Hartle-Thorne Metric Research Articles

An approximate Kerr–Newman-like metric endowed with a magnetic dipole and mass quadrupole

Approximate all-terrain spacetimes for astrophysical applications are presented. The metrics possess five relativistic multipole moments, namely, mass, rotation, mass quadrupole, charge, and magnetic dipole moment. All these spacetimes approximately satisfy the Einstein–Maxwell field equations. The first metric is generated using the Hoenselaers–Perjés method from given relativistic multipoles. The second metric is a perturbation of the Kerr–Newman metric, which makes it a relevant approximation for astrophysical calculations. The last metric is an extension of the Hartle–Thorne metric that is important for obtaining internal models of compact objects perturbatively. The electromagnetic field is calculated using Cartan forms for locally non-rotating observers. These spacetimes are relevant for inferring properties of compact objects from astrophysical observations. Furthermore, the numerical implementations of these metrics are straightforward, making them versatile for simulating potential astrophysical applications.

A coupling prescription for post-Newtonian corrections in quantum mechanics

The interplay between quantum theory and general relativity remains one of the main challenges of modern physics. A renewed interest in the low-energy limit is driven by the prospect of new experiments that could probe this interface. Here we develop a covariant framework for expressing post-Newtonian corrections to Schrödinger’s equation on arbitrary gravitational backgrounds based on a 1/c^21/c2 expansion of Lorentzian geometry, where cc is the speed of light. Our framework provides a generic coupling prescription of quantum systems to gravity that is valid in the intermediate regime between Newtonian gravity and General Relativity, and that retains the focus on geometry. At each order in 1/c^21/c2 this produces a nonrelativistic geometry to which quantum systems at that order couple. By considering the gauge symmetries of both the nonrelativistic geometries and the 1/c^21/c2 expansion of the complex Klein-Gordon field, we devise a prescription that allows us to derive the Schrödinger equation and its post-Newtonian corrections on a gravitational background order-by-order in 1/c^21/c2. We also demonstrate that these results can be obtained from a 1/c^21/c2 expansion of the complex Klein-Gordon Lagrangian. We illustrate our methods by performing the 1/c^21/c2 expansion of the Kerr metric up to \mathcal{O}(c^{-2})𝒪(c−2), which leads to a special case of the Hartle-Thorne metric. The associated Schrödinger equation captures novel and potentially measurable effects.

Testing no-hair theorem by quasi-periodic oscillations: thequadrupole of GRO J1655–40

We perform an observational test of no-hair theorem usingquasi-periodic oscillations within the relativistic precession model. Twowell motivated metrics we apply are Kerr-Q and Hartle-Thorne metrics inwhich the quadrupole is the parameter that possibly encodes deviations fromthe Kerr black hole. The expressions for the quasi-periodic frequencies arederived before comparing the models with the observation. We encounter adegeneracy in constraining spin and quadrupole parameters that makes itdifficult to measure their values. In particular, we here propose a noveltest of no-hair theorem by adapting the Hartle-Thorne metric. We complement our study by doing a model comparison. We findthat there is strong evidence that Kerr black hole is the source of the central object in GROJ1655–40 given the present observational precisions.

К ОБРАТНОЙ ЗАДАЧЕ НЕБЕСНОЙ МЕХАНИКИ

The link between exterior solutions to the Einstein gravitational field equations such as the exact Erez-Rosen metric and approximate Hartle-Thorne metric is established here for the static case in the limit of linear mass quadrupole moment (Q) and second order terms in total mass (M). To this end, the Geroch-Hansen multipole moments are calculated for the Erez-Rosen and Hartle-Thorne solutions in order to find the relationship among the parameters of both metrics. The coordinate transformations are sought in a general form with two unknown functions in the corresponding limit of ~Q and ~M^2. By employing the perturbation theory, the approximate Erez-Rosen metric is written in the same coordinates as the Hartle-Thorne metric. By equating the radial and azimuthal components of the metric tensor of both solutions the sought functions are found in a straightforward way. It is shown that the approximation ~Q and ~M^2, which is used throughout the article, is physical and suitable for solving most problems of celestial mechanics in post-Newtonian physics. This approximation does not require the use of the Zipoy-Voorhees transformation, which is a necessary strict mathematical requirement in the ~Q approximation, i.e. when no other approximations are made. This implies that the explicit form of the coordinate transformations depends entirely on the approximation that is adopted in each particular case. The results obtained here are in agreement with the previous results in the literature and can be applied to different astrophysical goals. The paper pursues not only pure scientific, but also academic purposes and can be used as an auxiliary and additional material to the special courses of general theory of relativity, celestial mechanics and relativistic astrophysics.

Three-dimensional general relativistic Poynting-Robertson effect. IV. Slowly rotating and nonspherical quadrupolar massive source

We consider a further extension of our previous works in the treatment of the three-dimensional general relativistic Poynting-Robertson effect, which describes the motion of a test particle around a compact object as affected by the radiation field originating from a rigidly rotating and spherical emitting source, which produces a radiation pressure, opposite to the gravitational pull, and a radiation drag force, which removes energy and angular momentum from the test particle. The gravitational source is modeled as a nonspherical and slowly rotating compact object endowed with a mass quadrupole moment and an angular momentum and it is formally described by the Hartle-Thorne metric. We derive the test particle's equations of motion in the three-dimensional and two-dimensional cases. We then investigate the properties of the critical hypersurfaces (regions where a balance between gravitational and radiation forces is established). Finally, we show how this model can be applied to treat radiation phenomena occurring in the vicinity of a neutron star.

Approximate spacetime for neutron stars

An approximate realistic metric representing the spacetime of neutron stars is obtained by perturbing the Kerr metric. This metric has five parameters, namely the mass, spin or angular momentum, mass quadrupole, spin octupole and mass hexadecapole. Moreover, a version of the Hartle-Thorne metric containing these parameters is constructed by means of a series transformation between these spacetimes and solving the Einstein field equations. The form of the Pappas metric in Schwarzschild spherical coordinates is found. The three relativistic multipole structures are compared.

On the post-linear quadrupole-quadrupole metric

The Hartle-Thorne metric defines a reliable spacetime for most astrophysical purposes, for instance simulations of slowly rotating stars. Solving the Einstein field equations, we added terms of second order in the quadrupole moment to its post-linear version in order to compare it with solutions found by Blanchet in the multi-polar post-Minkowskian framework. We first derived the extended Hartle-Thorne metric in harmonic coordinates and then showed agreement with the corresponding post-linear metric from Blanchet. We also found a coordinate transformation from the post-linear Erez-Rosen metric to our extended Hartle-Thorne spacetime. It is well known that the Hartle-Thorne solution can be smoothly matched with an interior perfect fluid solution with appropriate physical properties. A comparison among these solutions provides a validation of them. It is clear that in order to represent realistic solutions of self-gravitating (axially symmetric) matter distributions of perfect fluid, the quadrupole moment has to be included as a physical parameter.

Geodesics in the field of a rotating deformed gravitational source

We investigate equatorial geodesics in the gravitational field of a rotating and deformed source described by the approximate Hartle-Thorne metric. In the case of massive particles, we derive within the same approximation analytic expressions for the orbital angular velocity, the specific angular momentum and energy, and the radii of marginally stable and marginally bound circular orbits. Moreover, we calculate the orbital angular velocity and the radius of lightlike circular geodesics. We study numerically the frame dragging effect and the influence of the quadrupolar deformation of the source on the motion of test particles. We show that the effects originating from the rotation can be balanced by the effects due to the oblateness of the source.

Approximate Kerr-Like Metric with Quadrupole

A new approximate metric representing the spacetime of a rotating deformed body is obtained by perturbing the Kerr metric to include up to the second order of the quadrupole moment. It has a simple form, because it 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, which guarantees its validity to be useful in studying compact object, and it is possible to find an inner solution.

A RAY-TRACING ALGORITHM FOR SPINNING COMPACT OBJECT SPACETIMES WITH ARBITRARY QUADRUPOLE MOMENTS. II. NEUTRON STARS

A moderately spinning neutron star acquires an oblate shape and a spacetime with a significant quadrupole moment. These two properties affect its apparent surface area for an observer at infinity, as well as the lightcurve arising from a hot spot on its surface. In this paper, we develop a ray-tracing algorithm to calculate the apparent surface areas of moderately spinning neutron stars making use of the Hartle-Thorne metric. This analytic metric allows us to calculate various observables of the neutron star in a way that depends only on its macroscopic properties and not on the details of its equation of state. We use this algorithm to calculate the changes in the apparent surface area, which could play a role in measurements of neutron star radii and, therefore, in constraining their equation of state. We show that whether the spinning neutron star appears larger or smaller than its non-rotating counterpart depends primarily on its equatorial radius. For neutron stars with radii ~10 km, the corrections to the Schwarzschild spacetime cause the apparent surface area to increase with spin frequency. In contrast, for neutron stars with radii ~15 km, the oblateness of the star dominates the spacetime corrections and causes the apparent surface area to decrease with increasing spin frequency. In all cases, the change in the apparent geometric surface area for the range of observed spin frequencies is < 5% and hence only a small source of error in the measurement of neutron star radii.

Neutrino oscillations in the field of a rotating deformed mass

The neutrino oscillations in the field of a rotating deformed mass is investigated. The phase shift is evaluated in the case of weak field limit, slow rotation and small deformation. To this aim the Hartle–Thorne metric is used, which is an approximate solution of the vacuum Einstein equations accurate to second order in the rotation parameter a/M and to first order in the mass quadrupole moment q. Implications on atmospheric, solar and astrophysical neutrinos are discussed.

Global solution for a viscous accretion disc around a rotating compact object: pseudo-general-relativistic study

We study the solution for a viscous accretion disc around a rotating compact/central object having a hard surface, i.e. a neutron star, a strange star or any other highly gravitating object. We choose a pseudo-Newtonian approach to describe the relativistic accretion disc. For this purpose, a new pseudo-Newtonian potential is established that is applicable to describing the relativistic properties of a star and its disc. As we know, the Hartle–Thorne metric can describe the geometry of a star as well as the space–time outside it, so we use this metric to establish our potential. Our potential reproduces the marginally stable orbit exactly as does general relativity. It also reproduces the marginally bound orbit and specific mechanical energy at the marginally stable orbit with at most 4 and 10 per cent error, respectively. Using this potential, we study the global parameter space of the accretion disc. Thus we find the physical parameter regime for which a stable accretion disc can be formed around a gravitating object with a hard surface. We also study how the fluid properties get changed with different rotations of the central star. We show that, with a change in rotation of the central object, the valid disc parameter region changes dramatically. We also show the effect of viscosity on the fluid properties of the disc. Subsequently, we give a theoretical prediction of kilohertz QPO frequencies, at least for one of a pair, for a fast-rotating compact object such as 4U 1636-53.