Here we follow the mainstream of thinking about physical equivalence of different representations of a theory, regarded as the consequence of invariance of the laws of physics -- represented by an action principle and the derived motion equations -- under given transformations; be it coordinate, gauge or conformal transformations. Accordingly the conformal transformations' issue is discussed by invoking the assumed invariance of the laws of physics -- in particular the laws of gravity -- under conformal transformations of the metric. It is shown that Brans-Dicke and scalar-tensor theories are not well-suited to address physical equivalence of the conformal frames since the corresponding laws of gravity are not invariant under the conformal transformations or Weyl rescalings. The search for conformal symmetry leads us to explore the physical consequences of Weyl-invariant theories of gravity, that represent a natural arena where to discuss on physical equivalence of the conformally related representations. We show that conformal invariance of the action of a (supposedly conformal invariant) theory and of the derived motion equations is not enough to ensure actual Weyl invariance. It is required, also, that the underlying geometrical structure of the background spacetime be, at least, Weyl-integrable. Otherwise, if assume (as usual) spacetimes of (pseudo)Riemannian geometrical structure, the resulting -- apparently conformal invariant -- theory is anomalous in that, only massless matter fields can be consistently coupled. Gauge freedom, a distinctive feature of actually Weyl-invariant theories of gravity, leads to very unusual consequences.
It is well-known that the conformal structure of a relativistic spacetime is of profound physical and conceptual interest. In this note, we consider the analogous structure for Newtonian theories. We show that the Newtonian Weyl tensor is an invariant of this structure.
We examine the question of scale vs. conformal invariance for the linearized Einstein-Hilbert action, which describes the IR fixed point of quantum gravity. In $D = 4$, although the action is not conformally invariant in the usual sense, we explicitly show that the theory is a conformal field theory at the level of correlation functions. In higher dimensions, we show that the theory is scale but not conformally invariant, but can be embedded into a larger non-unitary conformal field theory, analogous to what has been found for Maxwell theory in $D>4$. We give evidence that similar statements a…
Conformal Coupling and Invariance in Different Dimensions
比较了以下几种共形变换：（1）共形坐标变换，（2）D维几何的拉格朗日模型的共形变换，由度量g为任意签名的黎曼流形M给出，（3）（迷你）超空间几何的共形变换。对于这种变换下的共形不变性，分别给出了以下应用：（1）多维几何的自然时间规范，（2）耦合到空间齐次标量场的几何的共形等价Lagrange模型，（3）多维几何的小超空间的$n$维流形$M上的共形Laplace算子和Wheeler de Witt方程.共形耦合常数xi_c是严格区别于任意耦合xi，对于D维几何的拉格朗日模型和n维minisuperspace上的共形几何都是等价的。对于维数D= 3，4，6或10，临界数xi_c={D-2}/{4（D-1）}特别简单，是一个有理分式。
In this paper, we prove that lightlike geodesics of a pseudo-Finsler manifold and its focal points are preserved up to reparametrization by anisotropic conformal changes, using the Chern connection and the anisotropic calculus and the fact that geodesics are critical points of the energy functional and Jacobi fields, the kernel of its index form. This result has applications to the study of Finsler spacetimes.
Conformal invariance can ameliorate or eliminate the singularities residing in the black holes, and may still exist in the strong gravity regimes close to these black holes. In this paper, we try to probe this conformal invariance by looking into the wave absorption and scattering by the nonsingular static spherical black holes. The partial and total absorption cross section, as well as the differential scattering cross section, are presented for black holes with different choices of conformal parameters. Although the photon trajectories are unchanged from the Schwarzschild case since the spac…
