This is the distinction of the "near" and the "far" field of a radio antenna. The border is a fuzzy line at about one radian.
The quadrupole formula
The formula is:
where I is the quadrupole moment tensor. Wikipedia says that the metric perturbation h-overline is "trace reversed". The quadrupole moment tensor I is traceless, because of the -1/3 r² δij term.
Two masses M orbiting each other on a single circular orbit
R • observer
| \
| \
α
^
| M
● r × r ●
M | ω angular vel.
v
^ y
|
-----> x
Two masses M orbit each other on a single circular orbit whose center (0, 0) is marked with ×. The radius is r. The angular velocity is ω. The angle α is center ×, observer, M.
Let us calculate what is the stretching of the x metric g₁₁ at an y distance R from the binary star of the masses M. We assume that we can sum the Schwarzschild metric perturbations of each M at the observer.
For each M, the Schwarzschild radial metric
grr = 1 + rs / R,
where
rs = 2 G M / c².
The radial metric stretches the x metric g₁₁ by a factor
g₁₁ = 1 + (grr - 1) * α²
= 1 + 2 G M / c² * r² / R³.
The quadrupole moment and h
If the rightmost mass M has turned an angle β = ω t relative to the diagram, then the traceless quadrupole moment tensor for a single mass M is
I =
2/3 M r² cos²(β) ... 0
+ 1/3 M r² sin²(β)
... -1/3 M r² cos²(β) 0
- 2/3 M r² sin²(β)
0 0 -1/3 M r² cos²(β)
+ 1/3 M r² sin²(β)
The second derivative
d²I₁₁ / dt² = d²(1/3 M r² cos²(ω t)) / dt²
= 1/3 M r² ω² * -cos(2 ω t),
which for t = 0 is
1/3 M r² ω² = 1/3 M v²,
where v is the velocity of M. The corresponding value for
h₁₁-overline = 2 G / c⁴ * 1 / R
* 1/3 M r² ω²,
which for the distance of one radian R = c / ω is
2/3 G / c⁵ * M r² ω³.
The stretching of g₁₁ at one radian
One radian corresponds to a distance
R = c / ω.
The perturbation of g₁₁ is
2 G M / c² * r² / R³
= 2 G / c⁵ * M r² ω³.
The value of g₁₁ varies between 1 and 1 + the perturbation. We must compare one half of that:
G / c⁵ * M r² ω³.
to h₁₁-overline. The difference is in the factor 2/3.
At least qualitatively, the method of simply calculating the Schwarzschild spatial metric perturbation caused by the two masses M, agrees with the linearized Einstein equations, from which the quadrupole formula is derived.
Harvesting energy at the border of the "far" field
What about the energy density of the gravitational wave? Can we somehow show that the energy density of the quadrupole formula agrees with the energy density derived with other methods?
The coupling of matter to a metric g is very natural. Let us put some device D to the distance of one radian from the quadrupole, and use it to harvest energy from the gravity field. We can interpret this in two ways:
1. D harvests energy from the gravitational wave, or
2. D harvests energy from the (almost static) field of the quadrupole.
In the gravitational wave interpretation 1, the device D cancels a part of the wave through a destructive interference. The destructed part gives its energy to D.
In the interpretation 2, the devide D draws its energy directly from the quadrupole, through (almost) static forces.
In case 2, can we harvest an essentially limitless amount of energy if we put an extremely rigid spring there to utilize the oscillating spatial metric there?
"Energy" of a spatial metric change
k spring constant
M ● /\/\/\/\/\ ● M <--- • ---> m
1 meter R
If we put an extremely rigid spring between two very large masses M, can we extract a lot of energy if we move a small mass m back and forth?
If we pull m farther, then there will be a positive pressure in the spring. Can this pressure stretch the spatial metric enough, so that the spring almost retains its proper length?
Let us assume that the spring is one cubic meter.
The stretching of the Schwarzschild radial metric at the spring is
x = 1/2 (grr - 1)
= G m / c² * 1 / R.
The force is
F = k x
= k G m / c² * 1 / R.
The associated Komar mass m' of the positive pressure in the spring is very roughly
m' ~ k G m / c⁴ * 1 / R.
The energy of the spring stretching is
E = 1/2 k x²
= 1/2 k G² m² / c⁴ * 1 / R².
If
k = c⁴ / G,
then the Komar mass of the pressure can very roughly cancel the shrinking of the spatial metric when we pull m away. What is the energy which we were able to harvest?
E = 1/2 G m² / R².
The energy is as if we were pulling m from a "mirror image" of itself located inside the spring. The energy is very small if m is small.
The power of a quadrupole
Let us then calculate the power for the quadrupole above, of the two M orbiting each other. We assume that at the distance of one radian, we have a half wavelength thick extremely rigid shell. How much power can the wall extract from the tangential fluctuation of the spatial metric? We ignore the radial component fluctuation of the metric because gravitational waves cannot be longitudinal.
The tangential stretching at a distance of one radian, c / ω, by one M is
g₁₁ - 1 = G / c⁵ * M r² ω³.
Let us look at one square radian of a solid angle. The pressure inside the shell in the volume associated with that square radian should compensate for the change in the tangential metric as M moves through an angle of 90 degrees.
What kind of a Komar mass would compensate the stretching above? If the mass is m, then the stretching at the distance c / ω is
G m / c² * ω / c
= G / c³ * m ω.
The stretching has to be equal to
G / c⁵ * M r² ω³.
We obtain:
m c² = M r² ω².
That is the "Komar pressure amount". The work associated with that pressure is very roughly the pressure amount times the stretching of the metric:
E = G / c⁵ * M r² ω³ * M r² ω²
= G / c⁵ * M² r⁴ ω⁵.
The system will emit that amount of energy ω / (2 π) times per second. The power is
P = 1 / (2 π) * G / c⁵ * M² r⁴ ω⁶.
The quadrupole formula gives
where
d³I₁₁ / dt³ = d³(1/3 M r² cos²(ω t)) / dt³
= 2/3 M r² ω³ * sin(2 ω t).
The power from the quadrupole formula is qualitatively the same as we obtained by calculating how much power we can, at the maximum, harvest from the fluctuating tangential spatial metric, at the distance of one radian, or c / ω.
Our harvesting process creates pressures whose own gravitational waves, presumably, cancel the gravitational wave of the quadrupole. That is, the energy in the quadrupole wave is harvested totally.
Does the electromagnetic lagrangian rule out waves whose speed is larger than c?
We showed in our blog posts on August 4 and 9, 2024 that the Einstein-Hilbert action has serious problems in creating a gravitational wave, even though the linearized version can calculate the power of the wave correctly.
We have to check if the electromagnetic lagrangian, or action, has similar problems in creating electromagnetic waves. One particular question is if it allows waves which propagate faster than light. Or do we have to prohibit faster-than-light signals explicitly in histories?
If the electromagnetic action creates a wave, is the history really a stationary point of the action? Does one have to impose separately the condition that the wave must propagate at the speed of light?
In newtonian mechanics, the action, presumably, can calculate the speed of sound waves in an elastic material. Can the electromagnetic action calculate c as the speed of light?
Conclusions
We showed that one can, qualitatively, obtain correct gravitational waves by assuming that the tangential field of a quadrupole is "detached" from the (almost) static field at the distance of one radian, if the wavelength is taken to be 2 π. The detachment at one radian is analogous to electromagnetic waves.
We know that the energy density of a gravitational wave is 16-fold, compared to an analogous electromagnetic wave. The reason for the large energy density is that one can harvest a lot of energy from the stretching of the tangential spatial metric at the distance of one radian.
We want to construct gravitational waves which only increase proper spatial distances in Minkowski space, never decrease them. That prevents time travel paradoxes. If a gravitational wave would allow a signal to have a coordinate speed > c in Minkowski space, then we would be able to send signals to our own past. Obviously, that has to be blocked.
Sketch of a model. The "fundamental" gravitational wave is similar to the analogous electromagnetic wave. The stretching of the spatial metric is a side-effect of the "electric field" of the wave. The stretching always increases proper distances.
m • --> r ● M
<--------------------------
gravitational energy
shipped to m over r
In this blog we were able to explain the Schwarzschild radial metric by assuming extra inertia from "energy shipping" to the test mass m from the distance r to the spherical mass M. Moving the energy increases the radial inertia of the test mass m and makes radial distances to appear longer.
gravity pull
of the wave
^
| | gravitational energy
| | shipped to m over
| | distance c t
| | ^
| | |
| | • m ● ●
v quadrupole
gravity pull
of the wave
(analogous to
electric field)
In the case of a gravitational wave, the energy shipping distance cannot be the distance to the quadrupole. Rather, the energy has to be shipped from a distance of at most c t, where t is a fraction of the cycle time of the wave. For example, t might be cycle time divided by 2 π.
In the diagram, m has a larger inertia in the vertical direction. Proper distances in the vertical direction appear longer.
How to write this model into a lagrangian or an action?
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