UPDATE January 9, 2022: The reasoning that we can "balance" the shipping of energy using negative pressure in a simple way is incorrect.
The negative pressure should be positioned at the same location as a positive mass for this to work. In a gravitational wave, the location of "positive mass" in ambiguous.
A massless very rigid wall does not stop or reflect a gravitational wave. The wall stretches less than the stretching of the metric in empty space. The energy is transmitted as elastic energy of the wall. The wall is like a high refractive index material for light: the electric field inside the material is less, but the energy is transmitted.
To reflect a gravitational wave, we need a massive rigid wall, or pairs of large masses where the pair is connected with a very rigid rod. The movement of the masses changes the quadrupole moment of the pair. The changing quadrupole moment causes destructive interference to the passing wave => some energy is reflected back.
----
test mass m
• ->
● <-------- r --------> ●
black hole black hole
<----------------------- energy of gravity
In the diagram, the black holes pull the test mass to opposite directions. If we let the test mass to move a short distance to the right, then energy is shipped from the black hole on the right to the left one.
The inertia of the test mass increases because when we move the test mass a short distance, we also ship energy over a long distance.
The obvious way to cancel the shipping of energy is to put negative pressure close to both black holes. When the test mass moves right, it expands the spatial volume on the right. It has to do work to expand the volume. Thus, energy is shipped from the left to the right.
Hypothesis. If we balance the shipping of energy to the left and to the right, then the extra inertia is canceled. The spatial metric becomes 1 again.
field of black hole 1 field of black hole 2
g g
2 r <----------------------- • -----------------------> 2 r
test mass
In the previous blog post we were able to explain the stretched metric inside a gravitational wave when we guessed that both fields ship energy over the distance 2 r. The field strength of both fields is g.
Why is the energy shipped over 2 r, and not over r?
The reason might be that if we harvest energy from the electric field of a wave, we must harvest the corresponding energy from the magnetic field, too. That might explain the extra factor 2.
The gravity of a volume of pressure far away from a test mass; the Komar mass
Let us calculate the force that the negative pressure -p in a volume V exerts on the test mass m. We denote the distance of V from m by R.
m <-------- R --------> O
V, -p
pressure in the direction of R
--------------------------------> x
The test mass causes the metric in the direction of R to stretch by a factor
1 + 1/2 r_m / R
= 1 + G / c² * m / R
where r_m is the Schwarzschild radius of m.
The energy of the negative pressure grows by
W = p V G / c² * m / R
= G m * (p V / c²) / R
= G m M / R.
We see that W is just like the gravitational potential for two objects: one of mass m, and the other of mass
M = p V / c².
If we have pressure into all three orthogonal directions, then the potential at the center of V is three times the potential of pressure just to one direction. That looks like the Komar mass of uniform pressure p in a volume V:
3 p V / c².
Calculating the effect of harvesting energy through negative pressure
We want to extract energy from negative pressure which the stretching of the metric causes. The extraction process attenuates the gravitational wave - but how much?
s
-->
O <------ 2 r ------> • <------ 2 r ------> O
V, -p m V, -p
Let us put pockets V of negative pressure -p on both sides of the test mass m, at the distance 2 r from it. The pressure is only in the direction of g, not to orthogonal directions.
Let m move a short distance s to the right. The energy displacement to the left, by the two fields g, is
C = 4 r * m g s.
Let the stretching of the metric be 1 + d. Then
d = C / (m c² * s)
= 1 / c² * 4 r g
<=>
g = c² / 4 * d / r.
The negative pressure -p cancels a part of the energy displacement C because when we move m to the right, then the left pocket V contracts and the right pocket V expands. Energy is shipped over a distance 4 r to the right.
Let us denote by f the "field" of each pocket V of negative pressure -p.
The field strength is
f = G / c² * p V / (4 r²).
Let us use the fields f to cancel a part of the fields g. Let us cancel, say, 1% close to the pressure system.
f = 1% * g
<=>
p V = 4% * c² / G * r² g.
Let the maximum stretching of the metric be
1 + d
along the x axis. The pressure system harvests the energy
E = 2 d p V
= 8% * c² / G * r² g * 1 / c² * 4 r g.
= 32% * 1 / G * r³ g²
The "size" of our negative pressure system is 4 r. It causes destructive interference within a certain zone. How large is the zone?
The cross section of a typical isotropic receiving antenna in radio technology is
~ λ² / (4 π).
However, our pressure system does not try to collect the maximum energy, but only 2%. The cross section might be much larger, up to λ².
Let us put (λ / 2) / (4 r) pressure systems in a "series". The system removes 2% of the wave energy over the volume λ³.
Note that the claim that a "series" works this way is yet another guess.
We have
E * (λ / 2) / (4 r) = 2% * D λ³,
where D is the energy density of the gravitational wave. Then
D = 50/8 E / r * 1 / λ²
= 1 / (2 G) * r² g² * ω² / (π² c²)
= 1 / (2 π²) * 1 / (G c²) * ω² r g²
= 1 / (2 π²) * 1 / (G c²) * ω² r²
* c⁴ / 16 * d² / r²
= 1 / (32 π²) * c² / G * ω² d²
= 1 / (8 π²) * c² / G * ω² H²,
where d = 2 H, and H is the maximum value of h+.
Kostas D. Kokkotas (2002) gives the above formula for t₀₀^GW, which is the energy density in the stress-energy pseudotensor.
The formula looks similar to our formula for the energy density D, except for a constant factor. Our calculation was extremely crude. It is pure luck that the constant factor is close to our value in D.
Do the fields of the two black holes really exist separately?
We have been explaining the increase in the inertia of the test mass by claiming that the fields of the two black holes in the gravitational wave "exist" individually, and pull the test mass m to opposite directions.
However, it might be that the increase of inertia, after all, is due to the sum of the fields. The sum is the quadrupole field. The field is quite weak, and complicated in form. The field of the test mass m does strengthen and weaken the quadrupole field at faraway locations. If we move the test mass, energy is shipped over great distances in the quadrupole field. This shipping might be enough to explain the increase in inertia.
We have to study this. The separate existence of fields is an ugly assumption.
s
-->
● • ●
mass 1 test mass mass 2
In the case of static fields, in the diagram we have a typical configuration. Moving the test mass the distance s ships energy from the field of the mass 2 to the field of the mass 1. We do not need to assume the separate existence of the fields for the two masses. Moving the test mass ships energy from the field behind the mass 2 to the field behind the mass 1.
It might be that the tug-of-war on the test mass m is correctly described by either
1. each mass i (i = 1 or i = 2) pulling the test mass with the newtonian gravity force, and storing the energy in the mass i itself, or
2. calculating the energy of the sum of fields of the masses 1 and 2, and calculating how moving the test mass m ships energy around in the field.
A wave is a greatly stretched "image" of the above configuration. If the test mass is at a distance R from the source, energy may be shipped over a very great distance, ~ 2 R. The quadrupole field may be strong enough to explain the increase in inertia.
In the above sections, we assumed that the fields ship energy over a distance 2 r, where r is the separation of the black holes. The distance scale 2 r is strangely short, since the fields at a distance of R span a length ~ 2 R. Maybe 2 R is the relevant distance scale, and the fields are weaker because of the destructive interference in a quadrupole field.
Conclusions
We were able to derive a formula for the energy density of a gravitational wave. The formula looks a lot like the corresponding formula of general relativity. Is this a coincidence? We do not think so. The energy of a gravitational wave seems to be in the stretching of the transverse metric.
To explain the transverse metric we had to make a major assumption: the fields of the black holes exist separately and ship energy over a distance 2 r, where r is the separation of the black holes. Alternatively, the fields do not exist separately, and energy is shipped over great distances in the quadrupole field.
In the calculation of the field energy we had to make another major assumption: a "series" of receiving antennas of gravitational waves behaves like antennas in radio technology.
The latter assumption might be wrong. If we ship field energy to opposite directions, why would these shipments cancel the extra inertia which is caused by shipping energy over large distances? How does Nature know how to optimize shipments so that canceling of extra inertia happens?
The calculations above were extremely crude. We should refine them. Also, we could calculate the effect of harvesting energy with two test masses connected with a rigid rod.
We believe that the extra inertia happens also inside an electromagnetic wave for a test charge. This implies that we are studying general field theory, rather than just gravity.
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