Can quadropoles slow down gravitational radiation?

[pervect]

This is sort of an offshoot of some other recent threads.

We know that E&M radiation interacts with dipoles (Example: (+...-), plus being a positive charge and - a negative charge) and that dipoles can "slow down" E&M radiation - i.e. light travels slower than 'c' in a dieletric medium.

Do we have any idea whether quadropoles (+...+) being an example of a quadropole, have any similar (and probably very very small) effect on gravitational radiation?

 

[Chris Hillman]

Rephrase the question?

Hi, pervect,

Do we have any idea whether quadropoles (+...+) being an example of a quadropole, have any similar (and probably very very small) effect on gravitational radiation?

You didn't specify whether you were talking about an isolated concentration of electric charge/current with possessing a nonzero electrical quadrupole moment or (as I assume) isolated concentrations of mass-energy possessing a nonzero mass quadrupole moment, but I guess that your real question might be this:

EM radiation (light) slows down when it passes through a dielectric medium: does gravitational radiation slow down when it passes through matter?

 

[pervect]

Hi, pervect,
You didn't specify whether you were talking about an isolated concentration of electric charge/current with possessing a nonzero electrical quadrupole moment or (as I assume) isolated concentrations of mass-energy possessing a nonzero mass quadrupole moment, but I guess that your real question might be this:

EM radiation (light) slows down when it passes through a dielectric medium: does gravitational radiation slow down when it passes through matter?

Right - specifically, does (or can) passing through matter with a non-zero mass quadropole moment slow down gravity waves.

A sub-point is that it's not uncommon for matter to have a quadropole moment - any diatomic molecule will have some (very small) mass quadropole moment.

Related points are that we could artifically engineer a "media" to have a larger mass quadropole moment than naturally occurring matter - for instance, consider a structure with a large number of solid rotating bars - this could have a much larger quadropole moment than would be present in a diatomic gas. While it wouldn't be strictly "continuous", if the size of the bars was smaller than the size of the gravitational waves, apprxoimating it as a continuous media should be a reasonable approximation.

 

[tehno]

This is sort of an offshoot of some other recent threads.

We know that E&M radiation interacts with dipoles (Example: (+...-), plus being a positive charge and - a negative charge) and that dipoles can "slow down" E&M radiation - i.e. light travels slower than 'c' in a dieletric medium.

Do we have any idea whether quadropoles (+...+) being an example of a quadropole, have any similar (and probably very very small) effect on gravitational radiation?

For EM radiation (propagation) in dielectrics it's the matter of interacting EM radiation with matter.We have absorption and remmission impact on time in the path of propagation of the radiation.
But what would "absorb" gravitational waves in the case of your quadrupoles?

 

[JesseM]

For EM radiation (propagation) in dielectrics it's the matter of interacting EM radiation with matter.We have absorption and remmission impact on time in the path of propagation of the radiation.
But what would "absorb" gravitational waves in the case of your quadrupoles?

The absorption/emission picture may be a rough way of understanding the slowing of light at the level of quantum field theory, but my understanding is that it can also be understand using purely classical EM, where electromagnetic waves aren't made of photons at all. Something to do with the incoming EM wave causing the dipoles to oscillate and emit their own waves, I think...maybe the sum of the original waves and the new waves generated by the oscillating dipoles looks like a wave moving slower than c?

 

[tehno]

The absorption/emission picture may be a rough way of understanding the slowing of light at the level of quantum field theory, but my understanding is that it can also be understand using purely classical EM, where electromagnetic waves aren't made of photons at all. Something to do with the incoming EM wave causing the dipoles to oscillate and emit their own waves, I think...maybe the sum of the original waves and the new waves generated by the oscillating dipoles looks like a wave moving slower than c?

Rough picture:
EM wave's phase velocity is slowed in a material because the electric field creates a disturbance in the charges of each atom (primarily the electrons) proportional to the permittivity. The charges will, in general, oscillate slightly out of phase with respect to the driving electric field. The charges thus radiate their own electromagnetic wave that is at the same frequency but with a phase delay. The macroscopic sum of all such contributions in the material is a wave with the same frequency but shorter wavelength than the original, leading to a slowing of the wave's phase velocity. Most of the radiation from oscillating material charges will modify the incoming wave, changing its velocity.
Well.. I'm missing what would be analogous mechanism for the gravitaional waves?

 

[ZapperZ]

This is sort of an offshoot of some other recent threads.

We know that E&M radiation interacts with dipoles (Example: (+...-), plus being a positive charge and - a negative charge) and that dipoles can "slow down" E&M radiation - i.e. light travels slower than 'c' in a dieletric medium.

Do we have any idea whether quadropoles (+...+) being an example of a quadropole, have any similar (and probably very very small) effect on gravitational radiation?

Er... shouldn't electric quadrupole be like this:

- ... +

+ ... -

etc?

This, actually is a very common lattice. For example, in a simple cubic lattice, you have the ions at the lattice points, and you have the electron could making the covalent bonds. When you have unpolarized light going through the solid, you have what is sometime known as a "breathing" mode, where the cube expands and contracts. Depending on the phase and the lattice symmetry, you sometime have all the axes expanding at the same time, or you can them be 180 degrees out of phase.

But as to how something like this could slow down gravity, I have no idea. Don't you have to first of all show that gravity does interact with electric dipole?

Zz.

 

[Stingray]

Er... shouldn't electric quadrupole be like this:

- ... +

+ ... -

etc?

Yes, but I'm pretty sure that pervect was referring to a mass quadrupole, not a charge quadrupole. Since changing electric dipoles radiate electromagnetic waves and changing mass quadrupoles radiate gravitational waves, you might expect analogous effects (to a point).

This is actually correct. An incident gravitational wave could drive a system to oscillate in such a way that the induced radiation would reduce the overall phase velocity. Kip Thorne has some notes which discuss this. See sect. 2.4.3 in http://elmer.caltech.edu/ph237/week6/g.pdf" .

As for engineering materials, here's an interesting paper: http://www.springerlink.com/content/m024122721657715/". Very amusing things are possible in principle, but it seems unlikely that any appropriate materials could be made (and there might be good fundamental reasons for that). As Kip points out in the notes I referenced above, the effect is extremely small in naturally-occurring systems.

 

[ZapperZ]

Yes, but I'm pretty sure that pervect was referring to a mass quadrupole, not a charge quadrupole. Since changing electric dipoles radiate electromagnetic waves and changing mass quadrupoles radiate gravitational waves, you might expect analogous effects (to a point).

But do we actually even have a "mass dipole" in the first place before we can actually go to a mass quadrupole?

Zz.

 

[Stingray]

But do we actually even have a "mass dipole" in the first place before we can actually go to a mass quadrupole?

Yes, you can define any multipole moment you like. In a Newtonian context, the dipole moment of a mass distribution $\rho(\bf{x})$ is
$$D^{i} = \int d^{3}\bf{x} \rho(\bf{x}) x^{i} ~.$$
Similarly, the quadrupole moment is
$$Q^{ij} = Q^{ji} = \int d^{3}\bf{x} \rho(\bf{x}) x^{i} x^{j} ~.$$

These objects obviously depend on the choice of origin. This is also true for electromagnetic multipole moments (though the choice of origin does drop out in special cases). It's not a problem. It just means that the statements we've been making in this thread would have to be restated more precisely if you were to try to do anything with them.

Incidentally, you might notice that the origin is at the object's center-of-mass if its mass dipole vanishes. This condition is actually used to define the center-of-mass in more sophisticated formalisms applicable in strongly curved spacetimes.

 

[pervect]

Re: dipoles vs quadropoles

Here's the way I understand the difference:

+...- pure dipole

pure quadropole - the same as your diagram, but rotated through an angle:

   -
   .
   .
+...+
   .
   .
   -

Now if we add an equal charge to the four charges above, we get this, which eliminates the negative masses, and adds only a monople term (plus probably some higher order terms).

+...+

This actually represents a quadrople plus a monople (plus some higher order terms, I think). But without negative masses, this is as close as we can come to the "pure" quadropole above. It's got a monople moment, no dipole moment, a quadropole moment, plus (I think) some small amount of higher-order moments.

[add]
Why do we care specifically about quadropoles? We care because gravity doesn't have an dipole interaction with matter - the lowest order interaction is the quadropole. On the WWW, see for instance

http://math.ucr.edu/home/baez/physics/Relativity/GR/grav_speed.html

I think I've given the textbook references in MTW which say the same thing before (but they won't be useful to anybody without that textbook). Most books on GR that talk about gravitational radiation should mention this. It's usually discusssed in the context of dipole terms not existing in sources due to the conservation laws making all the dipoole terms zero, but it is also true AFAIK that dipoles just don't interact gravitationally.

 

[pervect]

Yes, but I'm pretty sure that pervect was referring to a mass quadrupole, not a charge quadrupole. Since changing electric dipoles radiate electromagnetic waves and changing mass quadrupoles radiate gravitational waves, you might expect analogous effects (to a point).

This is actually correct. An incident gravitational wave could drive a system to oscillate in such a way that the induced radiation would reduce the overall phase velocity. Kip Thorne has some notes which discuss this. See sect. 2.4.3 in http://elmer.caltech.edu/ph237/week6/g.pdf" .

As for engineering materials, here's an interesting paper: http://www.springerlink.com/content/m024122721657715/". Very amusing things are possible in principle, but it seems unlikely that any appropriate materials could be made (and there might be good fundamental reasons for that). As Kip points out in the notes I referenced above, the effect is extremely small in naturally-occurring systems.

Great references! Thank you very much, Stingray.

 

[pervect]

But as to how something like this could slow down gravity, I have no idea. Don't you have to first of all show that gravity does interact with electric dipole?

Zz.

Sorry I wasn't very clear. What I'm actually trying to say is that it has been shown that gravity interacts with mass quadropoles. Which is why I singled them out as being of interest.

I haven't properly introuduced the analogy I was using, which goes something like this:
charge (E&M) is analogous to mass (gravity)

coulomb's force law (E&M) is analogous to Newton's law (gravity).

Magnestism (E&M) is analogous to gravitomagnetism.

At low velocities, one can use the gravitomagnetic analogy

http://en.wikipedia.org/wiki/Gravitoelectromagnetism

to come up with almost an exact equivalent of Maxwell's equations based on the above substitions. Unfortunately this approach isn't good enough for dorect use with gravity waves, because the velocites aren't low enough. But one can still usefully, to some extent, cross identify charge with mass as an analogy in an effort to understand gravity waves by building on one's understanding of electromagnetic waves.

In electromagnetism, the main interaction with matter is through electric dipoles. For gravity waves, there is no analogous dipole interaction, the lowest order is quadropole.

 

[pervect]

Rough picture:
EM wave's phase velocity is slowed in a material because the electric field creates a disturbance in the charges of each atom (primarily the electrons) proportional to the permittivity. The charges will, in general, oscillate slightly out of phase with respect to the driving electric field. The charges thus radiate their own electromagnetic wave that is at the same frequency but with a phase delay. The macroscopic sum of all such contributions in the material is a wave with the same frequency but shorter wavelength than the original, leading to a slowing of the wave's phase velocity. Most of the radiation from oscillating material charges will modify the incoming wave, changing its velocity.
Well.. I'm missing what would be analogous mechanism for the gravitaional waves?

Gravitational waves causes a disturbance in the masses it travels through, causing them to move, much as the electric field causes electrons to move. The electric field acts on charges, the gravitational field acts on masses.

This motion of matter (actually, what is needed is the third time derivative of position, not just motion) causes gravity waves. These gravity waves emitted because of the induced motion interfere with the original gravity waves with similar results to the E&M case.

Note in the E&M case you need only that the second time derivative of position be present to cause E&M waves, but the third time derivative of position must be nonzero for gravity waves. this is an important difference between gravity and E&M.