Kinetic energy and galaxy arm rotation rate

Question

The higher than expected rotational velocity of stars and gas clouds in the outskirts of galaxies is explained today by invoking dark matter that supplies not only the additional gravitational mass explaining the increased rotational speed in the arms, but explains the amount of gravitational lensing of more distant galaxies.

Furthermore, dark matter distribution about spiral galaxies places it on the outside of galaxies and not so much on the inside.

Certainly the higher than expected velocity on the outside of galaxies also translates into higher than expected kinetic energy. Should the extra kinetic energy also increase the gravitational stress energy tensor in that region of space.

If so, do our models already factor in kinetic energy and it's gravitational stress energy tensor, overlook it, or the effect is just too minuscule to be of any importance?

I don't expect the extra kinetic energy to be a replacement for dark matter, and suspect the effect may be too small to be of much importance, but the kinetic energy distribution around a galaxy, intuitively on the surface, appears to have the right distribution, so I'm asking generically, do astrophysics factor this in their models, or not, and if not, should it?

Answer 1

It is miniscule. General Relativity is not required to understand the dynamics of galaxies. Motions are non-relativistic, a few 100 km/s at most, so the kinetic energy of objects is always much smaller than their rest mass energy.

Another way of seeing this is to calculate $GM/Rc^2$, a ratio which tells you the relative size of errors you will get in ignoring GR in any calculation.

For our Galaxy, we might say there is about $M\sim 10^{11} M_{\odot}$ within $R\sim 15$ kpc, and the ratio is then $3\times 10^{-7}$, which indicates Newtonian gravity is fine for most purposes.

Note that gravitational potential energy is negative and is somewhat larger in magnitude than the kinetic energy for an object in a bound orbit. This means that adopting a General Relativistic approach would mean that the "effective gravitational mass" is actually less than adding up the masses of all the Galaxy's components. But as I said above, the effect is negligibly small.

Note also that the distribution of the inferred dark matter extends way beyond the location of most of the visible matter. Thus the distribution of kinetic energy density of the things we can see does not meet the requirements.