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If a large star goes supernova, but not enough mass collapses to form a black hole, it often forms a neutron star. My understanding is that this is the densest object that can exist because of the Pauli exclusion principle: It's made entirely of degenerate matter, each particle of which cannot occupy the same quantum state of any other.

So these objects are so massive that they gravitationally lens light. If you make them more massive, they bend the light more. Keep going and going until they bend the light so much that light passing near the surface can barely escape. It's still a neutron star. Add a bit more mass, just enough that light passing just over the surface cannot escape. Now it's a black hole with an event horizon (I think?). Does this mean the neutron star has become a singularity? Isn't it still just a neutron star just beneath the event horizon?

Why are black holes treated as having a singularity instead of just an incredibly massive neutron star at its center? Does something happen when an event horizon is "created?"

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Strictly speaking, all objects gravitationally lens light. One of the main techniques we use to find extra-solar planets is micro-lensing. See en.wikipedia.org/wiki/… –  Alasdair Allan Jun 2 '11 at 9:43
    
I remember reading somewhere that for ordinary degenerate matter, the event horizon is inside the object, so it is not a black-hole or a singularity. I think some basic calculations could help here. For instance, a neutron star is several kms in radius, while the event horizon will be a few hundred meters. Also, it is believed that there are more degenerate states, for instance quark matter. –  Jus12 Jul 29 '11 at 16:40
    
@Jus12: Pedantically, the event horizon would form if the mass were compressed to a radius inside the current size of the ordinary or degenerate body. For the Earth the critical size in on order of 1 centimeter. –  dmckee Mar 9 '12 at 18:49
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@Jus12: there is no event horizon anywhere in ordinary matter. Once an event horizon forms, all of the matter inside must collapse to a black hole. This is the key difference between a neutron star and a black hole--a neutron star is made up of (very dense) ordinary matter in a stable configuration. A black hole is essentially a pointlike or ringlike concentration of mass with infinite density. There is no way to perturb a neutron star and get a stable black-hole like configuration. –  Jerry Schirmer Aug 7 '12 at 14:44
    
@Jerry Schirmer. My error.. I meant "the event-horizon for a blackhole of equivlent mass", not the event horizon itself. –  Jus12 Aug 11 '12 at 6:02
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Short answer is yes.

But if you want to nit pick, I could argue that when a star collapses to form a BH, it first forms a horizon before the singularity forms (cannot form a "naked singularity"). And since time inside the horizon is essentially frozen with respect to that of an observer outside, the singularity NEVER forms. Yet from the point of view of the collapsing star, the singularity forms in about a millisecond after the horizon.

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hmm I see, I forgot about the time bit. –  Carson Myers Jun 1 '11 at 21:26
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In classical General Relativity, once an event horizon forms, every particle inside that event horizon will inevitably travel in the direction of the center of the black hole. This is what is meant by "gravitational collapse" and how matter comes to form a singularity in the center- no matter how small it is, or how close to the center it is, nothing can prevent it from approaching ever closer to the center. From the point of view of the object itself, it does reach the center in a finite time.

In some more exotic theories of physics, such as string theory or loop quantum gravity, the quantized nature of space and time comes to the rescue and prevents a singularity from forming, so a maximum, finite density is reached and an equilibrium is maintained in the center. This is similar conceptually to what you describe, but still a more exotic and much, much denser material than neutron star-stuff.

The density we're talking about here would be approximately one Planck Mass per cubic Planck Length, in other words 2.176 51 × 10^−8 kg / (1.616 199 × 10^−35 m)^3 ~= 5.15556^96 kg/m^3, where neutron star material is "only" (roughly) 10^18 kg/m^3.

In either case, however, outside the event horizon, the black hole can be treated mathematically and observationally as a simple singularity, so for observational calculations, there is no "value added" in worrying about the inner workings of the black hole. The theorem describing this is colloquially called "Black holes have no hair." This theorem was proven and coined by John Wheeler, the same physicist who coined the phrase "black hole" in the first place.

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This is only true for neutral uncharged black holes. Nothing massive can reach the center for rotating or charged black holes, only light. –  Ron Maimon Aug 9 '12 at 8:09
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Wikipedia seems to indicate they they all do:

At the center of a black hole as described by general relativity lies a gravitational singularity, a region where the spacetime curvature becomes infinite. For a non-rotating black hole this region takes the shape of a single point and for a rotating black hole it is smeared out to form a ring singularity lying in the plane of rotation. In both cases the singular region has zero volume. It can also be shown that the singular region contains all the mass of the black hole solution. The singular region can thus be thought of as having infinite density.

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Sure, but is this just because we don't know more about them? Is it just for mathematical simplicity or does the matter of a neutron star actually collapse to zero volume when it becomes massive enough for an event horizon to form? –  Carson Myers Jun 1 '11 at 20:12
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If there is a trapped null surface, and one of the energy conditions like the null energy condition, or the weak energy condition is satisfied, and space outside is noncompact, there has to exist either a singularity or closed timelike curve inside the black hole.

See Penrose and Hawking.

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"Pensose and Hawking" --> "Hawking and Ellis"? Or en.wikipedia.org/wiki/… –  Ben Crowell May 5 '13 at 2:34
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The maximum mass for a neutron star is the Tolman–Oppenheimer–Volkoff limit and is thought to be between 1.5 and 3 solar masses, the range being due to uncertainties of the equation of state of matter at these extreme densities. If the mass of a neutron star exceeds this limit then implosion to a black hole is assumed to be inevitable, there is no force that can repel the collapse according to general relativity.

So black holes are distinct from neutron stars and an event horizon only forms around the black hole.

The quantum degeneracy pressure of electrons that you mentioned occurs inside white dwarfs. In neutron stars the quantum degeneracy pressure of neutrons is responsible.

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So, you are hoping a neutron star's edge could be just under the event horizon and be stable there - in other words, the neutron star would not collapse into the singularity that would form.

That is not possible. See my answer here: Why can't you escape a black hole? It has a nice picture that explains that once you pass the event horizon, the curvature of space-time essentially rotates the time direction to point into a spatial direction towards the center of the black hole. So just as you cannot resist moving forward in time, the edge of the neutron star cannot resist falling into the center of the black hole - that is it's future time direction.

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This. John Rennie also elaborated on this in physics.stackexchange.com/questions/122941/…. +1. –  Renan Jul 3 at 20:37
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This is a question for the Physics forum.

The honest answer is that we don't know for sure. General relativity is a classic (non-quantum) theory, so it should fall apart at very small scales and very high energy densities - exactly what's supposed to happen in a singularity. We're still waiting for the quantum relativity theory to emerge; if and when that happens, we'll know a lot more about singularities.

WRT black holes, it is perhaps prudent to say that, the deeper in you go, the less we actually know what's going on.

We know a lot about the region surrounding the event horizon; we're pretty confident we got that right, and actually we have observations nowadays matching the theory.

We believe we know a little about stuff happening within the event horizon, but things are getting a bit foggy there.

We can't in all honesty say that we know much about what happens down at the bottom, in the very singularity; that's where general relativity divides by zero and goes belly up.

So, take everything with a grain of salt and keep an open mind.

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It is true that we do not know what happens at the singularity or very near the singularity, but for astrophysical black holes, classical General Relativity should hold well everywhere in the vicinity of the event horizon. –  FrankH Mar 9 '12 at 13:33
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All black holes contain singularities, however not all singularities involve black holes. A neutron star may be dense, matter the size of a pinhead can weigh as much as the earth, but there seems to be a mathematical cut-off point beyond which a black hole is formed. The first step for this is the formation of the even horizon, and everything within the event horizon is your singularity. If a neutron star's mass increases in relation to its radius to form its critical circumference (a star 10x heavier than our sun would have a critical circumference of around 110 miles, or a 20 mile radius), it undergoes gravitational collapse and you have your black hole. Beyond it, the matter is so infinitely dense that the gravitational pull sucks every photon of light into its centre. At this point you have your singularity, where the infinite density mean space and time as we know it, cease to exist. You find yourself in a constant state of chaotic equilibrium; like the ultimate drag car using up hundreds of kilos of fuel every second.

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What does it mean that "not all singularities involve black holes"? Have you disproved cosmic censorship? It is false that inside a black hole everything is sucked to the center-- it's just not true for rotating or charged solutions. There is no consensus on what happens in those. –  Ron Maimon Aug 9 '12 at 8:08
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