Third derivative of quadrupole moment as source of radiation

[bcrowell]

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:

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 P\propto |d^3Q/dt^3|^2. 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 P\propto |d^2D/dt^2|. (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 |d^3Q/dt^3|^2 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 |d^3Q/dt^3|^2 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, |d^3Q/dt^3|^2 produces the correct (zero) result, so the colliding sheets provide a plausibility argument for this expression.

 

[bcrowell]

Looking through some books, I'm failing to find any explicit demonstration that the appropriate generalization of radiated power from the sinusoidal case to the nonsinusoidal case is obtained by converting the factor of ω²ⁿ into the nth derivative, which then gets squared.

Jackson only discusses the sinusoidal case in sections 9.3 and 16.7. He never writes anything in terms of an nth derivative, unless I'm missing it somewhere.

MTW section 36.1 simply asserts as a standard result that electric quadrupole radiation should depend on the square of the third derivative of the quadrupole moment.

Wald finds an expression for the linearized gravitational field on p. 83, and it depends on ω². He then gets it in terms of the second derivative of the quadrupole moment, which makes sense because he's Fourier analyzing it, and for a sinusoidally varying component of the Fourier analysis, clearly taking a second derivative is the same as pulling down a factor of ω². Then on p. 86 he says, "A lengthy calculation ... yields the final result ..." which has the third derivative of the quadrupole moment in it. Not very illuminating.

Is the justification for the nth-derivative generalization simply that if you want to find the instantaneous power, it has to depend on instantaneous properties of the source, and derivatives are the only properties of the source that can be defined on an instantaneous basis?

 

[PAllen]

This paper is out of my league, but I wonder if it helps you at all, esp. eg. appendix A:

http://www.davis-inc.com/relativity/grav-rad1.pdf

 

[bcrowell]

This paper is out of my league, but I wonder if it helps you at all, esp. eg. appendix A:

http://www.davis-inc.com/relativity/grav-rad1.pdf

Hmmm...thanks for the suggestion, but I guess I don't really follow it. The third derivative shows up, seemingly out of nowhere, at equation 2.70. The fact that it's Brans-Dicke and not just GR also makes it more complicated.

 

[bcrowell]

OK, maybe I'm starting to understand this better.

In one dimension, Taylor and Wheeler's argument is that a body feels the retarded force from another body, and this leads to nonconservation of mechanical energy. E.g., if two bodies are falling toward one another, they each feel the weaker force from the other's earlier position, so less work is done than "should" be done in Newtonian terms. The only way to restore conservation of energy is to add in gravitational radiation.

If there is a *constant* force, then the time delay has no effect. Therefore you don't get any gravitational radiation. The conclusion is that a body doesn't radiate gravitational waves unless its acceleration is varying, i.e., d³x/dt³ is nonzero. We know that radiation always varies as the square of the amplitude of the oscillation, so we get power varying as (d³x/dt³)².

From general arguments about multipole radiation, we know that quadrupole radiation from a sinusoidally varying source must give a radiated power that varies as Q²ω⁶. To generalize this to the case of a non-sinusoidal source, we know that the instantaneous power can only depend on the derivatives of Q, which are the only things that are measurable about the function Q(t) at a given instant in time. To match the power of ω, we have to use (d³Q/dt³)². This is consistent with the conclusion of the preceding paragraph.