34 pages, 21 figures. Presentation at the International Conference on Two Cosmological Models, Universidad Iberoamericana, Mexico City, November 17-19, 2010. Updated final version, contains many new footnotes. To appear in Foundations of Physics
We review some recent developments in the conformal gravity theory that has been advanced as a candidate alternative to standard Einstein gravity. As a quantum theory the conformal theory is both renormalizable and unitary, with unitarity being obtained because the theory is a $PT$ symmetric rather than a Hermitian theory. We show that in the theory there can be no a priori classical curvature, with all curvature having to result from quantization. In the conformal theory gravity requires no independent quantization of its own, with it being quantized solely by virtue of its being coupled to a quantized matter source. Moreover, because it is this very coupling that fixes the strength of the gravitational field commutators, the gravity sector zero-point energy density and pressure fluctuations are then able to identically cancel the zero-point fluctuations associated with the matter sector. In addition, we show that when the conformal symmetry is spontaneously broken, the zero-point structure automatically readjusts so as to identically cancel the cosmological constant term that dynamical mass generation induces. We show that the macroscopic classical theory that results from the quantum conformal theory incorporates global physics effects that provide for a detailed accounting of a comprehensive set of 138 galactic rotation curves with no adjustable parameters other than the galactic mass to light ratios, and with the need for no dark matter whatsoever. With these global effects eliminating the need for dark matter, we see that invoking dark matter in galaxies could potentially be nothing more than an attempt to describe global physics effects in purely local galactic terms. Finally, we review some recent work by 't Hooft in which a connection between conformal gravity and Einstein gravity has been found.
We consider the gravitational radiation in conformal gravity theory. We perturb the metric from flat Mikowski space and obtain the wave equation after introducing the appropriate transformation for perturbation. We derive the effective energy-momentum tensor for the gravitational radiation, which can be used to determine the energy carried by gravitational waves.
We extend Maldacena's argument, namely, obtaining Einstein gravity from Conformal Gravity, to six dimensional manifolds. The proof relies on a particular combination of conformal (and topological) invariants, which makes manifest the fact that 6D Conformal Gravity admits an Einstein sector. Then, by taking generalized Neumann boundary conditions, the Conformal Gravity action reduces to the renormalized Einstein-AdS action. These restrictions are implied by the vanishing of the traceless Ricci tensor, which is the defining property of any Einstein spacetime. The equivalence between Conformal an…
Conformal symmetries appear in many parts of physics and play a unique role in exploring the Universe. In this work, we consider the possibility of constructing conformal theories of gravity in the Symmetric Teleparallel Gravity framework, where gravitation is expressed through non-metricity rather than curvature or torsion. We demonstrate that it is possible to formulate a family of conformal theories with second-order field equations and have the metric tensor as the fundamental variable. Moreover, we consider the addition of a scalar field to the theory, and compare the results to the Telep…
We consider holography of two pp-wave metrics in conformal gravity, their one point functions, and asymptotic symmetries. One of the metrics is a generalization of the standard pp-waves in Einstein gravity to conformal gravity. The holography of this metric shows that within conformal gravity one can have realised solution which has non-vanishing partially massless response (PMR) tensor even for vanishing subleading term in the Fefferman-Graham expansion (i.e. Neumann boundary conditions), and vice-versa.
We provide a new derivation of the Hawking mass and Willmore energy functionals for asymptotically AdS spacetimes, by embedding Einstein-AdS gravity in Conformal Gravity. By construction, the evaluation of the four-dimensional Conformal Gravity action in a manifold with a conical defect produces a codimension-2 conformal invariant functional $L_Σ$. The energy functionals are then particular cases of $L_Σ$ for Einstein-AdS and pure AdS ambient spaces, respectively. The bulk action is finite for AdS asymptotics and both Hawking mass and Willmore energy are finite as well. The result suggests a g…
We present a new class of solutions for static spherically symmetric wormhole spacetimes in conformal gravity and outline a detailed method for their construction. As an explicit example, we construct a class of traversable and non-traversable wormholes that are locally conformal to Schwarzschild-(anti) de Sitter spacetimes. These wormhole spacetimes are exact vacuum solutions in, but not being limited to, Weyl gravity and conformal scalar-tensor theories. Importantly, the method implies that every conformal theory of gravity with static spherically symmetric solutions will trivially contain w…
We study the gravity action built from two gauge fields corresponding to the generators of the conformal group. Starting with the action from which one can obtain Einstein gravity and conformal gravity upon imposing suitable constraints, we keep two independent gauge fields and integrate out the field corresponding to the generator of Lorentz transformations. We identify the two gauge fields with two vierbeins and perturb them around an Anti--de Sitter space. This gives the linearized equations that differ from both, Einstein gravity and conformal gravity linearized equations. We also study th…
The solutions of the Einstein equation are a subset of the solutions of conformal (Weyl) gravity, but the difference from the action means that the black hole thermodynamics of the two gravity theories would be different. In this paper we explore the thermodynamic phase structure for the conformal gravity in the four-dimensional AdS space-time. Special emphasis is put on the dependence on the parameter $c_1$ in the linear-$r$ term in the metric. The thermodynamic phase structure of the conformal gravity is very rich, including two branches of equations of states, negative thermodynamic volume,…
We investigate the black holes in the new massive conformal gravity which is not invariant under conformal transformations because of the presence of the Einstein-Hilbert term. First, we show that the small Schwarzschild black hole is unstable against the $s$-mode of linearized Ricci tensor by solving the Lichnerowicz-Ricci tensor equation. This instability induces the appearance of the non-BBMB (Bocharova-Bronnikov-Melnikov-Bekenstein) black hole that has both Ricci tensor and conformal scalar hair.
Noncritical Conformal Gravity and Four-Dimensional Liouville Theory
National Taiwan University（台湾大学）；National Center for Theoretical Sciences（国家理论科学中心）；Nambu Yoichiro Institute of Theoretical and Experimental Physics (NITEP), Osaka Metropolitan University（大阪公立大学南阳一郎理论与实验物理研究所（NITEP））；Institute of Fundamental Physics and Quantum Technology, Department of Physics, School of Physical Science and Technology, Ningbo University（宁波大学物理科学与技术学院物理学与量子技术基础物理研究所）
The Janis-Newman-Winicour metric is a solution of Einstein's gravity minimally coupled to a real massless scalar field. The $γ$-metric is instead a vacuum solution of Einstein's gravity. These spacetimes have no horizon and possess a naked singularity at a finite value of the radial coordinate, where curvature invariants diverge and the spacetimes are geodetically incomplete. In this paper, we reconsider these solutions in the framework of conformal gravity and we show that it is possible to solve the spacetime singularities with a suitable choice of the conformal factor. Now curvature invaria…
