UPDATE October 21, 2023: We corrected calculation errors. Now the kinetic energy of M has "negative" gravitation: 1 / γ⁵ * M is the total gravity.
----
-V V
• <-- • • -->
m M₁ M₂
R = distance (m, M₁)
^ y
|
------> x
The masses M₁ and M₂ are close to each other relative to R, but not very close. The velocity V can be very large, so that the kinetic energy of M₁ and M₂ is even larger than their rest masses. The test mass m is initially static in the laboratory coordinates.
It is a conventional belief that we must include the kinetic energy of M₁ and M₂ into their gravitating mass. We may have a very hot star where much of the total energy is kinetic energy. A box full of photons would have exclusively kinetic energy, no rest mass at all.
Determining the laboratory frame acceleration of m due to M
Let us first assume that M₂ does not exist. We denote M₁ by simply M.
To determine the gravity field of M, let us switch to a comoving frame of M. We may assume that M is carrying a very long ruler which extends all the way to m. Let us have an observer standing on the ruler. The observer measures the acceleration of m due to the gravity of of M.
Assumption of Lorentz covariance. We assume that weak gravity forces are Lorentz covariant.
The observer sees m moving at a velocity V toward M in the Schwarzschild coordinates of M. To determine the acceleration in the Schwarzschild coordinates, we can use the equation
We can write the left side of the Wikipedia equation above as
1/2 m dr² / dt² * dt² / dτ²
= 1/2 m V² * dt² / dτ²
= 1/2 m V² * 1 / (1 - V'² / c²)
* 1 / (1 - r_s / r)
= A(V, r).
where V is the coordinate velocity, and V' is the velocity of m measured by a static observer:
V' = dr / sqrt(1 - r_s / r)
/
dt / sqrt(1 - r_s / r)
= V.
We denote
B(r) = G M m / r.
If r changes by dr, but V does not change, the left side of the equation changes by
dA(V, r) / dr
= -1/2 m V² * 1 / (1 - V² / c²)
* 1 / (1 - r_s / r)²
* r_s / r²
and the right side
dB(r) / dr = -G M m / r².
To keep the equation satisfied, V has to be adjusted so that
dA(V, r) / dV * dV
= ( -dA(V, r) / dr + dB(r) / dr ) * dr.
Let us calculate
dA(V, r) / dV
= m V * 1 / (1 - V² / c²) * 1 / (1 - r_s / r)
+ 1/2 m V² * -1 / (1 - V² / c²)²
* 1 / (1 - r_s / r) * -2 V / c²
= ( m V (1 - V² / c²) + m V³ / c² )
* 1 / (1 - V² / c²)² * 1 / (1 - r_s / r)
= m V / (1 - V² / c²)² * 1 / (1 - r_s / r).
The acceleration of m in the Schwarzschild coordinates in the comoving frame is
a' = dV / dt
= dV / dr * dr / dt
= V * dV / dr
= V ( -dA(V, r) / dr + dB(r) / dr)
/ dA(V, r) / dV
= V (
1/2 m V² * 1 / (1 - V² / c²)
* 1 / (1 - r_s / r)² * r_s / r²
- G M m / r²
)
/
(m V / (1 - V² / c²)² * 1 / (1 - r_s / r))
=
1/2 V² * (1 - V² / c²)
* 1 / (1 - r_s / r) * r_s / r²
- G M / r² * (1 - V² / c²)² * (1 - r_s / r)
= 1/2 V² / γ² * (1 + r_s / r) * r_s / r²
- G M / r² * 1 / γ⁴ * (1 - r_s / r),
where γ = 1 / sqrt(1 - V² / c²).
If M is small, we can drop the M² terms:
a' = 1/2 V² / γ² * 2 G M / c² * 1 / r²
- G M / r² * 1 / γ⁴
= G M / r² * (-1 / γ⁴ + V² / c² * 1 / γ²).
We have
1 / γ² + -1 / γ² + V² / c² * 1 / γ²
= 1 / γ² - 1 / γ⁴.
Thus,
a' = G M / r² * (1 / γ² - 2 / γ⁴).
For a small V,
1 / γ² - 2 / γ⁴ = 1 - V² / c² - 2 + 4 V² / c²
= -1 + 3 V² / c²
= -1 / γ⁶.
Since m is initially static in the laboratory frame, we have
a' = a / γ³.
Also,
r = γ R.
Thus,
a = -1 / γ⁵ * G M / R².
Discussion
If we sum the metric perturbations caused by M₁ and M₂, the kinetic energy of them does not gravitate. But in the Einstein field equations, in the stress energy tensor, kinetic energy is included in the gravitating mass. This is a contradiction.
Let us recapitulate what we assumed to arrive at a contradiction:
1. The Schwarzschild metric is correct;
2. gravity is Lorentz covariant;
3. there is no strange nonlinearity in the Einstein field equations where even a very small mass M₁ can fundamentally change the field of another very small mass M₂ far away.
The nonlinearity mentioned in item 3 is extremely improbable.
Conclusions
The gravity is reduced when M has kinetic energy. This is surprising. We have to double-check the result.
Yet another aspect is the status of Birkhoff's theorem. If we have a mass shell exploding outward, is its gravitational mass reduced relative to a static shell?
We will study in subsequent blog posts if a "metric" or a "field" can capture the dynamics of a many body system. Our previous post suggests that defining an "electric" field at the location of a test charge q may be impossible. This is because the inertia of q may depend in a complex way on the sources of the electric field Q₁, Q₂, Q₃, ... .
Since the Schwarzschild solution seems to work well both for gravity and for the Coulomb field, we have a good chance to develop a unified field theory.
If the two masses M₁ and M₂ move tangentially relative to m, the problem with the gravity of kinetic energy remains. In a comoving frame, M₁ does not possess kinetic energy: the kinetic energy cannot gravitate. Should we declare the frame of the test mass the canonical frame where energy has to be measured? That might spoil the orbits of planets. We may declare the center of mass of M₁ and M₂ the canonical frame. That breaks Lorentz covariance?
A possible solution: Lorentz transform the newtonian gravity field like the Coulomb field. A tangentially moving charge Q has a stronger electric field E toward the test charge q because the electric field is squeezed.
A radially moving charge Q is a problem. We showed in the first blog post on September 9, 2023 that the electric field E_x is hard to define in such a situation because changes in the inertia of the test charge q confuse things. In the current blog post we showed that radial kinetic energy does not gravitate. If we have random movement, then 2/3 of the kinetic energy does gravitate (or has a Coulomb attraction), but 1/3 does not. Strange.
Our observation that 2/3 of moving charges exhibit increased Coulomb attraction clashes with Gauss's law:
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