Thank you Charles for the quick response. But since an observer takes a finite time and hence a finite distance to reach the singularity at r=0, can (s)he not tell us about the radial distance covered and hence the volume?

Secondly, considering that the radial coordinate is imaginary, we can take its magnitude to represent the physically real distance and hence can get the volume.

wouldn't Charles Francis' topo(theo)logy imply that a closed hull that has no "meaningful interiors" allows for a worm hole? now worm holes and time machines are very neat indeed albeit somewhat fictional...

now let's assume an infinitesimal distance into that hull's radius... can we bend the rules and workout the volume? can we add those infinitesimals adiabatically to a closed cavity?   

https://en.wikipedia.org/wiki/Schwarzschild_radius simply tells Vs = (4*PI/3)*(rs)3, with rs = 2MG/c2.

Rajat Pradhan the spatial volume inside of a Schwarzschild Black hole is exactly zero by consequence of the Einstein Field Equations and  Schwarzschild's method. Gravity compresses space, which is the true meaning of curvature. When mass resides inside a closed surface, the volume inside the surface is always less than classical geometry would predict. This is what Ricci Tensor does in GR. The trace of the Ricci Tensor is just a way to make sure the boundary conditions agree with observations when stress energy is zero. Abraham Pias gives a reasonably good derivation in his book "Subtle Is The Lord."

Inside a real event horizon space is compressed to one Planck Volume which appears as large as the calculated event horizon when viewed by a distant observer because of relativistic transformations. This is how recent opinions of black holes differ from Schwarzschild.

Cosmic background microwaves were discovered long after the early theories of black holes were written. Roger Penrose and Stephen Hawking attempted to reconcile theories with blue shifted microwaves, resulting in a new black hole theory where the holes are not completely black, but are hotter than the microwaves and in thermal equilibrium with distant space. They give off black body radiation by quantum effects.

I withdraw the comment "Schwarzschild black hole is historic but obsolete," in which the following comment by Asher Klatchko is technically correct.

@Jerry Decker Schwarzschild radius is not obsolete 

Schwarzschild radius is not obsolete, but the original concepts of black hole dynamics is distant history. Much of science has not reconciled microwaves blue shifted to infinite frequency and zero wave length continually pouring into a black hole until Planck Temperature is reached.

Penrose and Hawking are the only two writers I know of who have tackled the problem. It is a rather obvious deficiency in much of what is said about black holes. Hawking proposed thermodynamic equilibrium from quantum events. It means as much energy coming out as going in, after a short time during which the black hole heats up to an equilibrium temperature and becomes a black body radiator.

I differ from Hawking about what temperature is required. He gets a low temperature from quantum calculations. I get a very high Planck Temperature based on out coming heat that is red shifted to 2.7 degrees Kelvin at great distance. 

At Planck Temperature, the old concept of zero volume doesn't apply. The Planck volume does.

@Rajat Pradhan if BHs are products of a collapse of celestial objects then you want to move to the Kerr's BH. there you have an ergosphere and your question can be answered as the radius is a function of the angle. please see attached link.   

http://eagle.phys.utk.edu/guidry/astro490/lectures/lecture490_ch13.pdf

I agree that other models of black holes are more realistic about things like spin and magnetic fields. When there are two event horizons the answers get more complicated about curvature, temperature, and volume.

Consider a heavy neutron star that is about to acquire additional mass and become a black hole. The neutron star is probably spinning quiet fast and generating a strong magnetic field. It seems likely that further collapse would increase the spin and magnetic field strength.

Then the case occurs in which part of the event horizon might be rotating near light speed while bulging at the equator.  It gets rather far from your original question in which the mathematical answer is zero volume, but the physical situation is different.

While the existence and possible nature of black holes is still matter of controversial discussions such as this one here, how can the discovery of gravitational waves reported recently be justified by a black hole merger event as a natural source ?

"... the further collapse is slowed down and never actually completes."

Thanks, Charles, for your kind response ! In your understanding, should the slowing down effect be distinguishable in the gravitational wave (GW) profile as compared to the effect of other GW sources received on Earth ?

Dear Sir : I do not have much knowledge about it but may I suggest you a following link ?

http://arxiv.org/pdf/0801.1734v1...

 Thank you very much Jerry, Asher, Johan, Arun and Charles for your responses and attempts to answer this one. I am surprised to note that When experts tell us about the interior of the SBH, that nothing is known about the geometry etc. and then they go on to tell us about the existence of a real singularity at r=o.  How contradictory! r=Rs is meaningful, r=0 is meaningful, but in between, the region 0

To define it consistently isn't easy-the reason being that the horizon is a light-like surface, not a space-like surface, like the surface that surrounds a volume in space. Since the area of a black hole horizon does have a meaning as a thermodynamic quantity, it might be useful to discuss its volume in the same context. Cf. https://arxiv.org/pdf/1012.2888v2.pdf for instance and, of course,  the followup work.

Thanks a lot Stam for the comment and the reference.

Dear All,

What I learn from the references cited by Arun and Stam is that the volume depends upon the choice of the coordinate. It can be any number zero to infinity and even can be time dependent for certain choices. For equilibrium thermodynamics we can't accept a time-dependent volume. The time-independent volume is not unique. The zero volume that we get in the original Schwarzschild system and the Eddington-Finkelstein system is still a queer thing to interpret, considering that the surface area is finite. 

Any further explanation on this matter is welcome.

No, the spacetime volume doesn't depend on the coordinate system: the integral of d^Dx sqrt(|g(x)|) is invariant under general coordinate transformations, up to surface terms-that's one way of realizing that the cosmological constant term respects the symmetries of general relativity. However there can be more than one way of defining it: the usual, geometric, way, just given, or using thermodynamics, when discussing black holes. The two ways don't need to give the same answer, that's all. But these are, then, physical differences, not coordinate artifacts. So the question is which definition can be more relevant, depending on what issue one wants to discuss. I'd recommend studying the paper  https://arxiv.org/pdf/1012.2888v2.pdf  in detail-I think it clears up a lot of things.

If we're talking about space-time, then can we really be discussing volume in isolation from time?  If space becomes time-like, and time becomes space-like, does that imply that there are 3 temporal dimensions, and does that imply that space-time would have a 6 dimensional 'volume'?  In our non-BH spacetime, we would see the vector sum of the 3 temporal dimensions.

In any case, if the BH is absorbing or emitting any mass/energy, then the 6D volume would be both space and time dependant.  And there wouldn't be any real examples where it wouldn't be absorbing or emitting any mass/energy, so there would always be a family of curves that would be defining the , rather than a single constant answer.

No we can't ``isolate'' space form time:  the volume is a spacetime volume-just like the area of the black hole horizon is a spacetime area.  The space-like volume isn't invariant under general coordinate transformations.

In the interior of the black hole the signature doesn't change: there is only one time-like direction, only it's not the same as the time-like direction outside the black hole.  So spacetime is, always, four-dimensional. 

Stam, the time dependence of volume in some coordinates is derived in: arxiv.org:/0801.1734v1 cited above in the response by Arun V. Panat.

the singularity is a result of the geometric optics of GR's geodesics, so is the wormhole. before moving to QM consider the departure to wave phenomena. albeit nonlinear medium waves develop in spacetime, they diffract and the resulting pattern obscures the geodetic caustic. getting rid of the singularity makes the Hawking information paradox obsolete.   

One should, indeed, be careful, whether one is referring to the spatial volume or the spacetime volume. The former, of course, depends on the coordinates, since, by construction, it isn't invariant under general coordinate transformations. Therefore it doesn't make sense focusing on it. 

So, is the 4-volume of a black hole would be measured in m3-s?  What would the mass of a black hole have to be to have 4-volume of 1 cubic light year - year?  How would that be solved? 

Suvendu Giri, Please check the references cited above esp: the work of M. K. Parikh, Tehani K Finch and that of B. DiNunno and R. A. matzner.

 Stam, "Therefore it doesn't make sense focusing on it." This old view does not help any more. We don.t know the statistical basis of holography yet and we don't know anything about the bulk DOF(if any) in the interior (if any) of a Black hole (if any). So it is worth while discussing this point and it is only in the last five six years that attention has been focused on this issue by researchers. 

No: the ``old'' statement (which is still correct) is that a quantity, that isn't invariant under general coordinate transformations, isn't physical-nor mathematical-within the context of gravity. That's all. It's just an auxiliary quantity, that only helps in computing the invariant quantities, e.g. the spacetime volume. It makes as much sense discussing the meaning of a spatial volume in the context of gravity, as in arguing  over the meaning of any  particular coordinate.

And the consistency of theoretical extensions of general relativity can be checked by showing that they do lead to the calculation of quantities that are invariant under general coordinate transformations, whatever the intermediate steps are.

Stam, please see:  M. Christodoulou and C. Rovelli, Phys. Rev. D 91, 064046 (2015).[arXiv:1411.2854 [gr-qc]].

Once more: the spatial volume, by itself, doesn't have any meaning. It's only a means to an end, to quantities that are invariant under general coordinate transformations. So one should focus on them. So the question that must be answered is, whether the definition, proposed in the paper referred to, is, indeed,  invariant under general coordinate transformations. And just what time dependence means. 

Stam, The paper is already published in a very reputed international journal and is getting lots of references and I think what objections you are raising are already answered in that paper and the ones that have followed it.. the volume defined by them is not invariant and GCT, but is pretty meaningful. The invariants are not the only things to bother about. The differences in different frames are also interesting and meaningful in their own right. Is the entropy invariant under GCT? 

Yes the entropy is invariant under general coordinate transformations. The area is in spacetime, not space, that's why.  

The invariants are the only ``things to worry about''. Their calculation involves dealing with non-invariant quantities; but the non-trivial statement is the existence of invariant quantities. The statement that a quantity is not invariant under general coordinate transformations, in the context of gravity, is the same, in different words, with the statement, that it isn't meaningful. Only the invariant quantities, that can be constructed from it, are-and if none can be constructed, then this is a way of showing that a calculation doesn't make sense-even, though, obviously, it will give some result. 

The content of a paper matters, not the sociology. So it doesn't do it justice to focus on the latter and not on the former.  It's not fair to misrepresent it. Either one discusses its contents, or not. Just quoting it isn't useful.

@ Rajat

Rs = 2GM/c^2

and

V= 4/3 pi Rs^3

The volume depends on the mass M

That's the spatial volume of the star-not the spatial volume of the black hole. The difference between the two is that, for the star, the surface r=Rs is a space-like surface, whereas, for the black hole, it's a light-like surface and a coordinate artifact.  The black hole and the star are two completely different objects.

Another way of understanding that the spatial volume of a black hole doesn't make sense, is to recall that the horizon is a coordinate artifact. It's possible to use Kruskal coordinates, that are regular, everywhere, except at the singularity. In general relativity and its extensions, there isn't any way to define the position of the horizon, or what it would mean to cross it.

And by drawing the Carter-Penrose diagram of a collapsing star one immediately understands what makes sense and what doesn't.

The only meaningful statements are that there exist spacetime observers that can avoid the black hole singularity-it's not in their future; and spacetime observers that will reach it in finite proper time-it is in their future. 

I'm not convinced that the horizon is purely a coordinate artifact, if space becomes time-like and time becomes space-like.

Particle pairs, for instance, lose contact across the EH - the EH separates 'here' from 'there' in a meaningful way.

For a single free-falling particle, there may be no effect as one's light-cones get shifted perpendicularly, but for free-falling particles travelling together, their physical separation in space becomes a separation in 'time-like', and particles that were closer radially to the BH become ahead in 'time-like' of the particles that were farther radially.  Since particles in the future have no way to affect particles in the past, they will cease to communicate.  Unless, of course, field effects extend forward and backward in time, which would be very interesting - could explain dark matter quite nicely.

The statements that space ``becomes time-like'' and ``time becomes space-like'' don't mean anything-they can't be given any meaning. 

The statement that two particles are in free fall ``together'' is incomplete, since what ``together'' means must be specified. If it means that both follow geodesics in this spacetime, then there's no issue. Just take two geodesics-both end up at the singularity and nothing happens to either at the horizon. That they are both free falling means that there isn't any relative force between them. Both will end up at the singularity at finite proper time, that's what matters. 

@ Stam

By the way it was not me that down voted you, 

But you may have overlooked the obvious- that is that the Schwarzschild radius is the effective radius of a black hole event horizon- and not a star

PS.

This answers the question which is about a Schwarzschild black hole

No-the surface of a star is a space-like surface, the black hole horizon is a light-like surface. They don't have anything in common.  A star and its surface are somewhere-a black hole horizon isn't anywhere in particular; and the black hole singularity is a future event not a place.

The correct statement is that, if any object  has size less than or equal to  its Schwarzschild radius, then it cannot escape gravitational collapse to a black hole.

When one makes statements about astrophysical black holes one is referring to the stars, whose collapse led to the black holes. 

So while the spatial volume of a star does make sense, because its surface is space-like, the spatial volume of a black hole doesn't make sense, because there doesn't exist a space-like surface that can be defined as its boundary. 

Stam;

I agree that if they have no forces acting between them, then they can be each considered as a single particle. I made the point that they were falling together intentionally - there are forces acting between them, but together as a group they are in free fall. Then I'm not sure your simplification applies.

@Stam /rAJAT

don't  believe that nonsense about tensor calculus.

A Schwarzschild radius, is a radius and as everyone knows has a space like area, A

The absence of space like features of the interior would account for the absence of apparent volume.

But the absence of interior space like and time like structure is just a quirk of the maths of tensor calculus and the recurrent terms it produces.

So there is a volume and it is the standard volume of the sphere as given above.

Once more: the spatial volume doesn't have any meaning for a black hole-it does for a star. The reason is that a star is an object in space, with a surface that's space-like; a black hole isn't, because it can't be defined by a space-like surface. 

The exterior and interior of a black hole are regions in spacetime, not space, that's why. 

So there's simply no point in using notions about stars to discuss notions about black holes, that, by construction, can't be associated to stars. 

Hi Stam

You are amazingly persistant at following the party line even if it is completely illogical.

I have already explained there is a quirk in  maths of tensor calculus when it comes to black holes which produces a singulariy with no volume and where no time passes -  and yet it has a radius the schwarzschild radius- completely illogical.

There isn't any quirk-nor any notion of a party line. A black hole spacetime is a spacetime that contains a singularity for certain observers-those that move along a geodesic defined by the black hole metric-and doesn't contain a singularity for others-some of those that don't. (There do exist observers, that don't move along geodesics of the black hole spacetime, that will nonetheless, experience the singularity in finite proper time.)  The former, as well as some of the non-inertial observers,  can predict when they will experience the singularity, even if, in practice, as it were, they will be destroyed by tidal forces (if they're not a single point) long before. The latter-a subset of the non-inertial observers-can predict what they have to do in order to avoid the experience of the singularity, unless they're eternal. But there isn't any ``place'' that designates the event horizon of a black hole-or the singularity. That's why the word ``event'' is used and is appropriate: the horizon is a locus of events and the singularity is an event. So one can ask what happens at the horizon and the answer is, simply, that nothing happens. One can, similarly, ask, what happens at the singularity and the answer to that is that something happens, but the details are not known. Any observer is dead long before. 

When a star collapses to a black hole, the spacetime geometry as a whole changes, that's the point. That's why the notion of the spatial volume of a black hole doesn't make sense.

@ Stam

illogical dogma produced by the quirky maths of tensor calculus.

In Einstein's own words "since the mathematicians invaded my theory I no longer understand it".

While mathematics can be counterintuitive,  it is, by construction,  logical. And it's meaningless to quote scientific texts, because the words don't matter, their meaning does. Personal opinions are just that. They don't matter, since they can't be addressed by impersonal arguments. The technical work of Einstein led to his equations and one class of solutions was found by Schwarzschild and it's straightforward to check that these are solutions and what their properties are, in an impersonal way. That's all that matters. 

So, once more: a black hole is different from a star. The latter has a spatial volume, the former doesn't. The astrophysical objects that are *called* black holes are the result of gravitational collapse of stars. In this case one is using the term ``star'', if one is interested in properties of the star that depend on the stuff it's made of-the technical term is ``equation of state''; one is using the term ``black hole'' to stress that one is not interested in properties that depend on the stuff the star is, or was, made of, its equation of state. What's interesting is that one can, indeed, show that the result of the gravitational collapse is an object, whose properties do not depend on the equation of state of the matter that had collapsed,or was ejected. Only mass, charge and angular momentum are relevant-though there are now some examples, where certain features of the equation of state, might, indeed, be relevant. The causal properties of the spacetime after collapse are a different issue.  

A star doesn't have an event horizon, for instance, nor does it have a singularity.  

So the spatial volume of a star that collapses to form an object, some of whose properties can be described by the  Schwarzschild black hole solution, does depend on the star-on its equation of state. It's not a property of the result of the collapse, that's why it doesn't make sense to discuss it in relation to the black hole. 

Cf. for example: https://arxiv.org/abs/astro-ph/0607429