Clocks slow down in a deep gravitational potential. We explained that by claiming that all particles acquire a "package" of extra inertia from the newtonian gravity field. The extra inertia might be explained by "field energy" moving around.
The gravity field of the central mass M is static. When a charge moves in a static electric field, we can ignore the magnetic field. That suggests that we can ignore "magnetic gravity" effects in the treatment below.
A modified newtonian effective potential
The effective potential in the Schwarzschild metric is
Let us assume that the mass m of the test particle is very small compared to the mass M of the central mass. Then
μ = m
and
M + m = M.
We can write
L = p r,
where p is the momentum of the particle in the tangential direction.
Then the centrifugal potential in newtonian gravity is
L² / (2 m r²) = p² / (2 m).
A faraway observer sees the particle to carry an inertia of
m / sqrt(1 - r_s / r),
where r_s is the Schwarzschild radius.
Let us assume that only the momentum p' associated with the original inertia m "takes part" in the generation of the fictitious centrifugal force:
p' = p * sqrt(1 - r_s / r).
The centrifugal potential in this modified newtonian gravity is
p² / (2 m) * (1 - r_s / r)
= L² / (2 m r²) * (1 - r_s / r)
= L² / (2 m r²) - L² * 2 G M / (2 m r² c² r)
= L² / (2 m r²) - G M L² / (c² m r³).
The formula for the centrifugal potential agrees with the one for the Schwarzschild metric.
The tangential velocity of the particle, as seen by a faraway observer
The above formula gives a constant of motion for the particle.
Let us assume that the particle is on a very eccentric orbit. What is its Schwarzschild coordinate velocity at the perihelion?
We assume that m is much smaller than M, which implies μ = m.
The proper time dτ is measured by an observer riding on the particle. To get the proper time dτ from the coordinate time dt we have to multiply dt by the factor
1 / γ = sqrt(1 - v² / c²),
where v is the velocity measured by a local static observer. We also have to multiply dt by
sqrt(|g₀₀|) = sqrt(1 - r_s / r),
where r_s is the Schwarzschild radius.
We have
L = m r² dφ / dt * 1 / sqrt(1 - r_s / r)
* 1 / sqrt(1 - v² / c²).
Let us try to interpret the formula. The coefficient
1 / sqrt(1 - r_s / r)
we explained with the model where the extra inertia acquired by the particle steals its fair share of the angular momentum.
The Lorentz factor is easy to understand:
γ = 1 / sqrt(1 - v² / c²).
The kinetic energy of the particle gets its fair share of the angular momentum.
Question. Why the Lorentz factor does not show up in the effective potential formula? Could it be a result of the stretching of the radial metric? The stretching of the metric balances the fact that the tangential speed of the particle is reduced when kinetic energy takes its fair share of the tangential momentum?
The centrifugal force and newtonian gravity in circular orbits
Above we calculated that effective potential is like in a modified newtonian gravity where extra inertia has stolen some momentum from the particle, and only the momentum associated with the particle mass m takes part in the fictitious centrifugal force.
The user "ProfRob" calculates the coordinate speed for a circular orbit in the Schwarzschild metric. It agrees with newtonian gravity.
Conclusions
The above results support our view that general relativity is actually newtonian gravity where we take into account also complex, non-intuitive effects of the newtonian gravity field – particularly the extra inertia that a test body acquires in a low potential.
For example, a black hole captures any particle which endeavours below 1.5 Schwarzschild radii. This is because the extra inertia of the particle takes its fair share of the momentum of the particle, and the particle moves then so slowly (measured in Schwarzschild global coordinates) that the centrifugal force can no more win the newtonian gravity attraction.
We may imagine that there is "syrup" in low potentials. The particle is slowed down by the syrup, so that newtonian gravity wins the centrifugal force.
This may help us in analyzing what happens when we overspin a black hole. Newtonian gravity is much easier to analyze than general relativity.
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