Robert C. Hilborn (2017) gives the formulae for the polarizations + and × of a gravitational wave from a binary black hole (equation (83) in the paper):
z
^
|
| θ_i • observer
| /
| /
| / R
| /
| /
● r ● black holes
|
-----------------------------------> x
Let the black holes orbit in the xy plane around the z axis. The distance of the black holes from each other is r, and the distance of the observer is R. The angular velocity of the orbit is ω. The angle of R from the z axis is θ_i. The global time of the observer is t, and t_R is the retarded time at the observer, that is,
t_R = t - R / c.
The stretching of the spatial metric is due to increased inertia to one direction; the sum rule of inertia
In this blog we claim that the stretching of the spatial metric is due to the inertia of a test mass being larger to one direction than the other. The extra inertia is a consequence of energy being moved over large distances in the combined newtonian gravity field of the test mass and other masses around it.
● <-- • -->
mass test mass
If we have just one mass, then the inertia of a test mass is larger to the radial direction, because energy from the field far away is shipped over a large distance to the test mass when we move the test mass radially.
What if we have several masses around the test mass?
Then we obviously have to sum the extra inertia due to each mass. Even if the combined newtonian gravity force on the test mass would be zero, one still has to ship energy of the gravity field from one faraway location to another.
Robert C. Hilborn mentions at equation (83), which we copied above, that the electromagnetic analogue of the binary black hole would have no radiation to the direction of the z axis, though the gravitational wave has the largest amplitude to that direction.
analogue field of black hole 1
--------->
<---------
analogue field of black hole 2
The electromagnetic analogue field of the wave on the z axis is zero because the field vectors point to opposite directions for each black hole.
Linearized Einstein equations calculate the sum of the metric perturbations for each black hole. One can add the stretching of the metric due to the black hole 1, and due to the black hole 2.
We must assume that the field of each mass exists individually?
Let the observer be at the z axis far away from the binary black hole. Because of the symmetry, the combined newtonian gravity field is very weak in the gravitational wave generated by the black holes, just like we argued in the previous section for the analogue electromagnetic wave.
If we want to explain the stretching of the spatial metric at the z axis at an observer far away, we cannot do it based on the sum of the newtonian fields in the vicinity of the observer. We must assume the separate existence of the field for each black hole. We can then calculate the stretching of the metric for each black hole, and sum the values.
In this blog we have been sceptical of the existence of "the electromagnetic field" in a spatial volume V; that is, sceptical of the existence of a "pool field" which would hold the fields of all charges, as well as hold all electromagnetic waves.
A photon seems to be born in the field of an individual accelerated charge. If there exists "the electromagnetic field" of a spatial volume V, when does the photon leave the field of the charge and enter the "pool field"?
Inertia in electromagnetics
If we move a test charge in an electric field, we expect to observe all the same effects about inertia as we observe in gravity.
We probably must sum the extra inertia due to each charge.
The problems that we face in determining the extra inertia inside an electromagnetic wave are probably identical to gravity.
We need to develop a framework for determining inertia in various fields.
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