Similarly, a spherical shell of light should collapse fast into a black hole. Observers close to the shell see it collapsing faster than light in their local frame.
What does general relativity say about the collapse of a shell of light or very fast massive particles?
How does general relativity treat faster-than-light particles?
If one could move faster than light in the Minkowski space, then we would face the paradoxes of time travel. We do not think such travel is possible.
On the other hand, if the speed of photons is locally low because of some interaction, then there is no paradox if something else moves at the speed of light in vacuum. An example is a block of glass where photons move slowly, but Cherenkov electrons move fast.
How does general relativity treat faster-than-light particles?
That is a tricky question. The stress-energy tensor is the view of a local observer at a static position according to the spatial coordinates. What does he measure as the energy of a faster-than-light particle?
If the observer were in the Minkowski space, he might think that the energy is infinite?
Should we use comoving coordinates, so that the particle does frame-dragging and does not move faster than light in those coordinates? This would be like the Alcubierre drive.
Maybe particles cannot move faster than light in general relativity?
A collapse of a shell consisting of a single layer of particles
Let us have a shell of dust collapsing in newtonian gravity. We assume that there is no mass inside the shell. The shell probably becomes thinner and thinner because the innermost particles do not feel any pull of gravity.
Let us assume that there is only a single layer of particles. Gravity squeezes the particles in the radial direction. If they are point particles, this is no problem.
Outside the shell we have the Schwarzschild metric
Let r be the radial coordinate of the shell. The radial coordinate speed of light is
(1 - r_s / r) * c
where c is the value far away. Clocks tick slowly, and radial distances have grown. That is why the radial coordinate speed of light is slow.
Inside the shell we have the flat Minkowski metric. Clocks run slowly down there, but the spatial metric is the coordinate metric. The radial coordinate speed of light is
sqrt(1 - r_s / r) * c.
The speed of light is faster inside the shell.
Now we face a dilemma: if the particles of the shell are moving almost at the speed of light, then an observer, who is static outside the shell, will think that their energy is several times m c², where m is the mass of the particle. However, an observer inside thinks that the particles move significantly slower than light. Their energy might be just a little over m c².
What does a static observer just at the shell think about the energy?
Which observer is right? Whose stress-energy tensor should we use in general relativity?
Let us imagine that the observers suddenly stop the particles. They can harvest some amount of energy. The observers must agree on the amount. Should we use this value in the stress-energy tensor?
But in general relativity, the stress-energy tensor is a local thing. The tensor cannot be aware of what happens if we stop the entire shell from advancing.
The Oppenheimer-Snyder collapse of a uniform ball of dust
The Oppenheimer-Snyder 1939 model is not a collapse of a shell, but it is the best researched model of a collapse.
Let us assume that the dust ball is initially static. A clock at the center of the ball ticks slower than a clock at the surface, and observers can measure this. Thus, it is not the FLRW metric, even though several papers claim it is. In the FLRW metric, observers can verify that their clocks run at the same rate.
Oppenheimer and Snyder believed that they must match the metric inside the collapsing ball to the Schwarzschild metric outside the ball.
Let us look at an infinitesimally thin shell of dust at the surface of the dust ball. If the shell obeys the Schwarzschild metric which is outside the ball, it will never reach the event horizon in the Schwarzschild coordinate time.
This means that the ball of dust never becomes smaller than its Schwarzschild radius.
That does not sound right. We would expect the dust ball to contract much smaller. The hamiltonian or lagrangian of the theory is strange if it can bring a system to a total standstill, though the system is clearly in an unstable state.
We argued in the previous blog post that using Eddington-Finkelstein coordinates to jump over the infinity of Schwarzschild time is incorrect physics. New coordinates do not come to the rescue.
A solution to the Oppenheimer-Snyder problem: do not match the metric inside the dust ball to the surrounding Schwarzschild metric
Cherenkov radiation in water in the Reed Research Reactor
Consider a block of glass. Light propagates slowly in glass, but Cherenkov electrons can move considerably faster.
Assume that the refractive index of the glass varies from place to place. Observers living inside glass can map the geometry with rays of light, and calculate a metric.
Then Cherenkov electrons arrive, and the observers are perplexed. They see electrons passing them faster than light.
If Cherenkov electrons would be observers, they would think that the block of glass has a very different metric from what glass observers see.
It does not make sense to match the metric of glass observers to Cherenkov observers.
In the case of a collapsing shell, we have claimed that the shell does not acquire extra inertia, and "sees" a metric which is very different from the metric that small point masses see in the Schwarzschild solution. Point masses acquire huge extra inertia close to the horizon.
It may be that there is no sense in matching the metric of a collapsing shell to the Schwarzschild metric.
In our Minkowski & newtonian model the "geometry of spacetime" is something that the fields simulate for small point masses, in certain cases. A collapsing shell sees a different "metric". We should not try to match it to the Schwarzschild metric.
The word "metric" is misleading if we have to calculate the paths of different objects using a toolkit of various rules. When one calculates paths in electromagnetism, one does not use the word metric.
Conclusions
In literature there is debate about the behavior of collapsing shells in general relativity. Do they freeze at the event horizon or not?
The confusion is due to inherent problems in general relativity. It is not clear who should measure the stress energy tensor and how. Neither it is clear how one should match metrics in different zones.
We claim that the basic problem in general relativity is that it assumes the existence of a "metric" which is supposed to describe the behavior of all objects. The phenomena associated with electromagnetism or gravity cannot be reduced to a "metric". Rather, one must calculate energy flow in the fields to determine the inertia of various objects, and proceed from that.
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