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I have read these questions:

Why does a black hole have a finite mass?

Are black holes an infinite source of energy?

Black Hole: mass density or energy density?

Do all black holes have the same mass density?

Supermassive black holes with the density of the Universe

If the observable universe were compressed into a super massive black hole, how big would it be?

where it says :

We obtained the result that even if we only count the "particulate" component of the mass of the Universe, we find out that the Universe is actually smaller than the black hole.

Supermassive black holes with the density of the Universe

What is exactly the density of a black hole and how can it be calculated?

where it says:

There are an awful lot of constants in that equation, and it might be a bit easier to grasp if we write it in the form:

$$ \rho \approx 1.85 \times 10^{19} \frac{1}{m^2} $$

where now $m$ is the mass of the black hole in solar masses i.e. units where $1$ means the same mass as the Sun. With this equation we can see immediately that a black hole with the same mass as the Sun would have the (enormously high) density of $1.85 \times 10^{19}$ kg/m$^3$. Alternatively, a super supermassive black hole with the mass of 4.3 billion Suns would have a density equal to one i.e. the same density as water.

And it made me curious.

I understand that there are different types of black hole, some with singularities some with none, and there are Schwarzshild black holes.

None of these answers talks about if black holes would have a maximum mass/stress-energy limit. They all say that black holes have a finite mass/energy. And they have zero volume.

Now according to Heisenberg uncertainty principle, it assert a fundamental limit to precision with which we can know certain pairs of physical properties of a particle, such as position and momentum.

According to the Pauli exclusion principle, two or more fermions cannot occupy the same quantum state within the same quantum system. That is true for all quarks and leptons.

Now a black hole can increase its mass by getting more and more fermions too not just bosons.

The questions say too that not all black holes are singularities.

Now if a black hole keeps increasing its mass/stress-energy, will there be a certain limit where either:

  1. if it is a singularity, the black hole will not be able to increase more its mass/stress-energy without increasing its volume (so stop being a singularity)

  2. if it is not a singularity, the black hole will increase its volume and thus after a while become too big, where its radius will be more then the Schwarzshild radius and seize to be a black hole

Question:

  1. Is there a maximum limit to the mass/stress-energy of a black hole that can stay in the singularity (if it is a singularity) or is there a maximum amount of mass/stress energy that can be in a certain volume (if not a singularity) that will keep the radius below the Schwarzschild radius?

  2. Do we know of any physical processes/limitations that would limit the mass/stress-energy of a black hole? Why haven't we observed black holes that would be bigger then a whole mass/stress-energy of a galaxy cluster (the biggest black holes have still much smaller mass/stress-energy then a galaxy cluster)?

A supermassive black hole (SMBH or SBH) is the largest type of black hole, on the order of hundreds of thousands to billions of solar masses (M☉), and is found in the centre of almost all currently known massive galaxies.[2][3] In the case of the Milky Way, the SMBH corresponds with the location of Sagittarius A*.[4][5]

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First, physicists don’t believe that Schwarzschild black holes are existing, because general relativity breaks down at $r = 0$. The singularity is a point in time and does not belong to the manifold. This requires infinite density and curvature which is unphysical. Instead it is believed that the mass is in the center, whereby its state is unknown and subject of a future theory of quantum gravity. Such black holes are sometimes called “real” or “astrophysical” black holes.

Is there a maximum limit to the mass/stress-energy of a black hole that can stay in the singularity (if it is a singularity) or is there a maximum amount of mass/stress energy that can be in a certain volume (if not a singularity) that will keep the radius below the Schwarzschild radius?

There is no such maximum limit for $M$ of a Schwarzschild black hole. Note that the stress-energy tensor is identical zero in this case, because the Schwarzschild metric is a vacuum solution. The mass “in a certain volume” (meaning a real black hole) is due to quantum gravity as mentioned above. So it makes no sense to speculate about said limit.

Do we know of any physical processes/limitations that would limit the mass/stress-energy of a black hole?

We don’t, because we don’t have a theory of quantum gravity yet.

For the same reason the Pauli principle is not applicable. It is restricted to known physics.

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