Slow Rotation Approximation Research Articles

Detection of Simultaneous Quasiperiodic Oscillation Triplets in 4U 1728-34 and Constraining the Neutron Star Mass and Moment of Inertia

We report simultaneous detection of twin kHz and ∼40 Hz quasiperiodic oscillations (QPOs) in the time-resolved analysis of the AstroSat/LAXPC observation of the neutron star low-mass X-ray binary, 4U 1728-34. The frequencies of the multiple sets of triplets are correlated with each other and are consistent with their identification as the orbital, periastron precession, and twice the nodal precession frequencies. The observed relations, along with the known spin of the neutron star, put constraints on the mass and the ratio of the moment of inertia to the mass of the neutron star to be M⊙*=1.92±0.01 and I45/M⊙*=1.07±0.01 under the simplistic assumption that the metric is a Kerr one. We crudely estimate that the mass and moment of inertia values obtained may differ by about 1% and 5%, respectively, if a self-consistent metric is invoked. Using the Tolman–Oppenheimer–Volkoff equations for computing the moment of inertia of a neutron star in slow rotation approximation, having different equations of state, we find that the predicted values of neutron star parameters favor stiffer equations of state. We expect more stringent constraints would be obtained using a more detailed treatment, where the equation of state–dependent metric is used to compute the expected frequencies rather than the Kerr metric used here. The results provide insight into both the nature of these QPOs and the neutron star interior.

Accuracy of the slow-rotation approximation for black holes in modified gravity in light of astrophysical observables

Accuracy of the slow-rotation approximation for black holes in modified gravity in light of astrophysical observables

A Tight Universal Relation between the Shape Eccentricity and the Moment of Inertia for Rotating Neutron Stars

Universal relations that are insensitive to the equation of state are useful in reducing the parameter space when measuring global quantities of neutron stars (NSs). In this paper, we reveal a new universal relation that connects the eccentricity to the radius and moment of inertia of rotating NSs. We demonstrate that the universality of this relation holds for both conventional NSs and bare quark stars (QSs) in the slow-rotation approximation, albeit with different relations. The maximum relative deviation is approximately 1% for conventional NSs and 0.1% for QSs. Additionally, we show that the universality still exists for fast-rotating NSs if we use the dimensionless spin to characterize their rotation. The new universal relation will be a valuable tool to reduce the number of parameters used to describe the shape and multipoles of rotating NSs, and it may also be used to infer the eccentricity or moment of inertia of NSs in future X-ray observations.

Quasinormal modes of slowly rotating Kerr-Newman black holes using the double series method

We calculate the spectrum of quasinormal modes of slowly rotating Kerr-Newman black holes. Using a perturbative double expansion method, second order in rotation and first order in non-radial perturbations, we obtain the system of equations that describe polar-led and axial-led perturbations. We analyse gravitational, electromagnetic and scalar fundamental modes, focusing on the $\mathrm{l}=2$ perturbations. We reproduce previous results and check that isospectrality between axial and polar-led perturbations is approximately satisfied with good accuracy. Our results show that the slow rotation approximation can be used to estimate with reasonable precision the spectrum of configurations up to 50-60$\%$ of the extremal angular momentum.

Spinning (A)dS black holes with slow-rotation approximation in dynamical Chern-Simons modified gravity

One of the most crucial areas of gravity research, after the direct observation of gravitational waves, is the possible modification of General Relativity at ultraviolet and infrared scales. In particular, the possibility of parity violation should be considered in strong field regime. The Chern-Simons gravity takes into account parity violation in strong gravity regime. For all conformally flat spacetimes and spacetimes with a maximally symmetric two-dimensional subspace, Chern-Simons gravity is identical to General Relativity. Specifically, the Anti-de Sitter (A)dS-Kerr/Kerr black hole is not a solution for Chern-Simons gravity. The slow-rotating BH and the quadratic order in spin solutions are some of the known solutions to quadratic order in spin and they are rotating solutions in the frame of dynamical Chern-Simons gravity. In the present study, for the (A)dS slow-rotating situation (correct to the first order in spin), we derive the linear perturbation equations controlling the metric and the dynamical Chern-Simons field equation corrected to the linear order in spin and to the second order in the Chern-Simons coupling parameter. We show that the black hole of the (A)dS-Kerr solution is stronger (i.e. more compact and energetic) than the Kerr black hole solution and the reason for this feature comes form contributions at Planck scales. Moreover, we calculate the thermodynamical quantities related to this black hole. Finally, we calculate the geodesic equation and derive the effective potential of the black hole.

Is a black hole shadow a reliable test of the no-hair theorem?

Capturing the image of the shadow cast by the event horizon of an illuminated black hole is, at the most basic level, an experiment of extreme light deflection in a strongly curved spacetime. As such, the properties of an imaged shadow can be used to probe the general relativistic Kerr nature of astrophysical black holes. As an example of this prospect, it is commonly asserted that a shadow can test the validity of the theory's famous ``no hair theorem'' for the black hole's mass and spin multipole moments. In this paper, we assess this statement by calculating the shadow's equatorial radius in spacetimes with an arbitrary multipolar structure and within a slow rotation approximation. We find that when moments higher than the quadrupole are taken into account, the shadow acquires a high degree of degeneracy as a function of the deviation from the Kerr multipole moments. The results of our analysis suggest that dark objects with strongly non-Kerr multipolar structure could nevertheless produce a Kerr-like shadow with its characteristic quasicircular shape.

QNMs of slowly rotating Einstein–Bumblebee black hole

We have studied the quasinormal modes (QNMs) of a slowly rotating black hole with Lorentz-violating parameter in Einstein–Bumblebee gravity. We analyse the slow rotation approximation of the rotating black hole in the Einstein–Bumblebee gravity, and obtain the master equations for scalar perturbation, vector perturbation and axial gravitational perturbation, respectively. Using the matrix method and the continuous fraction method, we numerically calculate the QNM frequencies. In particular, for scalar field, it shows that the QNMs up to the second order of rotation parameter have higher accuracy. The numerical results show that, for both scalar and vector fields, the Lorentz-violating parameter has a significant effect on the imaginary part of the QNM frequencies, while having a relatively smaller impact on the real part of the QNM frequencies. But for axial gravitational perturbation, the effect of increasing the Lorentz-violating parameter ell is similar to that of increasing the rotation parameter {tilde{a}}.

Nonanalytic Relativistic r-Modes of Slowly Rotating Nonbarotropic Neutron Stars

We show that the r-modes of slowly rotating nonbarotropic neutron stars are described by nonanalytic functions of stellar angular velocity, which makes the perturbation techniques, used so far in the r-mode theoretical studies, inapplicable. In contrast to those studies and in accordance with numerical calculations beyond the slow rotation approximation, the obtained r-mode spectrum is discrete, which resolves the continuous spectrum problem, lasting since 1997. Our findings imply that the relativistic r-modes in slowly rotating neutron stars dramatically differ from their Newtonian cousins, which may have important implications for the detectability of r-mode signatures in observations, in particular for the r-mode excitation efficiency during the neutron star inspirals.

Radio Pulsars in an Electromagnetic Universe

The vacuum and plasma magnetosphere of neutron stars (NSs) in an electromagnetic universe (EU) were studied. First, we found vacuum solutions of the Maxwell equation for the electromagnetic field of rotating magnetized NSs in the slow-rotation approximation and analyzed the effects of the EU parameter on magnetic field components. It has been shown that in the presence of the EU field, the magnetic field lines near the star become denser and stronger compared with the case of GR. The Goldreich–Julian (GJ) charge density as a source of induced electric field was calculated. Our analyses showed that the GJ charge density increases with the increase in the EU parameter, while the size of the polar cap of NSs decreases. The solutions of Poisson equation for the scalar electric field have also been obtained at near and far zones. It has been shown that the parallel accelerating electric field increases in presence of the EU. We have also analyzed the effects of the EU on the death line for radio pulsars and shown that the position of the death line in the P−P˙ diagram shifts up. Finally, we investigated the energy losses of rotating NSs due to electromagnetic radiation. It was obtained that at a critical value of the EU parameter, which depends on the compactness parameter, the luminosity of electromagnetic radiations of magnetospheric radiation increases about 106 times.

Tidal effect on the gyroscopic precession around a compact star

General relativistic effects around massive astrophysical objects can be captured using a test gyro orbiting the object in a circular geodesic. This paper discusses how the tidal field due to a companion object affects the spin precession frequency and orbital angular velocity of a spinning gyro orbiting around a compact astrophysical object. The precession frequency is studied in a region of space around the central object using a perturbative approach. The central object is either a neutron star or a white dwarf in this study. The test gyro is any planetary or asteroid-like object orbiting a neutron star or a white dwarf. Moreover, the companion object that causes the tidal field can be a neutron star, white dwarf or a stellar black hole. It is seen that the tidal effect significantly affects the spacetime around the central object, which affects the gyro precession frequency and the orbital angular velocity. Slow rotation approximation has been considered for the central object, which creates negligible deformation. The change in the gyro’s precession frequency and the orbital angular velocity due to the tidal field increases with an increase in the companion object’s mass and decreases as the separation between the central star and the companion star increases. The tidal effect also varies with the stiffness of the equation of state of matter describing the host star. The lower the compactness of the host star, the greater is the tidal response; thus the greater is the change in the gyro’s precession and angular velocity of the geodesic.

Slowly rotating black holes and their scalarization

We study scalarization of slowly rotating black holes in the Einstein-scalar-Gauss-Bonnet (GB)-Chern-Simons (CS) theory. In the slow rotation approximation of $a\ll1$ with rotation parameter $a$, the GB term is given by a term for Schwarzschild black hole, whereas the CS term takes a linear term of $a$. The tachyonic instability for slowly rotating black holes represents the onset of spontaneous scalarization. We use the (2+1)-dimensional hyperboloidal foliation method to show the tachyonic instability for slowly rotating black holes by considering the time evolution of a spherically symmetric scalar mode. A threshold (existence) curve is obtained from the constant scalar modes under time evolution, which means the boundary between stable and unstable black holes. It is found that the slowly rotating black holes turn out to be unstable against a spherically symmetric scalar-mode propagation for positive coupling $\alpha$. However, we could not find tachyonic instability and any $a$-bound for scalarization for negative coupling $\alpha$.

Scalarization of slowly rotating black holes

It is interesting to note that most black holes are born very slowly rotating. We investigate scalarization of slowly rotating black holes in the Einstein-scalar-Chern–Simons (EsCS) theory. In the slow rotation approximation, the CS term takes a linear form of rotation parameter [Formula: see text] which determines the tachyonic instability. The tachyonic instability for slowly rotating black holes represents the onset of spontaneous scalarization. It is shown that the slowly rotating black holes are unstable against a spherically symmetric scalar-mode perturbation for positive coupling [Formula: see text], whereas these black holes are unstable for negative coupling without any [Formula: see text]-bound.

Petrov type, principal null directions, and Killing tensors of slowly rotating black holes in quadratic gravity

The ability to test general relativity in extreme gravity regimes using gravitational wave observations from current ground-based or future space-based detectors motivates the mathematical study of the symmetries of black holes in modified theories of gravity. In this paper we focus on spinning black hole solutions in two quadratic gravity theories: dynamical Chern-Simons and scalar Gauss-Bonnet gravity. We compute the principal null directions, Weyl scalars, and complex null tetrad in the small-coupling, slow rotation approximation for both theories, confirming that both spacetimes are Petrov type I. Additionally, we solve the Killing equation through rank 6 in dynamical Chern-Simons gravity and rank 2 in scalar Gauss-Bonnet gravity, showing that there is no nontrivial Killing tensor through those ranks for each theory. We therefore conjecture that the still-unknown, exact, quadratic-gravity, black-hole solutions do not possess a fourth constant of motion.

Slowly rotating neutron stars in scalar torsion theory

In this present paper, a slowly rotating stat is investigated in shift symmetric scalar torsion theory framework using a nondiagonal tetrad that gives an axially symmetric spacetime. We present the general equations for a general Lagrangian in a spherical symmetric space time and then in an axially symmetric spacetime. The obtained equations will allow us to study the behaviour of a specific model at the center of the star and at large distance. We find that this particular model affects the behaviour at the center but it is not case for large value of the radial coordinate r. The integration of the equations of motion, for different realistic equations of state (EoS), confirms that the mass, the radius as well as the moment of inertia are effected by varying the parameters of the model. Finally, we examine the universal relation of normalized moment of inertia and the stellar compactness of neutron star in slow rotation approximation. We showed that for all values of parameters present in the model leads to a deviation from GR for all EoS with a relative deviation below 10%.

Painlevé–Gullstrand form of the Lense–Thirring Spacetime

The standard Lense–Thirring metric is a century-old slow-rotation large-distance approximation to the gravitational field outside a rotating massive body, depending only on the total mass and angular momentum of the source. Although it is not an exact solution to the vacuum Einstein equations, asymptotically the Lense–Thirring metric approaches the Kerr metric at large distances. Herein we shall discuss a specific variant of the standard Lense–Thirring metric, carefully chosen for simplicity, clarity, and various forms of improved mathematical and physical behaviour, (to be more carefully defined in the body of the article). We shall see that this Lense–Thirring variant can be viewed as arising from the linearization of a suitably chosen tetrad representing the Kerr spacetime. In particular, we shall construct an explicit unit-lapse Painlevé–Gullstrand variant of the Lense–Thirring spacetime, one that has flat spatial slices, a very simple and physically intuitive tetrad, and extremely simple curvature tensors. We shall verify that this variant of the Lense–Thirring spacetime is Petrov type I, (so it is not algebraically special), but nevertheless possesses some very straightforward timelike geodesics, (the “rain” geodesics). We shall also discuss on-axis and equatorial geodesics, ISCOs (innermost stable circular orbits) and circular photon orbits. Finally, we wrap up by discussing some astrophysically relevant estimates, and analyze what happens if we extrapolate down to small values of r; verifying that for sufficiently slow rotation we explicitly recover slowly rotating Schwarzschild geometry. This Lense–Thirring variant can be viewed, in its own right, as a “black hole mimic”, of direct interest to the observational astronomy community.

Disformal transformation of physical quantities associated with relativistic stars

We investigate how physical quantities associated with relativistic stars in the Jordan and Einstein frames are related by the generalized disformal transformations constructed by the scalar and vector fields within the slow-rotation approximation. We consider the most general scalar disformal transformation constructed by the scalar field, and by the vector field without and with the $U(1)$ gauge symmetry, respectively. At the zeroth order of the slow-rotation approximation, by imposing that both the metrics of the Jordan and Einstein frames are asymptotically flat, we show that the Arnowitt-Deser-Misner mass is frame invariant. At the first order of the slow-rotation approximation, we discuss the disformal transformations of the frame-dragging function, angular velocity, angular momentum, and moment of inertia of the star. We show that the angular velocity of the star is frame invariant in all the cases. While the angular momentum and moment of inertia are invariant under the scalar disformal transformation, they are not under the vector disformal transformation without and with the $U(1)$ gauge symmetry.

Trapping of null geodesics in slowly rotating spacetimes

Extremely compact objects containing a region of trapped null geodesics could be of astrophysical relevance due to trapping of neutrinos with consequent impact on cooling processes or trapping of gravitational waves. These objects have previously been studied under the assumption of spherical symmetry. In the present paper, we consider a simple generalization by studying trapping of null geodesics in the framework of the Hartle–Thorne slow-rotation approximation taken to first order in the angular velocity, and considering a uniform-density object with uniform emissivity for the null geodesics. We calculate effective potentials and escape cones for the null geodesics and how they depend on the parameters of the spacetimes, and also calculate the “local” and “global” coefficients of efficiency for the trapping. We demonstrate that due to the rotation the trapping efficiency is different for co-rotating and retrograde null geodesics, and that trapping can occur even for R>3GM/c^2, contrary to what happens in the absence of rotation.

Probing Einstein-dilaton Gauss-Bonnet gravity with the inspiral and ringdown of gravitational waves

Gravitational waves from extreme gravity events such as the coalescence of two black holes in a binary system fill our observable universe, bearing with them the underlying theory of gravity driving their process. One compelling alternative theory of gravity -- known as Einstein-dilaton Gauss-Bonnet gravity motivated by string theory -- describes the presence of an additional dilaton scalar field coupled directly to higher orders of the curvature, effectively describing a "fifth force" interaction and the emission of scalar dipole radiation between two scalarized black holes. Most previous studies focused on considering only the leading correction to the inspiral portion of the binary black hole waveforms. In our recent paper, we carried out inspiral-merger-ringdown consistency tests in this string-inspired gravity by including corrections to both the inspiral and ringdown portions, as well as those to the mass and spin of remnant black holes, valid to quadratic order in spin. We here extend the analysis by directly computing bounds on the theoretical coupling constant using the full inspiral-merger-ringdown waveform rather than treating the inspiral and merger-ringdown portions separately. We also consider the corrections valid to quartic order in spin to justify the validity of black hole's slow-rotation approximation. We find the quasinormal mode corrections to the waveform to be particularly important for high-mass events such as GW170729, in which the dilaton fields' small-coupling approximation fails without such effects included. We also show that future space-based and multiband gravitational-wave observations have the potential to go beyond existing bounds on the theory. The bounds presented here are comparable to those found in via the inspiral-merger-ringdown consistency tests.

Gravitational waves from non-radial perturbations in neutron stars

In the present work, we explore the Rossby mode perturbations in case of neutron stars as sources of continuous gravitational waves. The intensity and time evolution of the emitted gravitational waves in terms of the amplitude of the strain tensor are estimated in the slow rotation approximation using β-equilibrated neutron star matter obtained from density dependent M3Y effective interaction. For a wide range of neutron star masses, the fiducial gravitational and various viscous time scales, the critical frequencies and the time evolutions of the frequencies are calculated. Unlike other non-radial perturbations, amplitude of Rossby mode perturbation increases due to the emission of gravitational waves which carry angular momentum of the star to infinity. Hence, existence of such perturbation in a star implies a positive rate of change in the angular momentum transfer. It, therefore, implies that for a particular neutron star, the intensity of gravitational wave emission increases with time until the angular frequency reduces down to a critical value below which the star stops emitting radiation.

Particle acceleration and electromagnetic field of deformed neutron stars

Neutron stars (NS)s are astrophysical objects with strong gravitational and electromagnetic fields. Since there are several effects on radiation processes around the star, it is impossible to consider whole effects all together. One way to study the processes is by considering them one by one as a toy model. In this paper, we have investigated the effects of spacetime deformation on the surface magnetic field of the slowly rotating neutron star and its plasma magnetospheric processes, such as the plasma magnetosphere formation around the star. At first, the approximate vacuum solutions of the Maxwell equations for the electromagnetic fields of a magnetized neutron star in a slowly rotating deformed spacetime metric have been obtained. It has been shown that the positive deformation parameter leads to an increase in the value of the (surface) magnetic field at the near zone of the neutron star, while the effect of the negative deformation parameter is vice versa. We have also considered the electric field of the slowly rotating neutron star in the spacetime. In the slow rotation approximation, we have studied the particle acceleration in the polar cap zone, considering the effect of deformation of spacetime on the [Formula: see text]-Lorentz factor of a relativistic charged particle. It is shown that in the case of the positive deformation, an additional gravity occurs around the NS. The effects of spacetime deformation on magneto-dipolar radiation of radio pulsars and polar cap size have also been studied and shown that negative deformation of spacetime increased the radiation luminosity and as positive deformation increases, the luminosity decreases. Size of polar cap region of a neutron star, where magnetic field lines open, increases with increasing the value of the deformation parameter [Formula: see text]. Moreover, we have studied the influence of the spacetime deformation on the death line for radio pulsar, which separates the region in [Formula: see text] [Formula: see text] diagram, where the pulsar can or cannot radiate in radio band (create pair production) through inverse compton scattering (ICS). It is shown that the negative (positive) deformation shifts upward (downward) the death line, which means that even a small negative (positive) deformation of spacetime may cause to be radio-quite (be radio load) the radio pulsar which is lying on the death line (in [Formula: see text] diagram) in the GR frame with its corresponding parameters.