In this paper, we develop a new class of models for a compact star with anisotropic stresses inside the matter distribution. By assuming a linear equation of state for the anisotropic matter composition of the star we solve the Einstein field equations. In our approach, for the interior solutions we use a particular form of the ansatz for the metric function $g_{rr}$. The exterior solution is assumed as Schwarzschild metric and is joined with the interior metric obtained across the boundary of the star. These matching of the metrices along with the condition of the vanishing radial pressure at the boundary lead us to determine the model parameters. The physical acceptability of the solutions has verified by making use of the current estimated data available from the pulsar 4U1608-52. Thereafter, assuming anisotropy due to tidal effects we calculate the Love numbers from our model and compare the results with the observed compact stars, viz. KS 1731- 260,4U 1608- 52,4U 1724- 207,4U 1820- 30,SAX J1748.9-2021 and EXO 1745-268. The overall situation confirms physical viability of the proposed approach,which can shed new light on the interior of the compact relativistic objects.
Tidal deformability of a coalescing neutron star subjected to an external tidal field plays an important role in our probe for the structure and properties of compact stars. In particular, the tidal love number provides valuable information about the external gravitational field responsible for deforming the star. In this article, we compute the tidal love number of a particular class of anisotropic stars and analyze the impacts of anisotropy and compactness on the tidal love number.
In Newtonian gravitational theory, a tidal Love number relates the mass multipole moment created by tidal forces on a spherical body to the applied tidal field. The Love number is dimensionless, and it encodes information about the body's internal structure. We present a relativistic theory of Love numbers, which applies to compact bodies with strong internal gravities; the theory extends and completes a recent work by Flanagan and Hinderer, which revealed that the tidal Love number of a neutron star can be measured by Earth-based gravitational-wave detectors. We consider a spherical body defo…
The metric outside a compact body deformed by a quadrupolar tidal field is universal up to its Love numbers, constants which encode the tidal response's dependence on the body's internal structure. For a non-rotating body, the deformed external geometry is characterized by the familiar gravitational Love numbers $K_2^{\text{el}}$ and $K_2^{\text{mag}}$. For a slowly rotating body, these must be supplemented by rotational-tidal Love numbers, which measure the response to couplings between the body's spin and the external tidal field. By integrating the interior field equations, I find that the …
We examine the tidal deformation of a nonrotating compact body (material body or black hole) in general relativity. The body's exterior metric is calculated in a simultaneous expansion in powers of the ratio between the distance to the body and three distinct length scales: the radius of curvature of the external spacetime in which the body is inserted, the scale of spatial inhomogeneity of the curvature, and the scale of temporal variation. The metric is valid in the body's immediate neighborhood, which excludes the external matter responsible for the tidal environment. The body's tidal respo…
One of the macroscopically measurable effects of gravity is the tidal deformability of astrophysical objects, which can be quantified by their tidal Love numbers. For planets and stars, these numbers measure the resistance of their material against the tidal forces, and the resulting contribution to their gravitational multipole moments. According to general relativity, nonrotating deformed black holes, instead, show no addition to their gravitational multipole moments, and all of their Love numbers are zero. In this paper we explore different configurations of nonrotating compact and ultracom…
In this present work, we suggest studying neutron star with a conformal coupling by using the analytical expressions of realistic equations of state. The equations of perturbed and nonperturbed metrics, scalar field and the pressure of the matter are derived in order to observe the effects of the conformal coupling on the mass, radius and tidal love number of polar and axial types. The numerical and the analytical analysis show a different comportment of our model from GR by varying the value of the parameters in the model. We showed also for a particular case that the scalar field becomes van…
Axial tidal Love numbers of black holes in matter environments
Institute of Space Sciences (ICE, CSIC)（空间科学研究所）；Gran Sasso Science Institute (GSSI)（格兰萨索科学研究所）；INFN, Laboratori Nazionali del Gran Sasso（意大利国家核物理研究所，格兰萨索国家实验室）
The gravitomagnetic tidal Love number of a slowly rotating body was calculated previously under the assumption that the velocity perturbation created by the tidal field consists of an induction piece proportional to the vector potential, and a rotational piece that scales with $Ω$, the body's angular velocity. The second part of this assumption is wrong: the rotational piece of the velocity perturbation scales in fact like $Ω^0 = 1$. The previous calculations are therefore incorrect, and the purpose of this paper is to repair the mistake. To keep the technical difficulties to a minimum, the tr…
The deformation of a nonrotating body resulting from the application of a tidal field is measured by two sets of Love numbers associated with the gravitoelectric and gravitomagnetic pieces of the tidal field, respectively. The gravitomagnetic Love numbers were previously computed for fluid bodies, under the assumption that the fluid is in a strict hydrostatic equilibrium that requires the complete absence of internal motions. A more realistic configuration, however, is an irrotational state that establishes, in the course of time, internal motions driven by the gravitomagnetic interaction. We …
In this note, we analyze black holes solutions under R3 corrections, which is the leading correction induced by quantum corrections in four-dimensions. We showed that perturbations around this black hole background will lead to non-zero tidal Love number (TLN). This further accentuates the \unnaturalness" of the vanishing TLN for Schwarzschild black hole under Einstein-Hilbert action.
A sum rule for tidal Love number is derived from quantum field theory computations, which relates tidal susceptibility of a spinless body to transition rates of its single graviton emission processes. An analogous sum rule for electromagnetism is given as an example, which is substantiated by comparing to the solved problem of the hydrogen atom. Based on the semiclassical Hawking radiation spectrum, a finite nonvanishing value for quantum corrections to the Love number of Schwarzschild black holes in general relativity is computed using the sum rule, which is known to classically vanish.
Observations of gravitational waves from inspiralling neutron star binaries---such as GW170817---can be used to constrain the nuclear equation of state by placing bounds on stellar tidal deformability. For slowly rotating neutron stars, the response to a weak quadrupolar tidal field is characterized by four internal-structure-dependent constants called "Love numbers." The tidal Love numbers $k_2^\text{el}$ and $k_2^\text{mag}$ measure the tides raised by the gravitoelectric and gravitomagnetic components of the applied field, and the rotational-tidal Love numbers $\mathfrak{f}^\text{o}$ and $\…
Gravitational-wave measurements of the tidal deformability of neutron stars could reveal important information regarding their internal structure, the equation of state of high-dense nuclear matter and gravity in strong field regime. In this work, we extend the relativistic theory of the tidal deformability of neutron stars to Rastall gravity. Both the electric-type and magnetic-type quadrupole tidal Love numbers are calculated for neutron stars in the polytrope model. It is found that neutron star's tidal Love numbers in Rastall gravity is significantly smaller than those in general relativit…
An important physical phenomenon that manifests itself during the inspiral of two orbiting compact objects is the tidal deformation of each under the gravitational influence of its companion. In the case of binary neutron star mergers, this tidal deformation and the associated Love numbers have been used to probe properties of dense matter and the nuclear equation of state. Non-spinning black holes on the other hand have a vanishing (field) tidal Love number in General Relativity. This pertains to the deformation of the asymptotic gravitational field. In certain cases, especially in the late s…
We calculate the tidal Love numbers of black holes and neutron stars in the presence of higher dimensions. The perturbation equations around an arbitrary static and spherically symmetric metric for the even parity modes are presented in the context of an effective four-dimensional theory on the brane. This subsequently leads to the sought expression for the tidal Love number for black holes in the presence of extra spatial dimensions. Surprisingly, these numbers are non-zero and, more importantly, negative. We extend our method to determine the tidal Love number of neutron stars in a spacetime…
The coupling between the angular momentum of a compact object and an external tidal field gives rise to the "rotational" tidal Love numbers, which affect the tidal deformability of a spinning self-gravitating body and enter the gravitational waveform of a binary inspiral at high post-Newtonian order. We provide numerical evidence for a surprising "hidden" symmetry among the rotational tidal Love numbers with opposite parities, which are associated to perturbations belonging to separate sectors. This symmetry, whose existence had been suggested on the basis of a Lagrangian description of the ti…
Dynamical tidal Love numbers of black holes under generic perturbations: Connecting black hole perturbation theory with effective field theory
School of Physical Sciences, Indian Association for the Cultivation of Science（印度科学文化协会物理科学学院）；Max-Planck-Institute for Gravitational Physics (Albert-Einstein-Institute)（引力物理学马普研究所（爱因斯坦研究所））；International Centre for Theoretical Sciences, Tata Institute of Fundamental Research（塔塔基础国际理论科学中心）；Institute for Theoretical Physics, Utrecht University（乌得勒支大学理论物理研究所）
Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical Physics of Gansu Province, School of Physical Science and Technology, Lanzhou University（兰州大学物理科学与技术学院理论物理甘肃省重点实验室兰州理论物理中心）
Hořava-Lifshitz gravity is an alternative theory to general relativity which breaks Lorentz invariance in order to achieve an ultraviolet complete and power-counting renormalizable theory of gravity. In the low energy limit, Hořava-Lifshitz gravity coincides with a vector-tensor theory known as khronometric gravity. The deviation of khronometric gravity from general relativity can be parametrized by three coupling constants: $α$, $β$, and $λ$. Solar system experiments and gravitational wave observations impose stringent bounds on $α$ and $β$, while $λ$ is still relatively unconstrained ($λ\les…
The moment of inertia, the spin-induced quadrupole moment, and the tidal Love number of neutron-star and quark-star models are related through some relations which depend only mildly on the stellar equation of state. These "I-Love-Q" relations have important implications for astrophysics and gravitational-wave astronomy. An interesting problem is whether similar relations hold for other compact objects and how they approach the black-hole limit. To answer these questions, here we investigate the deformation properties of a large class of thin-shell gravastars, which are exotic compact objects …
In this brief note, we clarify certain aspects related to the magnetic (i.e., odd parity or axial) tidal Love numbers of a star in general relativity. Magnetic tidal deformations of a compact star had been computed in 2009 independently by Damour and Nagar and by Binnington and Poisson. More recently, Landry and Poisson showed that the magnetic tidal Love numbers depend on the assumptions made on the fluid, in particular they are different (and of opposite sign) if the fluid is assumed to be in static equilibrium or if it is irrotational. We show that the zero-frequency limit of the Regge-Whee…
We construct the external metric of a slowly rotating, tidally deformed material body in general relativity. The tidal forces acting on the body are assumed to be weak and to vary slowly with time, and the metric is obtained as a perturbation of a background metric that describes the external geometry of an isolated, slowly rotating body. The tidal environment is generic and characterized by two symmetric-tracefree tidal moments E_{ab} and B_{ab}, and the body is characterized by its mass M, its radius R, and a dimensionless angular-momentum vector χ^a << 1. The perturbation accounts for all c…
The open question of whether a black hole can become tidally deformed by an external gravitational field has profound implications for fundamental physics, astrophysics and gravitational-wave astronomy. Love tensors characterize the tidal deformability of compact objects such as astrophysical (Kerr) black holes under an external static tidal field. We prove that all Love tensors vanish identically for a Kerr black hole in the nonspinning limit or for an axisymmetric tidal perturbation. In contrast to this result, we show that Love tensors are generically nonzero for a spinning black hole. Spec…
Proca stars (arXiv:1508.05395) are everywhere regular, asymptotically flat self-gravitating solitons composed of complex, massive vector bosons, forming a macroscopic, star-sized, Bose-Einstein condensate. They have been suggested as a classical, dynamical model of exotic compact objects, black hole mimickers and dark matter candidates. In spherical symmetry, they are qualitatively similar to their scalar cousins, the standard (scalar) boson stars in all respects studied so far. Here we study the tidal deformability and the quadrupolar tidal Love numbers, both electric-type and magnetic-type, …
The tidal deformability of a self-gravitating object leaves an imprint on the gravitational-wave signal of an inspiral which is paramount to measure the internal structure of the binary components. We unveil here a surprisingly unnoticed effect: in the extreme-mass ratio limit the tidal Love number of the central object (i.e. the quadrupole moment induced by the tidal field of its companion) affects the gravitational waveform at the leading order in the mass ratio. This effect acts as a magnifying glass for the tidal deformability of supermassive objects but was so far neglected, probably beca…
By extending our recent framework to describe the tidal deformations of a spinning compact object, we compute for the first time the tidal Love numbers of a spinning neutron star to linear order in the angular momentum. The spin of the object introduces couplings between electric and magnetic distortions and new classes of spin-induced ("rotational") tidal Love numbers emerge. We focus on stationary tidal fields, which induce axisymmetric perturbations. We present the perturbation equations for both electric-led and magnetic-led rotational Love numbers for generic multipoles and explicitly sol…
For a variety of fully relativistic polytropic neutron star models we calculate the star's tidal Love number k2. Most realistic equations of state for neutron stars can be approximated as a polytrope with an effective index n~0.5-1.0. The equilibrium stellar model is obtained by numerical integration of the Tolman-Oppenheimer-Volkhov equations. We calculate the linear l=2 static perturbations to the Schwarzschild spacetime following the method of Thorne and Campolattaro. Combining the perturbed Einstein equations into a single second order differential equation for the perturbation to the metr…
Matching Tidal Deformability (Wilson) Coefficients to Black Hole Love Numbers in Higher-Curvature Gravity
Perimeter Institute for Theoretical Physics（Perimeter理论物理研究所）；University of Waterloo（滑铁卢大学）；UNESP - Universidade Estadual Paulista（圣保罗州立大学）；King’s College London（伦敦国王学院）；ICTP South American Institute for Fundamental Research（国际理论物理研究所南美基础研究所）
We comment on the paper of S. Postnikov et al. in Phys. Rev. D 82, 024016 (2010) and give a modified formula that needs to be taken into account when calculating the tidal Love number of neutron stars in case a first order phase-transition occurs at non-zero pressure. We show that the error made when using the original formula tends to zero as $p \rightarrow 0$ and we estimate the maximum relative error to be $\sim 5\%$ if the density discontinuity is at larger densities.
We investigate the properties of relativistic stars made of dark energy. We model stellar structure assuming i) isotropic perfect fluid and ii) a dark energy inspired equation of state, the generalized equation of state of Chaplygin gas, as we will be calling it. The mass-to-radius profiles, the tidal Love numbers as well as the ten lowest radial oscillation modes are computed. Causality, stability and energy conditions are also discussed.
We study the tidal deformations of various known black hole and wormhole solutions in a simple context of warped compactification -- Randall-Sundrum theory in which the four-dimensional spacetime geometry is that of a brane embedded in five-dimensional Anti-de Sitter spacetime. The linearized gravitational perturbation theory generically reduces to either an inhomogeneous second-order ODE or a homogeneous third-order ODE of which indicial roots associated with an expansion about asymptotic infinity can be related to Tidal Love Numbers. We describe various tidal-deformed metrics, classify their…
We extract the black hole (BH) static tidal deformability coefficients (Love numbers) and their spin-0 and spin-1 analogs by comparing on-shell amplitudes for fields to scatter off a spinning BH in the worldline effective field theory (EFT) and in general relativity (GR). We point out that the GR amplitudes due to tidal effects originate entirely from the BH potential region. Thus, they can be separated from gravitational non-linearities in the wave region, whose proper treatment requires higher order EFT loop calculations. In particular, the elastic scattering in the near field approximation …
Moment of inertia ($I$), rotational~(tidal) Love number ($λ^{\rm (rot)}$) and quadrupole moment ($Q$) of slowly rotating massive neutron star~(NS) with holographic multiquark~(MQ) core are computed in comparison to pure MQ star. The Chiral Effective Theory~(CET) stiff equation of state~(EoS) is used in the crust of the neutron star. The dimensionless multipole moments $\bar{I},\barλ^{\rm (rot)}, \bar{Q}$ are found to be independent of the rotation parameters and determined completely by the zeroth-order star profile. Universal ``I-Love-Q'' relations found by Yagi and Yunes [1,2] are mostly pre…
Gravitational waves from the final stages of inspiralling binary neutron stars are expected to be one of the most important sources for ground-based gravitational wave detectors. The masses of the components are determinable from the orbital and chirp frequencies during the early part of the evolution during which tidal effects provide small correction; however, during this phase the signal is relatively clean. The accumulated phase shift due to tidal corrections is characterized by a single quantity, the Love number, which is sensitive to the compactness parameter M/R and the star's internal …
