Schwarzschild Spacetime Research Articles

Gravitational collapse and odd-parity black hole perturbations in minimal theory of bigravity

In the former part, we study the gravitational collapse of pressureless dust and find special solutions, where, in both the physical and fiducial sectors, the exterior and interior spacetime geometries are given by the Schwarzschild spacetimes and the Friedmann-Lema\^itre-Robertson-Walker universes dominated by pressureless dust, respectively, with specific time slicings. In the case where the Lagrange multipliers are trivial and have no jump across the matter interfaces in both the physical and fiducial sectors, the junction conditions across them remain the same as those in general relativity (GR). For simplicity, we foliate the interior geometry by homogeneous and isotropic spacetimes. For a spatially flat interior universe, we foliate the exterior geometry by a time-independent flat space, while for a spatiallycurved interior universe, we foliate the exterior geometry by a time-independent space with deficit solid angle. Despite the rather restrictive choice of foliations, we find interesting classes of exact solutions that represent gravitational collapse in MTBG. In the spatially flat case, under a certain tuning of the initial condition, we find exact solutions of matter collapse in which the two sectors evolve independently. In the spatially closed case, once the matter energy densities and the Schwarzschild radii are tuned between the two sectors, we find exact solutions that correspond to the Oppenheimer-Snyder model in GR. In the latter part, we study odd-parity perturbations of the Schwarzschild$-$de Sitter solutions written in the spatially flat coordinates. For the higher-multipole modes $\ell\geq2$, we find that, in general, the system reduces to that of four physical modes, where two of them are dynamical and the remaining two are shadowy, i.e., satisfying only elliptic equations.

A review on analytical studies in gravitational lensing

In this study, we review some current studies on gravitational lensing for black holes, mainly in the context of general relativity. We mainly focus on the analytical studies related to lensing with references to observational results. We start with reviewing lensing in spherically symmetric Schwarzschild spacetime, showing how to calculate deflection angles before moving to the rotating counterpart, the Kerr metric. Furthermore, we extend our studies for a particular class of newly proposed solutions called black-bounce spacetimes and discuss throughout the review how to explore lensing in these spacetimes and how the various parameters can be constrained using available astrophysical and cosmological data.

On the non-minimal coupling of magnetic fields with gravity in Schwarzschild spacetime

In this work we study the effects of non-minimal coupling between electromagnetism and gravity, which are motivated by quantum effects such as vacuum polarization. We investigate the modification of both asymptotically dipole and uniform magnetic fields around a Schwarzschild black hole that come as the result of non-minimal coupling. While in both cases the magnetic field gets enhanced or suppressed with respect to the case of minimal coupling, depending on the sign of non-minimal coupling parameter, in the case of a background uniform magnetic field the direction of the magnetic field also alters in the vicinity of the black hole horizon. We have discussed the possible astrophysical and cosmological sources for which the vacuum polarization may be at play, while also discussing the observational effects, in particular the possibility of synchrotron radiation from the vicinity of a black hole. We conclude that such observations could be used to constrain the value of the non-minimal coupling parameter.

Aspects of Einstein’s general relativity: A classical deformation of Schwarzschild spacetime

In this work, we explore the Schwarzschild geometry in a spherically symmetric gravitational field. We build the non-commutative equations of motion with the aid of the Hamiltonian function and modified algebra. We then study the implications of the non-commutative geometry on the trajectory of a light ray, traveling in null and particles geodesics. Also, we interpret the effect of non-commutativity in both the bending of light and the perihelion advance of Mercury. Therefore, introducing a non-commutative parameter provides a slight correction to the results of general relativity.

Rotation in vacuum and scalar background: Are there alternatives to Newman–Janis algorithm?

The Newman–Janis (NJ) algorithm is the standard approach to rotation in general relativity which, in vacuum, builds the Kerr metric from the Schwarzschild spacetime. Recently, we have shown that the same algorithm applied to the Papapetrou antiscalar spacetime produces a rotational metric devoid of horizons and ergospheres. Though exact in the scalar sector, this metric, however, satisfies the Einstein equations only asymptotically. We argue that this discrepancy between geometric and matter parts (essential only inside gravitational radius scale) is caused by the violation of the Hawking–Ellis energy conditions for the scalar energy–momentum tensor. The axial potential functions entering the metrics appear to be of the same form both in vacuum and scalar background, and they also coincide with the linearized Yang–Mills field, which might hint at their common nongravitational origin. As an alternative to the Kerr-type spacetimes produced by NJ algorithm we suggest the exact solution obtained by local rotational coordinate transformation from the Schwarzschild spacetime. Then, comparison with the Kerr-type metrics shows that the Lense–Thirring phenomenon might be treated as a coordinate effect, similar to the Coriolis force.

Blow-up and lifespan estimate to a nonlinear wave equation in Schwarzschild spacetime

We study the semilinear wave equation with power type nonlinearity and small initial data in Schwarzschild spacetime. If the nonlinear exponent p satisfies 2≤p<1+2, we establish the sharp upper bound of lifespan estimate, while for the most delicate critical power p=1+2, we show that the lifespan satisfiesT(ε)≤exp⁡(Cε−(2+2)), the optimality of which remains to be proved. The key novelty is that the compact support of the initial data can be close to the event horizon. By combining the global existence result for p>1+2 obtained by Lindblad et al. (2014) [25], we then give a positive answer to the interesting question posed by Dafermos and Rodnianski (2005) [22, the end of the first paragraph in page 1151]: p=1+2 is exactly the critical power of p separating stability and blow-up.

Topological black holes in higher derivative gravity

We study static black holes in quadratic gravity with planar and hyperbolic symmetry and non-extremal horizons. We obtain a solution in terms of an infinite power-series expansion around the horizon, which is characterized by two independent integration constants – the black hole radius and the strength of the Bach tensor at the horizon. While in Einstein’s gravity, such black holes require a negative cosmological constant Lambda , in quadratic gravity they can exist for any sign of Lambda and also for Lambda =0. Different branches of Schwarzschild–Bach–(A)dS or purely Bachian black holes are identified which admit distinct Einstein limits. Depending on the curvature of the transverse space and the value of Lambda , these Einstein limits result in (A)dS–Schwarzschild spacetimes with a transverse space of arbitrary curvature (such as black holes and naked singularities) or in Kundt metrics of the (anti-)Nariai type (i.e., dS_2times S^2, AdS_2times H^2, and flat spacetime). In the special case of toroidal black holes with Lambda =0, we also discuss how the Bach parameter needs to be fine-tuned to ensure that the metric does not blow up near infinity and instead matches asymptotically a Ricci-flat solution.

Blow-up for the coupled system of wave equations with memory terms in Schwarzschild spacetime

This paper is concerned with blow-up results of solutions to the Cauchy problem for a coupled system of wave equations in Schwarzschild spacetime, where the nonlinear terms contain power-type memory terms, derivative-type memory terms, and combined-type memory terms, respectively. Upper-bound lifespan estimates of solutions to the Cauchy problem with small initial values and certain assumptions on the exponents in the memory terms are established by taking advantage of the Regge–Wheeler coordinate and iteration approach. Our main new contribution is that lifespan estimates of solutions to the problem are associated with the well-known Strauss conjecture and Glassey conjecture in Schwarzschild spacetime. To the best of the authors’ knowledge, the results in Theorems 1.1–1.3 are new.

Axisymmetric, stationary collisionless gas configurations surrounding Schwarzschild black holes

The properties of a stationary gas cloud surrounding a black hole are discussed, assuming that the gas consists of collisionless, identical massive particles that follow spatially bound geodesic orbits in the Schwarzschild spacetime. Several models for the one-particle distribution function are considered, and the essential formulae that describe the relevant macroscopic observables, like the current density four-vector and the stress–energy–momentum tensor are derived. This is achieved by rewriting these observables as integrals over the constants of motion and by a careful analysis of the range of integration. In particular, we provide configurations with finite total mass and angular momentum. Differences between these configurations and their nonrelativistic counterparts in a Newtonian potential are analyzed. Finally, our configurations are compared to their hydrodynamic analogues, the ‘polish doughnuts’.

Study of decoupled gravastars in energy–momentum squared gravity

In this paper, we generate an exact anisotropic gravastar model using gravitational decoupling technique through minimal geometric deformation in the framework of f(ℜ,T2) gravity. This novel model explains an ultra-compact stellar configuration whose internal region is smoothly matched to the exterior region. The developed stellar model satisfies some of the essential characteristics of a physically acceptable model such as a positive monotonically decreasing profile of energy density from the center to the boundary and monotonically decreasing behavior of the pressure. The anisotropic factor and Schwarzschild spacetime follows physically acceptable behavior. We find that all the energy bounds are satisfied except strong energy condition inside the ultra-compact stellar structure for the coupling constant of this theory, which is compatible with the regularity condition.

The fate of horizons under quantum corrections

In this paper, we study the physical effects of deforming the usual Einstein-Hilbert lagrangian with the Goroff-Sagnotti counterterm (the first which is nonvanishing on shell). Only spherically symmetric solutions are discussed. The facts that Schwarzschild's spacetime is not a solution to the corresponding equations of motion and Birkhoff's theorem validity are analyzed and some consequences worked out.

Gravitational time dilation in extended quantum systems: The case of light clocks in Schwarzschild spacetime

The precision of optical atomic clocks is approaching a regime where they resolve gravitational time dilation on smaller scales than their own extensions. Hence, an accurate description of quantum clocks has to take their spatial extension into account. In this article, as a first step toward a fully relativistic description of extended quantum clocks, we investigate a quantized version of Einstein's light clock fixed at a constant distance from a large massive object like the Earth. The model consists of a quantum light field in a one-dimensional cavity in Schwarzschild spacetime, where the distance between the mirrors is fixed by a rigid rod. By comparing a vertical and a horizontal clock, we propose an operational way to define the clock time when the clock resolves gravitational time dilation on scales smaller than its extension. In particular, we show that the time measured by the vertical light clock is equivalent to the proper time defined at its center. We also derive fundamental bounds on the precision of these clocks for measurements of proper time and the Schwarzschild radius.

Doubling of physical states in the quantum scalar field theory for a remote observer in the Schwarzschild spacetime

We discuss the problem of canonical quantization of a free real massive scalar field in the Schwarzschild spacetime. It is shown that a consistent procedure of canonical quantization of the field can be carried out without taking into account the black hole interior, so that in the resulting theory the canonical commutation relations are satisfied exactly, and the Hamiltonian has the standard form. However, unlike some papers, in which the expansion of the quantum field in spherical harmonics is used, here we use an expansion in scatteringlike states for energies larger than the mass of the field. This reveals a strange property of the resulting quantum field theory---doubling of the quantum states, which look as having the same asymptotic momentum to an observer located far away from the black hole. This purely topological effect cannot be eliminated by moving away from the black hole.

Direct and Exact Description of Null Geodesics in Schwarzschild Spacetime

Null geodesics for massless particles in Schwarzschild spacetime are obtained by direct integration of the trajectory equation in spatial coordinates without transformation to the inverse space. The results are classified following Chandrasekhar depending on the ratio of the impact parameter of the trajectory to its critical value. In the subcritical and supercritical cases the geodesics are expressed in terms of elliptic integrals of the first kind. Some results are formally different from the classical ones, but in fact equivalent to them, being at the same time more compact and descriptive.

Timelike and Null Geodesics in the Schwarzschild Space-time: Analytical Solutions

Timelike and Null Geodesics in the Schwarzschild Space-time: Analytical Solutions

Ascribing Quantum System to Schwarzschild Spacetime with Naked Singularity — ACS Quantization Method

Ascribing Quantum System to Schwarzschild Spacetime with Naked Singularity — ACS Quantization Method

Deflection of charged signals in a dipole magnetic field in a Schwarzschild background using the Gauss-Bonnet theorem

This paper studies the deflection of charged particles in a dipole magnetic field in Schwarzschild spacetime background in the weak field approximation. To calculate the deflection angle, we use Jacobi metric and Gauss-Bonnet theorem. Since the corresponding Jacobi metric is a Finsler metric of Randers type, we use both the osculating Riemannian metric method and generalized Jacobi metric method. The deflection angle up to fourth order is obtained and the effect of the magnetic field is discussed. It is found that the magnetic dipole will increase (or decrease) the deflection angle of a positively charged signal when its rotation angular momentum is parallel (or antiparallel) to the magnetic field. It is argued that the difference in the deflection angles of different rotation directions can be viewed as a Finslerian effect of the nonreversibility of the Finsler metric. The similarity of the deflection angle in this case with that for the Kerr spacetime allows us to directly use the gravitational lensing results in the latter case. The dependence of the apparent angles on the magnetic field suggests that by measuring these angles the magnetic dipole might be constrained.

Schwarzschild Spacetimes: Topology

This paper is devoted to the geometric theory of a Schwarzschild spacetime, a basic objective in applications of the classical general relativity theory. In a broader sense, a Schwarzschild spacetime is a smooth manifold, endowed with an action of the special orthogonal group SO(3) and a Schwarzschild metric, an SO(3)-invariant metric field, satisfying the Einstein equations. We prove the existence of and find all Schwarzschild metrics on two topologically non-equivalent manifolds, R×(R3∖{(0,0,0)}) and S1×(R3∖{(0,0,0)}). The method includes a classification of SO(3)-invariant, time-translation invariant and time-reflection invariant metrics on R×(R3∖{(0,0,0)}) and a winding mapping of the real line R onto the circle S1. The resulting family of Schwarzschild metrics is parametrized by an arbitrary function and two real parameters, the integration constants. For any Schwarzschild metric, one of the parameters determines a submanifold, where the metric is not defined, the Schwarzschild sphere. In particular, the family admits a global metric whose Schwarzschild sphere is empty. These results transfer to S1×(R3∖{(0,0,0)}) by the winding mapping. All our assertions are derived independently of the signature of the Schwarzschild metric; the signature can be chosen as an independent axiom.

A comparative study of new generic wormhole models with stability analysis via thin-shell

This analysis is devoted to exploring the interesting aspects of wormhole geometry. The energy conditions are checked for two different new generic shape functions, which satisfy the required wormhole properties. The presence of exotic matter is confirmed due to the violation of the energy conditions in the background of f(R, T) gravity as well as in the general relativity case. The traversable wormholes respecting the null energy conditions can be realized in both considered frameworks. A thin-shell around a wormhole geometry with two different generic shape functions is obtained by using the cut and paste approach taking Schwarzschild spacetime as an exterior manifold. The stability of thin-shell is explored with linearized perturbation along the equilibrium shell radius. Stable regions and the position of the expected event horizon depend on the choice of physical parameters. It is concluded that the number of expected event horizons increases for the second shape function.

Jacobi equations of geodetic brane gravity

We consider brane gravity as described by the Regge–Teitelboim geometric model, in any co-dimension. In brane gravity our spacetime is modelled as the time-like world volume spanned by a space-like brane in its evolution, seen as a manifold embedded in an ambient background Minkowski spacetime of higher dimension. Although the equations of motion of the model are well known, apparently their linearization has not been considered before. Using a direct approach, we linearize the equations of motion about a solution, obtaining the Jacobi equations of the Regge–Teitelboim model. They take a formidable aspect. Some of their features are commented upon. By identifying the Jacobi equations, we derive an explicit expression for the Morse index of the model. To be concrete, we apply the Jacobi equations to the study of the stability of a four-dimensional Schwarzschild spacetime embedded in a six-dimensional Minkowski spacetime. We find that it is unstable under small linear deformations.