium Oct 15 '18 at 14:23
• With some numerics and the Eddington limit hypothesis... – riemannium Oct 15 '18 at 14:44