What is the exact physical extent of a black hole singularity?

[nutgeb]

If a large (>1.5 SM) conventional star collapses to become a black hole, textbooks say that a singularity will form in the BH. A singularity is characterized by a gravitational field of infinite potential. (Or infinite spacetime curvature if you prefer that terminology).

1. As I understand it, a BH singularity does not possesses infinite mass. It possesses only the total mass of the original particles that collapsed to form it. Correct?

2. If the gravitational field is "infinite" at the "edge" of the singularity, how far must one be away from that edge for the gravitational field to be finite? In other words, is there a definition of the exact physical extent of the "infinite" singularity part of the BH? Is the boundary between infinite and finite gravitation zones in the BH crisp or indefinite? How can the potential of a gravitational field transition between infinite and finite merely as a consequence of a fairly small spatial relocation?

3. Assuming a BH singularity causes an infinite gravitational field (within it), it seems that the "infinite" gravitational energy is larger than the "finite" gravitational energy possessed by the star before its collapse. How is it possible for an object of finite mass (star) to increase its total gravitational energy from finite to infinite (by collapsing to a BH) without violating the conservation of energy? As I understand it, the "relativistic rest mass" of an object is equal to the rest mass of the individual particles that have collapsed together to form the object, minus the binding energy that would be needed to pull the particles back apart. The subtraction is called the "mass defect." Perhaps the answer to this question is that the infinite gravitational binding energy of the singularity should be subtracted from its infinite gravitational mass-energy, to yield a finite "net relativistic mass-energy?" I haven't seen it described this way so I would appreciate some help.

 

[Jonathan Scott]

If a large (>1.5 SM) conventional star collapses to become a black hole, textbooks say that a singularity will form in the BH. A singularity is characterized by a gravitational field of infinite potential. (Or infinite spacetime curvature if you prefer that terminology).

1. As I understand it, a BH singularity does not possesses infinite mass. It possesses only the total mass of the original particles that collapsed to form it. Correct?

2. If the gravitational field is "infinite" at the "edge" of the singularity, how far must one be away from that edge for the gravitational field to be finite? In other words, is there a definition of the exact physical extent of the "infinite" singularity part of the BH? Is the boundary between infinite and finite gravitation zones in the BH crisp or indefinite? How can the potential of a gravitational field transition between infinite and finite merely as a consequence of a fairly small spatial relocation?

3. Assuming a BH singularity causes an infinite gravitational field (within it), it seems that the "infinite" gravitational energy is larger than the "finite" gravitational energy possessed by the star before its collapse. How is it possible for an object of finite mass (star) to increase its total gravitational energy from finite to infinite (by collapsing to a BH) without violating the conservation of energy? As I understand it, the "relativistic rest mass" of an object is equal to the rest mass of the individual particles that have collapsed together to form the object, minus the binding energy that would be needed to pull the particles back apart. The subtraction is called the "mass defect." Perhaps the answer to this question is that the infinite gravitational binding energy of the singularity should be subtracted from its infinite gravitational mass-energy, to yield a finite "net relativistic mass-energy?" I haven't seen it described this way so I would appreciate some help.

1. Yes, when all forms of energy (including kinetic energy, heat and so on) are taken into account, the mass of the black hole is equal to the total mass that formed it. There is no additional correction needed for binding energy; if two masses fall together in Newtonian gravity theory, the binding energy (potential energy) is still there in the form of increasing kinetic energy until they collide, at which point it turns into heat or mechanical energy. It is only if that heat or mechanical energy is lost from the system that the overall energy changes. This also holds for a black hole, but any energy released after the falling object has passed the event horizon cannot escape, so the binding energy remains part of the mass of the black hole.

2. Your terminology is a little mixed up. The surface at which acceleration becomes inescapable is known as the event horizon, and the singularity is a point inside it. The event horizon is at least in theory a well-defined "crisp" surface.

3. There are no infinite energies involved. If you choose a frame where forces are infinite, they only last for infinitesimal time.

 

[nutgeb]

I understand what the event horizon is, but that's not what I was referring to. I'm referring to the physical location deep inside the event horizon where the black hole first becomes a singularity. I referred to that location as the "edge of the singularity" which seems like proper terminology to me (unless the singularity is comprised of a single point in space of zero physical extent.) My question is, how can a gravitational field transition smoothly from being finite at one location to being infinite at another location which is just an infintessimal distance away? Something is broken if that's the model.

If the gravitational field is truly infinite (as opposed to "approaching infinity") at the edge of the BH's singularity, then indeed the gravitational energy there is infinite, even if a physical object can experience that infinite energy only for an infintessimal time. I'm asking a pedantic question: How can a finite mass change its gravitational field from finite energy to infinite energy (by collapsing to a BH) without violating the conservation of energy?

 

[atyy]

Something is broken if that's the model.

Yes, it's universally agreed that singularities in GR indicate that the theory is broken.

MTW, 44.1, "Gravitational Collapse As The Greatest Crisis in Physics Of All Time"

 

[JesseM]

I understand what the event horizon is, but that's not what I was referring to. I'm referring to the physical location deep inside the event horizon where the black hole first becomes a singularity. I referred to that location as the "edge of the singularity" which seems like proper terminology to me (unless the singularity is comprised of a single point in space of zero physical extent.) My question is, how can a gravitational field transition smoothly from being finite at one location to being infinite at another location which is just an infintessimal distance away? Something is broken if that's the model.

If the gravitational field is truly infinite (as opposed to "approaching infinity") at the edge of the BH's singularity, then indeed the gravitational energy there is infinite, even if a physical object can experience that infinite energy only for an infintessimal time. I'm asking a pedantic question: How can a finite mass change its gravitational field from finite energy to infinite energy (by collapsing to a BH) without violating the conservation of energy?

I don't think the energy of the gravitational field ever becomes infinite; rather, it's the spacetime curvature that goes to infinity as you approach the singularity, as well as the tidal force. In GR these quantities would be finite at every finite distance from the singularity, though they would attain arbitrarily large values at arbitrarily small distances, much like the value of the function y = 1/x becomes arbitrarily large as you approach the point x=0, but is still finite at every finite distance from that point.

 

[DrGreg]

... (unless the singularity is comprised of a single point in space of zero physical extent.) ...

This is the cause of your confusion. According to General Relativity theory, the singularity really does consist of a single point with zero volume (and thus infinite mass density). Various quantities, e.g. potential, curvature, approach infinity as the "radius" decreases to zero, but at exactly zero the behaviour is undefined.

General Relativity theory insists that a black hole collapses to a single point. Quantum theory insists that collapse to a single point is impossible. So here we have two incompatible theories, and no-one has yet worked out how to resolve this. We just don't know what really happens at the very centre. Nevertheless, we can be pretty confident that, unless you are very close to the centre, where quantum effects become noticeable, the black hole behaves as if all its mass were concentrated at a single point.