# Relativistic mass of oscillation problem

• I
Let a mass oscillate with relativistic acceleration (sinusoidal) by means which are irrelevant. What does the gravitational field look like a distance R away?

pervect
Staff Emeritus
If one considers a sinusoidal mass oscillating in isolation, one finds that ##\nabla_a T^{ab}## is not equal to zero, while ##\nabla_a G^{ab} = 0##. As a consequence one cannot satisfy Einstein's field equations ##T^{ab} = 8 \pi G^{ab}## as taking the covariant derivative of each side yields the result that ##\nabla_a T^{ab} = \nabla_a G^{ab}##, but this is not possible.

Thus one is lead to the conclusion that the means by which the mass is made to osscilate cannot be ignored.. Another way of saying this that may be simpler - one needs the source to conserve energy-momentum (the precise mathematical statement of this idea is that ##\nabla_a T^{ab} = 0## ) in order to be able to apply Einstein's field equations in the first place. And an oscillating mass doesn't do that by itself, it needs help.

Orodruin
If one considers a sinusoidal mass oscillating in isolation, one finds that ##\nabla_a T^{ab}## is not equal to zero, while ##\nabla_a G^{ab} = 0##. As a consequence one cannot satisfy Einstein's field equations ##T^{ab} = 8 \pi G^{ab}## as taking the covariant derivative of each side yields the result that ##\nabla_a T^{ab} = \nabla_a G^{ab}##, but this is not possible.

Thus one is lead to the conclusion that the means by which the mass is made to osscilate cannot be ignored.. Another way of saying this that may be simpler - one needs the source to conserve energy-momentum (the precise mathematical statement of this idea is that ##\nabla_a T^{ab} = 0## ) in order to be able to apply Einstein's field equations in the first place. And an oscillating mass doesn't do that by itself, it needs help.
What if perhaps we had a mechanism that made it such that the mass oscillates with constant /omega

Ibix
2020 Award
The problem is that you need to specify the mechanism in detail. If I wave a charged body around there's a reaction that means that I wave slightly in the opposite direction. But I'm not charged, so for the purposes of electromagnetic fields we don't care about the details of how I'm waving the charge around.

However, if it's a mass I'm waving around and we want to know about gravitational fields then we can't ignore my mass and momentum. If the mass is big enough to be gravitationally significant I must be very big and strong, and I would also be a significant gravitational source. You can't ignore me without violating energy and momentum conservation which is "baked in" to Einstein's equations.

So there's no solution for "an oscillating mass", only for "a mass being oscillated by something".

edguy99
Gold Member
"So there's no solution for "an oscillating mass", only for "a mass being oscillated by something".

Very well put. Expansion/contraction is one of the easiest ways to model oscillation. It depends only on the forces holding it together internally and the local conditions around it. It varies on a periodic basis both time wise, and as it travels through space. It is "a mechanism that made it such that the mass oscillates with constant". You can model the oscillating wave properties of a photon (http://www.animatedphysics.com/games/photon_oscillator.htm) and immediately see the significance of the Planck constant.

I think of some examples
- two massive bodies connected by a spring, and
- heated material that contains atoms in vibration.

Kinetic energy should increase gravitational force than the cases of no motion.

edguy99