# Black hole singularity

I've been reading up a bit on black holes but I don't quite grasp why there should be a singularity at the center.

As far as I understand a singularity forms when all the mass of an object collapses to withing the schwarzschild radius, according to wikipedia http://en.wikipedia.org/wiki/Gravitational_collapse

"Within the event horizon, matter would have to be accelerated outwards faster than the speed of light in order to remain stable and avoid collapsing to the center. "

but as far as I know there's no upper limit to how fast something can accelerate is there?

So why assume that it collapses into a singularity instead of still having some kind of non infinite density?

## Answers and Replies

Nabeshin
Science Advisor
Poor wording on the part of the wikipedia article. They mean to say "Accelerated TO faster than the speed of light".

Also, it's difficult to say when the singularity forms. To an outside observer, the matter will just hang at the event horizon for all eternity. To an infalling observer, co-riding with the infalling shell, say, they hit a singularity in a (very!) finite amount of time. In my opinion, talking about a singularity forming is nonsense. Just refer to the EH forming.

Also, Penrose and Hawking mathematically proved that once you have matter inside the EH it is doomed to collapse to a singularity. If you want, you can think of this as saying that there is no force which could possibly withstand the matter from collapsing ad infinitum, or, to a singularity.

I think the singularities inside the black holes do not exist; nobody proved them anyway.
How the density inside the black hole is higher than Planck density to form a singularity?

Nabeshin
Science Advisor
I think the singularities inside the black holes do not exist; nobody proved them anyway.
How the density inside the black hole is higher than Planck density to form a singularity?

Well, sure. Quantum effects are something we don't understand yet. My post refers to the fact that within the classical theory of GR, a singularity must form (there are some assumptions with the theorems, and you may debate them if you like). As for what happens when quantum gravity comes into play is anyone's guess.

I don’t think there’s a singularity at the center of a black hole. It’s a mathematical artifact that comes about because GR neglects quantum mechanics. Whatever's at the center of a black hole should have properties similar to fermions. It should have mass, volume, charge, and rotation (i.e. spin).

No singularity, no event horizon...

No singularity, no event horizon, and possibly little mass transfer. Hardly anything goes out of a black hole and it’s possible that hardly anything goes in.

Nabeshin
Science Advisor
No singularity, no event horizon...

This statement is not at all true. One does not need a singularity to generate an event horizon, so even if future quantum gravity calculations remove the singularity, it is still likely we'll be left with an event horizon surrounding the remnant of stellar collapse.

Chronos
Science Advisor
Gold Member
It is easy to calculate the mass density required to generate an event horizon. The formula is R = 2GM/c^2 where R is the radius, G is the gravitational constant, M is mass and c is the speed of light. Plugging in the mass of the sun, for example [2e30 kilograms], yields a value of 3000 meters as the radius at which its density would be sufficient to generate an event horizon. No singularity required.

No singularity, no event horizon...

In other words, how can the singularity have a causal effect on the EH?

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Instead of thinking of a black hole as something that gobbles up everything in its path, what would happen if the center of a black hole was a single super massive particle, with properties similar to subatomic particles?

Nabeshin
Science Advisor
Instead of thinking of a black hole as something that gobbles up everything in its path,
Nobody thinks of it this way.

what would happen if the center of a black hole was a single super massive particle, with properties similar to subatomic particles?

You would still have an event horizon and to an outside observer there would be practically nothing different. Basically there would be no way to observe any differences short of falling into the black hole, at which point the EH prevents you from ever telling your pals outside what you've learned.

Ok, maybe I exaggerate with the gobbling up part :), but what if the center of a black hole was a single particle, something like a fuzzball, with four identified properties:

1) Mass
2) Volume
3) Charge
4) Spin

and these properties were quantized? For example, mass could have three different states:

1) Stellar black holes, some 20 times the mass of our sun, supernova remnant.
2) Intermediate black holes, about 100,000 times the mass of our sun, possibly at the center of some globular star clusters.
3) Supermassive black holes, the heavyweights at a billion to 20 billion times the mass of our sun, believed to be at the center of most galaxies.

and there would be very little in between. Volume could be related to mass, something like the Schwarzschild radius. Charge would probably be close to zero, but not necessarily. The most interesting property is spin, which doesn’t need to be at ± ½ as you would expect from fermions.

If a star collapses into a black hole, the gravitational field outside the black hole may be calculated entirely from the properties of the star and its external gravitational field before it becomes a black hole. Just as the light registering late stages in my fall takes longer and longer to get out to you at a large distance, the gravitational consequences of events late in the star's collapse take longer and longer to ripple out to the world at large.

This reference makes no mention of how a singularity may have a causal effect on the event horizon.

Nabeshin
Science Advisor
Ok, maybe I exaggerate with the gobbling up part :), but what if the center of a black hole was a single particle, something like a fuzzball, with four identified properties:

1) Mass
2) Volume
3) Charge
4) Spin

and these properties were quantized? For example, mass could have three different states:

1) Stellar black holes, some 20 times the mass of our sun, supernova remnant.
2) Intermediate black holes, about 100,000 times the mass of our sun, possibly at the center of some globular star clusters.
3) Supermassive black holes, the heavyweights at a billion to 20 billion times the mass of our sun, believed to be at the center of most galaxies.

and there would be very little in between. Volume could be related to mass, something like the Schwarzschild radius. Charge would probably be close to zero, but not necessarily. The most interesting property is spin, which doesn’t need to be at ± ½ as you would expect from fermions.

Then... nothing would appear different to an outside observer and the object would still be cloaked in an event horizon. Also, you cannot willy-nilly quantize mass like that. If anything, a quantization of mass for such an object would be on the quantum scale, so by the time you get to astrophysical masses the scale is essentially a continuum.

Basically you're engaging in groundless speculation. What if, instead of a singularity, there are very dense pink unicorns with only spin, charge, and mass at the center? The unicorns like to be a certain size so we preferentially get stellar mass, globular cluster mass, and supermassive mass holes.

Note: I'm being very harsh here. While theories regarding the removal of a singularity by some exotic quantum property of matter certainly are within the realm of hard science, they are very much in their infancy. So unless any of us happens to be on the forefront of such discussions, or we wish to start making references to peer-reviewed publications dealing with the subject, it's rather pointless to simply throw around "what if...?" statements.

Then... nothing would appear different to an outside observer and the object would still be cloaked in an event horizon. Also, you cannot willy-nilly quantize mass like that. If anything, a quantization of mass for such an object would be on the quantum scale, so by the time you get to astrophysical masses the scale is essentially a continuum.
On a subatomic scale, mass is quantized and very small:

Mass of an electron = 9.1093 x 10-31 kg
Mass of a proton = 1.6726 x 10-27 kg
Mass of a neutron = 1.6749 x 10-27 kg

For supermassive particles, like fuzzballs, the difference in mass could be quantized but that difference could be much larger than expected for subatomic particles.
Basically you're engaging in groundless speculation.
Maybe.
What if, instead of a singularity, there are very dense pink unicorns with only spin, charge, and mass at the center? The unicorns like to be a certain size so we preferentially get stellar mass, globular cluster mass, and supermassive mass holes.
I prefer pink elephants.
Note: I'm being very harsh here.
No you’re not. You are expressing your opinion on someone speculating about the center of a black hole. Expressing your opinion is at the core of the scientific process.
While theories regarding the removal of a singularity by some exotic quantum property of matter certainly are within the realm of hard science, they are very much in their infancy. So unless any of us happens to be on the forefront of such discussions, or we wish to start making references to peer-reviewed publications dealing with the subject, it's rather pointless to simply throw around "what if...?" statements.
Maybe :).

Nabeshin
Science Advisor
On a subatomic scale, mass is quantized and very small:

Mass of an electron = 9.1093 x 10-31 kg
Mass of a proton = 1.6726 x 10-27 kg
Mass of a neutron = 1.6749 x 10-27 kg

For supermassive particles, like fuzzballs, the difference in mass could be quantized but that difference could be much larger than expected for subatomic particles.

I'm only generally familiar with the fuzzball hypothesis, but I don't see how one can consistently have quantization of mass simultaneously solve the three-mass-scale problem you mentioned earlier. That is, we know there must be BH's of ~stellar mass. The unit of quantization, therefore, must be no larger than this. But then the situation is the same as I said earlier -- as one goes into thousands and millions of solar masses, the scale is essentially a continuum so you have not at all resolved the issue of why there seem to be black holes at these three mass regimes and nothing in between. So why posit quantization in the first place?

For subatomic particles:

Mass of an electron = 9.1093 x 10^-31 kg = Me
Mass of a proton = 1.6726 x 10^-27 kg = Mp

Mp/Me=1836 or about 2000.

It’s not necessarily the case that the mass a proton is 2 X the mass of an electron. It’s much greater than that, by about 1000. By analogy, if the center of a black hole is a fuzzball, and mass is quantized, then the difference between the mass of a stellar black hole and a super massive black hole is not necessarily 2 x the mass of a stellar black hole. That difference in mass can be much larger, by about a 100,000 to 20,000,000 more.

Nabeshin
Science Advisor
For subatomic particles:

Mass of an electron = 9.1093 x 10^-31 kg = Me
Mass of a proton = 1.6726 x 10^-27 kg = Mp

Mp/Me=1836 or about 2000.

For one, this number is just a coincidence. It doesn't have anything to do with quantization of mass or anything like that. For two, there is still no way to retain three different mass scales. There are three or four orders of magnitude between each of them, so any single unit of quantization will not suffice to explain all three. At most you can explain two of them.

stevebd1
Gold Member
On a slightly different note, if we consider that the singularity as Planck density as an absolute limit then we should also consider Planck pressure as a limit which produces an equation of state of 1:1 (i.e. the maximum possible). You might say that as the mass collapsed to Planck density, the pressure would also increase and based on $g=\rho c^2+3p$, 3/4 of the mass (assuming that pressure played little or no part in the original mass) would be converted to confined kinetic energy (i.e. pressure). possibly this EOS of 1:1 would halt the collapse or even cause a bounce.

On second thought, trying to quantize the mass of a black hole using only string theory (a.k.a fuzzballs) may be problematic. It’s like trying to show that the mass of subatomic particles is quantized, simply using quantum mechanics. Quantization of mass is a prerequisite, an initial condition, which comes about because of empirical observations. The mass of black holes seems to clump up in three regions, stellar, intermediate, and supermassive.

DLuckyE said:
So why assume that it collapses into a singularity instead of still having some kind of non infinite density?

The the Schwarzschild radius is the radius of the event horizon for a black hole, and all black holes are described by General Relativity. The singularity in General Relativity is intrinsic to the Einstein field equations as the Planck force, which is the ratio of electromagnetic energy per gravitational length:

Planck force:
$$F_P = \frac{c^4}{G}$$

The Planck force itself is mathematically intrinsic to General Relativity due to the integration of the initial General Relativity equations with Quantum Mechanics in absence of Quantum Gravitation.

Einstein field equation:
$$G_{\mu \nu} = \frac{8 \pi T_{\mu \nu}}{F_P} = 8 \pi \frac{G}{c^4} T_{\mu \nu}$$

This means that the maximum force that can be spherically symmetrically applied under General Relativity to the Einstein tensor and stress-energy tensor differential and is equivalent to the Planck force, which is a singularity:
$$F_P = \frac{c^4}{G} = 8 \pi \left( \frac{T_{\mu \nu}}{G_{\mu \nu}} \right)$$

Therefore, any mathematical model using an Equation of State described by General Relativity, absent any Quantum Gravity, must contain a Planck singularity.

If Planck pressure and Planck density are the maximum upper limits in the Universe, and both pressure and density functions both relativistically contribute to the total Equation of State for hydrostatic equilibrium inside a black hole, then the total integration for the differential pressure that describes the perfect fluid inside a black hole must be equivalent to the Planck pressure at the singularity core.

Relativistic Equation of State functions for hydrostatic equilibrium: (J = 0, Q = 0)
$$\frac{dP(r)}{dr} = - \frac{G}{r^2} \left( \rho(r) + \frac{P(r)}{c^2} \right) \left(M(r) + 4 \pi r^3 \frac{P(r)}{c^2} \right) \left( 1 - \frac{2 G M(r)}{c^2 r} \right)^{-1} \;$$

Note that this equation under General Relativity has a mathematical singularity as $$r \neq 0$$, because of metric geometry as $$r^{-1}$$ and $$r^{-2}$$.

Integration of the relativistic differential pressure Equation of State function for hydrostatic equilibrium for core pressure:
$$P_c = \int_{r_P}^{R_s} \left( \frac{dP(r)}{dr} \right) dr = \frac{c^7}{4 \pi \hbar G^2}$$

Where $$r_P$$ is the Planck radius and $$R_s$$ is the Schwarzschild radius.

Relativistic black hole singularity core pressure is equivalent to Planck pressure:
$$\boxed{P_c = P_P = \frac{c^7}{4 \pi \hbar G^2}}$$

Relativistic black hole singularity core density is equivalent to Planck density:
$$\boxed{\rho_c = \rho_P = \frac{3 c^5}{4 \pi \hbar G^2}}$$

Note that the core singularity inside a black hole under General Relativity does not have infinite dimensions and are limited to the Planck radius $$r_P$$ as the smallest spacial unit mathematically possible in any model in absence of Quantum Gravitation.

In physical cosmology, the Big Crunch is one possible scenario for the ultimate fate of the Universe, in which the metric expansion of space eventually reverses and the Universe recollapses, ultimately ending as a black hole singularity.

The Big Crunch cosmological theory is also based upon that premise that the Planck units are the maximum attainable limits in the Universe and cannot be exceeded without Quantum Gravity.

Reference:
http://en.wikipedia.org/wiki/Planck_force" [Broken]
http://en.wikipedia.org/wiki/Planck_density" [Broken]
http://en.wikipedia.org/wiki/Planck_pressure" [Broken]
http://en.wikipedia.org/wiki/Planck_length" [Broken]
http://en.wikipedia.org/wiki/Tolman%E2%80%93Oppenheimer%E2%80%93Volkoff_equation" [Broken]
http://en.wikipedia.org/wiki/Big_Crunch" [Broken]

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As a extension of this topic, an other theory can be considered: http://en.wikipedia.org/wiki/Big_Bounce" [Broken].

It is a whole Universe theory but maybe it can be applied to BH singularities too. I don't know if the "bounce" effect exist in this theory for ordinary BH and if such hypothetical effect will manifest in our Universe. But can be a theory that explain why singularities actually do not exist in Universe. If it is accurate, of course.

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Orion1,

when I posted about Planck density, the Planck units are the maximum attainable limits... last month I received a warning and nobody was interested!