Hilborn calculates the radiation for an orbiting binary system. Such a system is a quadrupole.
Why is the energy density in a gravitational wave 16-fold compared to its exact electromagnetic analogue?
The analogy between gravitational waves in linearized general relativity and electromagnetics looks very good, except for one thing: the energy drain from the binary system is 16-fold in general relativity! We need to find out why that happens.
For an oscillating (not rotating) electric quadrupole, the energy drain seems to be even less than for a rotating electric quadrupole.
Question 1. We know that the electromagnetic dipole wave is "detached" from the "quasi-static" electric field of the charge at the distance of roughly 1 radian. Where is the gravitational wave detached from the Schwarzschild metrics of the orbiting masses? (We use the word quasi-static because the "static" electric field of the charge is actually moving, and even accelerating.)
Question 2. What is the energy density of the wave if we use the electromagnetic model? The energy density does not need to be
ε₀ E²,
like for an electromagnetic wave. (A half of the energy resides in the magnetic field: that is why we do not have the coefficient 1/2 in front of the formula above.) What matters is how much work the wave can do, or equivalently, how much work is required to create the wave.
Harvesting energy from a wave
When an electric charge is accelerated, the charge has do work against the self-force which is imposed on it by its own quasi-static electric field. For a reason unknown to us, the formula ε₀ E² tells us the energy density of the wave. The formula could be different, since we do not completely understand the self-force.
We cannot pinpoint the exact location of the energy in an electromagnetic wave. Harvesting energy from a wave is a complex process, and it does not reveal us where the energy "exactly" was. The same holds for a gravitational wave: the pseudotensor tells the energy contained in a spatial volume whose size has to be several wavelengths.
(The diagram by MOBIe.
Energy is fed to a gravitational wave through a quadrupole, and people usually think that the energy has to be harvested in the same way, using a quadrupole.
The wave squeezes and stretches a ring of test masses. In the diagram we have a ring of test masses which has been squeezed horizontally by a + polarization gravitational wave. In principle, we could extract energy by putting rods between the test masses.
We can extract energy from an electromagnetic quadrupole wave with a dipole antenna where opposite charges travel to opposite directions. For gravity, we do not have that option.
Is the gravity field 4 times as strong in the wave, or is its energy for some other reason 16-fold?
If an electric field is 4 times as strong, then its energy density is 16-fold. If the gravity field somehow is detached from the quasi-static field at a distance of 1/4 radians, then the field would be 4 times as strong as in the electromagnetic analogue.
The bending of light close to the Sun comes 1/2 from the newtonian attractive force, and 1/2 from the stretching of the radial metric in the Schwarzschild solution. This suggests that the gravitational wave might have 2 times the amplitude of the electromagnetic analogue. But how do we get a factor of 4?
A wave as an image of the near-field behavior
If we have a ring of test masses very close to a binary star, we can harvest energy both from the movement of the masses and from the stretching of the radial metric in the Schwarzschild solution. We have to calculate how much do we get from each effect. Could that explain the 16-fold energy flow?
We believe that waves, in some sense, "transfer" the environment which is close to the source, close to the receiver. Can we find a reason why in the case of gravity, the receiver gets even closer to the source than in the case of a rotating electric quadrupole?
Conclusions
Our Minkowski & newtonian model must be able to explain the structure and the energy flux of gravitational waves. We were hoping that the simple analogue with electromagnetism would suffice. That is not the case. We need to investigate this.
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