The recursive growth of inertia close to an event horizon

We have barely touched the question why the speed of light drops to almost zero at a Schwarzschild black hole horizon. The inertia of a test mass m shoots up toward infinite.

The negative potential of a test mass m at the horizon is

       -m c²,

which can only explain a doubling of the inertia.

Recursive inertia

Let us have static observers who form a sequence toward the horizon.

The potential of a test mass m at the nth observer is

       (0.99ⁿ - 1)  *  m c²,

and the "remaining energy" of the test mass m is

       0.99ⁿ  *  m c².

The remaining energy is the energy which a distant observer would receive if we would convert the test mass to photons and send them to the distant observer.

             observer n        ●

 

             observer n + 1  ●    •  test mass m

                                                1% less mass-energy

                                                1% more inertia

           ------------------------------------- horizon

     

If the observer n + 1 is holding the test mass m, then the observer n thinks in his local frame that the mass-energy of the test mass is 1% less than if the observer n himself would be holding it.

Since the test mass carries along that 1% of negative potential energy, the observer n thinks that the inertia of the test mass in a horizontal movement is 1% larger than when the observer n himself is holding it.

The inertia is easiest to understand for observers 0 and 1. The observer 0 is far away in space, and the test mass close to him has the inertia m.

If the observer 0 lowers down the test mass to the observer 1, then the test mass will carry along negative potential energy -0.01 m. The inertia in a horizontal movement is 1.01 m.

The inertia grows exponentially as we go to ever lower observers:

       m / 0.99ⁿ.

How is the exponential growth possible? The reason must be that the extra inertia itself acquires ever more inertia as we go lower.

The extra inertia comes from energy flow in the newtonian gravity field. The lower we go toward the horizon, the more inertia a certain amount of energy in the field holds.

The inertia in the radial direction is 2% larger for the observer 1 than for the observer 0. We have to find a reason why also the radial inertia grows exponentially:

       m / 0.98ⁿ.