UPDATE September 19, 2023: We removed the claim that the Einstein field equations are not Lorentz covariant. Our recent observation that the Schwarzschild solution recognizes the force F₂, may save the covariance.
No, it makes the problem even worse. Now the acceleration is γ⁴ G M / r², while it should be γ G M / r²!
We have to find a new way to get rid of the excess acceleration associated with the kinetic energy of M.
----
Tangentially moving charge Q or mass M
^ v
|
• ● Q or M
q or m
test charge or mass charge or mass
The charge Q in the Lorentz transformation formulae above moves to the x direction. The electric field is squeezed in the x direction, which makes it stronger into the y and z directions. That is why we have the coefficient γ = 1 / sqrt(1 - v² / c²) there. If we instead of a charge Q would have a mass M, then the γ factor
γ M ≅ (1 + 1/2 v² / c²) M
would add the kinetic energy to the gravitating mass of M. We see that the gravity attraction and the Coulomb attraction are completely analogous here.
Radially moving charge Q or mass M
amplitude
2 s
<------------->
-v v
• <---- ● ---->
m M
r = average distance (m, M)
We wrote yesterday that the Lorentz transformation in the x direction, E_x' = E_x is incorrect. We will ignore the formula, and analyze this according to the lines of the second blog post yesterday. The test mass m and the mass M above could as well be electric charges. The treatment is the same.
Let us have the mass M moving back and forth. We calculated yesterday that the kinetic energy of M,
1/2 M v²,
does not gravitate.
But we did not analyze what happens when the mass M changes the direction of its velocity.
The test mass m shares inertia with M. When M switches its course, the shared part of the inertia follows. Thus, the turns in the path of M are directly reflected in the behavior of m.
In the diagram above, the average distance between m and M is r, and M oscillates radially, from a distance r - s to a distance r + s.
When M turns back at the near point to m, then m shares more inertia with M than at the far point. The net effect is that M pulls m more at r - s than it repulses at r + s. This may explain where the gravity of the kinetic energy of M disappeared: the gravity is "stored" until it reveals itself when M turns in its path.
If this is the correct explanation, then it also holds for electric charges. But does it? A fast moving electron most probably accelerates slower in an external electric field of Q since its inertial mass is larger.
A paradox of action and reaction
A new paradox:
Q₁ ● --->
<--- ● Q₂
In the comoving frame of Q₂, the electric field of Q₂ is weaker than that of Q₁. That would mean that Q₁ pulls more on Q₂ than vice versa, which is nonsensical.
The gravity of kinetic energy versus the Coulomb force of "kinetic charge"
● Q or M
^ m very fast
/
/
------------> q very fast
If we work in the frame of Q or M, then a test mass m and a test charge q take a different path. People interpret this that the gravity charge m grew to
γ m,
when it was accelerated to a fast speed v. But the electric charge did not grow.
However, this is frame-dependent. We may be able to preserve Lorentz covariance if we assume that the gravity charge of kinetic energy is twice the gravity charge of rest mass. This is an ugly hypothesis.
The Coulomb force seems to be Lorentz covariant if we correct the E_x' = E_x error. But gravity is harder. How to assign a gravity charge on kinetic energy without referring to a canonical frame?
In a typical calculation about gravity, the frame of M is taken as the canonical frame. This breaks Lorentz covariance.
A possible solution: assume that the spatial coordinates around M are like in the Einstein approximation formula which we heavily criticized in August 2023, except that this time the coordinates have the "pincushion" form above.
If the kinetic energy in M creates pincushion coordinates, then it is possible that m sees the gravity of the kinetic energy double, if m moves tangentially, just as we would like it to be.
If M consists of components M' which oscillate back and forth, the kinetic energy in them only affects through the squeezing of the field. That is, the effect is
1/2 M' v².
The effect of the spatial metric is "reversible". The same holds if m oscillates back and forth.
On the other hand, a consistent tangential movement of m relative to M receives an additional acceleration from the spatial metric. The effect of kinetic energy is doubled:
M v².
This would fix the problem for a slowly moving m, but what about a fast moving one, for which γ >> 1? We must increase the pincushion effect to be extreme?
Difference between gravity and the Coulomb force
The difference between gravity and the Coulomb force would be that gravity has pincushion canonical coordinates, while the Coulomb force does not. Why would it be that way?
Pincushion coordinates is a mechanism to implement the empirically observed behavior in a Lorentz covariant way. We observe that in laboratory coordinates, the kinetic energy of a test mass m seems to gravitate while the corresponding "kinetic charge" of a test charge q does not increase the Coulomb force.
The Schwarzschild solution
M
● ----> v
^ a_y
|
• m
^ y
|
-------> x
In the frame of m (the laboratory frame), let the acceleration be
a_y
upward and the associated "electric" gravity field upward E_g.
Then the Lorentz transformation of the acceleration in the comoving frame of M is
a_y' = a_y / γ²,
because the comoving observer sees clocks ticking at a 1 / γ pace in the laboratory frame, and the unit of acceleration is m / s².
We can calculate the acceleration a_y' in the comoving frame of M from the Schwarzschild metric, using the Schwarzschild coordinates. Then map it to the laboratory frame through a Lorentz transformation. Let E_g' be the "electric" gravity field measured at the location of m in the comoving frame. We have
F = m a_y
<=>
m E_g = m a_y,
because the gravity charge of the test mass is m in the laboratory frame and the inertia is also m.
In the comoving frame the gravity charge and the inertia of the test mass is γ m:
F' = γ m a_y'
<=>
γ m E_g' = γ m a_y / γ²
= m E_g / γ²
<=>
E_g = γ² E_g'.
This is not what we want. The gravity charge of a moving mass M should be γ M, not γ² M, so that the kinetic energy of M gravitates.
We used a Lorentz transformation to map a Schwarzschild acceleration to the laboratory frame, and got a result which is most probably wrong. We do not believe that kinetic energy should gravitate that much.
Previously, we used the Schwarzschild solution to calculate gravitomagnetic effects. It produced reasonable results. But on the gravitoelectric side Schwarzschild seems to be wrong.
Could our "light in a box" model save us here? No. The kinetic energy of a moving box is
1/2 M v²,
just like for any matter.
The Einstein field equations are not Lorentz covariant if there is no magical nonlinearity in them
If we use the Einstein field equation in the laboratory frame in the diagram above, then the stress-energy tensor T looks roughly like this:
M + 1/2 M v² / c² M v ...
M v 0
0 ...
The gravitating mass-energy is, quite correctly, M plus its kinetic energy. The acceleration of the test mass m, a_y, is correct, if there is no magical nonlinearity which would change it from the obvious value.
But above we showed that if we calculate the acceleration in the comoving frame of M, using the Schwarzschild solution, and transform it to the laboratory frame, then a_y has a different value!
Could the gravitational slowing down of time by M save Lorentz covariance? No. If the speed v is 0.1 c, then the discrepancy in a_t is 0.5%. It would require enormous masses to slow down time by, say, 0.25% – but we assumed that M is small.
In the comoving frame we have an exact solution of the Einstein field equations: the Schwarzschild metric. Suppose that magical nonlinearity in the laboratory frame makes the laboratory frame acceleration consistent with the comoving frame. In that case, the laboratory frame must count the kinetic energy twice as gravitating mass-energy, which is implausible. Thus, even magical nonlinearity produces an unsatisfactory solution.
Conclusions
We showed that the Schwarzschild metric has a strange consequence in gravitoelectric forces, that is, the plain old newtonian attraction of gravity. This means that the Schwarzschild solution probably does not describe gravity right. Our proof assumes that gravity is Lorentz covariant and that kinetic energy gravitates like any other form of energy.
We sketched the pincushion force which we can utilize to make gravity:
1. reasonable, so that kinetic energy gravitates like any other mass-energy, and
2. Lorentz covariant.
What does this mean? General relativity is seriously broken with respect to Lorentz covariance, and also the true-and-tested Schwarzschild solution needs fixing to make it Lorentz covariant.
No comments:
Post a Comment