Quadrupole Gravity

Most recent answer: 10/22/2007

What is the gravitational quadrupole moment? Can it change? (Please dumb it down for me)
- Larry (age 24)
Let me build up to that slowly.

An object's gravitational monopole is just the total amount of its mass.

An object's gravitational dipole is a measure of how much that mass is distributed  away from some center in some direction. It's a vector, since it had to convey not only how much the mass is off-center but also which way. Considering some object in the abstract, the natural 'center' to pick is the center of mass, which is the point around which the dipole is zero.

The quadrupole represents how stretched-out along some axis the mass is. A sphere has zero quadrupole. A rod has a quadrupole. A flat disk also has a quadrupole, with the opposite sign of the quadrupole of a rod pointing out from its flat sides. The rod is a sphere stretched along that axis and the disk is a sphere squashed along that axis. In general, objects can have quadrupole moments along three different axes at right angles to each other. (The quadrupole moment is something called a tensor.)

The quadrupole moment can definitely change. Think of two balls attached by a spring. If they are stretched apart and then allowed to oscillate, the quadrupole moment will get smaller and bigger in the oscillations.

The quadrupole moment does give gravitational fields, but they fall off much faster as you leave the object than does the main monopole field, which falls as the square of the distance from the center. The quadrupole field falls as the fourth power of the distance.

Mike W.

(published on 10/22/2007)

Follow-Up #1: Gravitational dipoles do not radiate

Would a gravitational dipole radiate, then, seeing as it's a vector?
- Devon (age 22)
Oscillating electric dipoles radiate electromagnetic waves.  But for gravitational radiation you need an oscillating quadrupole moment.  The difference is that electric charge comes in two varieties of charge, plus and minus.   When you interchange the two charges, as in an oscillating electric dipole, you get a change in the electric field distribution.  Gravitational mass, on the other hand, comes in only one sign: plus.   There are no minus values.    So if you interchange two masses you don't get a change in the gravitational field.   Hence, no dipole radiation. 


Another way to see this is as follows. Radiation comes from accelerating sources. When a quadrupole rotates, there is the usual acceleration associated with rotation. A dipole depends on the displacement of the center of mass from some fixed point. The velocity of the center of mass is simply given by the total momentum divided by the total mass. Since each of these quantities is conserved, the velocity of the center of mass doesn't change. In other words, it doesn't accelerate. So there's no dipole gravitational radiation. Mike W.

(published on 11/02/2008)

Follow-Up #2: Gravitational radiation from rotating mass

A spherical mass is rotating around on the end of a rod that is spinning on an axis through the other end. Would this produce a gravitational wave or would it require more than one mass? Is the moment of inertia mrr related to the power of the wave? Thanks!
- John St.Clair (age 64)
Madison, WI USA
Yes.  Electromagnetic radiation is proportional to the acceleration, or second time derivative of an electric dipole moment (or higher order terms like quadrupole, etc). Gravitational radiation, on the other hand, is proportional to the second time derivative of the mass quadrupole (plus higher order terms).  A mass on the end of a rod has a quadrupole moment with respect to the rotation axis and as the rod rotates this moment changes, giving rise to an acceleration.
 See for a definition of quadrupole moment.


p.s. As you guessed, the quadrupole moment is proportional to the moment of inertia for any particular shape, such as a rod or a dumbbell.  The constant of proportionality is of course different for different shapes.  Mike W.

(published on 09/09/2009)

Follow-Up #3: Do gravitational waves have mass?

Gravitational waves are generated when the mass quadrupole moment changes in time. We also know motion of mass contributes to its gravitation. Does the producing process of gravitational waves, which involves mass in accelerated motion, produce gravitation as well? If so, is it of less, equal or more magnitude than the gravitational waves being generated?
- Ranku
Just like electromagnetic waves, gravity waves have energy and travel at the same speed, the speed of light.  And just like light the gravitational equivalent of the photon has zero rest mass.  Again, like light, it has an 'effective' mass that is equal to its energy divided by c2.  This 'effective' mass is very small in comparison to the masses involved in generating the gravitational radiation. 


(published on 12/22/2009)

Follow-Up #4: Dipole gravity?

Would the presence of a cosmological constant mean that there is dipole gravity, due to a "positive" and "negative" component?
- Nicko (age 26)
No, a cosmological constant is just an extra (perhaps unnecessary) term in the Einstein equations of general relativity.  It affects the expansion rate of the universe but doesn't add any dipole asymmetry in the resulting matter distribution.
for some more information.


(published on 01/30/2010)