What is inertia? Inertia means that we must commit some amount of energy as the kinetic energy of an object A when we move the object. If the object A is attached with a rope to another object B, then we must commit more kinetic energy. The object B does not need to consist of atoms. B can also be energy which flows around in a field, or between atoms.
Energy that is harvested from an electromagnetic wave
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|
| r
| • q
| | s
| v
● Q
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v
Let us assume that a charge Q is moving up and down at a relativistic speed over a distance r. The charge Q produces a dipole wave.
The wavelength of the dipole wave might be
λ = π r
if Q moves very fast. The wave is "detached" from the local field at a distance ~ 1 radian or r / 2. Let us put the test charge q at the distance r / 2. Let us assume that in the diagram, the charges Q and q are opposite. Because of retardation, q "sees" Q quite far down, and the wave pulls q down.
If we move q a short distance s down, from where is the energy shipped to q? We can harvest the energy
F s,
where F is the force which the wave exerts on the test charge. From which location does that energy come from?
An equivalent dipole wave, but a different quadrupole moment?
Let us look at the electric field strength E which the oscillating charge produces. We can produce the same |E| and the wavelength λ either with:
1. a large charge Q which oscillates over a short distance r, or
2. a smaller charge Q which oscillates over a large distance r, and may move at a relativistic speed.
The oscillating dipole moment
Q r
is the same in both cases, but the oscillating quadrupole moment
Q r²
is larger in case 2.
In gravity, the stretching of the spatial metric in a gravitational wave is proportional to the quadrupole moment - it is not proportional to the dipole moment.
This follows from the definition of a metric. Let the metric be 1 to the direction of the x axis, and 1 + d to the direction of the y axis, where d > 1 is small. If we rotate the coordinate axes through an angle α, then the metric on the new x axis is
1 + sin²(α).
A small α is linearly proportional to r. Thus, r² is the relevant figure.
Linearized Einstein equations are wave equations for each component. The behavior of the metric close to the source must determine the amplitude of the wave.
We see that if we have almost planar dipole waves which have (almost) equal field strength, their effect on the metric depends on r, or on the quadrupole moment of the dipole.
We have been claiming that the stretching of the spatial metric is due to energy being shipped over a large distance. We have to show that this distance is linearly proportional to r. It does not depend on the wavelength λ.
In other words, the mechanism must be able to distinguish between almost planar waves which have almost the same field strength, but a different r or a quadrupole moment. How is this possible?
The distance over which energy is shipped: the Edward M. Purcell diagram
Edward M. Purcell derived the Larmor formula using the diagram below. The talk in the link and the diagram are due to Daniel V. Schroeder (1999).
A charge is suddenly moved a distance r to the left, and then stopped. The diagram shows the electric field lines after some time. The field lines far away are radial from the original position of the charge. Close to the charge, the field lines are radial from the new location. There are two zones, which are separated to a circular transition zone.
Imagine a very lightweight opposite test charge in the outer zone. It will be pulled to the left when the transition zone passes it. The following claim looks natural, looking at the diagram:
The energy to pull the test charge to the left comes from the field lines of the inner zone, from a distance r to the left from the test charge.
The quadrupole wave as two dipole waves separated by a distance r in a polarizable material
Suppose that we have two equal oscillating charges Q separated by a distance r. The test charge q has the opposite sign.
<--------------------- ~ R -------------------->
E'
----->
XXXX --------------------------------------> E2
E1 <--------------------------------------- XXXX
• --> s
q
^
|
|
R
|
|
v
● <--- r ---> ●
Q1 Q2
The charges Q oscillate in the horizontal direction over the distance r. They could also orbit each other, and the plane of the orbit would be normal to the screen. The test charge q oscillates over a short distance s at the same frequency as charges Q.
We have schematically drawn the electric fields E1 and E2 of the waves produced by the charges Q1 and Q2 at a distance R.
Note that our approximation of the form of E1 and E2 is extremely crude.
The fields do not overlap completely. There is a displacement of r in the fields. We have marked with XXXX the zones where there is no overlap.
The displacement obviously creates the quadrupole wave. There is no complete destructive interference of the fields in the XXXX zones.
How to model the electric fields E1 and E2? The fields are only λ / 2 thick but extend over a large distance R. We cannot create such narrow fields easily with static charges in empty space.
Let us instead assume that the electromagnetic wave propagates in a polarizable medium.
Also, let us change to a frame which moves at a relativistic speed up in the diagram. The wave then is redshifted and its wavelength can be many times R.
| q
v
C1 C2
- + + -
- + + -
- + + -
XXXX ------------------------------------> E2
E1 <---------------------------------- XXXX
<-----> <----->
r r
In the new frame, the test charge q moves at a relativistic speed downward.
The polarization of the medium carries both dipole waves. We have marked in the diagram the extra charge which the polarization for each field E1 and E2 have concentrated in certain areas. It is like capacitor plates.
Let us imagine that we move q a short distance s to the right and q is a negative charge. Then the charge of C2 pulls q and C1 repels q. Energy is shipped from C1 to C2. The energy moves over a distance r.
We did not need to assume that the fields of Q1 and Q2 "exist" separately. The sum field E1 + E2 was enough to explain the shipping of energy.
In a polarizable material, the energy of the wave flows between the electric field and the elastic energy of the "attachments" of the electrons to the atoms or molecules. In an electromagnetic wave in empty space, the energy flow is between the electric field and the magnetic field. Can we somehow extend the argument above to empty space?
Conclusions
We argued that inside an electromagnetic wave, moving a test charge q ships energy over a distance r, where r is the amplitude of the movement of the charges which generate the wave.
We have claimed that for a static field of a charge Q, the energy is shipped over the distance r of the test charge q from Q. In the last section above, we used our claim to derive the shipping distance in a wave in a polarizable material.
In empty space, the electric field is generated by a changing magnetic field. How could we show that the shipping distance is r? Maybe we have to take that as an axiom?
In a previous blog post we derived the gravitational wave metric by assuming that energy is shipped over 2 r (or actually, 4 r, because in the earlier blog post we counted the two dipoles separately). Why is it not over r? We have to find out what is the reason for the discrepancy.
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