# Do black holes have a limit of mass?

1. If you have a bunch of gas and increase its mass, gravity combines it into a planet.
2. If you have a planet and increase its mass, gravity forces the planet to undergo nuclear fusion, turning it into a star.
3. If you have a star and increase its mass, the gravitational collapse surpasses any thermal or degeneration pressure and turns the star into a black hole.
4. If you have a black hole and increase its mass, would gravity turn it into another thing? Do black holes have a limit of mass or they can go to infinity without undergoing any change (like nebula > planet > star > black hole > ...black hole?)?
• If we just think about mass density then the objects you described cover the full spectrum of possibilities from almost zero density to theoretical infinite density. – TheNaturalTanuki Feb 5 at 21:08
• Our Universe has a large mass, before the Big Bang it looks like a black hole. – Alex Trounev Feb 5 at 21:21
• @AlexTrounev: Our Universe has a large mass, before the Big Bang it looks like a black hole. Not true, see (1) physics.stackexchange.com/q/5150 , (2) physics.stackexchange.com/q/3294 – Ben Crowell Feb 5 at 21:29
• @BenCrowell It seems that the mainstream believes in one legend, and I in another. – Alex Trounev Feb 5 at 21:46

The rules of classical general relativity say that when you add mass to a black hole, you get a larger black hole. If you add angular momentum to a black hole at a greater rate than that at which you add mass, it would theoretically be possible to get a Kerr black hole with $$a \gt M$$, which would convert the black hole to a naked singularity, but the rules of black hole thermodyanamics say that a black hole with $$a = M$$ has zero temeperature, so creating a naked singularity in this way is believed to be impossible.
• Also, in a universe with a positive cosmological constant, if the mass of a black hole exceeds $\frac{1}{3\sqrt{\Lambda}}$ then the black hole would become naked. I suppose there should be some censorship argument so as to naturally prevent such a scenario in a generic situation. Edit: @Void, your comment appeared just when I pressed enter to post my comment :P Maybe you can also help me with a related old unanswered question of mine: physics.stackexchange.com/q/337760 – Dvij Mankad Feb 5 at 21:17
There is the tantalising possibility that adding more matter to a black hole can destroy it, by making its horizon disappear. Indeed a Kerr-Newman black hole, i.e. one that spins and that has an electric charge, must satisfy the relation $$m^2 \ge a^2 + e^2$$ where $$m$$ is the mass, $$e$$ the charge, and $$a=J/m$$ is the angular momentum per unit of mass (geometrised units G = c = 1 being used throughout). So as @Jerry-Shirmer stated, if the added matter contributes more spin or more charge than mass, then the black hole could be destroyed. There is a long history of research into proving that this cannot actually happen when properly modelling the motion of the matter falling into the black hole. The oldest result is, I think, that of Wald [1] about falling test particles (that is to say, loosely, bits of matter small enough that they do not affect the geometry of spacetime). The question of whether more realistic in-falling matter could destroy a blackhole has been discussed for many years. The latest publication I know of is another paper by Wald(!) [2] and the answer is no! You will find all the bibliography you want in that one. The only assumption is that the matter satisfies the so-called null energy condition, which is valid for all matter we have observed.