No Arabic abstract
We discuss the effect of a conformally coupled Higgs field on conformal gravity (CG) predictions for the rotation curves of galaxies. The Mannheim-Kazanas (MK) metric is a valid vacuum solution of CGs 4-th order Poisson equation only if the Higgs field has a particular radial profile, S(r)=S_0 a/(r+a), decreasing from S_0 at r=0 with radial scale length a. Since particle rest masses scale with S(r)/S_0, their world lines do not follow time-like geodesics of the MK metric g_ab, as previously assumed, but rather those of the Higgs-frame MK metric Omega^2 g_ab, with the conformal factor Omega(r)=S(r)/S_0. We show that the required stretching of the MK metric exactly cancels the linear potential that has been invoked to fit galaxy rotation curves without dark matter. We also formulate, for spherical structures with a Higgs halo S(r), the CG equations that must be solved for viable astrophysical tests of CG using galaxy and cluster dynamics and lensing.
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 $gamma$ 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 $gamma$. 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 $gamma$, i.e. by setting $gamma=0$, 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 $beta$ and $k$ of the Schwarzschild-like metric.
We propose a new formula to explain circular velocity profiles of spiral galaxies obtained from the Starobinsky model in Palatini formalism. It is based on the assumption that the gravity can be described by two conformally related metrics: one of them is responsible for the measurement of distances, while the other so-called dark metric, is responsible for a geodesic equation and therefore can be used for the description of the velocity profile. The formula is tested against a subset of galaxies taken from the HI Nearby Galaxy Survey (THINGS).
We show that conformal Chern-Simons gravity in three dimensions has various holographic descriptions. They depend on the boundary conditions on the conformal equivalence class and the Weyl factor, even when the former is restricted to asymptotic Anti-deSitter behavior. For constant or fixed Weyl factor our results agree with a suitable scaling limit of topologically massive gravity results. For varying Weyl factor we find an enhancement of the asymptotic symmetry group, the details of which depend on certain choices. We focus on a particular example where an affine u(1) algebra related to holomorphic Weyl rescalings shifts one of the central charges by 1. The Weyl factor then behaves as a free chiral boson in the dual conformal field theory.
We analyze conformal gravity in translationally invariant approximation, where the metric is taken to depend on time but not on spatial coordinates. We find that the field mode which in perturbation theory has a ghostlike kinetic term, turns into a tachyon when nonlinear interaction is accounted for. The kinetic term and potential for this mode have opposite signs. Solutions of nonlinear classical equations of motion develop a singularity in finite time determined by the initial conditions.
Weyl conformal geometry may play a role in early cosmology where effective theory at short distances becomes conformal. Weyl conformal geometry also has a built-in geometric Stueckelberg mechanism: it is broken spontaneously to Riemannian geometry after a Weyl gauge transformation (of gauge fixing) while Stueckelberg mechanism re-arranges the degrees of freedom, conserving their number ($n_{df}$). The Weyl gauge field ($omega_mu$) of local scale transformations acquires a mass after absorbing a compensator (dilaton), decouples, and Weyl connection becomes Riemannian. Mass generation has thus a dynamic origin, as a transition from Weyl to Riemannian geometry. We show that a gauge fixing symmetry transformation of the original Weyl quadratic gravity action in its Weyl geometry formulation immediately gives the Einstein-Proca action for the Weyl gauge field and a positive cosmological constant, plus matter action (if present). As a result, the Planck scale is an {it emergent} scale, where Weyl gauge symmetry is spontaneously broken and Einstein action is the broken phase of Weyl action. This is in contrast to local scale invariant models (no gauging) where a negative kinetic term (ghost dilaton) remains present and $n_{df}$ is not conserved when this symmetry is broken. The mass of $omega_mu$, setting the non-metricity scale, can be much smaller than $M_text{Planck}$, for ultraweak values of the coupling ($q$). If matter is present, a positive contribution to the Planck scale from a scalar field ($phi_1$) vev induces a negative (mass)$^2$ term for $phi_1$ and spontaneous breaking of the symmetry under which it is charged. These results are immediate when using a Weyl geometry formulation of an action instead of its Riemannian picture. Briefly, Weyl gauge symmetry is physically relevant and its role in high scale physics should be reconsidered.