# How do gravitational waves agree with Lorentz invariance?

Following is a simple but incorrect explanation for gravitational waves. My question is what is wrong with it?

I'd like to say that a gravitational wave is a periodic variation in the local gravitational field. For example, suppose the Earth is not rotating, for simplicity, and that the Moon orbits the Earth every 28 days. In this case, an observer on Earth would observe the Moon's gravitational field changing with a 28 day period, which seems to me would be an observation of a gravitational wave. The observer could also go sit on Pluto and measure the local gravitational field there changing from Earth's Moon. Again, this person would see it varying with a period of 28 days, but now delayed by about 5 hours due to the gravity transit time from Earth to Pluto. Again, this seems to me like an observation of a gravitational wave, but from a little farther away.

A problem with this explanation can be seen with the Earth measurement. From this explanation, I would expect the wave phase at Earth to be delayed by about 1 second from the Moon's position due to the fact that it takes light (and gravity) about 1 second to get from the Moon to the Earth. This seems reasonable on the surface, but it violates Lorentz invariance, which in this case states that the gravitational field direction for an object moving at constant velocity should point directly toward the object (see Wikipedia "Speed of gravity"). The same issue applies for the Pluto measurement, too. Intuitively, it seems hard to believe that there isn't a delay between the Moon's gravity and its measurement on Pluto, but that's what Lorentz invariance says. Admittedly, the Moon is accelerating very slowly, but that was not a central part of my explanation.

So, is my explanation some sort of "near-field" effect, and distinct from actual gravitational waves? Or am I missing something else?

Thanks for any replies.

-Steve

• Equations of motion are Lorentz invariant. – Avantgarde Nov 20 '18 at 22:37
• The Moon is not moving at constant velocity. If you want to say that it approximately is, you also need to allow that the position of the Moon one second ago is almost the same as its position now. – Javier Nov 20 '18 at 23:15
• The effect you're describing is not a radiation effect at all. Your effect is proportional to the moon's mass, but gravitational radiation from the earth-moon system would be proportional to the square of the moon's mass. I don't think this is any different from E&M. A very slowly rotating electric dipole will produce an oscillating electric dipole field, but it's not a radiation field, and in this limit of low frequency the Poynting vector goes to zero. – user4552 Nov 21 '18 at 1:50
• @BenCrowell. Thanks for the comment, which I agree with. I'm now thinking in terms of a gravitational near field. See my comment below. If you have insights on this, I'd be very interested. – user2419194 Nov 21 '18 at 5:34

Gravity

Due to the presence of mass (and energy) gravity distorts space time. The strength of the gravitational effect attenuates in proportion to $$\frac{1}{r^2}$$.

Gravitational Waves

Gravitational waves are a type of space time distortions that sustain and propagate at the speed of light. They are created by accelerating mass in certain circumstances (such as assymetric rotation). They attenuate more slowly, in proportion to $$\frac{1}{r}$$.

The effect of a gravitational wave is related to the size of the change in the gravitational field at the source, rather than the size of the gravitational field itself (although larger gravitational fields are more likely to produce larger gravitational changes, but not always). At relatively close range they are typically a much smaller than the effect of the gravity of the mass that creates it.

The earth moon system will create gravitational waves, but because of their small magnitude and our proximity to the originating gravitational mass they cannot be distinguished.

• Thanks for this answer. This reinforces my idea that the Moon's periodic effect on the Earth represents a gravitational near field. Like the EM near field, its effect dies off much more quickly than the radiative waves. Also like the EM near field, the "receiver" (the Earth) affects the "transmitter" (the Moon); in contrast, receiver and transmitter are decoupled for either EM or gravitational radiation. Can someone either support or refute this connection? – user2419194 Nov 21 '18 at 5:32
• A difference between EM radiation and gravitational radiation is that in EM case the simplest form of radiation is electric dipole whereas simplest gravitational radiation is mass quadrupole. One effect of quadrupole character is that for a binary system with period $T$ the fundamental radiated frequency is $1/(2T)$, not $1/T$. Just for fun, power radiated by Earth-Moon system amounts to $\simeq 7\,\mu W$. – Elio Fabri Nov 21 '18 at 15:15
• @ElioFabri. These are good points. For those who want a link to the 7 $\mu$W number and some more good discussion, I found it here. As an aside, searching the research literature shows that "gravitational near field" is not a new term, but is not used much. – user2419194 Nov 21 '18 at 19:03