Levels from infinity to close to the horizon
Let us have a straight line which extends from the horizon to infinity. The line is perpendicular to the horizon.
Let rₙ be such radii that the "remaining" energy of a static test mass m at that radius is
0.9ⁿ m c².
The remaining energy means m c² plus the potential energy.
In an earlier blog post we argued that the radii rₙ converge toward the horizon as n grows to infinity.
Let these rₙ define "layers" of the gravity field of the black hole.
• rₙ
• rₙ₊₁
● center of black hole
Let an observer sit at rₙ and lower a test mass m to rₙ₊₁. If the observer moves the test mass horizontally, he measures that the inertia of the mass is 1.1 -fold compared to if he would move the test mass at his own position.
o observer
|\
/\
| rope
|
• test mass
In earlier blog posts we explained that the inertia is larger because when the observer lowers the test mass with a rope, he gets 10% of the mass-energy of the test mass to the rope system. When he lifts the test mass above at a different location, he must use 10% of the mass-energy. The entire process moved the test mass m, and also moved an additional 10% of mass-energy from one location to another.
We assume that the process moves the test mass from a location X to a location Y, and induces a movement of "field energy" from some other location to some other location.
Let us try to build a model which explains why the inertia grows exponentially at lower levels.
..................................................................................
<----------- • field energy
level n . ------------> a little bit of field energy
..................................................................................
<------------ • field energy
level n + 1 ● ------------> test mass m
..................................................................................
X Y Y' X'
Let us look at the level n + 1.
The test mass m is moved from X to Y. The movement induces some field energy on the level n + 1 to move from X' to Y'.
Let us look at the level n.
Let us assume that the field energy on the level n + 1 itself is a source of gravity.
Then the movement of field energy on the level n + 1 from X' to Y' induces a movement of a little bit field energy on the level n, and this time from X to Y.
Also, the movement of the test mass m transports more field energy on the level n from X' to Y'.
When we look successively at n - 1, n - 2, ..., the movement of field energy from X to Y, and also from X' to Y', grows exponentially.
We have a simple model which might explain why the inertia of a test mass grows exponentially when it comes closer to the horizon. We assume:
1. The mass-energy (inertia) of the test mass alone is m on every level;
2. field energy is a source of gravity;
3. there is a cascading effect on various levels n: on each level, the field energy moved is 10% of the respective mass-energy moved on lower levels in the diagram;
4. a movement of mass-energy on a higher level in the diagram does not affect a lower level.
The field at the horizon and below it
Above we sketched a very crude model which allows us to inspect what happens when the test mass is at the horizon, or inside the horizon.
When we move the test mass horizontally just above the horizon, it causes a cascading effect of field energy movement on upper levels. It is obvious that the moved field energy cannot exceed M c², where M is the mass of the black hole.
Previously in this blog we have simply guessed that the inertia cannot exceed M. Now we have a crude model which supports our guess.
In principle, the movement of the test mass might cause field energy to move in a manner of a "gearbox". Certain field energy could move over large distances when the test mass moves over a short distance. Then the inertia could exceed M.
We conjecture that the inertia cannot exceed M, and is actually much less than M.
The inertia determines what is the speed of a photon as seen by a faraway observer. A local observer will feel gravity if the speed of a photon is less closer to the center of the black hole.
Since the inertia must be less than M, we can only define a finite number of levels n.
Conjecture. The speed of a photon is like in the Schwarzschild solution when we descend from infinity, until we are close to the horizon. At the horizon and inside the horizon, the speed of a photon slows down only moderately when we approach the center of the black hole.
The conjecture means that gravity is relatively weak at the horizon and inside the horizon.
Conclusions
We constructed an extremely crude model which may explain why gravity grows very strong close to the horizon.
The reason for the very large inertia close to the horizon is that a test mass makes field energy to move around, and field energy is a source of gravity, too. It is the "recursive" nature of gravity: any mass-energy carries a gravity charge.
We conjecture that the inertia only grows moderately when we travel from the horizon toward the center of the black hole.
Our model is about a pointlike test mass. We remarked in an earlier blog post that a symmetric spherical shell of mass will not gain more inertia and will collapse to the center very quickly.
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