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In a different thread, PAllen posted a link to an interesting paper by Carlip, http://arxiv.org/abs/gr-qc/9909087 . PAllen's summary reads, in part:
This relates to a question that has been bugging me for a while. In the case of a sinusoidally oscillating source, it's fairly easy to show that the power omitted by a monopole source (e.g., a loudspeaker) depends on ω2, a dipole gives ω4, and a quadrupole ω6. I make this argument here: http://www.lightandmatter.com/html_books/genrel/ch09/ch09.html#Section9.2 (subsection 9.2.5). Therefore it follows that gravitational radiation from a sinusoidally oscillating source depends on Q2ω6.
There is then the question of how to generalize this to a non-sinusoidal source. The form I've seen is [itex]P\propto |d^3Q/dt^3|^2[/itex]. This is consistent with the result above for the sinusoidal case. The Carlip paper is the first I've seen that attempts to lay out general physical reasons why the third derivative is reasonable. However, his argument for the gravitational case contains quite a bit of mathematical detail, as opposed to the case of electric dipole radiation, for which he gives a much simpler argument based on Lorentz invariance for generalizing D2ω4 to [itex]P\propto |d^2D/dt^2|[/itex]. (I'm using uppercase D for the dipole moment instead of his lowercase d, which looks like a differential.)
I'm interested in seeing whether there is a simple plausibility argument leading to the [itex]|d^3Q/dt^3|^2[/itex] result. Does the following work?
First consider the Newtonian-gravity situation where we have two rigid, parallel, planar sheets of mass, and we let them fall toward one another. If the gap between the sheets is small compared to their transverse dimensions, then the gravitational field experienced by each sheet is independent of distance. Therefore each sheet experiences a constant acceleration toward the other.
Now consider a semi-relativistic generalization of this, with fields that are fairly weak and velocities fairly small compared to c. We can't really have perfectly rigid bodies in GR, nor is the field of a sheet exactly uniform, but both of these are probably not showstopping issues in the appropriate semi-relativistic limit.
In the semi-relativistic case, we can infer the existence of gravitational radiation from conservation of energy plus the existence of time delays in the propagation of gravitational effects. Taylor and Wheeler give a nice argument to this effect in Spacetime Physics, and I've given a similar argument here http://www.lightandmatter.com/html_books/genrel/ch09/ch09.html#Section9.2 (subsection 9.2.1). In general, if a body is subjected to gravitational forces that are time-delayed, then conservation of mechanical energy fails, and the only way to restore conservation is by assuming that some power is radiated as gravitational waves. But in the case of the colliding sheets, there is no such effect, because the gravitational force is independent of the separation of the sheets, so time-delaying the forces has no effect. We therefore conclude that the colliding sheets do not radiate.
Now suppose we were trying to generalize from Q2ω6 in the sinusoidal case to [itex] |d^3Q/dt^3|^2[/itex] in the more general case. We could imagine doing something like taking the Fourier spectrum of Q and integrating the power over all frequencies. But this would produce a nonzero result in the case of the colliding sheets, and in any case it ignores the fact that the waves with different frequencies are actually coherent. On the other hand, [itex] |d^3Q/dt^3|^2[/itex] produces the correct (zero) result, so the colliding sheets provide a plausibility argument for this expression.
PAllen said:There is a significant, limited sense in which gravity appears to propagate instantaneously. The direction of attraction is to the quadratically extrapolated position of a gravitating source. This means that a gravitating body must have changing acceleration before you could (in principle) detect the finite propagation speed of gravity.
This relates to a question that has been bugging me for a while. In the case of a sinusoidally oscillating source, it's fairly easy to show that the power omitted by a monopole source (e.g., a loudspeaker) depends on ω2, a dipole gives ω4, and a quadrupole ω6. I make this argument here: http://www.lightandmatter.com/html_books/genrel/ch09/ch09.html#Section9.2 (subsection 9.2.5). Therefore it follows that gravitational radiation from a sinusoidally oscillating source depends on Q2ω6.
There is then the question of how to generalize this to a non-sinusoidal source. The form I've seen is [itex]P\propto |d^3Q/dt^3|^2[/itex]. This is consistent with the result above for the sinusoidal case. The Carlip paper is the first I've seen that attempts to lay out general physical reasons why the third derivative is reasonable. However, his argument for the gravitational case contains quite a bit of mathematical detail, as opposed to the case of electric dipole radiation, for which he gives a much simpler argument based on Lorentz invariance for generalizing D2ω4 to [itex]P\propto |d^2D/dt^2|[/itex]. (I'm using uppercase D for the dipole moment instead of his lowercase d, which looks like a differential.)
I'm interested in seeing whether there is a simple plausibility argument leading to the [itex]|d^3Q/dt^3|^2[/itex] result. Does the following work?
First consider the Newtonian-gravity situation where we have two rigid, parallel, planar sheets of mass, and we let them fall toward one another. If the gap between the sheets is small compared to their transverse dimensions, then the gravitational field experienced by each sheet is independent of distance. Therefore each sheet experiences a constant acceleration toward the other.
Now consider a semi-relativistic generalization of this, with fields that are fairly weak and velocities fairly small compared to c. We can't really have perfectly rigid bodies in GR, nor is the field of a sheet exactly uniform, but both of these are probably not showstopping issues in the appropriate semi-relativistic limit.
In the semi-relativistic case, we can infer the existence of gravitational radiation from conservation of energy plus the existence of time delays in the propagation of gravitational effects. Taylor and Wheeler give a nice argument to this effect in Spacetime Physics, and I've given a similar argument here http://www.lightandmatter.com/html_books/genrel/ch09/ch09.html#Section9.2 (subsection 9.2.1). In general, if a body is subjected to gravitational forces that are time-delayed, then conservation of mechanical energy fails, and the only way to restore conservation is by assuming that some power is radiated as gravitational waves. But in the case of the colliding sheets, there is no such effect, because the gravitational force is independent of the separation of the sheets, so time-delaying the forces has no effect. We therefore conclude that the colliding sheets do not radiate.
Now suppose we were trying to generalize from Q2ω6 in the sinusoidal case to [itex] |d^3Q/dt^3|^2[/itex] in the more general case. We could imagine doing something like taking the Fourier spectrum of Q and integrating the power over all frequencies. But this would produce a nonzero result in the case of the colliding sheets, and in any case it ignores the fact that the waves with different frequencies are actually coherent. On the other hand, [itex] |d^3Q/dt^3|^2[/itex] produces the correct (zero) result, so the colliding sheets provide a plausibility argument for this expression.
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