On Gravitational Waves

Question

I need someone to double check my thinking as I'm not sure I full understand General Relativity (GR) relating to gravitational waves.

imagine on a line there are three bodies, first a test particle 1 then a large mass 2 then a test particle 3.

Now imagine mass 2 is accelerated very sharply toward 3 (away from 1) to near the speed of light and then decelerated to near the same speed it was before.

The Gaussian shaped gravitational wave say (altered spacetime curvature wave) goes out toward particle 3... since GR would state that mass 2 gains weight at high speed 3 would then see and feel a sharp momentary pull (fall on the spacetime curve) toward 2.. but what would 1 feel when the wave hits it?

wouldn't 1 feel less pull toward 2 momentarily? would space time flatten? as the peak of the wave hits it? or would it still see a very heavy object when the wave hits it?

How would the picture change if 2 was a photon passing a particle 3, would particle 3 feel a step function initially as the gravitational wave from the photon hits it (imagining the impossible that the photon can go through 3), then after it's passed what kind of gravity would 3 feel from a photon moving away from it? any?

Answer 1

I think there are two misconceptions at work, so I'm not sure how to answer the question as stated.

First, relativity does not say "moving masses gain weight". In special relativity people used to define a quantity called the "relativistic mass" $m_r = \gamma m$, where $\gamma$ is the Lorentz factor (which increases with increasing speed). This formulation does not give the correct account of relativistic forces and is no longer used. In special relativity the mass of a particle is its rest mass $m$. That is the only mass. A particle's mass doesn't change according to its motion.

In GR the mass of a body is not well defined. If we want to calculate the gravitational field for a particular source, the stress-energy tensor has terms for the total energy density and the momentum density of the source. The motion of the source would contribute kinetic energy to the total energy density. To calculate the metric we would have to solve the Einstein equations for that stress energy tensor, which is no easy feat. Often we can simplify the process by taking certain limits of the full equations, for example weak field ($\frac{GM}{c^2 r} \ll 1$) or slow speed ($\frac{v}{c} \ll 1$).

Second, time varying gravitational potential is not the same as gravitational waves. The gravitational potential you experience if a source moves closer to or farther from you is a time varying gravitational potential, not a gravitational wave.

We usually consider gravitational waves far from their sources in the wave zone. The Newtonian-like gravitational field weakens rapidly ($\sim1/r^2$) with distance from the source, while the waves decays more slowly ($\sim1/r$). If an observer is experiencing gravitational waves they are not experiencing the plain-old gravitational field of the source. To calculate the gravitational waves from a particular source, the waves must be matched back to the time varying fields of the source zone in some intermediate region.

The same thing happens with EM waves in electromagnetism. The time varying potential experienced near moving charges, even accelerating charges, is not light.

I think your best bet is to map your problem to one with a known solution, for example the gravitational waves emitted by a simple harmonic oscillator. This would not be a constant acceleration motion, and the usual assumption is that the sources are moving slowly. The benefit is that you could look up how it works in a textbook.