Conformal gravity: light deflection revisited and the galactic rotation curve failure

Abstract

We show how Conformal Gravity (CG) has to satisfy a fine-tuning condition to describe the rotation curves of disk galaxies without the aid of dark matter. Interpreting CG as a gauge natural theory yields conservation laws and their associated superpotentials without ambiguities. We consider the light deflection of a point-like lens and impose that the two Schwarzschild-like metrics with and without the lens are identical at infinite distances from the lens. The energy conservation law implies that the parameter γ in the linear term of the metric has to vanish, otherwise the two metrics are physically inaccessible from each other. This linear term is responsible to mimic the role of dark matter in disk galaxies and gravitational lensing systems. Our analysis shows that removing the need of dark matter with CG thus relies on a fine-tuning condition on γ. We also illustrate why the results of previous investigations of gravitational lensing in CG largely disagree. These discrepancies derive from the erroneous use of the deflection angle definition adopted in General Relativity, where the vacuum solution is asymptotically flat, unlike CG. In addition, the lens mass is identified with various combinations of the metric parameters. However, these identifications are arbitrary, because the mass is not a conformally invariant quantity, unlike the conserved charge associated to the energy conservation law. Based on this conservation law and by removing the fine-tuning condition on γ, i.e. by setting γ = 0, the energy difference between the metric with the point-like lens and the metric without it defines a conformally invariant quantity that can in principle be used for (1) a proper derivation of light deflection in CG, and (2) the identification of the lens mass with a function of the parameters β and k of the Schwarzschild-like metric.

Dynamics of DiskMass Survey galaxies in refracted gravity

“…For example, neither f pRq nor conformal gravity appear to be able to yield the correct abundance of light elements (Azevedo & Avelino 2018;Elizondo & Yepes 1994). Conformal gravity also appears to suffer from a severe fine tuning in the description of the kinematics of disk galaxies (Campigotto et al 2019) and requires a too hot intergalactic medium in galaxy clusters (Horne 2006;Diaferio & Ostorero 2009). Modified Newtonian dynamics (MOND) has been proved to be the most successful phenomenological model on the scale of galaxies (Sanders & McGaugh 2002;McGaugh 2020), but attempts to build a covariant formulation of MOND remain challenging (Famaey & McGaugh 2012;Milgrom 2015;Hernandez et al 2019).…”

Modified general relativity

“…Λ must vanish and is replaced by Φ as proved in theorem 5.1. This is readily verified by restricting the dark energy variables to the constant values Λ d = −Λ and P d = c 2 Λ in (39) and (42). The Friedmann equations with the cosmological constant Λ are then recovered in accordance with theorem 5.1.…”

“…In [38], a theory of emergent gravity (EG) which claims a possible breakdown in general relativity, was introduced by Verlinde that provided an explanation for Milgrom's phenomenological fitting formula in reproducing the flattening of rotation curves. Campigotto, Diaferio and Fatibenec [39] showed conformal gravity cannot describe galactic rotation curves without the aid of dark matter. On the other hand, a logical analysis based on observational data was presented by Kroupa in [40] to support the conjecture that dark matter does not exist.…”

The generally covariant meaning of space distances

“…For example, in conformal gravity one has BH-like solutions depending on three parameters, one simulating cosmological constant (which is clearly not central mass). It is not very clear which combination of the other two parameters needs to be set to zero in order to define the solution with no central mass; see [21,22].…”