Are all black holes singularities identical?

Question

There seems to be a problem between a singularity and the event horizons size. My logic is this if u have two collapsing stars with different masses there horizons will be different in diameter yet both singularities are said to be the same infinite density points. This would make them to be identical structures since theres no difference between two infinite dense points. So how does one have a larger horizon. And another problem is that theory says that light coming from the collapsed star inside is curved back in what is converging light rays but if this is the case then why does the horizon become larger the more matter the black hole consumes. Shouldnt the light within curve even more since more mass is introduced and shouldnt the horizon shrink instead of grow?

Answer 1

singularities are said to be the same infinite density point

Not true. A black hole singularity is not a point in space. It's a portion of spacetime that is missing, not a point or set of points. We also can't define its dimensionality. And unlike the world-line of a point particle, it's spacelike, not timelike. GR also doesn't describe the mass of a black hole as residing at the singularity, because the singularity is a missing part of spacetime.

GR doesn't have a way of defining a local mass density at all, basically because we can't define a local energy associated with the gravitational field. (The newtonian expression for the energy density of the gravitational field is $\propto g^2$, but the equivalence principle tells us that in GR, there is no observable corresponding to $\textbf{g}$.) All we have are global definitions of mass, such as the ADM mass, which is an integral over all of space in an asympototically flat spacetime. In the ADM mass for a black hole, we'd be integrating only over the vacuum. The integral wouldn't include the singularity.

Because GR only talks about the spacetime manifold, and a singularity isn't present as part of the spacetime manifold, definitions of the properties of singularities often end up looking complicated and/or being nonunique. For example, even the definition of whether a singularity is timelike or spacelike is pretty complicated -- much more complicated than the definition for a point-set -- because it has to be phrased in terms of the nearby spacetime. For similar reasons, we can't define the temporal extent of a timelike singularity, or the spatial extent of a spacelike singularity such as the one in a black hole. To measure such a thing, we would need the metric, but the a singularity is a point where the metric misbehaves.

Re your second question, do you understand why the newtonian relation for escape velocity gives $r \propto m$ for a fixed escape velocity? If so, then I think the similar behavior for a black hole would seem pretty natural.

Answer 2

The radius of a black hole refers to the radius of the event horizon. The radius of the event horizon is given by the formula

$$r = \frac{2GM}{c^2}$$

Having a larger event horizon doesn't mean that the black hole is larger, it means that the black hole can consume stuff form a larger radius. The event horzion is bigger means that light can NOT escape from a larger distance.

Answer 3

My logic is this if u have two collapsing stars with different masses there horizons will be different in diameter yet both singularities are said to be the same infinite density points.

Density is mass divided by volume.

The mass can be different finite values and if divided by zero volume both will be infinitely dense.

Also note that we're not sure real singularities exist : general relativity is not a complete theory of the universe and we do not have a theory that describes what happens real matter at such extreme densities which incorporates general relativity and quantum theory (or similar).

So there may be no infinite density to worry about at all.

This would make them to be identical structures since theres no difference between two infinite dense points.

They are not identical because the mass of those points would be different. The density is not an issue.

So how does one have a larger horizon.

Again it's the mass that decides the radius of the event horizon, not the density.

You also need to understand that from outside the event horizon you get no information about the interior. The existence of a singularity is not required to get the same properties visible from outside. In fact there is a theorem called the No Hair Theorem which states that we can only know the mass, charge and angular momentum of a black black hole but nothing else. The interior is off-limits to us. The singularity is, from our point of view, no different from having the entire mass spread evenly over the entire volume - the effect outside the event horizon is the same.

And another problem is that theory says that light coming from the collapsed star inside is curved back in what is converging light rays but if this is the case then why does the horizon become larger the more matter the black hole consumes. Shouldnt the light within curve even more since more mass is introduced and shouldnt the horizon shrink instead of grow?

The only way to show this would be to derive the Schwarzchild radius from the metric. The simple fact is that as the mass grows, the event horizon grows.

Your confusion is in thinking that the path of light inside defines the event horizon - it does not.

The event horizon is a result of space-time being distorted. The effect the gravitational field has on light is a by-product of the distortion, not the cause.

The light is just "obeying the rules" inside the event horizon. This is a normal result of general relativity - light follows paths defined by the curvature of space-time (the technical term for these paths is null geodesic). This is why we say light is bent by gravitational fields (although the effect is not noticeable in our everyday world). All that happens inside an event horizon is that the distortion is extreme to beyond the point most people can visualize it all. But the light is still obeying normal rules.

Answer 4

**Your question concerning the ambiguities about BH singularities is quite justified, but not unique; there's widespread confusion regarding what happens to an initial BH mass as a Schwarzschild sphere forms around it. Current theory assumes that this mass is concentrated in a 'singularity' of undetermined gravity and dimensions. This assumption,however,is wrong. One can equally assume that as particle momentum approaches a critical value, mass is transformed into some other form of energy (radiation) by as-yet-unknown QM processes. From the standpoint of an external observer, this energy could be of any form, and a BH would 'look' the same...'sans' singularity. My view is that radiation energy would manifest itself in the region around the horizon. I realize this description is incomplete, and I welcome further discussion.

Answer 5

You should say centralarity. It's only a local Cartesian center. Are centralarities all the same? Only in a vague theoretical sense, they have a center. The term singularity has been expanded to imply a non-zero area. Physics has re-employed the word singularity to define that non-zero area. That dual employment of the concepts of centralarity, mass and non-zero doughnut that leads to confusion. Originally it was only a spacial parameter of a black hole, of the spacial center of mass, and that parameter is added to the mass, physics, rotation, energy, radiation, topology, age, dimension, homogenity, radiative process, the volume, the physical and quantum activity of the object.

Yes they are all different because they vary by every factor except for the vague notion of centralarity. All the other listed parameters and coeficcients of BH's vary from one BH to another.

You can suggest that singularities have an element of invariability if you think that there is no subatomic activity within them, because you can't see any.

Otherwise you can call them a squishylarity, a timularity, a multiplicity and an infinitularity as rotatularity, to label them from a different perspective.

Because singularity is a witty innuendo suggesting weirdness, it has been adopted by mainstream science to define the center of a black hole from a massive and energetic perspective as well as a local Cartesian one.