# Is there a maximum mass limit for black holes?

We all knows that neutron star and a black hole is a final end stage of a giant stars, and neutron stars can also become a black hole, we also knows that minimum mass limit for stars to become a black hole is at least 3 times mass of a sun. Then, is there a maximum mass limit for a black hole to remain a black hole? Where when a black hole eats another black hole and exceeds the maximum mass limit allowed to remain a black hole then something hypothetical might happens.

• Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer.
– Community Bot
Jul 5, 2022 at 22:07

In our universe with cosmological constant

$$\Lambda \approx 1.1 \cdot 10^{-52} \ \rm m^{-2}$$

if the mass of the supposed to be black hole was as high as

$$\rm M=\frac{c^2}{3 G \sqrt{\Lambda}}\approx4.3 \cdot 10^{52} \ \rm kg$$

the black hole and cosmic horizons would coincide at

$$\rm r=\frac{1}{\sqrt{\Lambda}} \approx 10 \ \rm Glyr$$

and if $$\rm M$$ were even higher (the average density sinks with increasing mass), the black hole horizon and the cosmic event horizon would pop each other, and no real solutions for $$\rm r$$ in

$$g_{\rm tt}=\rm c^2-\frac{2 G M}{r}-\frac{r^2 c^2 \Lambda}{3}=0$$

whose zeroes mark the horizons, exist when the mass gets larger than the critical value, so in that case no horizons would exist at all, see here:

If such a high mass were concentrated to a point it would have to be a naked singularity without a horizon around it.

In our current FLRW universe the average density is a little higher than the dark energy density $$\rm \rho_{\Lambda}=c^2 \Lambda/(8 \pi G)$$, so in the SSdS equations above you would effectively have to replace $$\rm \Lambda$$ with $$\rm 3 H^2/c^2$$ and the $$\rm M$$ would be a little lower, but not by much.

In the late universe when $$\rm H$$ asymptotically reaches a constant value we will basically have De Sitter and the equations become exact since $$\rm H$$ converges to $$\rm H \to H_0 \sqrt{\Omega_{\Lambda}} = c \sqrt{\Lambda/3}$$.

• Long before it approached the size of the cosmic horizon, when the black hole is as big as galactic clusters say, would the cosmological constant start to counteract the "structural integrity" of the black hole? Would the $\Lambda$ cause space to (try to) expand inside the event horizon, partially counteracting gravity? Maybe this is a topic for another question Jul 7, 2022 at 0:33
• @RC_23 - If the black hole gets to the size of the cosmic horizon, you won't be able to make it larger since you'd have to shove material from behind the cosmic horizon into it in order to make it heavier, which you can't since you can't get your hands on material that is behind an other horizon already. The stuff that is behind the cosmic horizon can never be brought to the black hole, just as the stuff behind the black hole horizon can't escape to infinity. Jul 7, 2022 at 11:07
• @RC_23 - about the question if Λ counteracts the gravity of M, the ȑ=-M/r² for the acceleration becomes ȑ=+Λr/3-M/r² (in natural units with G=c=1), see physics.stackexchange.com/questions/531426/… Jul 7, 2022 at 11:20
• In addition to answering the OP's question, this answer seems, together with the astronomical evidence for black holes and my assumption that Yukterez' math is entirely correct, to provide a reason for a continuing subdivision of a (possibly) "infinite & fully eternal" inflationary multiverse into "local universes": Views to the contrary would be much appreciated. Jul 7, 2022 at 17:20

No, there is no theoretical maximum limit for the mass of a black hole. In practice, one could argue that a black hole's mass can't exceed the total mass of the Universe, but defining what one actually means by "total mass of the Universe would be complicated". If two black holes merge, they simply form a larger black hole.

A way of proving this within classical General Relativity (which is the theory of gravity I'm assuming in this answer) could be through the so-called Area Theorem, also known as the Second Law of Black Hole Mechanics, which states that the total area of black holes in the Universe cannot decrease. If somehow two black holes merged to form something that is not a black hole, this theorem would be violated.

### Technical Discussion

More specifically, the area theorem states that (for more details, check Hawking & Ellis 1973 The Large Scale Structure of Spacetime, Prop. 9.2.7)

Given a black hole $$B$$ in a strongly future asymptotically predictable spacetime. Suppose $$R_{ab} k^a k^b \geq 0$$ holds for all null vector fields $$k^a$$, which is the case if the Einstein equations hold and the null energy condition is satisfied. Under these conditions, the area of the future horizon of the black hole never decreases.

The theorem can be modified to many black holes merging by simply adding the areas of all of the black holes at a given instant (the statement given in Hawking & Ellis takes that into account).

The assumptions made in the theorem are:

• strongly future asymptotically predictable spacetime This means the spacetime at infinity has a simple structure and has no matter, i.e., matter is confined to the "middle" of spacetime. This is what we're usually thinking of when we think of a black hole, which is a compact object, but does not hold if you start taking into account possible effects due to a cosmological constant.
• Einstein equations This means we're assuming the theory of gravity to be General Relativity, the best theory of gravity we have so far. Possible differences could arise with other theories (in which the theorem may or may not hold).
• null energy condition This means we're taking into account only standard forms of classical matter. The null energy condition is an imposition that all known forms of classical matter satisfy. The adjective classical is extremely important: once quantum effects, are taken into account, the null energy condition is often violated. That is why, when one considers quantum mechanics, black holes can evaporate. However, I'm assuming your question to be concerned with the classical theory mostly.

### Black Hole Thermodynamics

Inspired by the comments by Martin C. and Agnius Vasiliauskas that I had overlooked Hawking radiation, I think it is also interesting to add in a brief comment about black hole thermodynamics.

Due to the laws of black hole mechanics extremely resembling the ordinary laws of thermodynamics and the existence of Hawking radiation showing that the laws of black hole mechanics seem to be quite literally the laws of thermodynamics, there is wide belief that the area of a black hole is related to its entropy by the Bekenstein–Hawking expression $$S = \frac{A c^3}{4 G \hbar},$$ where $$c$$ is the speed of light, $$G$$ is Newton's constant, and $$\hbar$$ is the reduced Planck constant.

This expression also means that black holes have a lot of entropy.

As a consequence of this, if the merge of two black holes would result in something without an event horizon, we'd have an immense amount of entropy that would need to go somewhere, since entropy can't disappear (in this context, this is known as the Generalized Second Law of Thermodynamics). If somehow two black holes merged to form something that is not a black hole, that object would have an incredible amount of entropy (which I think is not necessarily a problem, but I found worth mentioning it).

• Black holes are known to decay through Hawking radiation, so I don’t think you could have a “proof” of this in the way you seem to mean. Jul 6, 2022 at 19:33
• Indeed, even at the same wiki page it is stated "This "law" was superseded by Hawking's discovery that black holes radiate, which causes both the black hole's mass and the area of its horizon to decrease over time." Jul 6, 2022 at 20:28
• @MartinC. Thanks for pointing it out. I added a more detailed discussion that explains what are the limitations of the theorem and, in particular, how Hawking radiation fits into the picture Jul 6, 2022 at 23:45
• @AgniusVasiliauskas This statement on the Wiki page seems a bit weird, especially because it makes it seem that the law has been destroyed, when it actually has been updated. A bit lower on the page the generalized second law is mentioned, which essentially states that the area of a black hole should be understood as entropy and the area theorem then fits in the second law of thermodynamics. I'll add it to my answer Jul 6, 2022 at 23:48
• @NíckolasAlves Could your updated argument be replaced by the observation that the black hole entropy is expected to match the Bekenstein bound? Jul 7, 2022 at 6:56

Physicists are more interested in a related question. "What is the biggest black hole we are likely to see?"

The centers of galaxies have truly massive black holes, millions to billions the mass of our sun. It is easy to see how they get so big. There are many stars close together at the center of a galaxy. Stars frequently pass near each other, sometimes leaving one star with a lot of kinetic energy and the other with little. Slow stars fall toward the center, and every so often one is captured.

Away from the center, it is harder to find ways that very large black holes would be made. Large stars collapse into black holes at the end of their life. But there is a limit to how big stars get. Large stars have a higher solar wind. The largest stars have such a high wind that they lose mass, literally blowing away.

In reasonable density parts of the galaxy, every so often to black holes merge, producing a larger black hole. But stars are so spread out that triple collisions are not significant.

It is like seeing 100 car pile ups on a freeway, where cars are close together. But airplanes flying around the Alaskan wilderness are so scarce that a collision between even two is rare.

Never the less, gravitational wave observations show larger black holes that people expected to find. Explaining where they come from will be interesting.

No (other than the mass of the entire universe as Níckolas states), but there are other pertinent limits on black holes you might be intrested in.

In General Relativity, when you have a charged, nonrotating black hole, the ratio of the black hole's mass, $$M$$ to its charge, $$e$$, actually radically alters the physics, and so in fact whether it fits the idea of a black hole we usually have.

Namely, if $$|e| > GM$$, we get a super-extremal black hole. One finds the field equations predict a singularity not enclosed within an event horizon. These are called naked singularities and are largely agreed to be unphysical.

If $$|e| < GM$$, we have a sub-extremal black hole. This is the type of black hole we know well; spacetime singularity at its center shielded by at least one event horizon (in fact two in this case!).

Finally, when $$|e| = GM$$, this is called an extremal black hole and can be shown to have some incredibly intresting geometry like the Einstien Rosen bridge: a wormhole between two flat regions of space.