We investigate the interior Einstein's equations in the case of a static, axially symmetric, perfect fluid source. We present a particular line element that is specially suitable for the investigation of this type of interior gravitational fields. Assuming that the deviation from spherically symmetry is small, we linearize the corresponding field equations and find several classes of vacuum and perfect fluid solutions. We find physically meaninful spacetimes by imposing appropriate matching conditions.
The Kerr metric is known to present issues when trying to find an interior solution. In this work we continue in our efforts to construct a more realistic exterior metric for astrophysical objects. A new approximate metric representing the spacetime of a charged, rotating and slightly-deformed body is obtained by perturbing the Kerr-Newman metric to include the mass-quadrupole and quadrupole-quadrupole orders. It has a simple form, because is Kerr-Newman-like. Its post-linear form without charge coincides with post-linear quadrupole-quadrupole metrics already found.
A new approximate metric representing the spacetime of a rotating deformed body is obtained by perturbing the Kerr metric to include til the second order of the quadrupole moment. It has a simple form, because is Kerr-like. Its Taylor expansion form coincides with second order quadrupole metrics with slow rotation already found. Moreover, it can be transformed to an improved Hartle-Thorne metric, this guarantees its validity to be useful in studying compact object, and that it is possible to find an inner solution.
Recently, spacetimes described by metrics with three parameters (mass, rotation and small quadrupole moment) was found, and in this paper, null geodesics for these metrics are calculated and visualized. Light scattering, as well as the role that the quadrupole moment plays in deforming these kinds of spacetimes are studied. This is a new application of a symbolic--numeric program that we previously used to study the Bonnor metric.
We present approximate exterior and interior solutions of Einstein's equations which describe the gravitational field of a static deformed mass distribution. The deformation of the source is taken into account up to the first order in the quadrupole.
Using the Post-Minkowskian formalism and considering rotation as a perturbation, we compute an approximate interior solution for a stationary perfect fluid with constant density and axial symmetry. A suitable change of coordinates allows this metric to be matched to the exterior metric to a particle with a pole-dipole-quadrupole structure, relating the parameters of both.
We study geometrically thick perfect-fluid tori with constant specific angular momentum, so-called "Polish doughnuts", orbiting deformed compact objects with a quadrupole moment. More specifically, we consider two different asymptotically flat, static and axisymmetric vacuum solutions to Einstein's field equation with a non-zero quadrupole moment, the q-metric and the Erez-Rosen spacetime. It is our main goal to find features of Polish doughnuts in these two spacetimes which qualitatively distinguish them from Polish doughnuts in the Schwarzschild spacetime. As a main result we find that, for …
In this contribution, we calculate the light deflection, perihelium shift, time delay and gravitational redshift using an approximate metric that contains the Kerr metric and an approximaction of the Erez-Rosen spacetime. The results were obtained directly using Mathematica. The results agree with the ones presented in the literature, but they are extended until second order terms of mass, angular momentum and mass quadrupole.
We present the Ernst potential and the line element of an exact solution of Einstein's vacuum field equations that contains as arbitrary parameters the total mass, the angular momentum, and the quadrupole moment of a rotating mass distribution. We show that in the limiting case of slowly rotating and slightly deformed configuration, there exists a coordinate transformation that relates the exact solution with the approximate Hartle solution. It is shown that this approximate solution can be smoothly matched with an interior perfect fluid solution with physically reasonable properties. This ope…
We investigate the Zipoy-Voorhees metric ($q-$metric) as the simplest static, axially symmetric solution of Einstein's vacuum field equations that possesses as independent parameters the mass and the quadrupole moment. In accordance with the black holes uniqueness theorems, the presence of the quadrupole completely changes the geometric properties of the corresponding spacetime that turns out to contain naked singularities for all possible values of the quadrupole parameter. The naked singularities, however, can be covered by interior solutions that correspond to perfect fluid sources with no …
The innermost stable circular orbit equation of a test particle is obtained for an approximate Kerr-like spacetime with quadrupole moment. We derived the effective potential for the radial coordinate by the Euler-Lagrange method. This equation can be employed to measure the mass quadrupole by observational means, because from this equation a quadratic polynomial for the quadrupole moment can be found. As expected, the limiting cases of this equation are found to be the known cases of Kerr and Schwarzschild.
Existing estimates of the gravitational-wave damping timescale of the dominant quadrupole oscillation mode in the case of rapidly rotating stars are based on using a Newtonian estimate for the energy of the mode, in combination with the lowest-order post-Newtonian quadrupole formula for estimating the gravitational-wave luminosity. We investigate a number of other choices for estimating the gravitational-wave damping timescale in the nonrotating limit and construct a highly accurate, empirically corrected formula that has a maximum relative error of only 3% with respect to the perturbative res…
The Neutron Star Interior Composition Explorer (NICER) mission has provided a unique opportunity to constrain the equation of state of neutron stars by using the technique of pulse-profile modelling. This technique requires accurate and efficient ray tracing, that in turn requires a robust representation of the spacetime around a neutron star. Several exact and approximate metrics have been proposed, and used, to perform ray tracing around neutron stars, with both moderate and fast rotation. In this paper, we perform a comparison between several of these metrics, when used for ray tracing. We …
A new approximate solution of vacuum and stationary Einstein field equations is obtained. This solution is constructed by means of a power series expansion of the Ernst potential in terms of two independent and dimensionless parameters representing the quadrupole and the angular momentum respectively. The main feature of the solution is a suitable description of small deviations from spherical symmetry through perturbations of the static configuration and the massive multipole structure by using those parameters. This quality of the solution might eventually provide relevant differences with r…
The multipolar-post-Minkowskian approach to gravitational radiation is applied to the problem of the generation of waves by the compact binary inspiral. We investigate specifically the third post-Newtonian (3PN) approximation in the total energy flux. The new results are the computation of the mass quadrupole moment of the binary to the 3PN order, and the current quadrupole and mass octupole to the 2PN order. Wave tails and tails of tails in the far zone are included up to the 3.5PN order. The recently derived 3PN equations of binary motion are used to compute the time-derivatives of the momen…
We derive a particular approximate solution of Einstein equations, describing the gravitational field of a mass distribution that slightly deviates from spherical symmetry. The deviation is described by means of a quadrupole parameter that is responsible for the appearance of a curvature singularity, which is not covered by a horizon. We investigate the motion of test particles in the gravitational field of this naked singularity and show that the quadrupole parameter affects the properties of Schwarzschild trajectories. By investigating radial geodesics, we find that no effects of repulsive g…
We investigate the influence of the quadrupole moment of a rotating source on the motion of a test particle in the strong field regime. For this purpose the Hartle-Thorne metric, that is an approximate solution of vacuum Einstein field equations that describes the exterior of any slowly rotating, stationary and axially symmetric body, is used. The metric is given with accuracy up to the second order terms in the body's angular momentum, and first order terms in its quadrupole moment. We give, with the same accuracy, analytic equations for equatorial circular geodesics in the Hartle-Thorne spac…
We obtain an approximate global stationary and axisymmetric solution of Einstein's equations which can be thought as a simple star model: a self-gravitating perfect fluid ball with a differential rotation motion pattern. Using the post--Minkowskian formalism (weak field approximation) and considering rotation as a perturbation (slow rotation approximation), we find approximate interior and exterior (asymptotically flat) solutions to this problem in harmonic coordinates. Interior and exterior solutions are matched, in the sense of Lichnerowicz, on the surface of zero pressure to get a global so…
A new Kerr-like metric with quadrupole moment is obtained by means of perturbing the Kerr spacetime. The form of this new metric is simple as the Kerr metric. By comparison with the exterior Hartle-Thorne metric, it is shown that it could be matched to an interior solution. This approximate metric may represent the spacetime of a real astrophysical object with any Kerr rotation parameter a and slightly deformed.
We investigate the hyperbolic scattering of test particles, spinning test particles and particles with spin-induced quadrupolar structure by a Kerr black hole in the ultrarelativistic regime. We also study how the features of the scattering process modify if the source of the background gravitational field is endowed with a nonzero mass quadrupole moment as described by the (approximate) Hartle-Thorne solution. We compute the scattering angle either in closed analytical form, when possible, or as a power series of the (dimensionless) inverse impact parameter. It is a function of the parameters…
We employ recently developed approximation methods in the hybrid quantization of the Gowdy $T^3$ model with linear polarization and a massless scalar field to obtain physically interesting solutions of this inhomogeneous cosmology. More specifically, we propose approximate solutions of the quantum Gowdy model constructed in such a way that, for the Hamiltonian constraint, they effectively behave as those corresponding to a flat homogeneous and isotropic universe filled with a perfect fluid, even though these quantum states are far from being homogeneous and isotropic. We analyze how one can ge…
We obtain an approximate global stationary and axisymmetric solution of Einstein's equations which can be considered as a simple star model: a self-gravitating perfect fluid ball with constant mass density rotating in rigid motion. Using the post-Minkowskian formalism (weak-field approximation) and considering rotation as a perturbation (slow-rotation approximation), we find approximate interior and exterior (asymptotically flat) solutions to this problem in harmonic and quo-harmonic coordinates. In both cases, interior and exterior solutions are matched, in the sense of Lichnerowicz, on the s…
We review the problem of describing the gravitational field of compact stars in general relativity. We focus on the deviations from spherical symmetry which are expected to be due to rotation and to the natural deformations of mass distributions. We assume that the relativistic quadrupole moment takes into account these deviations, and consider the class of axisymmetric static and stationary quadrupolar metrics which satisfy Einstein's equations in empty space and in the presence of matter represented by a perfect fluid. We formulate the physical conditions that must be satisfied for a particu…
We analyze in detail the spin precession equations in binary black hole systems, when the tidal torque on a Kerr black hole due to quadrupole-monopole coupling is taken into account. We show that completing the precession equations with this term reveals the existence of a conserved quantity at 2PN order when averaging over orbital motion. This quantity allows one to solve the (orbit-averaged) precession equations exactly in the case of equal masses and arbitrary spins, neglecting radiation reaction. For unequal masses, an exact solution does not exist in closed form, but we are still able to …
We investigate the properties of a closed-form analytic solution recently found by Manko et al. (2000) for the exterior spacetime of rapidly rotating neutron stars. For selected equations of state we numerically solve the full Einstein equations to determine the neutron star spacetime along constant rest mass sequences. The analytic solution is then matched to the numerical solutions by imposing the condition that the quadrupole moment of the numerical and analytic spacetimes be the same. For the analytic solution we consider, such a matching condition can be satisfied only for very rapidly ro…
We consider a hydrodynamic approach in which a quantum system of interacting quarks and gluons is approximated classically by representing it as a perfect fluid having intrinsic degrees of freedom. Every particle of such fluid is endowed with spin and non-Abelian color charge. The variational theory of such perfect spin fluid with color charge is constructed, the spin-polarization chromomagnetic effects in an external Yang-Mills field and in Riemann-Cartan space with curvature and torsion being taken into account.
We present new numerical algorithms for the coupled Einstein-perfect fluid system in axisymmetry. Our framework uses a foliation based on a family of light cones, emanating from a regular center, and terminating at future null infinity. This coordinate system is well adapted to the study of the dynamical spacetimes associated with isolated relativistic compact objects such as neutron stars. In particular, the approach allows the unambiguous extraction of gravitational waves at future null infinity and avoids spurious outer boundary reflections. The code can accurately maintain long-term stabil…
Astrophysical charge-free black holes are known to satisfy no-hair relations through which all multipole moments can be specified in terms of just their mass and spin angular momentum. We here investigate the possible existence of no-hair-like relations among multipole moments for neutron stars and quark stars that are independent of their equation of state. We calculate the multipole moments of these stars up to hexadecapole order by constructing uniformly-rotating and unmagnetized stellar solutions to the Einstein equations. For slowly-rotating stars, we construct stellar solutions to quarti…
