Our January 19, 2022 post about Birkhoff's theorem asked if the collapse of a spherical shell to a black hole happens extremely slowly. A shell cannot drag the frame.
It turns out that the collapse of a shell happens even faster, not slower.
A point negative charge approaches a central positive charge
Let us analyze the collapse in electromagnetism, so that general relativity does not confuse our thinking.
• ----> ● +
e-
If a single electron approaches a central positive charge, the inertia of the electron is larger than in empty space for two reasons:
1. The electron is transporting the negative potential energy it has in the field. More displacement of energy => more inertia.
2. When the electron approaches, more energy from the field is shipped over a large distance to become kinetic energy of the electron. The distance seems to be the distance between the electron and the positive charge.
A spherical shell of negative charge approaches a central positive charge
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- -
- -
- ---> ● + -
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Let the negative charge form a symmetric shell around the central positive charge. When the shell approaches the central charge:
1. It is not clear if the negative potential energy moves at all. The field outside the shell has lost energy. The outside field does not change when the shell moves closer.
2. The shell acquires kinetic energy by reducing the field at it, as the shell moves closer. A) Energy is shipped only over an infinitesimal distance? B) Alternatively, energy is shipped from the central positive charge?
It looks like the inertia of the shell does not grow at all, if 2. A is the case. The shell collapses fast. If 2. B is true, then inertia slows down the fall, but not as much as in the case of a falling point charge.
Our analysis suggests that the collapse of a spherical star into a black hole happens very quickly in the global Minkowski time, even faster than the merger of two equal-sized black holes. There is no "floating" of the shell close to the horizon. The floating effect only happens for a small point mass.
A symmetric shell seems to skip general relativity phenomena which are caused by increased inertia. As if there were no syrup.
If we send an expanding spherical shell outward, it will escape the black hole as easily. However, there is a problem. The shell moves faster than the local speed of a single photon. Is it possible to send the light shell?
General relativity in 1 + 1/2 dimensions
Working with spherical shells is doing physics with the time dimension and the positive half of the radial coordinate r. We could call this physics in 1 + 1/2 dimensions.
General relativity in 1 + 1/2 dimensions is presumably much simpler than in 1 + 3 dimensions.
We need to check what people have found out about general relativity in 1 + 1 dimensions. How much simpler are the phenomena which are due to increased inertia?
Galileo Galilei was wrong in 1638
Galileo Galilei claimed in his book in 1638 that all objects fall at the same speed in a gravity field, in vacuum. Our observation shows that he was wrong. A spherical shell, even of an infinitesimall mass, falls much faster into a black hole than a point mass does.
Drawing by Theresa Knott https://commons.wikimedia.org/wiki/File:Pisa_experiment.png#mw-jump-to-license
Also, our elephant in a syrup model shows that very heavy objects fall much faster than lightweight objects.
What is this about? This is about tidal effects. Tidal effects do not occur at all when a spherical shell falls. For a point mass, tidal effects have a huge influence on how it behaves in a very strong gravitational field. By tidal effects we mean anything that happens because of spherical asymmetry.
Since Birkhoff's theorem assumes full spherical symmetry, it is immune to tidal effects. However, one can ask what does the Schwarzshild metric in the theorem mean? Shells do not obey it at all. We have to investigate this.
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