I have heard that a wound clock weighs more than an unwound clock, because the energy stored in the winding gives the clock more mass: $E=mc^2$.
ChatGPT calculates that the rotational energy of a galaxy the size of the Milky Way is about 10^53 joules. Does this give the galaxy an additional mass = c^2/10^53? If yes, could this additional mass be the source of dark matter?
No. The rotational energy in the ordinary matter corresponds to under a millionth of the dark matter mass. Worse, it is more than compensated by the gravitational energy, leading the overall relativistic contribution to the galaxy's mass to be not positive but negative.
The largest orbital velocities around the Milky Way are of order 200 km/s. That's about $10^{-3}$ the speed of light. Kinetic energy is $\frac{1}{2}mv^2$ at leading order, so about $10^{-6}$ the rest mass. So if we generously assume that all of the mass orbits at that speed, then its kinetic energy contributes to the total mass at about one part in a million.
But the Milky Way is gravitationally bound, so the kinetic energy is necessarily less than the potential energy. Indeed, the virial theorem says that the gravitational potential energy is -2 times the kinetic energy. So the net relativistic contribution would reduce the galaxy's mass by the same figure of about one part in a million.
