Exploring the Density and Mass of Black Holes: An Insightful Analysis

[taylordnz]

if a black hole has infinite density wouldn't that mean it has infinite mass because its equation is

volume x mass = density

whats your opnion

 

[mathman]

if a black hole has infinite density wouldn't that mean it has infinite mass

That's a big if. In fact, the mass is finite. How it's distributed inside is an open question. Quantum theory and general relativity are in conflict.

 

[jcsd]

if a black hole has infinite density wouldn't that mean it has infinite mass because its equation is

volume x mass = density

whats your opnion

You've got the equation wrong: it's mass/volume = desnity

The mass of a black hole is finite, it reason why a black hole has infinite density is that it's mass is concentrated into a space of zero-volume. It may be that the singularity is avoided, but this is highly speculative.

 

[techwonder]

There's a lot of talk about the singularity in black holes. In general though, a theory coming up with infinities basically hints us that something is wrong. Or in other words - singularities do not exist.

 

[ArmoSkater87]

To make it quick and easy, black holes definately have finite mass. Before the black hole is formed it is a super massive star...which has finite mass and therefore finite density. So when it collapses, according to common sense it must still have finite density, but its just like jcsd said though, because mathematically the density becomes infinite since you would be dividing by zero volume.

 

[ArmoSkater87]

heh, but I am having a hard time conprehending how something with a volume can be crushed into no volume.

 

[Integral]

We have no concrete knowledge of what happens inside the event horizon of a black hole. Any comments on the physical dimensions of the mass inside the event horizon are pure speculation. We can only know the total mass from exterior measurements not the configuration of the mass, only that it is contained within the radius of the event horizon.

 

[ArmoSkater87]

yep...true true

 

[jcsd]

We have no concrete knowledge of what happens inside the event horizon of a black hole. Any comments on the physical dimensions of the mass inside the event horizon are pure speculation. We can only know the total mass from exterior measurements not the configuration of the mass, only that it is contained within the radius of the event horizon.

Ah yes, but the Scwarzchild solution can be extended into the event horizon right up to the singularity, so if we assume general relativity is correct we have a resoanble idea what's going on in there.

 

[Labguy]

Ah yes, but the Scwarzchild solution can be extended into the event horizon right up to the singularity, so if we assume general relativity is correct we have a resoanble idea what's going on in there.

No so. The only thing we can say today is exactly what Integral posted. There is no solution in either GR or QM that we can be certain of that yet "stands the test".

 

[jcsd]

No so. The only thing we can say today is exactly what Integral posted. There is no solution in either GR or QM that we can be certain of that yet "stands the test".

Yu can be certain that GRvstands the except when microscale physics become important. The problem is if you reject GR then there's not much point in talking about black holes anyway. It is reasonable to assume that GR holds within certain parameters.

 

[Integral]

I am sure that GR can be extended beyond the event horizon in a meaningful fashion. But we know that somewhere inside there Physics as we know it breaks down (The singularity) the question we cannot answer is WHERE the breakdown occurs. Thus I say, that any comments concerning the state of matter inside the event horizon are speculative.

 

[Orion1]

Schwarzschild Scheme...



The BH density is determined from its photon sphere and its event horizon and quantum singularity.

Although there is a radial Schwarzschild Solution in classical GR, BH density cannot be determined from this radial solution, because non-rotating event horizons do not exist.

The mass of a Chandresekhar BH is:
$$M_c = 1.457 M_o$$

The radial solution for a spherically symmetric Chandresekhar BH photon sphere is:
$$r_c = \sqrt[3]{ \frac{3 M_c}{4 \pi \rho_c}}$$

The radial solution for a spherically symmetric rotating gravitational BH event horizon is:
$$r_g = \frac{G M_c}{c^2}$$

radial criterion:
$$r_c = r_g$$

$$\sqrt[3]{ \frac{3 M_c}{4 \pi \rho_c}} = \frac{G M_c}{c^2}$$

Density solution for spherically symmetric rotational gravitational Chandresekhar BH:
$$\rho_c = \left( \frac{3c^6}{4 \pi G^3 M_c^2} \right)$$

This density formula does not violate QM or GR and exists at QM and GR 'quantum shutdown'.

For densities at $$r_c < r_g$$ a formula must be demonstrated that does not violate QM or GR.

College Physics 101 - Entrance Examination:

Based upon the Orion1 Equations:

What is the mass of a Chandresekhar BH?

What is the radius of a Chandresekhar BH photosphere?

What is the density of a Chandresekhar BH?


Sorry, but you failed to qualify for this course.

 

[Labguy]

Yu can be certain that GRvstands the except when microscale physics become important. The problem is if you reject GR then there's not much point in talking about black holes anyway. It is reasonable to assume that GR holds within certain parameters.

The "test" to which I was referring is at the "microscale" you mention. There is no way I would reject GR. In fact, I have all my money on the bet that all the "other experiments" about to be conducted (LIGO, etc.) will all confirm GR to the point where any doubters will have to concede. Since 1919 there have been too many confirmations on so many areas (predictions) of GR that I don't believe it is possible for it to fail, in as far as it went.

 

[jcsd]

The BH density is determined from its photosphere, not its event horizon.

Although there is a radial Schwarzschild Solution in classical GR, BH density cannot be determined from this radial solution, because non-rotating event horizons do not exist.

The mass of a Chandresekhar BH is:
$$M_c = 1.457 M_o$$

The radial solution for a spherically symmetric Chandresekhar BH photosphere is:
$$r_c = \sqrt[3]{ \frac{3 M_c}{4 \pi \rho_c}}$$

The radial solution for a spherically symmetric gravitational BH photosphere is:
$$r_g = \frac{G M_c}{c^2}$$

QM/GR shutdown:
$$r_c = r_g$$

$$\sqrt[3]{ \frac{3 M_c}{4 \pi \rho_c}} = \frac{G M_c}{c^2}$$

Density solution for spherically symmetric gravitational Chandresekhar BH:
$$\rho_c = \left( \frac{3c^6}{4 \pi G^3 M_c^2} \right)$$

This density formula does not violate QM or GR and exists at QM and GR 'quantum shutdown'.

For densities at $$r_c < r_g$$ a formula must be demonstrated that does not violate QM or GR.

College Physics 101 - Entrance Examination:

Based upon the Orion1 Equations:

What is the mass of a Chandresekhar BH?

What is the radius of a Chandresekhar BH photosphere?

What is the density of a Chandresekhar BH?


Sorry, but you failed to qualify for this course.

I think you're slightly confusing applications of the Scwarzchild solution. The Chandrasekhar limit is the maximum mass of a white dwarf and it comes from the degenracy pressure between electrons and nucleons, above this mass a white dwarf will become a neutron star (which has it's own limit).

The Scwarzchild solution can be used to describe the space around any spherically symmetric mass, which includes white dwarfs and neutron stars; classical black holes don't have photospheres, though they do have photon spheres.

 

[Orion1]

Schwarzschild Scheme...


I think you're slightly confusing applications of the Scwarzchild solution. The Chandrasekhar limit is the maximum mass of a white dwarf and it comes from the degenracy pressure between electrons and nucleons, above this mass a white dwarf will become a neutron star (which has it's own limit).



jcsd, what you have said is correct, and your confusion is understandable, by 'Chandrasekhar BH', I am referring to the Chandrasekhar Mass Value as applied to a BH, not the mass limit of a White Dwarf, although this may also be introduced as a comparative point.

However, since this topic has been introduced, what is the radius and density of a Chandrasekhar neutron star with a Chandrasekhar Mass?

Now for comparative purposes of density realization, compare these values to the Chandrasekhar BH radius and density demonstrated above?

Anyone, please post the values that you have calculated.

The Scwarzchild solution can be used to describe the space around any spherically symmetric mass, which includes white dwarfs and neutron stars; classical black holes don't have photospheres, though they do have photon spheres.


The Schwarzschild Solution is a solution in Classical GR that describes the event horizon around a spherically symmetric 'non-rotating' BH.

Applying the Schwarzschild Solution to objects with high spin is a poor description of the space-time around 'high spin' objects such as white dwarfs, neutron stars, however can be used to describe the space-time around 'non-rotating' classical BHs, which do not exist anyway.

The mass of a Chandresekhar BH is:
$$M_c = 1.457 M_o$$

The radial solution for a spherically symmetric Chandresekhar BH sphere horizon is:
$$r_c = \sqrt[3]{ \frac{3 M_c}{4 \pi \rho_s}}$$

The radial Schwarzschild Solution for a spherically symmetric gravitational BH event horizon is:
$$r_s = \frac{2 G M_c}{c^2}$$

QM/classical GR shutdown:
$$r_c = r_s$$

$$\sqrt[3]{ \frac{3 M_c}{4 \pi \rho_s}} = \frac{2 G M_c}{c^2}$$

Schwarzschild Density Solution for spherically symmetric gravitational Chandresekhar BH:
$$\rho_s = \left( \frac{3c^6}{32 \pi G^3 M_c^2} \right)$$

Based upon the Orion1 Equasions:

What are the radius and density values of a Chandrasekhar neutron star with a Chandrasekhar Mass?

What are the radius and density values of a Chandrasekhar BH with a Chandrasekhar Mass?

What are the classical Schwarzschild radius and density values for a Schwarzschild-Chandresekhar BH?

 

[jcsd]

Orion1, the value you've posted in the Chandrasekhar limit and it applies to white dwarves, It comes from electron degenracy, not neutron degenrancy. The Chandrasekhar limit is the maximum possible mass of a white dwarf, any white dwarf with a mass greater than this will collapse into a neutron star, not a black hole.

The value that is comparitve to the Chandrasekhar limit for a neutron star is about 4 solar masses, though realistically a neutron star cannot have a mass greater than 2 solar masses.

A black hole of a mass of about 1.4 solar masses could not realtiscally form via stellar evolution.

The photon sphere of a black hole is NOT analagous to the photosphere of a star. The photon sphere of a black hole refers to the last possible orbit around the black hole, whereas the photosphere of a star is where it is no longer transparent.

 

[Orion1]

Oppenheimer Oppression...

jcsd, what you have stated is true, however are still missing my point, I am only presenting a demonstration between the density of a BH and a neutron star with an equal mass. I have not inferred nor indicated that a neutron star with this mass will collapse into a BH. The mass selected is arbitrary only for convenience.
A black hole of a mass of about 1.4 solar masses could not realtiscally form via stellar evolution.

A BH with a Chandrasekhar mass can form through Hawking Radiation evaporation from a more massive BH, though the process is extremely slow.

The topic is the density of a black hole, not electron or neutron degeneracy.

The photon sphere of a black hole is NOT analagous to the photosphere of a star.

I was not attempting to make such an analogy, only stating that avoiding such contractions in physics is impossible. It is implied that a BH photosphere IS a photon sphere and NOT an analogy to a solar photosphere.

I was not attempting to demonstrate stellar evolution, only a demonstration of comparative mass densities.

The neutron degenerancy mass is called the Oppenheimer Mass Limit for a neutron star.

Oppenheimer Mass Limit:
$$M_o = 3M_\odot$$

The radial solution for a spherically symmetric Oppenheimer BH sphere horizon is:
$$r_o = \sqrt[3]{ \frac{3 M_o}{4 \pi \rho_o}}$$

The radial Schwarzschild Solution for a spherically symmetric gravitational BH event horizon is:
$$r_s = \frac{2 G M_o}{c^2}$$

QM/classical GR shutdown:
$$r_o = r_s$$

$$\sqrt[3]{ \frac{3 M_o}{4 \pi \rho_o}} = \frac{2 G M_o}{c^2}$$

Oppenheimer Density Solution for spherically symmetric gravitational Oppenheimer BH:
$$\rho_o = \left( \frac{3c^6}{32 \pi G^3 M_o^2} \right)$$

Based upon the Orion1 Equasions:

What are the radius and density values of a Oppenheimer neutron star with a Oppenheimer Mass?

What are the radius and density values of a Oppenheimer BH with a Oppenheimer Mass?

What are the classical Schwarzschild radius and density values for a Schwarzschild-Oppenheimer BH?

---
jcsd, this approach, although more realistic for stellar evolution, does not simplify my argument for comparative densities of equivalent masses.

 

[jcsd]

I don't see the relevnace tho' to the topic, the 'density' of a balck hole is usually measured from it's event horizon, out of interest the radius of a Scwarzchild black holes photon sphere is simply:

$$\frac{3GM}{c^2}$$

I'm not sure you can say that the black hole has any kind of photosphere as that is to do with the optical properties of a star, I suppose the venet horizon would be the nearest analogy to a photosphere.

 

[Labguy]

I don't see the relevnace tho' to the topic, the 'density' of a balck hole is usually measured from it's event horizon, out of interest the radius of a Scwarzchild black holes photon sphere is simply:

$$\frac{3GM}{c^2}$$

I'm not sure you can say that the black hole has any kind of photosphere as that is to do with the optical properties of a star, I suppose the venet horizon would be the nearest analogy to a photosphere.

The "density" of a BH isn't measured from its event horizon. We can't see or measure an event horizon radius. The only way we can arrive at an R_s is to know the BH mass. From the mass we use the classical $$\frac{2GM}{c^2}$$ to get the R_s, and, as you stated, $$\frac{3GM}{c^2}$$ for the photon sphere. As an aside, the "Oppenheimer" limit (there are several other names too) is 3.2 M_s instead of the 4 mentioned in an earlier post.

And, knowing the mass, therefore the EH and photon sphere radius, still tells us nothing at all about density of or in a BH since the EH is simply an "area of influence" defined by the math above. Density is and would be where there is a measurable quantity of matter (the mass) and a defined volume, Planck size or larger. As Integral mentioned in an earlier post, no sense guessing because we don't have the means to peer inside any Event Horizon for any information at all other than that it exists.

Lastly, since there can be no "static" (non-rotating) black holes, why do the excercises in static math, unless it is a fun or practice thing?

 

[jcsd]

The "density" of a BH isn't measured from its event horizon. We can't see or measure an event horizon radius. The only way we can arrive at an R_s is to know the BH mass. From the mass we use the classical $$\frac{2GM}{c^2}$$ to get the R_s, and, as you stated, $$\frac{3GM}{c^2}$$ for the photon sphere. As an aside, the "Oppenheimer" limit (there are several other names too) is 3.2 M_s instead of the 4 mentioned in an earlier post.

If you ever read in a textbook about the denisty of a black hole it means it's mass over the volume of the sphere formed by the event horizon. It is a term that is used. Yes I am awrae that 4 is just an approximation of the acutal limit which is nearer 3, but the absolute limit is affected by svereal factors

And, knowing the mass, therefore the EH and photon sphere radius, still tells us nothing at all about density of or in a BH since the EH is simply an "area of influence" defined by the math above. Density is and would be where there is a measurable quantity of matter (the mass) and a defined volume, Planck size or larger. As Integral mentioned in an earlier post, no sense guessing because we don't have the means to peer inside any Event Horizon for any information at all other than that it exists.

I don't think you uunderstand, the density, or average density if you prefer of any region of space is merely it's mass divided by it's volume. Talking about the density of a black hole does not imply anything about it's 'structure' or the distribution of mass in the region. As I said earlier you can be sure if someone is talking about it's black hole they are talking about the density of the region bounded by it's event horizon.

Lastly, since there can be no "static" (non-rotating) black holes, why do the excercises in static math, unless it is a fun or practice thing?

I really don't get your point, are you saying thaty the Schwarzschild solution has no practical applications? This is not the case as it can provide a good approximation in many cases where the angular momnetum is low. There's ceratinly no thepretica barrier to Schwarzschild black holes, it's just that we should expect to see Kerr black holes form as stars in just about all cases have angular momentum.

 

[Labguy]

If you ever read in a textbook about the denisty of a black hole it means it's mass over the volume of the sphere formed by the event horizon. It is a term that is used. Yes I am awrae that 4 is just an approximation of the acutal limit which is nearer 3, but the absolute limit is affected by svereal factors

I don't think you uunderstand, the density, or average density if you prefer of any region of space is merely it's mass divided by it's volume. Talking about the density of a black hole does not imply anything about it's 'structure' or the distribution of mass in the region. As I said earlier you can be sure if someone is talking about it's black hole they are talking about the density of the region bounded by it's event horizon.

But, Weinberg put forth that the pressure (P) must also be taken into account when trying to estimate a BH "density". I see no value for P anywhere above. Also, as pointed out in a past PF post, the "mass divided by volume" version doesn't work for a BH when you are just considering V to be the (R_s)³ because the "volume" in a BH is not just governed by our 3 dimensional spatial conditions, and a BH therefore has no meaningful volume! Your "any region of space" comment would not apply to a BH as the "region" is unknown. Exactly where is the GR "breakdown" that Integral mentioned? I don't know, do you?

I really don't get your point, are you saying thaty the Schwarzschild solution has no practical applications? This is not the case as it can provide a good approximation in many cases where the angular momnetum is low. There's ceratinly no thepretica barrier to Schwarzschild black holes, it's just that we should expect to see Kerr black holes form as stars in just about all cases have angular momentum.

Agreed, there are no "theoretical barriers to Schwarzschild black holes", just practical barriers since they don't exist, as you just acknowledged. Have you ever heard of any case where a star had no angular momentum?

My questions above are rhetorical and I wouldn't actually expect an answer. If you did answer I probably really wouldn't understand, any more than Orion1 really understands... :surprise:

 

[jcsd]

But, Weinberg put forth that the pressure (P) must also be taken into account when trying to estimate a BH "density". I see no value for P anywhere above. Also, as pointed out in a past PF post, the "mass divided by volume" version doesn't work for a BH when you are just considering V to be the (R_s)³ because the "volume" in a BH is not just governed by our 3 dimensional spatial conditions, and a BH therefore has no meaningful volume! Your "any region of space" comment would not apply to a BH as the "region" is unknown. Exactly where is the GR "breakdown" that Integral mentioned? I don't know, do you?

What kind of pressure i.e. some sort of link? the conventional model of a black hole necessarily has a singularity at it's centre there is no pressure in the sense that is comparable to the stra, perhaps you're referring to the Scwarzchild solution inside a star which does involve pressure.

I am not introducing some new or unheard of concept by talking about the volume of a black hole; yes space is warped, but there is still a good approximate relationship between it's mass and it's density.

The GR breakdown should occur approximatyely on the Planck scale.

there are no "theoretical barriers to Schwarzschild black holes", just practical barriers since they don't exist, as you just acknowledged. Have you ever heard of any case where a star had no angular momentum?

No, but it doesn't stop the Scwarzchild solution from having practical applications.

My questions above are rhetorical and I wouldn't actually expect an answer. If you did answer I probably really wouldn't understand, any more than Orion1 really understands... :surprise:

But that is beacuse you don't understand, I'm not introducing any new concepts and I really don't see the point to any of your posts as they're just based on nit-picks that are irrelevant to the my posts.

I mean is there really any point to this?

 

[Labguy]

What kind of pressure i.e. some sort of link?

It is a vacuum fluctuation pressure, not the "gas law" type. I'm sure there is a link somewhere but I wouldn't know. Sometime, I actually post things from memory without relying on the internet. I used to read a few books and have a "near" photographic memory. Just haven't read enough books to be an expert on anything...

No, but it doesn't stop the Scwarzchild solution from having practical applications.

To what little I know, what practical applications when applied to only a static BH?

But that is beacuse you don't understand, I'm not introducing any new concepts...

You have no way of knowing what I (or Orion1) do or do not understand. Are you suggesting that on this subject you are the only one who does understand? If so, collect your Nobel prize.

... and I really don't see the point to any of your posts as they're just based on nit-picks that are irrelevant to the my posts.

Yes, when generalities are thrown out as gospel I will tend to nit-pick. Stellar evolution is much more specific and complicated than the generalities you will find in almost any single book or website; they are simplified by necessity. Past examples are when I nit-picked on the statement that "accreting white dwarf stars will become a Type Ia supernova when the mass exceeds 1.44 solar masses". That isn't even close to accurate. Another was that "a BH can only emit one virtual particle from a pair since the other particle must fall back into the EH to return the borrowed energy". That isn't correct either. The (my) posts on these and other nit-picks can be found somewhere in past PF posts by a search, I assume. Ask Marcus and a few others if my nit-picking explanations were correct or not. They were.

I mean is there really any point to this?

No, there isn't. Especially when one party totally relies on just one point of view (theory as yet) out of several plausable theories that will all have merit for consideration until more research and information bolsters one and/or rejects others. I try to stay open-minded for now until we know more, much more, than today.

Just for fun (really, I don't want to always be an argumentative Labguy) I'll go ahead, totally off subject, and predict today that the concept of either a point singularity or even a planar ring singularity with zero volume will soon be dumped like a hot rock. Either of these, of course, leads to the necessity of a BH with a finite mass but "infinite density" somewhere within. The prediction is meaningless, of course, until and unless you someday remember the "Labguy prediction" when the dumping occurs. I'll also predict that, like Orion1, I will beg-out of this thread and post no more.

 

[jcsd]

Rightlabguuy I'll cut to the chase, our basic disagreement is about the validity of Penrose's singularity theorum. I'm quite happy to see it as our best description of what happens in a black hole while you are not.

 

[chroot]

None of my relativity textbooks bother to define either the density of a black hole or the volume of a black hole. You can certainly measure the radius event horizon, of course, using test photons or test particles. At the very least you could measure the radius of the photonsphere and infer the radius of the event horizon. I don't see any reason why you could not simply define a black hole's (average) density to be its mass divided by the volume enclosed by its event horizon. Spacetime does not break down there, and neither do our models of it. The Schwarzschild solution is valid all the way down to the singularity (or, shall we say, to the suspected very tiny quantum-mechanical object!), at least in Eddington-Finkelstein coordinates.

- Warren

 

[Orion1]

Kerr Quantum...


Black holes are predictions of Albert Einstein's theory of general relativity. In particular, they occur in the Schwarzschild Metric, one of the earliest and simplest solutions to Einstein's equations, found by Karl Schwarzschild in 1915. This solution describes the curvature of spacetime in the volume of a non-rotating (static) and spherically symmetric object:

Schwarzschild Metric:
$$ds^2 = - \left(1 - \frac{2M}{r} \right) dt^2 + \left(1 - \frac{2M}{r} \right)^{-1} dr^2 + r^2 d \Omega^2$$

Solid angle standard element:
$$d \Omega^2 = d \theta^2 + \cos^2 \theta d \phi^2$$

The neutron degenerancy mass is called the Oppenheimer Mass Limit for a neutron star.

Oppenheimer Mass Limit:
$$M_o = 3.2M_\odot$$

The radial solution for a spherically symmetric Oppenheimer BH sphere horizon is:
$$r_o = \sqrt[3]{ \frac{3 M_o}{4 \pi \rho_o}}$$

The radial Schwarzschild Solution for a spherically symmetric non-rotating gravitational BH photon sphere is:
$$r_s = \frac{3 G M_o}{2c^2}$$

radial criterion:
$$r_o = r_s$$

Schwarzschild-Oppenheimer Density Solution for spherically symmetric non-rotating gravitational Schwarzschild-Oppenheimer BH photon sphere:
$$\rho_o = \left( \frac{2c^6}{9 \pi G^3 M_o^2} \right)$$

The radial Schwarzschild Solution for a spherically symmetric non-rotating gravitational BH event horizon is:
$$r_s = \frac{2 G M_o}{c^2}$$

$$\sqrt[3]{ \frac{3 M_o}{4 \pi \rho_o}} = \frac{2 G M_o}{c^2}$$

Schwarzschild-Oppenheimer Density Solution for spherically symmetric non-rotating gravitational Schwarzschild-Oppenheimer BH:
$$\rho_o = \left( \frac{3c^6}{32 \pi G^3 M_o^2} \right)$$

The BH singularity surface and volume geometry is IDENTICAL to its BH event horizon surface and volume geometry and has a radius equal to Planck's Radius:

$$r_p = \sqrt{ \frac{ \hbar G}{c^3}}$$

QM/Classical GR criterion shutdown:
radial criterion:
$$r_s = r_p$$

Schwarzschild-Oppenheimer Density Solution for spherically symmetric non-rotating gravitational Schwarzschild-Oppenheimer Singularity:
$$\sqrt[3]{ \frac{3 M_o}{4 \pi \rho_o}} = \sqrt{ \frac{ \hbar G}{c^3}}$$

$$\rho_o = \frac{3M_o}{4 \pi} \left( \frac{c^3}{ \hbar G} \right)^{3/2}$$


Schwarzschild-Oppenheimer BHs do not exist.


The Kerr Metric is a metric discovered in 1963 which is an exact solution to the Einstein field equations. It describes the geometry of spacetime around a rotating black hole:

The Boyer-Lindquist form of the line element:
$$\inline{ds^2 = \rho \left( \frac{dr^2}{\Delta} + d \theta^2 \right) + \left( r^2 + a^2 \right) \sin^2 \theta d \phi^2 - dt^2 + \frac{2mr}{\rho^2} \left(a \sin^2 \theta d \phi - dt \right)^2}$$

$$\rho^2 = r^2 + a^2 \cos^2 \theta$$
$$\Delta = r^2 - 2mr + a^2$$

m - black hole mass
a - angular velocity, (as measured by a distant observer).

Note that r does not agree with the radial coordinate of the Schwarzschild Solution, except asymptotically.

For rotating black holes, the event horizon is predicted have an oblate spheroid (ellipsoid):

The radial solution for a oblate spheroid Oppenheimer BH sphere horizon is:

$$r_o^3 = r_a^2 r_c$$

$$r_o = \sqrt[3]{ \frac{3 M_o}{4 \pi \rho_o}} = \sqrt[3]{r_a^2 r_c}$$

Kerr radial solution for rotating oblate spheroid event horizon:
$$r_s = \frac{G M_o}{c^2}$$

$$r_o = r_s$$

$$\sqrt[3]{ \frac{3 M_o}{4 \pi \rho_o}} = \frac{G M_o}{c^2}$$

Kerr-Oppenheimer Density Solution for rotating oblate spheroid gravitational Kerr-Oppenheimer BH:
$$\rho_o = \left( \frac{3c^6}{4 \pi G^3 M_o^2} \right)$$

Kerr radial solution for rotating oblate spheroid Kerr Singularity:
$$r_p = \sqrt{ \frac{ \hbar G}{c^3}}$$

$$r_o = r_p$$

$$\sqrt[3]{ \frac{3 M_o}{4 \pi \rho_o}} = \sqrt{ \frac{ \hbar G}{c^3}}$$

Kerr-Oppenheimer Density Solution for rotating oblate spheroid gravitational Kerr-Oppenheimer Singularity:
$$\rho_o = \frac{3M_o}{4 \pi} \left( \frac{c^3}{ \hbar G} \right)^{3/2}$$

Kerr-Oppenheimer BHs exist.

BH Singularity Densities are extremely dense, however their densities are NOT infinite.


BH Singularity Density infinities do NOT exist.


Based upon the Orion1 Equasions:
What is the 'density value' for a Kerr-Oppenheimer Singularity?

Reference:
http://www.space.com/scienceastronomy/white_hole_030917.html
http://en.wikipedia.org/wiki/Black_hole
http://www.maths.soton.ac.uk/relativity/GRExplorer/singularities/singtheorems.htm
http://imagine.gsfc.nasa.gov/docs/science/know_l2/black_holes.html
http://www.gothosenterprises.com/black_holes/static_black_holes.html
http://en.wikipedia.org/wiki/Kerr_metric
http://mathworld.wolfram.com/OblateSpheroid.html

 

[Redfox]

How can BH mass be concentrated in a space of zero volume if it has what it seems to be visual borders and diameter, if it was a zero volume then we just would not see the thing... In the end no-one has ever been close to one...

 

[Orion1]

zero BH volume...



Classical Schwarzschild Singularity Dimension Number:
$$n = 1$$ - dimension #
$$dV = 2r_p$$ - volume
$$L = 0$$ - angular momentum

Solution for 'non-rotating' Classical Oppenheimer-Schwarzschild Singularity Density with one dimension:
$$\rho_o = \frac{M_o}{2} \sqrt{ \frac{ c^3}{\hbar G}}$$

Classical Oppenheimer-Schwarzschild Singularity Density value for a one-dimesional 'pointlike object':
$$\rho_o = 1.969*10^{65} kg*m^{-1}$$

'non-rotating' Classical Universe-Schwarzschild Singularity Density solution for a one-dimesional 'pointlike object':
$$\rho_u = \frac{M_u}{2} \sqrt{ \frac{ c^3}{\hbar G}}$$

$$\rho_u = 1.263*10^{85} kg*m^{-1}$$

The density values for these types of objects are a large values, however they are not infinite.