The effect of the stretched radial metric around M, in the comoving frame of M
M
●
r
v <---- • m
p_t'
^ y'
|
------> x'
We work in the Schwarzschild coordinates in the comoving frame of M. We learned today that the geodesic equation is aware of the (spurious) coordinate acceleration caused by the stretched radial metric around M. Let v << c.
When m moves to the left, some of its tangential momentum p_t' is converted into radial momentum p_r'. But the spatial metric is stretched in the radial direction. Coordinatewise, the test mass moves surprisingly slowly in the radial direction, though its momentum is as expected.
● M
^ v_t' r
\
\
• m
/
/
v v_r'
The radial velocity vector v_r' is surprisingly short after a while. The test mass appears to move up to the positive y direction. There is a coordinate acceleration up, caused by the stretched spatial metric.
Let us calculate this effect. Let m be right below M at the time 0. At a time t', m has moved a distance
x' = v t'
from directly below M. The radial velocity then is
v x' / r.
The radial velocity, coordinatewise, is surprisingly slow by the amount
1/2 r_s / r * v x' / r
= 1/2 * 2 G M / c² * v² t' / r²
= t' * G M v² / c² * 1 / r².
We have as the apparent acceleration to the y direction:
a_y' = γ² G M / r².
It is as if M would have the gravity of
M v² / c²
of additional mass, that is, twice the kinetic energy of M, if M would move at the speed v.
Calculation using the geodesic equation in the Schwarzschild metric
| | y
| |
v
<--- • m
------------------
x
Let us have the Schwarzschild metric around M. We take a small patch around m, and construct a y coordinate and an x coordinate in a way that the coordinates are orthogonal and their metric at m is
1 0
0 1
The coordinates of m are
(x, y) = (0, 0).
It is not exactly a flat metric because the r metric is stretched, or equivalently, the circumference of M at a radius r - y is "too long". If we would shrink the circumference of M at a radius r - y by a factor
1 - y / r * 1/2 r_s / r,
then it would be a flat metric. We conclude that in our x, y coordinates, the x metric g_xx at y is
g_xx(y) = 1 + y / r * r_s / r.
Since dy / dt = 0, in the geodesic equation, only the first term matters. There, μ = y, and α and β can be t or x, because dt / dt and dx / dt are non-zero.
Let us calculate the acceleration which is due to the changing spatial metric of x when we vary y:
Γ_yxx = -1/2 dg_xx / dy
= -1/2 * r_s / r².
Raising the y index has a negligible effect is M is not heavy. We obtain:
a_y'' = 1/2 * 2 G M / c² * 1 / r² * v²
= G M v² / c² * 1 / r².
It is the gravity of twice the kinetic energy of M, if the speed of M were v.
The geodesic equation agrees with our result in the previous section.
Switch back to the laboratory frame
M
● ----> v
r
^ a_y
|
• m
^ y
|
-----> x
The test mass m is initially static. The Lorentz transformation of accelerations tells us that
a_y' = a_y / γ².
We get a surprising result:
a_y = γ⁴ G M / r².
The extra gravity is as if the kinetic energy of M would gravitate fourfold. How can we explain such a large gravity?
We probably do not need to explain it. The natural frame to study the gravity is the comoving frame of M. If we switch to the comoving frame of m (= the laboratory frame), there a clock ticks slower by a factor 1 / γ, and the observer, cobsequently, measures the acceleration to be γ²-fold.
If the test mass m moves back and forth
● M
r
m
-v <---- • ---> v
|
| rope
|
| pull
v
^ y
|
------> x
We work in the Schwarzschild coordinates. The test mass m starts from directly under M, at coordinates
(t, x, y) = (0, 0, 0).
Let m initially possess a momentum
p(0) = γ m v,
which is purely to the positive x direction. Clocks tick slower close to M, which causes a "temporal" acceleration
a' = G M / r²
toward M. The associated force is
F' = γ m a',
and the cumulated momentum
p'(t) = t F',
which is almost exactly to the positive y direction. We divide the current momentum of m into parts:
p(t) = p'(t) + p''(t).
Let us imagine that we remove the momentum p'(t) by pulling m away from M with a rope. The remaining part of the momentum is
p''(t).
Its behavior depends on the spatial metric around M. Our goal is to prove that p''(t) varies cyclically with time, and there is no cumulation of p''(t) with time.
In our own Minkowski & newtonian gravity model, the apparent stretching of the radial metric is caused by the inertia of the field energy E which flows to m if m approaches M, and away from m if m recedes from M. The energy flows over a distance r.
● M
^ E energy flow
\
\ r
\
\
• ---> v
m
In the diagram, m is receding and its kinetic energy flows to back to the common field of M and m, over an average distance r.
The energy flow causes extra inertia to m in a radial motion. The inertia tries to resist m from moving farther from M.
If m moves back, the inertia resists m coming closer to M.
The process looks cyclic, with no net change in p''(t) over a cycle. We may imagine a system of levers which forces m to transport some weights over the distance r if it wants to move farther from M. If m wants to move closer, it has to move the weights back. If we apply a horizontal alternating force to m, the effects cancel each other and there is no cumulation of p''(t).
We conclude that only the temporal acceleration a' = G M / r² affects the y velocity of m in the long run.
The force caused by the temporal acceleration is
F' = G γ m M / r².
The gravity charge of m is γ m.
What if m is static and M moves back and forth?
We already calculated what happens if m moves back and forth. The gravity force is, on the average,
G γ m M / r².
The same should hold for M. Its gravity charge becomes γ M.
Conclusions
We hope that we finally got right the acceleration of m caused by a mass M moving monotonically to a certain direction. The acceleration has to be calculated in the comoving frame of M, using the Schwarzschild metric. The acceleration is surprisingly large, as if γ⁴ M were the gravity charge in the case where M moves tangentially relative to m.
If M moves tangentially back and forth, then the gravity charge is, on the average, γ M, as we would expect. We have to calculate the accelerations for a non-tangential motion of M later.
Our model is Lorentz covariant because the accelerations are calculated in the comoving frame of M and then Lorentz transformed to the desired frame.
We wrote on September 16, 2023 that there probably is no steepening of the gravity potential relative to the newtonian potential. The Schwarzschild metric should then be replaced with a "newtonian metric":
-1 + G / c² * M / r 0 0
0 1 + G / c² * M / r 0
0 1 + G / c² * M / r
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