Let us introduce a model which does not lose energy.
Let us have a binary black hole, or two equal electric charges orbiting each other. The fundamental reason why destructive interference does not cancel the entire wave is that the dipole wave produced by a single orbiting charge is asymmetric: the wave is stronger on the outer side than on the side of the center of the orbit. The weak wave is also in other ways deformed relative to the strong wave.
^
|
weak wave • ● strong wave
center Q
Let us then have two symmetric orbiting charges.
^
Q |
● <------ • ------> ●
| r Q
v
The quadrupole wave is only zero on the axis which is normal to the screen and goes through the center of the system. On the right side of the system it is the wave produced by the right Q which dominates.
Let us try to draw the asymmetric wave of Q.
E1 XXXX
---------------------------------------->
| X
| X E1
v
• ●
Q1
^
| X E1
| X
---------------------------------------->
E1 XXXX
We assume that Q1 is moving toward the screen. We assume that the wave is hugely asymmetric: there is no field at all on the far left side.
If we would draw the wave of the other charge Q2, it would cancel the wave of Q1, except in the zones marked with XXXX.
Now we see that the volume of the XXXX zones actually grows as R². Energy is no longer lost.
The asymmetry of the dipole wave
Let us construct the upper part of the quadrupole wave using polarization of a hypothetical material.
E1 field
strength
^
| -----------------
| / \
| / \
| / \
| / \
| •
center
---------------------|----------------------|-----> x
0 R
The diagram shows the strength of an asymmetric field E1. The strength of E2 is the mirror image around the center.
If we implement the fields using polarization, what is the energy shipping distance S?
It might be
R
S = ∫ |E1| - |E2| dx / max(|E1|).
0
Since E2(x) = E1(-x), the integral is a measure of the asymmetry of E1(x) with respect to the plane x = 0.
S is the "average translation" we have to do to convert the graph of |E1| to the graph of |E2|.
If Q1 and Q2 were static, then |E1| would be symmetric around the location of Q1, and S would be the separation r of Q1 and Q2. In a dynamic wave, S is not necessarily r. Could it be as large as 4 r?
How large is the asymmetry of a dipole wave?
reflection P symmetry
|
|
|
a
<--- ● Q1
-------------|--------------|--------> x
0 r / 2
Suppose that Q1 is orbiting around the center in the x-y plane. It is located at r / 2 on the x axis.
It creates a dipole wave whose absolute field strength |E1| is almost reflection symmetric relative to the plane P which goes through Q1 and is parallel to the y-z plane.
Assume then that Q2 is orbiting at x = -r / 2.
Assume first that |E1| would be perfectly symmetric relative to P. Then we could translate |E1| to |E2| by moving Q1 left the distance r. That is, S = r.
However, if |E1| is not completely symmetric, but stronger on the right side of P, then the average translation S is larger than r.
Let us try to calculate what S might be.
Assume that the static electric field rotates with Q1 around the center. We imagine that Q1 and its electric field are "glued" to a rod which is attached to the center and rotates around the center.
rotation
^
|
• ---------- ● ----------------------- rod
center r / 2 Q1
Since the electric field cannot move faster than light, the dipole wave is "detached" from the static field at the distance of 1 radian, or λ / (2 π).
The detachment distance on the left is the distance r farther from Q1 than on the right. The field of the wave on that side might be
2 r / [λ / (2 π)]
weaker because the static field is weaker at a larger distance, and the angular velocity of Q1 is smaller there. If r is 1% of 1 radian, then the static field is 2% weaker and the angular velocity is 1% less. Since the point of detachment is 1 % farther from Q1, and the wave field goes as 1 / R for the distance, we conclude that the dipole field is 2% + 1% - 1% weaker on the left side than on the right side, at the same distance R from Q1.
How much does the asymmetry contribute to S?
Let us calculate the contribution to S at the distance where the wave is "detached" from the static field at 1 radian.
Let us assume that 1 radian is one length unit, r < 1, and max(|E1|) = 1. Then the integral for a symmetric graph is
1
∫ |E1| - |E2| dx
0
≅ r.
If the field |E2| is only 1 - 2 r times the field |E1|, then asymmetry adds roughly 2 r to the above integral.
We conclude that the average translation S might be roughly 3 r. That is the distance over which energy is shipped if we move a test charge q at a distance 1 from the orbiting charges.
Our calculation is extremely crude. Mainly, it shows that the energy shipping distance S might be several times r.
The asymmetry of a dipole wave diminishes at large distances R
r <----
The diagram of the previous blog post, by Daniel V. Schroeder (1999), helps us to analyze the asymmetry of a dipole wave.
A charge is suddenly moved to the left a distance r, and then stopped. We have to connect the electric field lines in the circular region. The density of the lines in the region becomes large, which means a lot of energy concentrated there. The circular region is the electromagnetic wave which propagates at the speed of light. The radial lines are static electric fields.
Now we see that when the radius of the circle, R, grows, the wave becomes less asymmetric, relative to the size of the circle. The distance r is a kind of a measure of asymmetry, and r stays constant.
We did not find in literature calculations of the asymmetry of a dipole wave. The symmetry becomes (relatively) almost perfect at large distances R. That may be the reason why authors have ignored this question.
If the asymmetry would stay as is for large R, then our calculation of the integral |E1| - |E2| from 0 to R, would show that S grows without bounds, which makes no sense. If the asymmetry is ~ 1 / R, then S stays constant.
Conclusions
We used a model where an electromagnetic wave is created by polarization of a material. The energy flow is
the electric field <-> "elastic energy" of the material.
We do not yet know if we can generalize this to waves in a vacuum, where the energy flow is
the electric field <-> the magnetic field.
Calculating the energy shipping distance S in a quadrupole wave is hard, except if one uses a numerical computer calculation (?).
We argued that the energy shipping distance S in a quadrupole wave is several times r, but does not grow as the distance R from the source increases.
We argued that the relative asymmetry of a dipole wave grows smaller as R increases, probably by a formula 1 / R.
We were able to explain why a quadrupole wave transmits a constant power at large distances R, even though the energy shipping distance S stays constant.
We did not need to assume that the waves of Q1 and Q2 "exist" separately. We can calculate the effect from the sum of these waves.
Our arguments show that the energy shipping distance S might be 2 r or 4 r, which is required in our model to explain the metric perturbation of a gravitational wave. In this blog we claim that stretching of the spatial metric is caused by inertia which is larger in one direction than the other.
We need to analyze how a mechanical clock ticks inside a gravitational wave. That will tell us what is the perturbation of the metric of time.
In our blog we claim that any interaction increases the inertia of a test mass. This implies that a wave can only slow down time or make spatial distances larger. No wave can enable faster-than-light communication between points inside or outside the wave.
General relativity may allow faster-than-light communication inside a gravitational wave. If that is the case, we think that would be a fatal blow for general relativity, since that would bring all the paradoxes of time travel.
If general relativity allows clocks to run faster inside a gravitational wave than in the asymptotic Minkowski space, that is a breach of the weak energy condition. We need to check literature if anyone has studied this.
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