Understanding Black Hole Singularity

[DLuckyE]

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?

 

[Nabeshin]

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.

 

[universe11]

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]

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.

 

[Imax]

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).

 

[universe11]

No singularity, no event horizon...

 

[Imax]

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]

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]

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.

 

[skeptic2]

No singularity, no event horizon...

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

 

[Imax]

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]

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.

 

[Imax]

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.

 

[Mentz114]

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

This might help answer your question

http://www.math.ucr.edu/home/baez/physics/Relativity/BlackHoles/black_gravity.html

 

[skeptic2]

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]

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.

 

[Imax]

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]

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?

 

[Imax]

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]

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]

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.

 

[Imax]

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.

 

[Orion1]

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 spatial 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"
http://en.wikipedia.org/wiki/Planck_density"
http://en.wikipedia.org/wiki/Planck_pressure"
http://en.wikipedia.org/wiki/Planck_length"
http://en.wikipedia.org/wiki/Tolman%E2%80%93Oppenheimer%E2%80%93Volkoff_equation"
http://en.wikipedia.org/wiki/Big_Crunch"

 

[Skolon]

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

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.

 

[universe11]

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!

 

[Orion1]

The main idea behind the quantum theory of a Big Bounce is that, as density approaches infinity, the behavior of the quantum foam changes. All the so-called fundamental physical constants, including the speed of light in a vacuum, were not so constant during the Big Crunch, especially in the interval stretching 10−43 seconds before and after the point of inflection. (One unit of Planck time is about 10−43 seconds.)

If the fundamental physical constants were determined in a quantum-mechanical manner during the Big Crunch, then their apparently inexplicable values in this universe would not be so surprising, it being understood here that a universe is that which exists between a Big Bang and its Big Crunch.

One of the main problems with the Big Bang theory is that at the moment of the Big Bang, there is a singularity of zero volume and infinite energy. This is normally interpreted as the end of the physics as we know it; in this case, of the theory of General Relativity. This is why one expects quantum effects to become important and avoid the singularity.

However, research in loop quantum cosmology purported to show that a previously existing universe collapsed, not to the point of singularity, but to a point before that where the quantum effects of gravity become so strongly repulsive that the universe rebounds back out, forming a new branch. Throughout this collapse and bounce, the evolution is unitary.

Reference:
http://en.wikipedia.org/wiki/Big_Bounce"
http://en.wikipedia.org/wiki/Big_Crunch"
http://en.wikipedia.org/wiki/Planck_time"
http://en.wikipedia.org/wiki/Loop_quantum_cosmology"

 

[Dav333]

Well, sure. Quantum effects are something we don't understand yet.

Is this why people on here say virtual parties are not real? And what about the BH hawking radiation & the casimir effect? Very confused. :(

 

[kg4pae]

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?

The word "singularity" is a mathematical term that means that the function is non-analytic. Generally this comes about due to the fact that some law is applied where it is not defined. An example is the very singularity we are talking about:

g[0,0]=-(1-Rs/r)
where Rs=dthe Schwarzschild radius of the black hole

When r=0, then we are at the singularity: we are dividing by zero. This arises because of two problems. First, division by zero is not defined. Secondly, the metric associated with the exterior coordinates doesn't apply to the interior of the event horizon. We can cure the second case by switching to interior coordinates, but the first case still remains.

The only way out of the dilemma is to realize that something else must happen when r=0 (or its equivalent). Perhaps the very nature of the black hole changes. Perhaps the interior of the black hole is actually a quark-star, or made of bosonic matter. Who knows?

 

[kg4pae]

Is this why people on here say virtual parties are not real? And what about the BH hawking radiation & the casimir effect? Very confused. :(

Virtual particles are particles that seem to violate Heisenberg's Uncertainty Principle. That is because the distance over which they act is smaller than the wavelength corresponding to their momentum. As a result, they are particles that must exist due to the requirements of conservation laws, but can't be measured to exist.

Hawking radiation results from the fact that particle-antiparticle pairs always spontaneously appear in free space. Near a black hole there is a chance that one particle will fall into the black hole while the other escapes. Since the energy of the system must be conserved, according to Hawking some of the mass of the black hole must be radiated away with the escaping particle.

Casimir force results from the fact that we can associate a wavelength with each component of energy, which presumably consists of photons or photon-like quanta.When two conducting plates are parallel, some waves will fit inside the separation between the plates and others will not. The net result is that inside the separation of the plates will have less energy than the outside. As the plates are brought closer together, they will exhibit a force that arises completely due to this energy difference in the vacuum. Aparently this effect is real and has been observed.

 

[qraal]

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.

And what is goin to force that mass of particles to get so small? Gravity is opposed by degeneracy pressure - first by the electrons and then the nucleons - but pressure is energy per unit volume, which produces gravity too. A runaway into the conditions of a singularity occurs because the particles themselves are pushing too hard in opposition to their own gravity. There's an implied point of no return at which no known force can oppose runaway collapse, but whether that is at neutron star or quark star size no one presently knows. The properties of large balls of naked quarks isn't well constrained by current knowledge.

 

[stevebd1]

And what is goin to force that mass of particles to get so small?

I'm under the impression that collapse is inevitable once inside 2M because the matter now exists in space-like spacetime (i.e. r is temporal) and regardless of the forces involved, no stable radius is possible and matter will keep collapsing until r=0 or spacetime ceases to exist (which is suppose to be the case at Planck scale) or time-like spacetime is reinstated as in the case of rotating or charged black holes.

 

[Orion1]

Any model with a radius less than the Planck radius no longer obeys General Relativity and according to cosmologists, virtual particle turbulent Quantum Foam is encountered, and there is no evidence that Planck scale dimensions are capable of being exceeded without completely replacing General Relativity with Quantum Gravity. Also, the Planck radius is a stable radius.

Also, there may be other degeneracy conditions based upon degenerate quark matter and Quantum Gravity and other conservation laws that may exist inside a black hole to prevent the collapse to Planck and sub-Planck scales.

According to Loop Quantum Cosmology which is based upon Loop Quantum Gravity, all wave packets that reach a classical singularity as 'a crunch', 'bounce off'. The result is a state that oscillates between bouncing off the core singularity as 'a bounce' and rebounding back to the event horizon as 'a bang'. I have included a link to a video and references from Wikipedia that demonstrates this:
"[URL
[PLAIN]http://relativity.livingreviews.org/Articles/lrr-2005-11/tn_bouncestill.png

Reference:
http://en.wikipedia.org/wiki/Loop_quantum_gravity"
http://en.wikipedia.org/wiki/Loop_quantum_cosmology"
http://en.wikipedia.org/wiki/Quantum_foam"
http://en.wikipedia.org/wiki/Big_Crunch"
http://en.wikipedia.org/wiki/Big_Bounce"
http://en.wikipedia.org/wiki/Big_Bang"
http://en.wikipedia.org/wiki/Oscillatory_universe"
http://www.humnet.ucla.edu/humnet/phil/faculty/Smeenk/Bojowald_LQC.pdf"
http://relativity.livingreviews.org/Articles/lrr-2005-11/articlesu50.html"

 

[stevebd1]

While a radius (or reduced circumference) can be deemed a stable radius for matter that has collapsed to Planck density/pressure, r=0 would still apply to the outer edge of this radius as this would be the end of spacetime (though some kind of bounce would probably stop this occurring, the collapsed matter oscillating close to Planck properties), the volume of matter with Planck density/pressure contained within the radius would be without dimension, as with these properties, the energies that distinguish matter and space are supposed to combine. I'm tempted to say that the Planck matter would be supersymmetric but this would be speculating. It's also fair to say that no matter how little, spin would have some effect on the final collapse of the singularity which may go some way of stopping the matter from reaching absolute Plank properties, and as stated, high energy virtual particles would probably play some part also.

The following paper is a good overview of the different types of singularity-

http://www.unc.edu/~mgood/research/Singularity.pdf

The following papers are a bit maths heavy but insightful-

Singularities and Quantum Gravity
Authors: Martin Bojowald
http://arxiv.org/abs/gr-qc/0702144

Loop quantum gravity and black hole singularity
Authors: Leonardo Modesto
http://arxiv.org/abs/hep-th/0701239

On the black hole singularity issue in loop quantum gravity
Authors: A. DeBenedictis
http://arxiv.org/abs/0907.0826

The following isn't in a user friendly format but is also insightful-

Quantum Foam and de Sitter-like universe
Authors: P. A. Zizzi
http://arxiv.org/abs/hep-th/9808180

There's also the matter of the weak singularity at the inner horizon of rotating or charged BH's which if applies may render GR mute before even reaching the centre of the black hole.

 

[Chronos]

We already know GR is mute before the singularity.

 

[universe11]

Why so many physicists for years accepted GR beyond the Planck wall?