# Singularity - Black Hole or Naked

1. Jan 19, 2009

### mathman

In all descriptions of black holes or naked sigularities (latest issue of Scientific American) that I've seen, the assertion is made that because gravity is so strong the collapsing star ends up as a point of infinity density. However, it may be possible that internal pressure is so strong that it would balance gravity and the collapse will stop. Has any consideration been made to this idea?

2. Jan 19, 2009

### Nabeshin

The whole point of the concept of a black hole is that no known mechanism can provide enough pressure in order to halt the gravitational collapse, so we assume it continues collapsing without hindrance. Disregarding exotic forms of matter, neutron degenerate matter provides the highest pressures known, and even this only holds up to a few solar masses.

See: http://en.wikipedia.org/wiki/Tolman-Oppenheimer-Volkoff_limit

3. Jan 19, 2009

### Hurkyl

Staff Emeritus
They way I've always heard it described is that beyond a certain point, the increasing pressure actually accelerates the collapse: don't forget that pressure contributes to the stress-energy tensor!

4. Jan 19, 2009

### skeptic2

5. Jan 20, 2009

### stevebd1

Last edited by a moderator: May 3, 2017
6. Jan 20, 2009

### mathman

What is puzzling to me, particularly after reading Wikipedia descriptions of degeneracy, is why it is assumed that once a black hole is formed, i.e. the event horizon is greater than the radius of the (neutron or quark) star, it must automatically shrink to a singularity. Why not simply remain in the state it was in before event horizon got bigger?

I have noticed that although G.R. predicts singularity, the relationship with quantum theory may lead to a different conclusion.

7. Jan 20, 2009

### Nabeshin

The schwarzschild radius of a given star is constant, as it depends only on mass. Now, we have a star which is in a state of collapse. Assuming its mass is sufficient it will pass through the stages of electron, neutron, and perhaps quark, degeneracy pressure as it collapses. Now, at the moment the radius of the star becomes less than the schwarzschild radius (the moment it becomes a black hole), the star is in a state of collapse. Since we know of no form of matter which will halt the collapse, we assume that it proceeds to its logical conclusion.

8. Jan 21, 2009

### mathman

Could someone comment on the following scenario. Start with a neutron star with a Schwarzchild radius inside the star, so the escape velocity is less than c. Nearby is a red giant and the star is slowly accreting matter from the giant. At some point the Schwarzchild radius gets outside the neutron star. How would the neutron star know it is supposed to collapse at this point rather than just simply observe that the escape velocity is now greater than c?

9. Jan 21, 2009

### Nabeshin

In this situation, the Schwarzchild radius shouldn't change appreciably. Certainly not enough to engulf the entire neutron star. Rather, what changes is the mass of the neutron star and therefore the pressure necessary to halt gravitational collapse. The impending gravitational collapse once the mass is too large is what pushes the radius of the "star" below the Schwarzchild radius, not an expansion of the Schwarzchild radius.

This is simply the way the situation works out in the universe, because this critical mass will be reached to initiate collapse before the schwarzchild radius "expands" to engulf the star. Hypothetically speaking, it is possible that your situation could take place, and you're exactly right. The neutron star would simply reside within the event horizon.

10. Jan 21, 2009

### Chronos

I am betting on the Planck density as the ultimate state of degenerate matter. I can't 'show the math', but, my intuition insists that UP [uncertainty principle] forbids a true singularity.

11. Jan 22, 2009

### stevebd1

I have seen it mentioned in a number of papers that matter might collapse to 'at least' Planck density inside a BH. If matter was to collapse all the way to Planck density then Planck matter has a hypothetical equation of state of 1:1 (basically ρ=P), what effect might this have on the gravitational field of a black hole? (I understand that once inside a black hole, all bets are off when it comes to physics but pressure is just another form of energy (confined momentum) which might be captured by the black hole). In GR, the algebraic definition of gravity is g=ρ+3P/c^2 which would imply that based on Planck matter, the black hole might have an apparent mass of 4 times the original collapsed mass. While there are black holes out there that have been detected as being less than 10 sol mass, the majority of stellar-mass black holes detected are in the region of 10 to 15 sol mass which would coincide with an initial mass of 2.5 to 3 sol mass of say a neutron star (which is part of a binary system and has collapsed through accretion) collapsing to Planck density (though the increase from a hypothetical EOS of 1/6 for quark matter to 1/1 for Planck matter is a bit steep considering there's no other energy density but the star itself and what it's accreted, which may even go some way towards saying that Planck matter doesn't form in BH's due the inability to attain the 1:1 EOS which is synonymous with Pm). Maybe this increase in mass from 3 sol mass to 10+ sol mass is nothing more than the process of accretion but that would depend on the rate a black hole works through the accretion disk and how old the black hole is. There are also issues regarding the first law of thermodynamics and conservation of energy (how could a 2.5 sol mass object end up with a stress-energy tensor of 10 sol mass? Maybe only 1/4 of the original mass becomes Planck mass and the other 3/4 becomes 'confined momentum' which would mean there are no problems with the first law). Alternatively, maybe pressure is 'thrown off' when the black hole forms as gravitational waves or maybe pressure/mass/EOS has no meaning (or a different meaning) inside the black hole and the event horizon is based on the active mass just outside the event horizon at the moment of collapse.

http://en.wikipedia.org/wiki/Stellar_black_hole

Last edited: Jan 22, 2009
12. Jan 22, 2009

### Orion1

The TOV equation solution for a Planck singularity...

According to the General Relativity equation of state based upon the TOV equation, a super-critical TOV mass limit neutron star, collapses into a quantum singularity.

The TOV equation solution for a neutron star:
$$\frac{dP}{dr} = -(P(r) + \rho(r) c^2) \left( \frac{4 \pi G r^3 P(r)}{c^4} + \frac{r_s}{2} \right) \left[ r \left( r - r_s \right) \right]^{-1} \; \; \; (r > r_s) \; \; \; r \neq r_s$$

Planck sphere surface pressure:
$$P_p = \frac{c^7}{4 \pi \hbar G^2}$$

Planck sphere density:
$$\rho_p = \frac{3c^5}{4 \pi \hbar G^2}$$

Planck sphere density is equivalent to Planck sphere surface pressure:
$$\rho_p = \frac{3P_p}{c^2}$$
$$\frac{3c^5}{4 \pi \hbar G^2} = \frac{3}{c^2} \left( \frac{c^7}{4 \pi \hbar G^2} \right)$$

The TOV equation solution for a degenerate Planck_pressure and Planck density black hole:
$$\frac{dP}{dr} = - \left[ \frac{c^7}{4 \pi \hbar G^2} + \frac{3c^7}{4 \pi \hbar G^2} \right] \left( \frac{c^3 r^3}{\hbar G} + \frac{G m(r)}{c^2} \right) \left[ r \left( r - \frac{2G m(r)}{c^2}} \right) \right]^{-1} \; \; \; (r > r_p) \; \; \; r \neq r_p$$

The TOV equation solution for a Planck singularity:
$$\frac{dP}{dr} = - \left[ \frac{c^7}{4 \pi \hbar G^2} + \frac{3c^7}{4 \pi \hbar G^2} \right] \left( \frac{c^3 r^3}{\hbar G} + \frac{1}{2} \sqrt{\frac{\hbar G}{c^3}} \right) \left[ r \left( r - \sqrt{\frac{\hbar G}{c^3}} \right) \right]^{-1} \; \; \; (r > r_p) \; \; \; r \neq r_p$$

Why could not a degenerate Planck_pressure and Planck density prohibit a gravitational collapse at its singularity?

Reference:
https://www.physicsforums.com/showpost.php?p=1696682&postcount=18"
https://www.physicsforums.com/showpost.php?p=1696754&postcount=20"

Last edited by a moderator: Apr 24, 2017
13. Jan 22, 2009

### stevebd1

Hi Orion1

I don't entirely get that question. Are you asking why I suggest that matter might not collapse down to Planck density? To be honest, I like the idea but I'm wondering how Einstein's equation $g=\rho+3P/c^2$ fits into it all. If the EOS of Planck matter is 1:1, then surely not all of the neutron stars mass can collapse to Planck density as the gravitation field would increase 4-fold which doesn't seem right. Maybe only 1/4 of the stars original mass makes it to Planck density and the rest is either reduced to confined momentum flux (i.e. pressure) which gives the Planck matter its EOS of 1:1, or it's thrown off in some manner. In the case of the rotating black hole, maybe some of the stars mass hangs around the ring singularity in a cloud which may in some way explain mass-inflation and the Cauchy horizon. I suppose I'm saying that maybe the existing stress-energy tensor and the equation of state of Planck matter dictates how much of the stars original mass collapses to Planck density.

I think I raised a similar question in https://www.physicsforums.com/showthread.php?t=272606"

On a side note, regarding mathman's original question, there is something referred to as a 'red hole'- http://arxiv.org/abs/astro-ph/9908113v1

Last edited by a moderator: Apr 24, 2017
14. Jan 22, 2009

### Orion1

Black hole density solution...

In neutron stars, not all the degenerate neutron gas achieves neutron density at core:

$$r_p = 0.8757 \cdot 10^{-15} \; \text{m}$$ - Proton charge radius

$$\rho_n = \frac{3 m_n}{4 \pi r_p^3}$$

$$\boxed{\rho_n = 5.954 \cdot 10^{17} \; \frac{\text{kg}}{\text{m}^3}}$$

Neutron star Tolman VII density solution:
$$\rho(r) = \rho_c \left[1 - \left( \frac{r}{R} \right)^2 \right] \; \; \; \rho(R) = 0$$

Neutron star core density equivalent to neutron density:
$$\boxed{\rho_c = \rho_n}$$

$$\boxed{\rho(r) = \frac{3 m_n}{4 \pi r_p^3} \left[1 - \left( \frac{r}{R} \right)^2 \right]}$$

Planck sphere density:
$$\rho_p = \frac{3c^5}{4 \pi \hbar G^2}$$

$$\boxed{\rho_p = 1.230 \cdot 10^{96} \; \frac{\text{kg}}{\text{m}^3}}$$

Black hole core density equivalent to Planck sphere density:
$$\boxed{\rho_c = \rho_p} \; \; \; \boxed{R = r_s}$$

Black hole Tolman VII density solution:
$$\boxed{\rho(r) = \frac{3c^5}{4 \pi \hbar G^2} \left[1 - \left( \frac{r}{r_s} \right)^2 \right]}$$

Therefore, in black holes not all the degenerate QGP gas achieves Planck sphere density at core.

The result is that a super-critical TOV mass limit neutron star degenerate neutron density core collapses into a degenerate Planck density core.

Reference:
https://www.physicsforums.com/showpost.php?p=1789174&postcount=45"
https://www.physicsforums.com/showpost.php?p=1792334&postcount=47"
http://en.wikipedia.org/wiki/Degenerate_matter" [Broken]
http://en.wikipedia.org/wiki/Neutron_star" [Broken]
http://en.wikipedia.org/wiki/Black_hole" [Broken]

Last edited by a moderator: May 3, 2017
15. Jan 24, 2009

### stevebd1

Thanks for the reply. Correct me if I've got this wrong but while that's an interesting equation, it seems to be akin with an interior solution based on rs being the outside surface of a (hypothetical) Planck star, the density of which slowly increases to full Planck density at r=0 with ρ=0 at rs. I assume this is to demonstrate what you say that not all the original neutron stars matter collapses to Planck density. Is there a way of introducing a radius to replace rs that might represent the properties of a singularity better within the deep region of the black hole?

Regarding the issue of $g=\rho+3P/c^2$, based on a very simple model of the singularity, I'm leaning towards the idea that maybe ~1/4 of the neutron stars mass collapses completely to Planck density (induced by the space-like quality of spacetime within the event horizon) and the other ~3/4 of the stars mass might be reduced to http://en.wikipedia.org/wiki/Radiation_pressure" [Broken] in the Planck matter meaning the stress-energy tensor stays the same, which complies with the first law of TD, and the Planck matter has it's equation of state of 1:1.

Steve

Last edited by a moderator: May 3, 2017
16. Jan 24, 2009

### Orion1

Singularity Tolman VII density solution...

Black hole core density equivalent to Planck sphere density:
$$\boxed{\rho_c = \rho_p} \; \; \; \boxed{R = r_s}$$

Black hole Tolman VII density solution:
$$\boxed{\rho(r) = \frac{3c^5}{4 \pi \hbar G^2} \left[1 - \left( \frac{r}{r_s} \right)^2 \right]} \; \; \; 0 \leq r \leq r_s$$

$$r_s$$ - Schwarzschild radius

Planck sphere singularity core density equivalent to Planck sphere density:
$$\boxed{\rho_c = \rho_p} \; \; \; \boxed{R = r_p}$$

Planck sphere singularity Tolman VII density solution:
$$\boxed{\rho(r) = \frac{3c^5}{4 \pi \hbar G^2} \left[1 - \left( \frac{r}{r_p} \right)^2 \right]} \; \; \; 0 \leq r \leq r_p$$

$$r_p$$ - Planck radius

Reference:
http://en.wikipedia.org/wiki/Planck_length" [Broken]

Last edited by a moderator: May 3, 2017
17. Jan 26, 2009

### stevebd1

I'm assuming that when you refer to $r_p$ you imply Planck length $l_p$ which represents the radius of the singularity in all cases regardless of mass. This would imply that the Planck volume will always exceed Planck density (unless your using $r_p$ to represent the radius of the BH's core in units of Planck length). Maybe this is the case but another way to consider the black hole core is as a small sphere made up of Planck units $(\equiv l_p^3)$ either all with Planck density or maybe with varying degrees of density with the units at r=0 at Planck density.

In another thread entitled 'Mass-Radius relation of a neutron star', you make reference to the Tolman-Oppenheimer-Volkoff equation, particularly https://www.physicsforums.com/showpost.php?p=1718805&postcount=39"

Using the last equation in the post-

Total Tolman mass equation solution VII:

$$M_0 = \frac{8 \pi \rho_c R^3}{15}$$

re-arranging the equation-

$$R=\sqrt[3]{\frac{M_0\cdot 15}{8\pi\rho_c}}$$

where $M_0$ is the total mass as measured by the gravitational field felt by a distant observer, $\rho_c$ is Planck density and $R$ might be the radius of the collapsed Planck mass.

I'm aware this equation is specific to neutron degenerate matter, what changes (if possible) would be required for it to apply to Planck matter? (i.e. incorporating $P=\rho c^2$).

Based on the above equation, a 3 sol mass neutron star would collapse to a sphere of Planck matter with a radius of 8.84e-23 m (90 ym, about the effective cross section radius of a 1 MeV neutrino).

Last edited by a moderator: Apr 24, 2017
18. Jan 27, 2009

### Orion1

Tolman VII singularity...

Tolman total mass equation solution VII:
$$M_0 = \frac{8 \pi \rho_c R^3}{15}$$

The minimum neutron star total radius is equivalent to the Schwarzschild radius:
$$\boxed{R = r_s}$$

$$r_s = \frac{2G M_0}{c^2}$$

Integration by substitution:
$$M_0 = \frac{8 \pi \rho_c}{15} \left( \frac{2G M_0}{c^2} \right)^3$$

Schwarzschild-Tolman total mass equation solution VII:
$$\boxed{M_0 = \frac{c^3}{8} \sqrt{\frac{15}{\pi G^3 \rho_c}}}$$

Schwarzschild-Tolman total core density equation solution VII:
$$\boxed{\rho_c = \frac{15 c^6}{64 \pi G^3 M_0^2}}$$

Planck sphere singularity core density equivalent to Planck sphere density:
$$\boxed{\rho_c = \rho_p}$$

Planck sphere density:
$$\rho_p = \frac{3 c^5}{4 \pi \hbar G^2}$$

Integration by substitution:
$$M_0 = \frac{c^3}{8} \sqrt{\frac{15}{\pi G^3} \left( \frac{4 \pi \hbar G^2}{3 c^5} \right)} = \frac{1}{4} \sqrt{\frac{5 \hbar c}{G}} = \frac{\sqrt{5}}{4} m_ p$$

Schwarzschild-Tolman VII singularity total mass:
$$\boxed{M_0 = \frac{1}{4} \sqrt{\frac{5 \hbar c}{G}}}$$

Integration by substitution:
$$R = \frac{2G M_0}{c^2} = \frac{2G}{c^2} \left( \frac{1}{4} \sqrt{\frac{5 \hbar c}{G}} \right) = \frac{1}{2} \sqrt{\frac{5 \hbar G}{c^3}} = \frac{\sqrt{5}}{2} r_ p$$

$$\boxed{R = \frac{1}{2} \sqrt{\frac{5 \hbar G}{c^3}}}$$

Reference:
https://www.physicsforums.com/showpost.php?p=1718805&postcount=39"
http://en.wikipedia.org/wiki/Planck_mass" [Broken]
http://en.wikipedia.org/wiki/Planck_length" [Broken]

Last edited by a moderator: May 3, 2017
19. Feb 20, 2009

### stevebd1

Hi Orion1

If $M_0$ works out at $\sim 0.559\,m_p$ and $R$ at $\sim 1.118\,l_p$ (as per the equations in post #18), how is the critical density of Planck density attained (assuming $\rho_p=m_p/l_p^{\ 3}$ is the smallest scale possible)? Are you anticipated that there might be a scale smaller than Planck length? Also, how might these 'fixed' quantities apply to black holes of various mass? Are you assuming information is lost once Planck density has been attained and that these quantities for total radius & mass would apply to all static black holes regardless of initial mass?

The Tolman equations seem to imply there is a maximum mass possible based on a critical density which might not apply to the singularity (unless once Planck density is achieved, without speculating too much, the energy becomes supersymmetric and 'goes somewhere else').

There seems to be two schools of thought regarding what happens to matter that falls into a black hole, 1) it's lost 2) it's held in some kind of stasis until the black hole evaporates and is radiated back into the external universe again.

I have heard on more than one occasion that a bounce is likely to occur within about 40-50% of Planck density which might be what happens at the singularity.

Last edited: Feb 20, 2009
20. Feb 24, 2009

### Je m'appelle

True, as 'v' the volume of the star goes to 0, 'd' the density goes to infinity.

Is it possible to have enough pressure for that? Could elementary particles be under fusion process? Could it be that the pressure is so extreme that all elementary particles become one? Well, no one has actually made it (and never will) to the center of a singularity to answer it, so we are left with theories of what could be happening at the center. Remenber that we can't see what a black hole really is, perhaps it is matter in it's ultimate stage. Who said elementary particles behave inside a black hole at the same way as outside, i.e. around us.
I highly believe that we could figure out how elementary particles behave moments before the Big Bang if we could observe the inside of a singularity.