Using a framework based on the $1+3$ formalism we carry out a study on axially and reflection symmetric dissipative fluids, in the quasi--static regime. We first derive a set of invariantly defined "velocities", which allow for an inambiguous definition of the quasi--static approximation. Next we rewrite all the relevant equations in this aproximation and extract all the possible, physically relevant, consequences ensuing the adoption of such an approximation. In particular we show how the vorticity, the shear and the dissipative flux, may lead to situations where different kind of "velocities" change of sign within the fluid distribution with respect to theirs sign on the boundary surface. It is shown that states of gravitational radiation are not {\it a priori} incompatible with the quasistatic--regime. However, any such state must last for an infinite period of time, thereby diminishing its physical relevance.
We report a study on axially and reflection symmetric dissipative fluids, just after its departure from hydrostatic and thermal equilibrium, at the smallest time scale at which the first signs of dynamic evolution appear. Such a time scale is smaller than the thermal relaxation time, the thermal adjustment time and the hydrostatic time. It is obtained that the onset of non--equilibrium will critically depend on a single function directly related to the time derivative of the vorticity. Among all fluid variables (at the time scale under consideration), only the tetrad component of the anisotrop…
A study of proper affine vector fields in non static axially symmetric space-times is given by using holonomy and decomposability, the rank of the 6X6 Riemann matrix and direct integration techniques. It is shown that the special class of the above space-times admits proper affine vector fields.
Analytic Solutions for Geodesic Motion in Static Axially Symmetric Spacetime
Coordenação de Astronomia e Astrofísica, Observatório Nacional（天文与天体物理协调局，国家天文台）；Departamento de Física Teórica, Universidade do Estado do Rio de Janeiro (UERJ)（理论物理系，里约热内卢州立大学）；Sorbonne Université, UPMC Université Paris 06, LERMA, Observatoire de Paris-Meudon（索邦大学，巴黎第六大学，天体物理学研究实验室，巴黎-默东天文台）
We carry out a study of the exterior of an axially and reflection symmetric source of gravitational radiation. The exterior of such a source is filled with a null fluid produced by the dissipative processes inherent to the emission of gravitational radiation, thereby representing a generalization of the Vaidya metric for axially and reflection symmetric spacetimes. The role of the vorticity, and its relationship with the presence of gravitational radiation is put in evidence. The spherically symmetric case (Vaidya) is, asymptotically, recovered within the context of the $1+3$ formalism.
We consider the properties of a static axially symmetric wormhole described by an exact solution of Einstein's field equations and investigate how we can distinguish such a hypothetical object from a black hole. To this aim, we explore the motion of test particles and photons in the wormhole's space-time and compare it with the particle dynamics in the well known space-times of Schwarzschild and Kerr black holes. We show that precise simultaneous measurement of test particle motion and photon motion may provide the means to distinguish the wormhole geometry from that of a black hole.
We carry on a general study on axially symmetric, static fluids admitting a conformal Killing vector (CKV). The physical relevance of this kind of symmetry is emphasized. Next, we investigate all possible consequences derived from the imposition of such a symmetry. Special attention is paid to the problem of symmetry inheritance. Several families of solutions endowed with a CKV are exhibited.
In continuation of our previous study [Phys. Rev. D 99 (2019) 4, 044012], we investigate the motion of charged particles in the $γ$-metric. We provide some examples of curled trajectories in the equatorial plane and escape trajectories outside the equatorial plane. Finally, we consider harmonic oscillations due to small displacements from stable circular orbits on the equatorial plane and compute the epicyclic frequencies for different values of deformation parameter and magnetic field.
The sum of squared epicyclic frequencies of nearly circular motion ($ω_r^2+ω_θ^2$) in axially symmetric configurations of Newtonian gravity is known to depend both on the matter density and on the angular velocity profile of circular orbits. It was recently found that this sum goes to zero at the photon orbits of Schwarzschild and Kerr spacetimes. However, these are the only relativistic configurations for which such result exists in the literature. Here, we extend the above formalism in order to describe the analogous relation for geodesic motion in arbitrary static, axially symmetric, asympt…
A previously found definition of complexity for spherically symmetric fluid distributions [1], is extended to axially symmetric static sources. In this case there are three different complexity factors, defined in terms of three structure scalars obtained from the orthogonal splitting of the Riemann tensor. All these three factors vanish, for what we consider the simplest fluid distribution, i.e a fluid spheroid with isotropic pressure and homogeneous energy density. However, as in the spherically symmetric case, they can also vanish for a variety of configurations, provided the energy density…
A general procedure to find static and axially symmetric, interior solutions to the Einstein equations is presented. All the so obtained solutions, verify the energy conditions for a wide range of values of the parameters, and match smoothly to some exterior solution of the Weyl family, thereby representing globally regular models describing non spherical sources of gravitational field. In the spherically symmetric limit, all our models converge to the well known incompressible perfect fluid solution.The key stone of our approach is based on an ansatz allowing to define the interior metric in …
The exact axisymmetric and static solution of the Einstein equations coupled to axisymmetric and static gravitating scalar (or phantom) field is presented. The spacetimes modified by the scalar field are explicitly given for the so called $γ$-metric and Erez-Rosen metric with quadrupole moment $q$, influence of the additional deformation parameters $γ_*$ and $q_*$ generated by the scalar field is studied. It is shown that the null energy condition is satisfied for the phantom field, but it is not satisfied for the standard scalar field. The test particle motion in the both modified $γ$-metric …
Recently, several new characteristics have been introduced to describe null geodesic structure of strong gravitational field, such as photon regions, transversely trapping surfaces and some generalizations. They give an alternative and concise way to describe lensing and shadow features of compact objects with strong gravitational field without recurring to complete integration of the geodesic equations. Here we test this construction in the case of the Weyl metrics when geodesic equations are non-separable, and thus can not be integrated analytically, while the above characteristic surfaces a…
Quantum-mechanical motion of a half-spin particle was examined in the axially symmetric field of static naked singularities formed by mass distribution with quadrupole moment (q-metric). The analysis was performed by means of the method of effective potentials of the Dirac equation generalized for the case when radial and angular variables are not separated. As $-1<q<q_{lim}, |q_{lim}|<<1 $ the naked singularities do not except the existence of stationary bound states of Dirac particles for a prolate mass distribution in the q-metric along the axial axis. For the oblate mass distribution, the …
We carry on a systematic study of the physical properties of axially symmetric fluid distributions, which appear to be geodesic, shear--free, irrotational, non--dissipative and purely electric, for the comoving congruence of observers, from the point of view of the tilted congruence. The vanishing of the magnetic part of the Weyl tensor for the comoving congruence of observers, suggests that no gravitational radiation is produced during the evolution of the system. Instead, the magnetic part of the Weyl tensor as measured by tilted observers is non vanishing (as well as the shear, the four--ac…
We describe the transition of a graviationally radiating, axially and reflection symmetric dissipative fluid, to a non--radiating state. It is shown that very shortly after the end of the radiating regime, at a time scale of the order the thermal relaxation time, the thermal adjustment time and the hydrostatic time (whichever is larger), the system reaches the equilibrium state. This result is at variance with all the studies carried out in the past, on gravitational radiation outside the source, which strongly suggest that after a radiating period, the conditions for a return to a static case…
We consider strongly gravitating configurations consisting of coupled real Higgs scalar field and vector (Proca) field of mass $μ_P$. For such a system, we find static regular axially symmetric solutions describing asymptotically flat configurations which may be referred to as Proca-Higgs miniboson stars, since their total mass and spatial dimension are of order $M_{\text{Pl}}^2/μ_P$ and $μ_P^{-1}$, respectively.The system possesses an axially symmetric dipole field and may be regarded as a Proca dipole.
Applying the method of conformal metric to a given static axially symmetric vacuum solution of the Einstein equations, we have shown that there is no solution representing a cosmic ideal fluid which is asymtotically FLRW. Letting the cosmic fluid to be imperfect there are axially symmetric solutions tending to FLRW at space infinity. The solution we have found represents an axially symmetric spacetime leading to a spherically symmetric Einstein tensor. Therefore, we have found a solution of Einstein equations representing a spherically symmetric matter distribution corresponding to a spacetime…
Recent observations of the orbits of star clusters around Sgr $A^\star$, imaging of black holes and gravitational waveforms of merging compact objects require a detailed understanding of the general relativistic geodesic motion. We came up with a method to provide all the possible geodesics in an axially symmetric space-time. The Kerr metric is explicitly worked out, recovering the Schwarzschild geodesics in the static limit. We also found the most general Killing tensor and its associated constant of motion for an axisymmetric space-time. The relevance of these results is crucial to understan…
In this paper, we explore the source of gravitational radiation in the context of $f(R)$ gravity by considering axially symmetric dissipative dust under geodesic condition. We evaluate scalars associated with electric and magnetic parts of the Weyl tensor for both non-spinning (at the center) and spinning (in the surrounding of the center) fluids of the configuration. For this purpose, we use the evolution as well as constraint equations for kinematical quantities and Weyl tensor. Finally, we investigate the existence of gravitational radiation through super-Poynting vector. It is found that t…
On the basis of the Poincare-Weyl gauge theory of gravitation, a new conformal Weyl-Dirac theory of gravitation is proposed, which is a gravitational theory in Cartan-Weyl spacetime with the Dirac scalar field representing the dark matter model. A static approximate axially symmetric solution of the field equations in vacuum is obtained. On the base of this solution in the Newtonian approximation one considers the problem of rotation velocities in spiral components of galaxies.
We investigate the conditions for the (in)stability of the isotropic pressure condition in collapsing spherically symmetric, dissipative fluid distributions. It is found that dissipative fluxes, and/or energy density inhomogeneities and/or the appearance of shear in the fluid flow, force any initially isotropic configuration to abandon such a condition, generating anisotropy in the pressure. To reinforce this conclusion we also present some arguments concerning the axially symmetric case. The consequences ensuing our results are analyzed.
The recently proposed definition of complexity for static and spherically symmetric self--gravitating systems [1], is extended to the fully dynamic situation. In this latter case we have to consider not only the complexity factor of the structure of the fluid distribution, but also the condition of minimal complexity of the pattern of evolution. As we shall see these two issues are deeply intertwined. For the complexity factor of the structure we choose the same as for the static case, whereas for the simplest pattern of evolution we assume the homologous condition. The dissipative and non-dis…
In 1921 Bach and Weyl derived the method of superposition to construct new axially symmetric vacuum solutions of General Relativity. In this paper we extend the Bach-Weyl approach to non-vacuum configurations with massless scalar fields. Considering a phantom scalar field with the negative kinetic energy, we construct a multi-wormhole solution describing an axially symmetric superposition of $N$ wormholes. The solution found is static, everywhere regular and has no event horizons. These features drastically tell the multi-wormhole configuration from other axially symmetric vacuum solutions whi…
