For a spherically symmetric collapse of perfect dust we have a solution in general relativity, and a singularity forms at the center if we extend the spacetime as far as we can.
https://en.wikipedia.org/wiki/Lemaître_coordinates
Let us define coordinates using particles freely falling from an infinite distance to the singularity. The "time" coordinate is the proper time of a falling particle. The "radial" coordinate is the time delay between successive falling particles. These are the Lemaitre coordinates introduced in 1932.
The proper time of each particle ends in a finite time when it arrives at the center.
"time"
^ singularity
| #
| #
| # freely
|# ^ falling particle
| |
------------------------------------> "radius"
The hash symbols mark the line of the singularity, above which we cannot define the time coordinate.
The line of the singularity is spacelike. A particle which comes to the line cannot linger at the line but disappears completely.
A falling body stretches into a long line of particles before the particles one at a time disappear into the singularity. (Does the pressure grow at all among the particles or does the spatial volume stay constant or even grow for a freely falling dust ball?)
The light cone of particles within the Schwarzschild radius is such that when the proper time of the particle advances, the particle inevitably bumps into the line of the singularity.
If there is no material falling into the singularity, then a falling observer will see nothing on his way to the singularity. The space is "empty". Only the curvature of spacetime remains as a memory of the fallen matter.
The frozen star versus the singularity model
In the frozen star model, the proper time of a falling observer ends already when he reaches the event horizon. Or, since in that model the time of an outside observer is the canonical time coordinate, the falling observer will move ever slower, never quite reaching the horizon.
In the singularity model, the proper time of the observer ends somewhat later, at the line of the singularity. There is no obvious canonical time coordinate in this case.
Are naked singularities bad?
Since the light cones point to the line of the singularity, nothing at the line of the singularity can affect anything elsewhere in the diagram above. The singularity is very well behaved in that sense. It is a veiled singularity because it cannot affect anything around it.
If matter just disappears in a singularity, then even a "naked" singularity is ok, because its behavior is well defined. It is like things falling off the edge of a table. The singularity of the table edge is not veiled. One can go as near the edge as one wants and return back. If one falls off the edge, then one cannot disturb other things on the table any more. There is no indeterminism in this.
It turns out that singularities are not that bad as mathematical objects, provided that matter just disappears in them. If the singularity would be timelike, then matter could stay in the singularity. A naked singularity might have an unknown effect on its environment, and that would be a problem for physical theories.
If matter just disappears in a singularity, then even a "naked" singularity is ok, because its behavior is well defined. It is like things falling off the edge of a table. The singularity of the table edge is not veiled. One can go as near the edge as one wants and return back. If one falls off the edge, then one cannot disturb other things on the table any more. There is no indeterminism in this.
It turns out that singularities are not that bad as mathematical objects, provided that matter just disappears in them. If the singularity would be timelike, then matter could stay in the singularity. A naked singularity might have an unknown effect on its environment, and that would be a problem for physical theories.
Is unitarity broken in a singularity?
Unitarity means that we can reverse time and calculate the development of a physical system back in time, starting from a suitable hypersurface of spacetime.
A singularity devours information. Does it break unitarity?
If we require the hypersurface in the diagram above to avoid the line of the singularity, then we can calculate back in time, and unitarity holds.
One might want to use a hypersurface of a constant time in the diagram above. But the concept of a constant time is coordinate dependent. Why should we try to use a hypersurface which is not wholly in the well-defined spacetime, below the line of the singularity?
We conclude that a singularity does not break a reasonable definition of unitarity.
Should a theory of quantum gravity ban singularities?
Our analysis above did not reveal any reason why a theory of quantum gravity should do away with singularities.
A "popular science" image of a singularity is that matter is there squeezed into an infinite density, and we do not know how matter behaves under such conditions. Therefore, quantum gravity should somehow prevent the infinite density from happening.
Our analysis above tells a very different story: there is no matter at all in the singularity. There is no infinite density.
What about uncertainty relations? If a particle falls into a pointlike singularity, do we know its position and momentum simultaneously too precisely?
Does electromagnetic radiation gain or lose energy when the universe expands or contracts?
Let us consider a conical singularity where the space dimension is S_1, that is, a circle, and the the time dimension is along the height of the cone.
time
^
| /\
| / \
| / \
-----> space
Suppose that at some time t_0 we have a standing electromagnetic wave in S_1, such that the maximum E and B are E_0 and B_0.
If we let the spatial dimension S_0 contract to a point, what happens to the standing wave? Does its energy decrease, stay the same, or increase?
The converse development happens in a de Sitter universe. What happens to a classical electromagnetic wave as the universe expands? A common claim is that "each photon" loses energy when its wavelength increases, and therefore the energy of the wave decreases as the universe expands.
The common claim does not take into account the fact that the number of photons may vary as the universe expands. The number of photons is constant in a Minkowski space, but that does not mean it stays constant in an expanding universe.
The energy of a classical wave depends on the strength of the electric field E and the magnetic field B in it. How does a classical wave respond to an expanding spatial dimension?
What about a massive particle? Its wave function stretches as the universe expands. Should we normalize the wave function, so that there is just one electron also in the future?
Since the electromagnetic radiation energy density in the early universe was large and now it is low, the energy of an electromagnetic wave cannot increase much as the universe expands. It might be that the number of photons stays the same.
An observer in an expanding universe is, in a sense, accelerating away from the source of an electromagnetic wave. The measured energy of a wave depends on the observer. If all inertial observers are accelerating away from the source, then we may say that the wave has lost energy.
In a contracting universe, inertial observers are accelerating toward the source, and will see the energy ever higher.
In the diagram above, if we have a standing electromagnetic wave of, say, 5 wavelengths in S_1, then as the time progresses, S_1 grows shorter. The wavelength probably grows shorter at the same rate.
There is no problem with the uncertainty relation since the energy of a photon grows as S_1 becomes shorter.
At the point of singularity, the energy of a photon in the wave has grown infinite. We may say that the solution is not defined at the point. If we cut the point off from the diagram, then there might be no problems with physics. Of course, if the Planck length is a demarcation line for new physics, then something unexpected might happen.
Since the electromagnetic radiation energy density in the early universe was large and now it is low, the energy of an electromagnetic wave cannot increase much as the universe expands. It might be that the number of photons stays the same.
An observer in an expanding universe is, in a sense, accelerating away from the source of an electromagnetic wave. The measured energy of a wave depends on the observer. If all inertial observers are accelerating away from the source, then we may say that the wave has lost energy.
In a contracting universe, inertial observers are accelerating toward the source, and will see the energy ever higher.
In the diagram above, if we have a standing electromagnetic wave of, say, 5 wavelengths in S_1, then as the time progresses, S_1 grows shorter. The wavelength probably grows shorter at the same rate.
There is no problem with the uncertainty relation since the energy of a photon grows as S_1 becomes shorter.
At the point of singularity, the energy of a photon in the wave has grown infinite. We may say that the solution is not defined at the point. If we cut the point off from the diagram, then there might be no problems with physics. Of course, if the Planck length is a demarcation line for new physics, then something unexpected might happen.
Conclusions
A singularity might not be as bad a beast as one may think, without first analyzing what really happens.
In general relativity, the frozen star model stops the extension of the spacetime manifold at the forming horizon. It freezes the proper time within the star to the point when the horizon is forming.
The big flaw of the frozen star model is that how does the manifold inside the star know when to stops extending forward in time?
If we do not freeze, then at least for a spherically symmetric collapse, we have to stop extending the manifold at the singularity. Our analysis above reveals that that might be an acceptable solution.
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