I EFE: Deriving Speed of Gravity, Gravitational Waves?

[SamRoss]

Newton's law of gravitation cannot be compatible with relativity because the gravity from a massive object applies a force to all other masses infinitely fast. General relativity is supposed to correct this flaw by setting a speed limit on how fast the effect of gravity can reach a distant object. (1) How can one derive the speed of gravity from the EFE? (2) Is this speed c? (3) Is this what is meant by gravitational waves?

 

[phyzguy]

(1) This is usually derived by assuming that the metric is a flat-space Minkowski metric plus a small perturbation, and then inserting this into the EFE. See, for example, https://en.wikipedia.org/wiki/Linearized_gravity

(2) Yes

(3) Yes

 

[pervect]

Newton's law of gravitation cannot be compatible with relativity because the gravity from a massive object applies a force to all other masses infinitely fast. General relativity is supposed to correct this flaw by setting a speed limit on how fast the effect of gravity can reach a distant object. (1) How can one derive the speed of gravity from the EFE? (2) Is this speed c? (3) Is this what is meant by gravitational waves?

The eletromagnetic anaology to "gravity" is the coulomb force between charges. "Gravity" is similar to the repulsive force between charges. The sign is different for gravity, which always attracts, while charge comes in positive and negative variants and unlike charges attract in electromagnetism. Otherwise there is a lot of simlarity.

Basically, it usually leads to confusion to try to assign a "speed" to the coulomb force. Experimentally, one can't make a charge disappear and see what happens, because charge conservation is built into the laws of electormagnetism, Maxwell's equations.

There is a clear difference between the two. Light from the sun is aberrated by the Earth's orbital motion. Gravitational waves would be predeicted to aberrate in the same manner (if the Sun actually emitted any). Electric and gravitational fields from the sun are not aberrated in the same manner, and assuming that they are leads to wildly innacruate predictions that don't conserve angular momentum, and in the case of gravity would wind up with the Earth spiraling into the sun. For the full technical treatment, see for instance https://arxiv.org/abs/gr-qc/9909087 "Aberration and the Speed of Gravity".

Understanding the gravitational case is much easier if one understands the elctromagnetic case. It becomes clear that when one talks about the "speed of light", one talks about radiation, and not the direction of the coulomb force. The results for gravity are similar even though the details are different.

 

[Ibix]

As phyzguy says, you can derive gravitational waves by treating the metric as a flat background and adding a small perturbation. If you plug that into the vacuum form of the field equation the result turns out to be that the perturbation must satisfy what turns out to be the wave equation with a speed of c.

The LIGO people actually solved the full equations numerically because the "small perturbation" doesn't apply near anything massive and energetic enough to be emitting gravitational waves, but by the time they get here the waves are small. The results are consistent with GR predictions so far.

 

[SamRoss]

Thanks to everyone for your responses.