Charged fluids rotating around compact objects can form unique equilibrium structures when ambient large-scale electromagnetic fields combine with strong gravity. Equatorial as well as off-equatorial toroidal structures are among such figures of equilibrium with a direct relevance for astrophysics. To investigate their geometrical shapes and physical properties in the near-horizon regime, where effects of general relativity play a significant role, we commonly employ a scheme based on the energy-momentum conservation written in a standard representation. Here, we develop its interesting alternatives in terms of two covariant force representations, both based on a hypersurface projection of the energy-momentum conservation. In a proper hypersurface, space-like forces can be defined, following from a decomposition of the fluid four-acceleration. Each of the representations provides us with an insight into properties of the fluid flow, being well reflected in related conformal hypersurface geometries; we find behaviour of centrifugal forces directly related to geodesics of these conformal hypersurfaces and their embedding diagrams. We also reveal correspondence between the charged fluid flow world-lines from an ordinary spacetime, and world-lines determined by a charged test particles equation of motion in a conformal spacetime.
We study first BGG operators and their solutions on homogeneous conformal geometries. We focus on conformal Killing tensors, conformal Killing--Yano forms and twistor spinors in particular. We develop an invariant calculus that allows us to find solutions explicitly using only algebraic computations. We also discuss applications to holonomy reductions and conserved quantities of conformal circles. We demonstrate our result on examples of homogeneous conformal geometries coming mostly from general relativity.
Exact solutions of traversable wormholes were recently found under the assumption of spherical symmetry and the existence of a non-static conformal symmetry. In this paper, we verify that in the case of the conformally symmetric spacetimes with a non-static vector field generating the symmetry, the conformal factor $ψ$ can be physically interpreted in terms of a measurable quantity, namely, the tangential velocity of a massive test particle moving in a stable circular orbit in the spacetime. Physical properties of the rotational velocity of test particles and of the redshift of radiation emitt…
We seek exact solutions to the Einstein field equations which arise when two spacetime geometries are conformally related. Whilst this is a simple method to generate new solutions to the field equations, very few such examples have been found in practice. We use the method of Lie analysis of differential equations to obtain new group invariant solutions to conformally related Petrov type D spacetimes. Four cases arise depending on the nature of the Lie symmetry generator. In three cases we are in a position to solve the master field equation in terms of elementary functions. In the fourth case…
In this work, a general class of wormhole geometries in conformal Weyl gravity is analyzed. A wide variety of exact solutions of asymptotically flat spacetimes is found, in which the stress energy tensor profile differs radically from its general relativistic counterpart. In particular, a class of geometries is constructed that satisfies the energy conditions in the throat neighborhood, which is in clear contrast to the general relativistic solutions.
We treat the problem of defining, and characterising in a practical way, an appropriate class of distinguished curves for Poincaré-Einstein manifolds, and other conformally singular geometries. These "generalised geodesics" agree with geodesics away from the conformal singularity set and are shown to satisfy natural "boundary conditions" at points where they meet or cross the metric singularity set. We also characterise when they coincide with conformal circles. In the case of (Poincaré-)Einstein manifolds, we are able to provide a very general theory of first integrals for these distinguished…
We present an analysis of the classic wormhole geometries based on conformal Weyl gravity, rather than standard general relativity. The main characteristics of the resulting traversable wormholes remain the same as in the seminal study by Morris and Thorne, namely, that effective super-luminal motion is a viable consequence of the metric. Improving on previous work on the subject, we show that for particular choices of the shape and redshift functions the wormhole metric in the context of conformal gravity does not violate the main energy conditions at or near the wormhole throat. Some exotic …
This article is based on the study of wormhole geometries in the context of symmetric teleparallel gravity or $f(Q)$ gravity, where $Q$ is the non-metricity scalar, and it is responsible for the gravitational interaction. To discuss the wormhole solutions, we consider spherically symmetric static spacetime metric with anisotropic matter contents under well-known non-commutative distributions known as Gaussian and Lorentzian distributions with an extra condition of permitting conformal killing vectors (CKV). This work aims to obtain wormhole solutions under these distributions, and through we f…
It is shown that the Chern-Simons functional, built in the spinor representation from the initial data on spacelike hypersurfaces, is invariant with respect to infinitesimal conformal rescalings if and only if the vacuum Einstein equations are satisfied. As a consequence, we show that in the phase space the Hamiltonian constraint of vacuum general relativity is the Poisson bracket of the imaginary part of this Chern-Simons functional and Misner's time (essentially the 3-volume). Hence the vacuum Hamiltonian constraint is the condition on the canonical variables that the imaginary part of the C…
We provide five examples of conformal geometries which are naturally associated with ordinary differential equations (ODEs). The first example describes a one-to-one correspondence between the Wuenschmann class of 3rd order ODEs considered modulo contact transformations of variables and (local) 3-dimensional conformal Lorentzian geometries. The second example shows that every point equivalent class of 3rd order ODEs satisfying the Wuenschmann and the Cartan conditions define a 3-dimensional Lorentzian Einstein-Weyl geometry. The third example associates to each point equivalence class of 3rd o…
We propose that large quantum fluctuations of the conformal factor drastically modify classical general relativity at cosmological distance scales, resulting in a scale invariant phase of quantum gravity in the far infrared. We derive scaling relations for the partition function and physical observables in this conformal phase, and suggest quantitative tests of these relations in numerical simulations of simplicial four geometries with $S^4$ topology. In particular, we predict the form of the critical curve in the coupling constant plane, and determine the scaling of the Newtonian coupling wit…
The twisted geometries of spin network states are described by simple twistors, isomorphic to null twistors with a time-like direction singled out. The isomorphism depends on the Immirzi parameter, and reduces to the identity when the parameter goes to infinity. Using this twistorial representation we study the action of the conformal group SU(2,2) on the classical phase space of loop quantum gravity, described by twisted geometry. The generators of translations and conformal boosts do not preserve the geometric structure, whereas the dilatation generator does. It corresponds to a 1-parameter …
We study non-Einstein Bach-flat gravitational instanton solutions that can be regarded as the generalization of the Taub-NUT/Bolt and Eguchi-Hanson solutions of Einstein gravity to conformal gravity. These solutions include non-Einstein spaces which are either asymptotically locally flat spacetimes (ALF) or asymptotically locally Anti-de Sitter (AlAdS). Nevertheless, solutions with different asymptotic conditions exist: we find geometries that present a weakened AlAdS asymptotia, exhibiting the typical low decaying mode of conformal gravity. This permits to identify the simple Neumann boundary…
"Diagonal" spatially inhomogeneous (SI) models are introduced under the assumption of the existence of (proper) intrinsic symmetries and can be seen, in some sense, complementary to the Szekeres models. The structure of this class of spacetimes can be regarded as a generalization of the (twist-free) Locally Rotationally Symmetric (LRS) geometries without any global isometry containing, however, these models as special cases. We consider geometries where a six-dimensional algebra $\mathcal{IC}$ of Intrinsic Conformal Vector Fields (ICVFs) exists acting on a $2-$dimensional (pseudo)-Riemannian m…
Subtracted geometries are black hole solutions of the four dimensional STU model with rather interesting ties to asymptotically flat black holes. A peculiar feature is that the solutions to the Klein-Gordon equation on this subtracted background can be organized according to representations of the conformal group $SO(2,2)$. We test if this behavior persists for the linearized fluctuations of gravitational and matter fields on static, electrically charged backgrounds of this kind. We find that there is a subsector of the modes that do display conformal symmetry, while some modes do not. We also…
The gravitational measure on an arbitrary topological three-manifold is constructed. The nontrivial dependence of the measure on the conformal factor is discussed. We show that only in the case of a compact manifold with boundary the measure acquires a nontrivial dependence on the conformal factor which is given by the Liouville action. A nontrivial Jacobian (the divergent part of it) generates the Einstein-Hilbert action. The Hartle-Hawking wave function of Universe is given in terms of the Liouville action. In the gaussian approximation to the Wheeler-DeWitt equation this result was earlier …
We present a semi-analytic method for constructing holographic black holes that interpolate from anti-de Sitter space to hyperscaling-violating geometries. These are holographic duals of conformal field theories in the presence of an applied chemical potential, $μ$, at a non-zero temperature, $T$, and allow us to describe the crossover from `strange metal' physics at $T \ll μ$, to conformal physics at $T \gg μ$. Our holographic technique adds an extra gauge field and exploits structure of the Einstein-Maxwell system to manifestly find 1-parameter families of solutions of the Einstein-matter sy…
The near horizon aspects (and beyond) of a black hole metric, which belongs to a large class of static spherically symmetric black holes, are considered here. It has been realized recently that an atom falling into a black hole leads to the generation of acceleration radiation through virtual transitions. In recent studies, it has been argued that this acceleration radiation can be understood from the near horizon physics of the black hole. The near horizon approximation leads to conformal symmetry in the problem. We go beyond the near horizon approximation in our analysis. This breaks the con…
The near horizon geometries are usually constructed by implementing a specific limit to a given extreme black hole configuration. Their salient feature is that the isometry group includes the conformal subgroup SO(2,1). In this work, we turn the logic around and use the conformal invariants for constructing Ricci-flat metrics in d=4 and d=5 where the vacuum Einstein equations reduce to a coupled set of ordinary differential equations. In four dimensions the analysis can be carried out in full generality and the resulting metric describes the d=4 near horizon Kerr-NUT black hole. In five dimens…
Carroll-Cotton Tensors and Gravitational Radiation at Null Infinity
École polytechnique, Centre National de la Recherche Scientifique – Unité Mixte de Recherche 7644, Institut Polytechnique de Paris（巴黎综合理工学院，法国国家科学研究中心联合研究单位7644，巴黎理工学院）；Université Libre de Bruxelles and International Solvay Institutes（布鲁塞尔自由大学和国际索尔维研究所）；International School for Advanced Studies (SISSA)（高级国际研究学院（SISSA））；Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Trieste（意大利国家核物理研究所的里雅斯特分部）；Université de Mons（蒙斯大学）
By restricting the functional integration to the Regge geometries, we give the discretized version of the well known path integral formulation of 2--dimensional quantum gravity in the conformal gauge. We analyze the role played by diffeomorphisms in the Regge framework and we give an exact expression for the Faddeev--Popov determinant related to a Regge surface; such an expression in the smooth limit goes over to the correct continuum result.
The equations governing anti-self-dual and Einstein-Weyl conformal geometries can be regarded as `master dispersionless systems' in four and three dimensions respectively. Their integrability by twistor methods has been established by Penrose and Hitchin. In this note we present, in specially adapted coordinate systems, explicit forms of the corresponding equations and their Lax pairs. In particular, we demonstrate that any Lorentzian Einstein-Weyl structure is locally given by a solution to the Manakov-Santini system, and we find a system of two coupled third-order scalar PDEs for a general a…
