Naked singularities

[Gullible Jones]

Out of curiosity: am I the only guy who does not see how a naked singularity can exist?

Think about it. A singularity is an object of infinite density (or as close as you can get). This means that the curvature of space-time around it is infinitely strong, which in turn means that, within a certain distance from it, light can't escape - that would be the event horizon. If mass warps space-time, then how in the hell could a singularity possibly not have an event horizon? Is negative energy involved in hypothetical scenarios of singularity nudity?

[Kuroneko]

A singularity is an object of infinite density (or as close as you can get).

Perhaps a bit of caution is warranted here. Certain spacetimes, such as that of the standard Schwarzschild black hole, are vacuum solutions--there is no energy(mass)-momentum anywhere. Instead, they have certain geometrical parameters that we can interpret as the mass. Furthermore, the term 'singularity' doesn't exclusively mean curvature singularity: neglecting coordinate singularities, which are not physically significant, it can refer to the limits of geodesics (freefall paths of test particles) that terminate in finite proper time without any possible further extension.

This means that the curvature of space-time around it is infinitely strong, which in turn means that, within a certain distance from it, light can't escape - that would be the event horizon. ... Is negative energy involved in hypothetical scenarios of singularity nudity?

It doesn't mean that. The Reissner-Nordstrom black hole is a standard example: if it has more charge than mass (in geometric units), then it would have no event horizons while still maintaining a curvature singularity. Interestingly enough, electrons satisfy this criterion, with about twenty orders magnitude to spare, while being massive and point-like while having no event horizon. Of course, electrons aren't classical particles, so this is just an analogy.

If mass warps space-time, then how in the hell could a singularity possibly not have an event horizon?

The Chazy-Curzon particle is another interesting example--it is cylindrically symmetric, and at far enough distances, the field looks like that of Newtonian point-particle, but at the origin there is a directional curvature singularity: the scalar curvature diverges except for geodesics going along the axis of symmetry.

[Fingolfin_Noldor]

Perhaps a bit of caution is warranted here. Certain spacetimes, such as that of the standard Schwarzschild black hole, are vacuum solutions--there is no energy(mass)-momentum anywhere. Instead, they have certain geometrical parameters that we can interpret as the mass. Furthermore, the term 'singularity' doesn't exclusively mean curvature singularity: neglecting coordinate singularities, which are not physically significant, it can refer to the limits of geodesics (freefall paths of test particles) that terminate in finite proper time without any possible further extension.

Admittedly I have rudimentary knowledge about relativity beyond a undergrad level one, but what the Schwarzschild black hole that implies it is a vacuum solution? Is it because of the way it is defined where it is defined in terms of a modified version of the Minkowski metric?

[Kuroneko]

Admittedly I have rudimentary knowledge about relativity beyond a undergrad level one, but what the Schwarzschild black hole that implies it is a vacuum solution? Is it because of the way it is defined where it is defined in terms of a modified version of the Minkowski metric?

The simplistic answer is that it was constructed with this requirement. Einstein's field equation relates the stress-energy tensor T, which also describes things such as momentum and pressure, to the Ricci curvature tensor R (or actually the Einstein tensor G, which is a trace-reversed R); vacuum solutions are those with vanishing Ricci curvature, and consequently vanishing stress-energy tensor. One of the ways of arriving at the Schwarzschild metric is to explicitly plug it into the field equation with G = 0. Verifying that a given solution is a vacuum is a bit tedious, but I can describe the general procedure if you wish.

[Fingolfin_Noldor]

What do you mean by "Vacuum"? I recall the stress energy tensor provides for energy or matter in a certain space as you describe it roughly. Does that mean that the energy/matter content is zero in such a space?

[Kuroneko]

What do you mean by "Vacuum"? I recall the stress energy tensor provides for energy or matter in a certain space as you describe it roughly. Does that mean that the energy/matter content is zero in such a space?

Yes, at least in the sense that no matter where one looks on such a spacetime, there will be no mass/energy or momentum to be found. As said previously, we can still meaningfully associate a mass with the Schwarschild solution.

[Gullible Jones]

Wow. I think my mind just blew.

Uh, anyway... Regarding the classical Schwarzchild black hole - is it actually possible for one of those things to form from gravitational collapse, assuming the right conditions? Wouldn't that violate conservation of mass? Or is the mass of the black hole as seen from the outside the only thing that matters?

[Kuroneko]

I'm unsure what you mean. If you mean to ask whether it is possible for a Schwarzschild black hole to form from gravitational collapse, then yes. If you mean whether a naked singularity could form a Schwarzschild black hole through further collapse of possibly physically reasonable matter--then no. If you mean to ask whether a naked singularity could form from gravitational collapse of reasonable matter at all, then I don't know enough to say. There was a computer simulation done in the 90s that suggested it may be possible, but I'm not aware of the details.

[Gullible Jones]

I'm unsure what you mean. If you mean to ask whether it is possible for a Schwarzschild black hole to form from gravitational collapse, then yes. If you mean whether a naked singularity could form a Schwarzschild black hole through further collapse of possibly physically reasonable matter--then no. If you mean to ask whether a naked singularity could form from gravitational collapse of reasonable matter at all, then I don't know enough to say. There was a computer simulation done in the 90s that suggested it may be possible, but I'm not aware of the details.

The first (and I have no idea what you mean by "reasonable matter" - just a humble undergrad, sorry).

Wouldn't this violate conservation of mass though?

[Kuroneko]

The first (and I have no idea what you mean by "reasonable matter" - just a humble undergrad, sorry).

Matter satisfying certain energy conditions--non-negative energy density, speed of sound less than the speed of light, etc. By the way, I've tracked down some info on the aforementioned computer simulation, and it suggests that normal matter might be able to make a black hole with enough angular momentum to expose the singularity (the horizon shrinks the faster it spins).

Wouldn't this violate conservation of mass though?

In what sense? The collapse of isolated matter into a Schwarzschild black hole yields a vacuum, yes, but it doesn't mean it can't have meaningful mass-energy associated with it. It's not conceptually different from finding that electromagnetic field can make some effects that appear to violate conservation of energy and momentum and then defining energy/momentum associated with the field in such a way as to conserve those quantities. The mass-energy is not anywhere in spacetime, it will be in the way the spacetime is configured. (The are actually difficulties with doing this in general, but not in this case.)

[Gullible Jones]

Okay, I get it. Pretty damn cool.