Exterior Gravitational Field Research Articles

The complete exterior spacetime of spherical Brans-Dicke stars

We derive the complete expression for the Brans Class I exterior spacetime explicitly in terms of the energy and pressures profiles of a stationary spherisymmetric gravity source. This novel and generic expression is achieved in a parsimonious manner, requiring only a subset of the Brans-Dicke field equation and the scalar equation. For distant orbiting test particles, this expression promptly provides a simple, closed and exact formula of the γ Eddington parameter, which reads γexact=ω+1+(ω+2)Θω+2+(ω+1)Θ, where Θ is the ratio of the star's “total pressure” integral over its energy integral. This non-perturbative result reproduces the usual Post-Newtonian ω+1ω+2 expression in the case of a “Newtonian star”, in which the pressure is negligible with respect to the energy density. Furthermore, it converges to the General Relativity value (γGR=1) as the star's equation of state approaches that of ultra-relativistic matter (in which case Θ approaches 1), a behavior consistent with broader studies on scalar-tensor gravity. Our derivation underscores the essence of these results involving (1) the key relevant portion of the Brans-Dicke field equations, (2) the uniqueness of the Brans Class I vacuum solution for the non-phantom action, viz. ω>−3/2, and (3) the involvement of only two free parameters in this solution, hence requiring two quantities (energy and pressure integrals) of the mass source to fully characterize the solution. From a practical standpoint, it elucidates how a given stellar interior structure model determines the star's exterior gravitational field and impacts the motions of light objects (such as planets and accretion disks) orbiting it.

Gravitational fields of axially symmetric compact objects in 5D space–time–matter gravity

In the standard Einstein’s theory the exterior gravitational field of any static and axially symmetric stellar object can be described by means of a single function from which we obtain a metric into a four-dimensional space–time. In this work we present a generalization of those so called Weyl solutions to a space–time–matter metric in a five-dimensional manifold within a non-compactified Kaluza–Klein theory of gravity. The arising field equations reduce to those of vacuum Einstein’s gravity when the metric function associated to the fifth dimension is considered to be constant. The calculation of the geodesics allows to identify the existence or not of different behaviours of test particles, in orbits on a constant plane, between the two metrics. In addition, static solutions on the hypersurface orthogonal to the added dimension but with time dependence in the five-dimensional metric are also obtained. The consequences on the variation of the rest mass, if the fifth dimension is identified with it, are studied.

Observable Properties of Thin Accretion Disk in the γ Spacetime

We study matter accretion in a static, axially symmetric and vacuum geometry describing the exterior gravitational field of a black hole mimicker called the γ metric. We evaluate the thermal and optical properties of thin accretion disks, including the emission rate, luminosity and shadow, in the gamma spacetime. Also, we explore the radial accretion of polytropic matter fields onto the central source and evaluate the thermal and optical properties of the infalling gas, such as temperature and luminosity. The results are discussed in the context of evaluating the possibility that the true nature of astrophysical black hole candidates may not be a black hole but some exotic compact object possessing a non-vanishing mass quadrupole moment.

A shadows from the static black hole mimickers

In this work, we study shadows from the naked singularity spacetime. The most analytical solutions of black hole shadows only investigated the case that the geodesic equations for photons can separate variables. We review the spherical null naked singularity metric and this spherically symmetric naked singularity spacetime metric is the solution of Einstein equations with an anisotropic fluid source which has no photon sphere. We also review a static, axially-symmetric singular solution of the vacuum Einstein’s equations without an event horizon which is can be used to describe the exterior gravitational field of a mass distribution with quadrupole moment. Moreover, the corresponding spacetime is characterized by the presence of naked singularities. It is theoretically known that not only a black hole can cast shadow, but other compact objects such as naked singularities, gravastar or boson stars can also cast shadows. We present the analytical calculation of shadows for both naked singularities spacetime and compare with the shadow of Schwarzschild static black hole, we show that this can serve as a black hole mimicker. Keywords: compact object, naked singularity, shadow.

Charged dust in higher curvature geometry

We analyze the configuration of charged dust in the context of the higher dimensional and higher curvature Einstein–Gauss–Bonnet–Maxwell theory. With the prescription of dust, there remains one more prescription to be made in order to close the system of equations of motion. The choice of one of the metric potentials appears to be the only viable way to proceed. Before establishing exact solutions, we examine conditions for the existence of physically reasonable charged dust fluids. It turns out that the branches of the Boulware–Deser metric representing the exterior gravitational field of a neutral spherically symmetric Einstein–Gauss–Bonnet distribution, serve as upper and lower bounds for the spatial potentials of physically reasonable charged dust in Einstein–Gauss–Bonnet–Maxwell gravity. Some exact solutions for 5 and 6 dimensional charged dust hyperspheres are exhibited in closed form. In particular the Einstein ansatz of a constant temporal potential while defective in 5 dimensions actually generates a model of a closed compact astrophysical object in 6 dimensions. A physically viable 5 dimensional charged dust model is also contrasted with its general relativity counterpart graphically.

Gravitomagnetism in the Lewis cylindrical metrics

The Lewis solutions describe the exterior gravitational field produced by infinitely long rotating cylinders, and are useful models for global gravitational effects. When the metric parameters are real (Weyl class), the exterior metrics of rotating and static cylinders are locally indistinguishable, but known to globally differ. The significance of this difference, both in terms of physical effects (gravitomagnetism) and of the mathematical invariants that detect the rotation, remain open problems in the literature. In this work we show that, by a rigid coordinate rotation, the Weyl class metric can be put into a ‘canonical’ form where the Killing vector field ∂ t is time-like everywhere, and which depends explicitly only on three parameters with a clear physical significance: the Komar mass and angular momentum per unit length, plus the angle deficit. This new form of the metric reveals that the two settings differ only at the level of the gravitomagnetic vector potential which, for a rotating cylinder, cannot be eliminated by any global coordinate transformation. It manifests itself in the Sagnac and gravitomagnetic clock effects. The situation is seen to mirror the electromagnetic field of a rotating charged cylinder, which likewise differs from the static case only in the vector potential, responsible for the Aharonov–Bohm effect, formally analogous to the Sagnac effect. The geometrical distinction between the two solutions is also discussed, and the notions of local and global staticity revisited. The matching in canonical form to the van Stockum interior cylinder is also addressed.

On testing CDM and geometry-driven Milky Way rotation curve models with Gaia DR2

ABSTRACT Flat rotation curves (RCs) in disc galaxies provide the main observational support to the hypothesis of surrounding dark matter (DM). Despite of the difficulty in identifying the DM contribution to the total mass density in our Galaxy, stellar kinematics, as tracer of gravitational potential, is the most reliable observable for gauging different matter components. From the Gaia second data release catalogue, we extracted parallaxes, proper motions, and line-of-sight velocities of unprecedented accuracy for a carefully selected sample of disc stars. This is the angular momentum supported population of the Milky Way (MW) that better traces its observed RC. We fitted such data to both a classical, i.e. including a DM halo, velocity profile model, and a general relativistic one derived from a stationary axisymmetric galaxy-scale metric. The general relativistic MW RC results statistically indistinguishable from its state-of-the-art DM analogue. This supports the ansatz that a weak gravitational contribution due to the off-diagonal term of the metric, by explaining the observed flatness of MW’s RC, could fill the gap in a baryons-only MW, thus rendering the Newtonian-origin DM a general relativity-like effect. In the context of Local Cosmology, our findings are suggestive of the Galaxy’s phase space as the exterior gravitational field in equilibrium far from a Kerr-like inner source, possibly with no need for extra matter to account for the disc kinematics.

Connecting the exterior gravitational field with the energy\u2013momentum tensor of axially symmetric compact objects

A method to construct interior axially symmetric metrics that appropriately match with any vacuum solution of the Weyl family is developed in Hernández-Pastora et al. (Class Quantum Gravity 33:235005, 2016). It was shown, for the case of some vacuum solutions, that the simplest solution for the interior metric leads to sources with well-behaved energy conditions. Now, we integrate the field equations to obtain the interior metric functions in terms of the anisotropies and pressures of the source. As well, the compatible equations of state for these global models are calculated. The interior metric and the suitable energy–momentum tensor describing the source are constructed in terms of the exterior metric functions. At the boundary of the compact object, the behaviour of a pressure T_m, defined from the energy–momentum tensor, is shown to be related with the exterior gravitational field. This fact allows us to explore the differences arising at the matter distribution when the spherical symmetry of the global metric is dropped. Finally, an equation derived from the matching conditions is obtained which allows us to calculate the Weyl coefficients of the exterior metric as source integrals. Hence the Relativistic Multipole Moments of the global model can be expressed in terms of the matter distribution of the source.

Use of Tetrahedral Finite Element Method for Computing the Gravitation of Irregular-Shaped Asteroid

Asteroids are one of the most important targets for deep space exploration. In previous asteroid missions, the accurate estimate of gravity has proved to have a strong influence on the design of the approach orbit and navigation strategy. The wield gravitational field of an asteroid is mainly caused by the irregular overall shape and possible heterogenous mass distribution the interior. We propose to use the finite element method to compute the gravity of irregularly shaped asteroids; this method combines the advantages of the conventional mascon method and the polyhedral method. The tetrahedral meshes can be generated following the conventional division technique. Taking asteroid 216 Kleopatra as an example, we calculate the exterior gravitational field using the above mentioned methodology. We then compare the results from the finite element method and the polyhedral method under a degenerated case, i.e., with constant density. Then, four different density distribution assumptions are given, and the gravitational fields are calculated respectively. The comparative study and the density distribution assumptions indicate that the proposed method is suitable for modeling an arbitrary asteroid with nonuniform mass distribution. This method is expected to provide reliable gravity data for the design of guidance, navigation, and control systems in future asteroid missions.

Quasinormal modes of a black hole with quadrupole moment

We analytically determine the quasinormal mode (QNM) frequencies of a black hole with quadrupole moment in the eikonal limit using the light-ring method. The generalized black holes that are discussed in this work possess arbitrary quadrupole and higher mass moments in addition to mass and angular momentum. Static collapsed configurations with mass and quadrupole moment are treated in detail and the QNM frequencies associated with two such configurations are evaluated to linear order in the quadrupole moment. Furthermore, we touch upon the treatment of rotating systems. In particular, the generalized black hole that we consider for our extensive QNM calculations is a completely collapsed configuration whose exterior gravitational field can be described by the Hartle-Thorne spacetime [Astrophys. J. 153, 807-834 (1968)]. This collapsed system as well as its QNMs is characterized by mass $M$, quadrupole moment $Q$ and angular momentum $J$, where the latter two parameters are treated to first and second orders of approximation, respectively. When the quadrupole moment is set equal to the relativistic quadrupole moment of the corresponding Kerr black hole, $J^2/(Mc^2)$, the Hartle-Thorne QNMs reduce to those of the Kerr black hole to second order in angular momentum $J$. Using ringdown frequencies, one cannot observationally distinguish a generalized Hartle-Thorne black hole with arbitrary quadrupole moment from a Kerr black hole provided the dimensionless parameter given by $|QMc^2-J^2|c^2/(G^2M^4)$ is sufficiently small compared to unity.

Стационарное вакуумное решение уравнений Эйнштейна

We investigate a stationary generalization of the static metric. The static -metric is a variant of the Zipoy-Voorhees metric and simplest generalization of the Schwarzschild metric, containing a quadrupole parameter. In the present work, we introduce the stationary version of the -metric, and this stationary metric find by using the complex Ernst potential . The metric function determined by two first-order differential equations that can be integrated by quadratures once  is known. To obtain an explicit form of the new Ernst potential, we use the solution generation techniques that allows us to generate stationary solutions from a static solution. It possesses three independent parameters related to the mass, quadrupole moment and angular momentum. We investigate the geometric and physical properties of this exact stationary solution of Einstein’s vacuum equations and show that it can be used to describe the exterior gravitational field of rotating, axially symmetric, compact objects. According to the relativistic invariant Geroch definition, we analyze multipole structure using the corresponding Ernst function and we compute the lowest ten relativistic multipole moments for the static quadrupole metric. The particular choice of parameters we obtain the known solutions. i.e., the exterior Schwarzschild Solution find with the vanishing quadrupole and rotating parameter, Corresponding static -metric find with the vanishing rotating parameter and non-zero quadrupole parameter. The multipole moments of the well-known Kerr solution are given by the vanishing quadrupole parameter and non-zero rotating parameter.

Higher-dimensional black holes with Dirac–Born–Infeld (DBI) global defects

It is well-known that the exact solution of non-linear $\sigma$ model coupled to gravity can be perceived as an exterior gravitational field of a global monopole. Here we study Einstein's equations coupled to a non-linear $\sigma$ model with Dirac-Born-Infeld (DBI) kinetic term in $D$ dimensions. The solution describes a metric around a DBI global defects. When the core is smaller than its Schwarzschild radius it can be interpreted as a black hole having DBI scalar hair with deficit conical angle. The solutions exist for all $D$, but they can be expressed as polynomial functions in $r$ only when $D$ is even. We give conditions for the mass $M$ and the scalar charge $\eta$ in the extremal case. We also investigate the thermodynamic properties of the black holes in canonical ensemble. The monopole alter the stability differently in each dimensions. As the charge increases the black hole radiates more, in contrast to its counterpart with ordinary global defects where the Hawking temperature is minimum for critical $\eta$. This behavior can also be observed for variation of DBI coupling, $\beta$. As it gets stronger ($\beta\ll1$) the temperature increases. By studying the heat capacity we can infer that there is no phase transition in asymptotically-flat spacetime. The AdS black holes, on the other hand, undergo a first-ordered phase transition in the Hawking-Page type. The increase of the DBI coupling renders the phase transition happen for larger radius.

On relativistic multipole moments of stationary space-times.

Among the known exact solutions of Einstein's vacuum field equations the Manko–Novikov and the Quevedo–Mashhoon metrics might be suitable ones for the description of the exterior gravitational field of some real non-collapsed body. A new proposal to represent such exterior field is the stationary q-metric. In this contribution, we computed by means of the Fodor–Hoenselaers–Perjés formalism the lowest 10 relativistic multipole moments of these metrics. Corresponding moments were derived for the static vacuum solutions of Gutsunayev–Manko and Hernández–Martín. A direct comparison between the multipole moments of these non-isometric space–times is given.

Forward calculation of gravity and its gradient using polyhedral representation of density interfaces: an application of spherical or ellipsoidal topographic gravity effect

A density interface modeling method using polyhedral representation is proposed to construct 3-D models of spherical or ellipsoidal interfaces such as the terrain surface of the Earth and applied to forward calculating gravity effect of topography and bathymetry for regional or global applications. The method utilizes triangular facets to fit undulation of the target interface. The model maintains almost equal accuracy and resolution at different locations of the globe. Meanwhile, the exterior gravitational field of the model, including its gravity and gravity gradients, is obtained simultaneously using analytic solutions. Additionally, considering the effect of distant relief, an adaptive computation process is introduced to reduce the computational burden. Then features and errors of the method are analyzed. Subsequently, the method is applied to an area for the ellipsoidal Bouguer shell correction as an example and the result is compared to existing methods, which shows our method provides high accuracy and great computational efficiency. Suggestions for further developments and conclusions are drawn at last.

Exterior field of slowly and rapidly rotating neutron stars: Rehabilitating spacetime metrics involving hyperextreme objects

The 4-parameter exact solution presumably describing the exterior gravitational field of a generic neutron star is presented in a concise explicit form defined by only three potentials. In the equatorial plane, the metric functions of the solution are found to be given by particularly simple expressions that make them very suitable for the use in concrete applications. Following Pappas and Apostolatos, we perform a comparison of the multipole structure of the solution with the multipole moments of the known physically realistic Berti-Stergioulas numerical models of neutron stars to argue that the hyperextreme sectors of the solution are not less (but possibly even more) important for the correct description of rapidly rotating neutron stars than the subextreme sector involving exclusively the black-hole constituents. We have also worked out in explicit form an exact analog of the well-known Hartle-Thorne approximate metric.

Spheroidal models of the exterior gravitational field of Asteroids Bennu and Castalia

Gravitational field of small bodies can be modeled e.g. with mascons, a polyhedral model or in terms of harmonic functions. If the shape of a body is close to the spheroid, it is advantageous to employ the spheroidal basis functions for expressing the gravitational field. Spheroidal harmonic models, similarly to the spherical ones, may be used in navigation and geophysical tasks. We focus on modeling the exterior gravitational field of oblate-like Asteroid (101955) Bennu and prolate-like Asteroid (4769) Castalia with spheroidal harmonics. Using the Gauss–Legendre quadrature and the spheroidal basis functions, we converted the gravitational potential of a particular polyhedral model of a constant density into the spheroidal harmonics. The results consist of (i) spheroidal harmonic coefficients of the exterior gravitational field for the Asteroids Bennu and Castalia, (ii) spherical harmonic coefficients for Bennu, and (iii) the first and second-order Cartesian derivatives in the local spheroidal South-East-Up frame for both bodies. The spheroidal harmonics offer biaxial flexibility (compared with spherical harmonics) and low computational costs that allow high-degree expansions (compared with ellipsoidal harmonics). The obtained spheroidal models for Bennu and Castalia represent the exterior gravitational field valid on and outside the Brillouin spheroid but they can be used even under this surface. For Bennu, 5 m above the surface the agreement with point-wise integration was 1% or less, while it was about 10% for Castalia due to its more irregular shape. As the shape models may produce very high frequencies, it was crucial to use higher maximum degree to reduce the aliasing. We have used the maximum degree 360 to achieve 9–10 common digits (in RMS) when reconstructing the input (the gravitational potential) from the spheroidal coefficients. The physically meaningful maximum degree may be lower (≪360) but its particular value depends on the distance and/or on the application (navigation, exploration, etc.).

Multipole structure of compact objects

We analyze the applications of general relativity in relativistic astrophysics in order to solve the problem of describing the geometric and physical properties of the interior and exterior gravitational and electromagnetic fields of compact objects. We focus on the interpretation of exact solutions of Einstein's equations in terms of their multipole moments structure. In view of the lack of physical interior solutions, we propose an alternative approach in which higher multipoles should be taken into account.

Core-envelope and regular models in Einstein-Maxwell fields

New classes of exact solutions which could serve as sources for the Reissner-Nordstrom metric representing the exterior gravitational field of an isolated charged sphere are derived. Firstly, we sacrifice the requirement that the stellar centre is free of a singularity and then obtain a core-envelope model. The charged fluid envelope is matched suitably to the neutral core and the vacuum exterior solution. Next we investigate models of charged stars that are regular at the stellar centre. The Einstein-Maxwell system of partial differential equations is reduced to the study of a single first-order differential equation in which two of the matter or geometrical variables must be specified at the outset. In each case mentioned above, new exact models are found by choosing functional forms for the electric field intensity and one of the gravitational potentials. A Riccati equation is then solved to obtain the remaining potential. The charged spherical shell model as well as the non-singular models are shown to display necessary qualitative features that are demanded for physical acceptability. It is shown that the regular model has a vanishing pressure-free hypersurface. The density and pressure profiles are positive and monotonically decreasing outwards from the centre of the sphere for a chosen set of parameters. The weak strong and dominant energy conditions are also satisfied. A drawback of the model is that the causality criterion is not satisfied within the fluid boundary.

Gravity of a noncanonical global monopole: conical topology and compactification

We obtain solutions of Einstein's equations describing gravitational field outside a noncanonical global monopole with cosmological constant. In particular, we consider two models of k-monopoles: the Dirac-Born-Infeld (DBI) and the power-law types, and study their corresponding exterior gravitational fields. For each model we found two types of solutions. The first of which are global k-monopole black hole with conical global topology. These are generalizations of the Barriola-Vilenkin solution of global monopole. The appearance of noncanonical kinetic terms does not modify the critical symmetry-breaking scale, $\eta_{crit}$, but it does affect the corresponding horizon(s). The second type of solution is compactification, whose topology is a product of two $2$-dimensional spaces with constant curvatures; ${\mathcal Y}_4\rightarrow {\mathcal Z}_2\times S^2$, with ${\mathcal Y}, {\mathcal Z}$ can be de Sitter, Minkowski, or Anti-de Sitter, and $S^2$ is the $2$-sphere. We investigate all possible compactifications and show that the nonlinearity of kinetic terms opens up new channels which are otherwise non-existent. For $\Lambda=0$ four-dimensional geometry, we conjecture that these compactification channels are their (possible) non-static super-critical states, right before they undergo topological inflation.

Improved universality in the neutron star three-hair relations

No-hair like relations between the multipole moments of the exterior gravitational field of neutron stars have recently been found to be approximately independent of the star's internal structure. This approximate, equation-of-state universality arises after one adimensionalizes the multipole moments appropriately, which then begs the question of whether there are better ways to adimensionalize the moments to obtain stronger universality. We here investigate this question in detail by considering slowly-rotating neutron stars both in the non-relativistic limit and in full General Relativity. We find that there exist normalizations that lead to stronger equation-of-state universality in the relations among the moment of inertia and the quadrupole, octopole and hexadecapole moments of neutron stars. We determine the optimal normalization that minimizes the equation-of-state dependence in these relations. The results found here may have applications in the modeling of X-ray pulses and atomic line profiles from millisecond pulsars with NICER and LOFT.