Max Keon wrote:

[...]

Centrifigal forces change at the rate of orbital speed squared,
so if Mercury was traveling at an average of 24000 m/sec instead
of the average 48000 m/sec, it would be restrained from falling
at the full rate by 24000^2 / 48000^2 = .25 of the .0395 m/sec^2
gravity rate. The fall rate is .25 * .0395 = 9.875e-3 m/sec^2.

[...]

The gravity force is pointing directly at the Sun, so unless
Mercury falls closer to the Sun on average its orbital speed
cannot be increased. Adding a new force does not change the pull
direction, so orbital speed cannot change from the normal unless
the average radial length changes. Can you now see that?

Have you asked yourself why the centrifugal force is (m v^2 / r)?

In spacetime with the Schwarzschild metric, the Euler-Lagrange
equation associated with r can simply be derived as the following
where the orbital motion is confined to the equatorial plane of the
gravitating body.

d^2r/ds^2 - (1 - 3 U) r (dO/ds)^2 = - U / r

Where

** O = Longitude
** s = Spacetime
** U = G M / c^2 / r

Say the orbit is circular. That mean the following.

d^2r/ds^2 = 0

We have the following.

(1 - 3 U) r (dO/ds)^2 = U / r

Or

(1 - 3 U) r^2 (dO/ds)^2 = U

Or

v^2 / c^2 = U / (1 - 3 U)

Where (through BS but another chapter of discussion)

** v / c = r dO/ds

It actually takes a little bit higher speed than Newtonian result to
complete an orbit.

Thus back to the Euler-Lagrange equation associated with r, we must
have the centrifugal force identified as the following.

m c^2 (1 - 3 U) r (dO/ds)^2

The curvature in Schwarzschild spacetime actually causes a weaker
centrifugal force. In order to keep an equilibrium orbit, the speed
must be a little higher than the Newtonian result. If not, it will
fly spirally away.

Thus, the advance of Mercury's perihelion must have two components
identified as follows.

** Orbital geometric anomaly (dwelled on by physicists)
** Orbital speed anomaly (eluded physicists sigh)

GR prediction based on orbital geometric anomaly alone somewhat
satisfies the observation. However, including the orbital speed
anomaly, it falls far short. This of course is based on the
speculation that the geodesic motion follows the path of maximum
accumulated spacetime. If the model of geodesics is to follow the
path of least accumulated time or the principle of least time, you
will find the orbital geometric anomaly describes almost the result of
observation while the orbital speed anomaly is null. The net result
fits the observation almost perfectly. But don't celebrate the
achievement of GR yet, because under this model of geodesics it
predicts photons would travel beyond the speed of light due to the
Euler-Lagrange equation associated with r. Dealing with second order
effects, everything must fit in place before betting on a new
hypothesis. GR does not.