1. we calculate the Schwarzschild acceleration of m for each part of M, and sum the accelerations;
2. we sum the metric perturbations for each part of M and then calculate the acceleration of m using the summed metric?
If M, for example, is a rotating disk, its parts are under a constant centripetal acceleration. We suspected on September 3, 2023 that the acceleration of the parts spoils the method 2 above.
We did get consistent result for a moving cylinder on August 28, 2023, using the methods 1 and 2 above. The cylinder moved at a constant speed, no part of it was under an acceleration.
An accelerating M and a test mass m by its side
M
● -----> a_x
r
• --> a_x'
^ y m
|
-------> x
Both M and m are initially static. The mass M starts to accelerate right. What happens to m?
Our own Minkowski & newtonian gravity model suggests that M pulls m along with it. The masses M and m share some inertia. The inertia starts to accelerate with M.
The mass m has gained the extra inertia
m * 1/2 r_s / r
in the field of M. This extra inertia resists movements of m in the field of M. If M starts to accelerate, then we can guess that the inertia tries to keep m moving along with M.
If m is very close to the horizon of M, then the speed of light there relative to M can be tiny, as observed by a faraway observer. The test mass m must move along with M – otherwise it would break the speed limit.
Does the metric of general relativity say anything about the problem?
Let us imagine that M has already gained a tiny velocity v, and m is somehow "thrown" into the metric of M. The test mass m "enters" the field of M at the opposite velocity -v.
If m enters the field of M from far away, the slow metric of time close to M slows down the movement of m, as shown by the constants of motion for the Schwarzschild metric: if the proper time of m slows down, so does its velocity.
We can guess that m attains an acceleration
a_x' = a_x * (1 - 1 / sqrt(1 + r_s / r))
in the diagram above. If r is large, then the value is
a_x' = a_x * 1/2 r_s / r.
The test mass m behind M
m
• --> r ● -----> a_x
a_x' M
-----> x
In the radial direction, m has double the extra inertia of the tangential direction. Both m and M are initially static. Then M starts to accelerate to the right. What is the acceleration a_x' of m?
The formula might be
a_x' = a_x * 2 (1 - 1 / sqrt(1 + r_s / r))
= a_x * r_s / r,
for large r.
Can general relativity suggest the same formula? Yes. If we "throw" m to the metric of M, m loses coordinate velocity because its proper time τ slows down, and because the radial metric is stretched. We obtain a double effect relative to the "by the side" case.
Does this affect the field of a rotating disk?
<-- ω
____
/ \
\_____/
^ V
|
• m
Every part of the disk accelerates toward the center of the disk. The parts close to m share more inertia with m. There might be an additional force which pulls m toward the disk if m is static? We are not sure.
Suppose then that m approaches the disk at a velocity V. Then m starts to share more inertia with the near part of the disk than with the far part. This is the traditional magnetic force which we have always included in our calculations.
Could there be a subtler mechanism through which the acceleration of the parts of the disk pulls m to the side?
We wrote on September 3, 2023:
"If we let the disk parts fly loose, so that there is no acceleration in their paths, then the parts on the left of the disk really start a collective movement toward the test mass, and our argument on August 10 that the left parts are "approaching" the test mass faster, is true.
But in a rotating disk, the collective field of the left side of the disk is time-independent. The collective field is not approaching the test mass faster on the left than on the right."
The discrepancy between the August 10, 2023 calculation of the sum of the accelerations, and the August 29, 2023 sum of the metric perturbations probably came from the fact that the acceleration calculation lets the parts of the disk "fly loose", without being accelerated toward the center of the disk. The situation is "dynamic".
The sum of the metrics, on the other hand, takes into account that the parts cannot fly away, they are stuck to their position in the disk. The metric becomes time-independent.
Which is the correct way to calculate? Probably neither one. They both ignore the acceleration of m when its extra inertia bound to a part of disk accelerates as the disk turns.
The magnetic effect of a simple mass flow
Let us analyze a simpler configuration:
1 3
\ / ^
\ / / v
\_______/ mass flow
2
^ V ^
| | F' force
• m
\
v F force
\ F''
v
^
\ F'''
^ y
|
-------> x
We tune V and v so that the mass in the part 3 of the mass flow is almost static relative to the test mass m. The parts 1 and 3 point directly toward m.
When m moves toward the mass flow, it acquires inertia from the mass flowing at the part 1 of the mass flow. The acquired inertia pushes m down to the right with a force F.
The acquired inertia is "carried" by the mass flow from the part 1 to the part 3. The inertia accelerates upward when it takes the turns as it moves from 1 to 3. This exerts another force F' which pushes m up.
The force F' partially cancels the y component of F. The net force
F + F'
pushes m to the right, and also down.
We conclude that the parts 1 and 2 do exert a "magnetic", horizontal force on m. If we let the parts 1 and 2 "fly loose", so that they do not turn at the bends of the mass flow above, then the effect is essentially the same as for the configuration where the mass flow takes the turns (= accelerates at the turns).
Let us then analyze the extra inertia which m acquires as it approaches the horizontal part 3. The extra inertia pushes m to the right and down with a force F'' but when the extra inertia turns to the part 3, it accelerates to the left and and pushes m up and to the left with a force F'''. Note that if we naively calculate the metric of the mass flow, it is not aware of the force F'''.
The sum
F + F' + F'' + F'''
pushes m down and to the right.
Since the part 1 is length contracted, gravity pulls m more to the left. Does this cancel the entire magnetic effect of the mass flow? Probably not. If V is much larger than v, then we can ignore all effects ~ v².
Let us analyze under the assumption V >> v.
1. F_x ~ v V,
2. F_x' ~ v V,
3. F_x'' ~ v V,
4. F_x''' ~ v V.
All the forces are relevant.
We have yet another force F'''' which is caused by the part 3 slowing down m as m approaches 3.
The force which the part 1 exerts on the test mass m is, to a larige extent, due to the slowing down of the time close to 1. If we imagine that the part 1 "flies loose", then m approaches the slow time zone of 1 at a velocity ~ V + v, while it only approaches the slow time zone of 3 at a velocity ~ V - v. The configuration is asymmetric horizontally.
However, the metric close to 1, 2, 3 is time-independent, and the time runs at about the same rate close to 1 and 3. The metric does not reveal the time asymmetry between 1 and 3.
Accelerations probably are not too relevant, after all
<--- ω
____
/ \ M
\_____ /
• m
^ y
|
------> x
Consider again the case of a rotating disk. The configuration is almost symmetric in the x direction. On the left side the parts are accelerating to the right. Some inertia of m accelerates right, causing a force F on m to the positive x direction.
But the force F is canceled by the corresponding effect on the right side.
The acceleration of the far side pushes m down and the acceleration of the near side pulls m up. The sum of forces pulls m upward, but there is no horizontal force.
Conclusions
We now have a very simple hypothetical formula which tells us how the acceleration of M affects a test mass m.
The discrepancy in our calculations of the magnetic effect of a rotating disk on August 10 and August 29, 2023 is not explained by the centripetal acceleration of the parts of the disk.
The discrepancy, probably, is due to the fact that the acceleration of m has to be calculated from its "private" interaction with each part of the disk. If we first sum the metric perturbations for each part, we lose information which is required in calculating the acceleration. This means that the concept of a "metric" is flawed, as we have suspected for the past two months. We will write a new blog post which examines in detail this question.
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