We wrote about "private" interactions on September 23, 2023.
The metric g₀₂ is not aware of handedness and loses information
The problem seems to be that the metric of general relativity "loses information", which would be required to calculate the magnetic effect.
v
<----- rotation
-----------
/ \
| × | rotating circle M or
\ / current loop, charge Q
• -----------•
| dm' or dq'
v vy
mass element
dm or
electron dq
^ V
|
• test mass m or
test charge q
× × B magnetic field
× ×
^ y
|
------> x
When we calculate the magnetic field B at the test charge q, we use the vector (q, dq) in the calculation. In the diagram, the magnetic field dB caused by vy points into the screen (marked with ×), because dq is approaching, at it is to the left of q.
Also the electron dq' creates a magnetic field dB which points into the screen at q.
In the case of the metric g produced by the rotating circle M, the cross term g₀₂ of
dt dy
is not aware if the contribution came from the left or from the right. The cross term does not know the "handedness". Since the contributions from each side cancel out each other in g₀₂, they do not produce any gravitomagnetic force on m.
The contribution of vy cancels a large part of the magnetic field B produced by vx. This explains, partially, why gravitomagnetism is stronger by a factor 4 than the analogous magnetism.
Should gravity be aware of handedness?
Is general relativity wrong in ignoring the handedness?
Gauss's law seems to require that for "free" fields, gravitomagnetism must be identical to the analogous magnetism. Can we allow that it differs for a rotating circle?
Could electromagnetism have a 4-fold magnetic field B for a closed current loop?
The magnetic field in electromagnetism is probably dictated by special relativity, and Gauss's law.
For a rotating current loop, there is no outgoing radiation. It might be that the Gauss's law could be satisfied with different strenghts of the magnetic field B. Also, it might be that special relativity would allow the same.
ω rotation
<---
___ Q charge
/
| × center
\____
B
• • •
| | | E
Suppose that we have a charged half-circle rotating around the center. The magnetic field in this case has to make sure that Gauss's law is satisfied. The lines of of force of (partially induced by dB / dt) the electric field E must not break.
The half-circle gives out electromagnetic radiation.
Let us then combine the half-circle with another half-circle. We have a full circle now, and there no longer is electromagnetic radiation going out.
Could it be that B can change significantly for a full circle?
If that is the case, then nature must somehow know that the circle is now full, and it can strengthen B. We could imagine that if there is no radiation out, then B can somehow grow stronger. But this sounds far-fetched.
Empirically we know that magnetism works the same way for a half-circle and a full circle.
Gravitomagnetism has different fluxes inside the circle and outside it
In electromagnetism, vy reduces the magnetic field outside the current loop, because dB from vy points to the opposite direction to dB from vx.
But inside the loop, these dB point to the same direction.
The metric in gravity loses the information about vy outside the loop. The magnetic field is 4X because of that.
What about inside the loop?
^ ω
\
| |
| ^ v |
| | |
| • |
m
^ y
|
----> x
The information about vy is lost also inside the loop.
In electromagnetism, the effect of vy is to strengthen the magnetic field B inside the loop. This means that the gravitomagnetic field is surprisingly weak inside the loop.
It looks like the hypothetical gravitomagnetic field lines do not close in gravitomagnetism in general relativity!
This may be yet another fatal flaw in general relativity.
If magnetic field lines break, then we have "gravitomagnetic monopoles" in empty space. Does this break the Einstein equations there?
Geodesic deviation equation
^ ^ test masses turn up
| |
Bg
... ----------
× × × test masses m
... ----------
× × × going into the screen
... ----------
^
|
gravitomagnetic
field lines break here
^ z
|
•-----> x
y points into the screen
Suppose that gravitomagnetic field lines break in empty space.
It looks ugly, but we are not able to prove that the Ricci tensor R then necessarily differs from zero.
If "gravitoelectric" field lines would break, then a cube of initially static test masses will increase in volume, and R₀₀ is then non-zero. But breaking magnetic lines do not seem to cause the same effect.
Conclusions
For a rotating circle, disk, or sphere, the metric of general relativity loses information, compared to the magnetic field B of the analogous electric current loop.
This makes the gravitomagnetic field ugly: gravitomagnetic field lines break. We conclude that general relativity probably has a wrong approach to the problem.
We believe that the correct gravitomagnetic field of a rotating object is fully analogous to the magnetic field of the analogous electric current, but we are not yet able to prove that that is the case.
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