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Gravitational waves by analogy with Maxwell's equations

by johne1618
Tags: analogy, equations, gravitational, maxwell, waves
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johne1618
#1
Apr21-11, 10:34 AM
P: 373
According to the Wikipedia article on Gravitomagnetism:

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

There is a gravitational analog of maxwell's field equations that is valid for weak gravitational fields.

Basically all you have to do is replace eps_0 in maxwell's equations with -1/4 pi G.

My question is could one understand gravitational waves using this analog?

For instance the Larmor formula gives the total power, P, radiated in electromagnetic waves by a charge e with an acceleration a as:

P = e^2 a^2 / 6 pi eps_0 c^3

Could one argue by analogy that a mass m with an acceleration a will radiate gravitational waves with a total power P :

P = 2 G m^2 a^2 / 3 c^3

just by substituting m for e and 1 / 4 pi G for eps_0?

John
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Naty1
#2
Apr21-11, 11:29 AM
P: 5,632
"My question is could one understand gravitational waves using this analog?"


You'll probably need a better answer from someone who knows the detailed mathematics of GR, but there are some "intuitive" hints here :

http://en.wikipedia.org/wiki/Gravita...tational_waves

"Some more detailed examples : (of gravitational radiation sources)

Two objects orbiting each other in a quasi-Keplerian planar orbit (basically, as a planet would orbit the Sun) will radiate.
A spinning non-axisymmetric planetoid — say with a large bump or dimple on the equator — will radiate.
A supernova will radiate except in the unlikely event that the explosion is perfectly symmetric.
An isolated non-spinning solid object moving at a constant speed will not radiate. This can be regarded as a consequence of the principle of conservation of linear momentum.
A spinning disk will not radiate. This can be regarded as a consequence of the principle of conservation of angular momentum. However, it will show gravitomagnetic effects.
A spherically pulsating spherical star (non-zero monopole moment or mass, but zero quadrupole moment) will not radiate, in agreement with Birkhoff's theorem.."

So it seems it would be really difficult.
Bill_K
#3
Apr21-11, 12:39 PM
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It's true there's an 'analogy' between Einstein's Equations and Maxwell's Equations, and it's true that general relativity contains a magnetic-like effect. However the Wikipedia article you refer to drastically overstates the situation, and contains highly questionable material. The primary difference is that electromagnetism is described by a pair of vector fields E, B in three dimensions, or a rank two tensor Fμν in four dimensions, whereas gravitation is described by a rank four tensor Rμνστ, the Riemann tensor. Approximately one can write a three-dimensional description of gravity using a pair of fields E and B, but they will be rank two symmetric traceless tensors, not vectors. E comprises the first five components Ri0j0, and is generated by stationary masses, while B comprises the remaining five components Rijkl, and is generated by moving masses or 'mass currents'.

Your radiation example is a good illustration of the difference between the two theories. You can have dipole radiation in electromagnetism, but not in gravitation because the dipole moment of a gravitational source is a conserved quantity. That is, you can't wave a mass back and forth without violating momentum conservation. There is indeed a formula for radiated power in linearized gravity analogous to the Larmor formula, but it is for a source with an oscillating quadrupole moment.

johne1618
#4
Apr21-11, 05:49 PM
P: 373
Gravitational waves by analogy with Maxwell's equations

That is, you can't wave a mass back and forth without violating momentum conservation.
Thanks for the reply.

I guess that if I tried to wave a mass back and forth in space I would move in the opposite direction so that the centre of mass of the combined system would not change. Maybe any gravitational radiation that the mass gives off would be counteracted by an opposite radiation given off by me moving in the opposite direction.
Cleonis
#5
Apr23-11, 07:52 AM
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Quote Quote by johne1618 View Post
[...]
There is a gravitational analog of maxwell's field equations that is valid for weak gravitational fields.
[...]
My question is could one understand gravitational waves using this analog?
It is my understanding that the analogy is rather weak.

The Hulse-Taylor binary is losing orbital energy, consistent with the GR prediction that celestial bodies orbiting each other will radiate gravitational waves.
However, the energy loss is far less than in an analogous electromagnetic case. Charged particles in a cyclotron emit cyclotron radiation, and they lose energy very fast.

The energy loss of celestial bodies is far less than that and Steve Carlip has discussed that in his article Aberration and the speed of gravity.

Carlip describes that when charged particles are being accelerated relative to each other (which includes the case of charged particles accelerating each other), then the direction of the force that each particle experiences is not precisely in the direction towards the other particle. This force aberration gives rise to change of angular momentum.

The amount of energy that is radiated away is correlated with the degree of aberration. The total energy of the system is conserved


Carlip describes that in the case of gravitational interaction there is much more extensive cancelation than in the case of electromagnetic interaction. The result of the cancelations is that the corresponding gravitational force is very close to pointing towards the other celestial body. There is very little gravitational aberration, hence only very little loss of orbital energy to gravitational radiation.


As far as the math is concerned I'm out of my depth here.
Actually, I would love to see the result of a computation that compares energy loss of orbiting celestial bodies to energy loss of charged particles along a circular trajectory, such as in a cyclotron.
Or a comparison of how small the gravitational aberration is compared to what the electromagnetic aberration would be.
bcrowell
#6
Apr23-11, 06:22 PM
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Although the OP's original idea doesn't work, it is nevertheless possible to do a pretty good job of estimating the strength of gravitational waves without using fancy math, or without even having a detailed knowledge of the mathematics of GR. See http://www.lightandmatter.com/html_b...tml#Section9.2 (subsection 9.2.5).


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