UPDATE January 6, 2022: We updated the text concerning whether we should count the effect of a dipole twice or once. The guess is S = 2 r or 4 r depending on that.
UPDATE December 27, 2021: We realized that g₁₁ - g₂₂ varies by 4 h+, and updated the text accordingly.
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The electric field of a dipole wave
From the link we find the electric field strength in a wave generated by a dipole source:
^
| a
| angle θ
● Q <-------------- R ----------------> • observer
Let us denote the r in the formula by our own symbol R.
We assume that a charge Q accelerates upward and the acceleration is a. An observer is at a distance R. The angle between a and R is θ.
The electric field E is transverse: it is normal to R. Since
μ₀ = 1 / (c² ε₀),
we have
E = 1 / (4 π ε₀) * Q a / (c² R) * sin(θ).
The analogous formula for the newtonian gravity force field is
g = G m a / (c² R) * sin(θ).
The newtonian field in a wave generated by a binary black hole
If the components of a binary black hole do not orbit at a relativistic speed, then the binary creates two dipole waves whose field strengths almost completely cancel each other in destructive interference. The dipoles have a 180 degree phase shift.
However, it looks like the stretching of the spatial metric is not canceled. On the contrary, we have to sum the stretching for the two dipoles.
z
^
| • observer
| /
| / R
| /
● <---------> ●
r
binary black hole
Let us have two black holes, each of a mass M. They move in a circular orbit and their separation is r.
The centripetal acceleration is
a = G M / r².
We have
g = G² M² / (r² c²) * 1 / R * sin(θ)
as the field strength of a single dipole. We can take θ = 90 degrees, because in the movement, the projection of each black hole on a normal of R is a harmonic oscillation whose amplitude is r.
The stretching of the spatial metric
Let us move the test mass m to the direction of a short distance s. The field g increases the inertia of a test mass m by a factor
(m s + m g s / c² * S) / m s
= 1 + g / c² * S,
where S is the (long) distance over which field energy is shipped.
Let us guess S = 2 r.
The increase in the inertia is by a factor
1 + 2 / R * G² M² / (r c⁴),
and the stretching of the spatial metric has the same factor. The stretching is in the direction of g.
For two dipoles, the stretching d is
1 + d = 1 + 2 / R * 2 G² M² / (r c⁴)?
Or should we only count the effect once? If only once, then we should set S = 4 r.
^
| θ_i • observer
| /
| / R
| /
● <---------> ●
r
binary black hole
The metric perturbation h+ is
0 0 0 0
0 h+ 0 0
0 0 -h+ 0
0 0 0 0
Let us compare our result to the metric given by Robert C. Hilborn (2017), above.
Note that the stretching d is 1/2 of the perturbation of the metric because, e.g., g₁₁ is the square of the stretching in the x direction.
Let us denote the maximum value of h+ by H. The value g₁₁ - g₂₂ varies by 4 H. In our own formula, d = 2 H.
We conclude that our result agrees with that of Hilborn in the case θ_i = 90 degrees.
What about θ_i = 0 degrees?
Let us in the diagram have the x axis pointing out of the screen. Our observer measures the metric horizontally, which is his x direction, and vertically, which is his y direction.
When θ_i = 90 degrees, there is stretching in the metric in the x direction, but none in the y direction. Let the stretching be d. The difference
g₁₁ - g₂₂
varies between 0 and 2 d.
If θ_i = 0 degrees, there is stretching in the x metric as well as in the y metric. The difference g₁₁ - g₂₂ varies between -2 d and 2 d.
Robert C. Hilborn uses a traceless gauge. The relevant figure is then g₁₁ - g₂₂, and it has a double amplitude in the case θ_i = 0 compared to θ_i = 90 degrees.
We conclude that our formula agrees with his.
What about an arbitrary θ_i?
The field g to the y axis direction is proportional to cos(θ_i).
Now we have to make another guess: the distance s over which the field energy is shipped in the general case is r cos(θ_i).
After this second guess, our formula agrees with that of Robert C. Hilborn.
Conclusions
We were able to reproduce the amplitude of the + polarization gravitational wave from the newtonian field and the electromagnetic analogue.
We had to assume that we can sum the metric perturbations for each dipole source of the wave. For this, we have to assume that the field of each mass exists separately: the fields cannot be summed and treated as a one whole in the "pool field" of a spatial volume V.
Also, we had to make two guesses about the distance over which the field energy is shipped when we move a test mass. In the future, we must give good grounds for those guesses.
Our result is very encouraging. Our result also shows that the "true" metric of space is always stretched, never contracted. That prevents superluminal communication.
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