Can General Relativity be used to study galaxy rotation curves instead of using the usual Newtonian gravity and mechanics? Does it give a different result?
We measure the velocities of stars and gas in galaxies around their center, and how generally the rotation curves are flat instead of following the expected decay of velocities at large radii from Newton, the so-called Keplerian falloff. The Newtonian calculation basically consists on assuming Newton’s shell theorem as an approximation (even though the theorem only holds for spherically symmetric systems), taking the mass of the galaxy enclosed by the radius r, and estimating the rotational velocity by Newton’s gravitational force and second law of inertia. Different potentials that the point-mass ones, such as for flattened systems, should be used for more accuracy.
One might be tempted to conclude that using Newtonian mechanics and gravity here, even at weak gravitational regimes and low velocities with respect to the speed of light, is too much of an approximation and not accurate enough. So, let’s try to study galaxies with Einstein’s theory of General Relativity.
The linear approximation to General Relativity is given by Gravitoelectromagnetism, which is a second-order, “post-Newtonian” effect. For slowly moving matter, we start with a metric solution similar to the Schwarzschild metric,
in which Phi is the usual Newtonian gravitational potential and A is the so-called gravitomagnetic vector potential. Now we can use the same formalism as in classical electrodynamics. We can also define the gravitoelectric vector potential E. The gravitoelectromagnetic equivalent of the Lorentz-force, acting on a particle with mass m, moving with velocity v, results in an acceleration value (independent of the mass of the particle to satisfy the equivalence principle) that depends on the gravitoelectric potential E, the velocity of the mass, and the gravitomagnetic potential B. The contribution of the gravitomagnetic potential B to the equations of motion is responsible for the effect known as Lense–Thirring precession effect. We could think that maybe the gravitomagnetism effect of B alters the orbits of stars and gas around the galaxy and explains the rotation curves.
Using the Milky Way’s visible mass with no dark matter, we can roughly estimate its angular momentum at a typical distance from the core and we get a radial acceleration contribution from the gravitomagnetic vector potential which is more than five orders of magnitude smaller than the Newtonian centrifugal acceleration using only the Newtonian potential. Thus, the gravitomagnetic effect is quite negligible, and using the linear approximation to General Relativity, is not enough to explain the galaxy rotation curves. And this is no surprise, because the velocity discrepancies between the measured velocities and the expected from Newton are really big!
We borrow these calculations from Viktor T. Toth, which we found on his blog, and he also uploaded a preprint to ArXiv about it, which are available in the references at the end of this post. Viktor has worked together with John Moffat in MOG or Scalar-tensor-vector gravity as a modified gravity theory attempting to solve galaxy rotation curves, the dynamics and lensing of clusters, and cosmological observations without dark matter.
At the end of his preprint, Viktor also addresses a question I’ve been thinking about for some time, which is: what if by assuming asymptotic flatness and flat Minkowski metric at infinity we are not considering boundary conditions which are the ones that explain the galaxy dynamics? In the end, the modification required to solve galaxy rotation curves shows that the departure from General Relativity or Newtonian gravity must kick in at very low field intensities, in the weak gravitational regime of the outer part of galaxies, where the effects of boundary conditions could make a difference. But I thought this could not be the case, because General Relativity satisfies Birkhoff’s theorem and the strong equivalence principle, and it looks like one needs to break with them in a modified theory of gravity that can explain the rotation curves.
Nevertheless, Viktor uses McVittie’s metric, for a compact, spherically symmetric source of gravitation, with mass M embedded in a Friedmann–Lemaitre–Robertson–Walker cosmological background. This basically accounts for the expansion of the universe, and although the contribution from the gravitomagnetic term is not small, it’s a pure radial term and its curl and contribution to B vanishes, so Viktor concludes that boundary conditions can be ignored and that they cannot resolve the problem of galaxy rotation curves.
In his blog, he also states that the universe at large is measured to be almost flat, and in galactic scales, the error introduced in assuming that spacetime is flat at the large scale must be very small. But we know curvature can be very small near the event horizon of a very big black hole, while acceleration is very strong. Small curvature often implies weak tidal forces, but it doesn’t automatically mean weak gravitational accelerations.
Thus, I’m not entirely convinced that the universe at large doesn’t significantly modify local dynamics in galaxies with pure General Relativity. It makes sense that the expansion of space in the Friedmann–Lemaitre–Robertson–Walker metric doesn’t, but the gravitational potential of the observable universe is very big, and we know that deviations from Newton or Einstein always take place below a certain acceleration scale, which matches the gravitational field intensity of the observable universe. This observation is the true achievement of MOND. And thank you Viktor T. Toth for such an interesting calculation.
References:
Dynamics and Astrophysics of Galaxies free Book by Jo Bovy (for Newtonian potentials for galaxies)