Can GR Really Be Reasonably Approximated With Newtonian Dynamics In Galaxies?

Newtonian v. post-Newtonian Analysis of Galaxy Scale Physics

It is common practice in N-body simulations of gravity in galaxies to assume that the relativistic effects that distinguish General Relativity from Newtonian gravity are negligible and to use Newtonian gravity as a result.

But, a tool called post-Newtonian theory, which approximates the non-Newtonian effects of General Relativity by perturbing Newtonian gravitational expectations with Post-Newtonian terms, has been applied in a recent paper to suggest that maybe those post-Newtonian effects aren't so negligible after all. Basically, post-Newtonian theory approximates a lot of the discrepancy between Newtonian gravity and General Relativity, but not all of it.

This is particularly notable because post-Newtonian theory has consistently punched above its weight class, (see, e.g. here and here) showing fewer discrepancies between this perturbative approximation and either full fledge General Relativity from first principles or experimental observations than a naive theoretical error estimate would suggest.

The paper is as follows:

The gravitational stability of a two-dimensional self-gravitating and differentially rotating gaseous disk in the context of post-Newtonian (hereafter PN) theory is studied. Using the perturbative method and applying the second iterated equations of PN approximation, the relativistic version of the dispersion relation for the propagation of small perturbations is found. We obtain the PN version of Toomre's local stability criterion by utilizing this PN dispersion relation. In other words, we find relativistic corrections to Toomre's criterion in the first PN approximation. 

Two stability parameters η and μ related to gravity and pressure are introduced. We illustrate how these parameters determine the stability of the Newtonian and PN systems. Moreover, we show that, in general, the differentially rotating fluid disk is more stable in the context of PN theory relative to the Newtonian one. Also, we explicitly show that although the relativistic PN corrections destabilize non-rotating systems, they have the stabilizing role in the rotating thin disks. Finally, we apply the results to the relativistic disks around hypermassive neutron stars (HMNSs), and find that although Newtonian description predicts the occurrence of local fragmentations, PN theory remains in agreement with the relevant simulations, and rules out the existence of local fragmentations.

Ali Kazemi, Mahmood Roshan, Elham Nazari "Post-Newtonian corrections to Toomre's criterion" (August 17, 2018) (accepted in ApJ).

Previous papers by overlapping authors include a paper exploring topics similar to this one, and a post-Newtonian Jeans analysis.

Modified Gravity and post-Newtonian Theory

One of the authors has a paper looking at the emergence of stellar bars in Moffat's MOG theory. MOG and PN both induce stability in disk galaxies relative to Newtonian approximations in a paper from earlier this year:

We study the stellar bar growth in high resolution numerical galaxy models with and without dark matter halos. In all models the galactic disk is exponential and the halos are rigid or live Plummer spheres. More specifically, when there is no dark matter halo, we modify the gravitational force between point particles. To do so we use the weak field limit of an alternative theory of dark matter known as MOG in the literature. The galaxy model in MOG has the same initial conditions as in galaxy models with dark matter halo. On the other hand, the initial random velocities and the Toomre's local stability parameter are the same for all the models. 

We show that the evolution and growth of the bar in MOG is substantially different from the standard cases including dark matter halo. More importantly, we find that the bar growth rate and its final magnitude is smaller in MOG. On the other hand, the maximum value of the bar in MOG is smaller than the Newtonian models. It is shown that although the live dark matter halo may support the bar instability, MOG has stabilizing effects. Furthermore, we show that MOG supports fast pattern speeds, and unlike in the dark matter halo models pattern speed does not decrease with time. Theses differences, combined with the relevant observations, may help to distinguish between dark matter an modified gravity in galactic scales.

Mahmood Roshan, "Stellar Bar evolution in the absence of dark matter halo" (January 25, 2018).

Another MOG study with overlapping authorship is here.

MOG is more elaborate than MOND, but is relativistic and consistently performs well in areas like CBM predictions and galactic clusters, where the toy model theory of MOND does not.

How Accurately Can Scalar Graviton Theories Reproduce GR and Observations?

The classical Post-Newtonian scheme has some similarities to that of Deur's quantum gravity efforts which is a quantum static massless scalar graviton approximation to true quantum gravity, arranged the terms of his quantum gravity equation so that the first term is equivalent to Newtonian gravity, and the next two or three higher order terms give rise to quantum gravity effects that produce dark matter and dark energy phenomena. In this case, scalar gravitons are used merely as an approximation that is easier to calculate with and not because the theory presumes that real gravitons are spin-0 rather than spin-2.

Another comparison of post-Newtonian theory to scalar graviton models that consider self-interaction in light of the PPN (parameterize post-Newtonian) formalism can be found here:

We construct a general stratified scalar theory of gravitation from a field equation that accounts for the self-interaction of the field and a particle Lagrangian, and calculate its post-Newtonian parameters. Using this general framework, we analyze several specific scalar theories of gravitation and check their predictions for the solar system post-Newtonian effects.

Diogo P. L. Bragança, José P. S. Lemos "Stratified scalar field theories of gravitation with self-energy term and effective particle Lagrangian" (June 29, 2018).

The conclusion to this paper notes that:

In this paper, we presented a general stratified scalar field theory of gravitation in a Minkowski background. Then, we calculated two post-Newtonian parameters from three general parameters of the theory B, C and k, concluding that it is perfectly possible for such a scalar theory to explain the four solar system tests. Finally, we used this general theory to rapidly compute the PPN parameters β and γ for a set of scalar theories of gravitation to verify if they agree with the experimental tests of gravitation in the solar system. Therefore, with this formalism, one can directly find those two PPN parameters only from the field equation and the particle Lagrangian of a given scalar theory of gravitation. Although this is a very efficient method to calculate β and γ for a given theory, it does not allow one to compute the other PPN parameters. It would be interesting to generalize this approach to efficiently calculate the remaining PPN parameters for scalar theories and verify if it is possible for such a theory to explain every phenomenon predicted by general relativity. 

The stratified theories that were analyzed (Page and Tupper’s, and Ni’s) yielded the correct PPN parameters relevant for solar system tests. One could wonder whether this indicates that they are valid theories, and the answer to that relies in analyzing the remaining PPN parameters. This analysis was done by Nordtvedt and Will [60] and Ni [50] and the conclusion was that stratified theories cannot account for Earth-tide measurements due to the motion of the solar system relative to the preferred frame (defined by the distant stars). 

The conformal theories that were analyzed did not yield the correct γ parameter even in very general cases. This motivates future work on the analysis of a relativistic scalar theory including a derivative coupling in the Lagrangian, of the type T ab(∂aΦ)(∂bΦ). Such a theory would not have preferred frame effects (it would respect Lorentz symmetries), so if it predicted the correct parameters β and γ it would not have the problem of Earth-tide measurements. 

If such a scalar theory correctly predicts the outcome of every weak field gravity experiment, then we can only rule it out using strong gravity experiment results (e.g. LIGO, neutron star binaries, cosmology). Note also that a scalar theory of gravity is much simpler than general relativity, since it describes gravity with one function instead of ten. In such theories, unlike general relativity, it is generally possible to define a local gravitational energy-momentum tensor, which is always an attractive feature, and is still a problem in general relativity.