I know that gravitational waves are generated by accelerated masses, but do all accelerated masses generate gravitational waves, even small masses?

Question

In other words, is there an accelerated mass that is too small to generate gravitational waves?

Answer 1

I know that gravitational waves are generated by accelerated masses, …

This is not precisely true, better think that gravitational waves are produced by a system and not by individual masses comprising such a system. Note, that in order to have accelerating masses the system must consist of at least two masses. And in order to determine if a given system generates gravitational waves one has to consider its evolution as a whole.

… but do all accelerated masses generate gravitational waves, even small masses?

Weak gravitational fields are well described by linearized version of Einstein field equations, so if we replace a system that we know to emit gravitational waves with a version with smaller masses it would still be emitting gravitational radiation but appropriately scaled down. If we keep lowering masses, eventually we would need quantum mechanics to describe dynamics of the system and instead of classical gravitational waves we would need to speak about emission of gravitons.

However, smallness of the mass is not the key factor here. There are many examples of systems that contain accelerating constituent parts and yet do not produce gravitational waves. Here are a few of them:

• Spherically symmetric system. For example, a spherically symmetric nebula could coalesce into a planet (or even collapse into a black hole). Matter comprising such nebula would be moving with acceleration, but as long as spherical symmetry remains undisturbed, no gravitational waves are produced.

• Axially symmetric body rigidly rotating about its symmetry axis with constant angular velocity. Its constituent parts are moving in circles and thus accelerating, but there is no gravitational radiation.

• Kinnersley's `photon rocket'. This is a solution of Einstein's equations describing a mass (varying with time) moving along a given trajectory due to the recoil from anistropic emission of photon stream (or, to be more precise, null fluid). The source of gravitational field here contains two quite different parts: central mass and the stream of photons, but taken as a whole the system does not produce gravitational waves.

And finally, one should keep in mind, that the source of gravitational field in general relativity is not just mass (or energy) density but the entire stress–energy–momentum tensor, so one could devise a system with (approximately) static distribution of energy, that has dynamic momenta and stresses that would emit gravitational radiation.

Answer 2

No. All masses do. But our detectors are only sensitive enough at this time to detect waves from the largest collisions (black holes, neutron stars).