Schwarzschild Case Research Articles

Harrison Metrics for the Schwarzschild Black Hole

Based on the hidden conformal symmetry, some authors have proposed a Harrison metric for the Schwarzschild black hole. We give a procedure which can generate a family of Harrison metrics starting from a general set of SL(2, R) vector fields. By analogy with the subtracted geometry of the Kerr black hole, we find a new Harrison metric for the Schwarzschild case. Its conformal generators are also investigated using the Killing equations in the near-horizon limit.

Weak gravitational lensing by compact objects in fourth order gravity

We discuss weak lensing characteristics in the gravitational field of a compact object in the low-energy approximation of fourth order f(R) gravity theory. The particular solution is characterized by a gravitational strength parameter $\sigma $ and a distance scale $r_{c}$ much larger than the Schwarzschild radius. Above $r_{c}$ gravity is strengthened and as a consequence weak lensing features are modified compared to the Schwarzschild case. We find a critical impact parameter (depending upon $r_{c}$) for which the behavior of the deflection angle changes. Using the Virbhadra-Ellis lens equation we improve the computation of the image positions, Einstein ring radii, magnification factors and the magnification ratio. We demonstrate that the magnification ratio as function of image separation obeys a power-law depending on the parameter $\sigma $, with a double degeneracy. No $\sigma \neq 0$ value gives the same power as the one characterizing Schwarzschild black holes. As the magnification ratio and the image separation are the lensing quantities most conveniently determined by direct measurements, future lensing surveys will be able to constrain the parameter $\sigma $ based on this prediction.

Geodesic motion in the space-time of a noncompact boson star

We study the geodesic motion of test particles in the space-time of noncompact boson stars. These objects are made of a self-interacting scalar field and---depending on the scalar field's mass---can be as dense as neutron stars or even black holes. In contrast to the former these objects do not contain a well-defined surface, while in contrast to the latter the space-time of boson stars is globally regular and can---however---only be given numerically. Hence, the geodesic equation also has to be studied numerically. We discuss the possible orbits for massive and massless test particles and classify them according to the particle's energy and angular momenta. The space-time of a boson star approaches the Schwarzschild space-time asymptotically; however, it deviates strongly from it close to the center of the star. As a consequence, we find additional bound orbits of massive test particles close to the center of the star that are not present in the Schwarzschild case. Our results can be used to make predictions about extreme-mass-ratio inspirals, and we hence compare our results to recent observational data of the stars orbiting Sagittarius ${A}^{*}$---the radio source at the center of our own Galaxy.

A Central Mass in a Stationary Vacuum without Spherical or Axial Symmetry

A vacuum spacetime with a central mass is derived as a stationary solution to Einstein's equations. The vacuum metric has a geodesic, shear-free, expanding, and twisting null congruence k and thus is algebraically special. The properties of the metric are calculated. In particular, it is shown that the spacetime has an event horizon inside which there is a black hole. The metric is neither spherically nor axially symmetric. It is therefore in interesting contrast with the majority of metrics featuring a central mass which have one or more of these symmetry properties. The metric reduces to the Schwarzschild case when a certain parameter is set to zero.

Conditional symmetries and the canonical quantization of constrained minisuperspace actions: The Schwarzschild case

A conditional symmetry is defined, in the phase space of a quadratic in velocities constrained action, as a simultaneous conformal symmetry of the supermetric and the superpotential. It is proven that such a symmetry corresponds to a variational (Noether) symmetry. The use of these symmetries as quantum conditions on the wave function entails a kind of selection rule. As an example, the minisuperspace model ensuing from a reduction of the Einstein–Hilbert action by considering static, spherically symmetric configurations and r as the independent dynamical variable is canonically quantized. The conditional symmetries of this reduced action are used as supplementary conditions on the wave function. Their integrability conditions dictate, at the first stage, that only one of the three existing symmetries can be consistently imposed. At a second stage one is led to the unique Casimir invariant, which is the product of the remaining two, as the only possible second condition on Ψ. The uniqueness of the dynamical evolution implies the need to identify this quadratic integral of motion to the reparametrization generator. This can be achieved by fixing a suitable parametrization of the r-lapse function, exploiting the freedom to arbitrarily rescale it. In this particular parametrization the measure is chosen to be the determinant of the supermetric. The solutions to the combined Wheeler–DeWitt and linear conditional symmetry equations are found and seen to depend on the product of the two “scale factors”.

Local decay of waves on asymptotically flat stationary space-times

In this article we study the pointwise decay properties of solutions to the wave equation on a class of stationary asymptotically flat backgrounds in three space dimensions. Under the assumption that uniform energy bounds and a weak form of local energy decay hold forward in time we establish a $t^{-3}$ local uniform decay rate for linear waves. This work was motivated by open problems concerning decay rates for linear waves on Schwarzschild and Kerr backgrounds. In the Schwarzschild case, such a decay rate has been heuristically derived by Price. Our results apply to both of these cases.

Casadio–Fabbri–Mazzacurati black strings and braneworld-induced quasars luminosity corrections

This paper aims to evince the corrections on the black string warped horizon in the braneworld paradigm, and their drastic physical consequences, as well as to provide subsequent applications in astrophysics. Our analysis concerning black holes on the brane departs from the Schwarzschild case, where the black string is unstable to large-scale perturbation. The cognizable measurability of the black string horizon corrections due to braneworld effects is investigated, as well as their applications in the variation of quasars luminosity. We delve into the case wherein two solutions of Einstein’s equations proposed by Casadio, Fabbri and Mazzacurati, regarding black hole metrics presented a post-Newtonian parameter measured on the brane. In this scenario, it is possible to analyze purely the braneworld corrected variation in quasars luminosity, by an appropriate choice of the post-Newtonian parameter that precludes Hawking radiation on the brane: the variation in quasars luminosity is uniquely provided by pure braneworld effects, as the Hawking radiation on the brane is suppressed.

Gravitational energy of a magnetized Schwarzschild black hole: a teleparallel approach

We investigate the distribution of gravitational energy on the spacetime of a Schwarzschild black hole immersed in a cosmic magnetic field. This is done in the context of the {\it Teleparallel Equivalent of General Relativity}, which is an alternative geometrical formulation of General Relativity, where gravity is describe by a spacetime endowed with torsion, rather than curvature, with the fundamental field variables being tetrads. We calculate the energy enclosed by a two-surface of constant radius - in particular, the energy enclosed by the event horizon of the black hole. In this case we find that the magnetic field has the effect of increasing the gravitational energy as compared to the vacuum Schwarzschild case. We also compute the energy (i) in the weak magnetic field limit, (ii) in the limit of vanishing magnetic field, and (iii) in the absence of the black hole. In all cases our results are consistent with what should be expected on physical grounds.

Quasi-normal modes, area spectra and multi-horizon spacetimes

We suggest an interpretation for the highly damped QNM frequencies of the spherically symmetric multi-horizon spacetimes (Reissner-Nordstrom, Schwarzschild-deSitter, Reissner-Nordstrom-deSitter) following Maggiore's proposal about the link between the asymptotic QNM frequencies and the black hole thermodynamics. We show that the behavior of the asymptotic frequencies is easy to understand if one assumes that all of the horizons have the same equispaced area spectra. The QNM analysis is then consistent with the choice of the area spectra to be the one originally proposed for the black hole's horizon by Bekenstein: A=8\pi n (in Planck units). The interpretation of the highly damped QNM frequencies in the multi-horizon case is based on the similar grounds as in the single horizon (Schwarzschild) case, but it has some new features that are discussed in the paper.

Do we know the mass of a black hole? Mass of some cosmological black hole models

Using a cosmological black hole model proposed recently, we have calculated the quasi-local mass of a collapsing structure within a cosmological setting due to different definitions put forward in the last decades to see how similar or different they are. It has been shown that the mass within the horizon follows the familiar Brown-York behavior. It increases, however, outside the horizon again after a short decrease, in contrast to the Schwarzschild case. Further away, near the void, outside the collapsed region, and where the density reaches the background minimum, all the mass definitions roughly coincide. They differ, however, substantially far from it. Generically, we are faced with three different Brown-York mass maxima: near the horizon, around the void between the overdensity region and the background, and another at cosmological distances corresponding to the cosmological horizon. While the latter two maxima are always present, the horizon mass maxima is absent before the onset of the central singularity.

Relativistic Accretion into a Reissner-Nordström Black Hole Revisited

The accretion of relativistic and nonrelativistic fluids into a Reissner-Nordström black hole is revisited. The position of the critical point, the flow velocity at this point, and the accretion rate are only slightly affected with respect to the Schwarzschild case when the fluid is nonrelativistic. On the contrary, relativistic fluids cross the critical point always subsonically. In this case, the sonic point is located near the event horizon, which is crossed by the fluid with a velocity less than the light speed. The accretion rate of relativistic fluids by a Reissner-Nordström black hole is reduced with respect to those estimated for uncharged black holes, being about 60% less for the extreme case (charge-to-mass ratio equal to one).

GRAVITATIONAL LENSING IN AN ENERGY-DEPENDENT SPACETIME METRIC

In this article, we explore the consequences in the calculation of deflection angle of photons, caused by a gravitational source of mass M, for the case of an energy dependent metric. We analyze the corrections to the standard Schwarzschild case for the weak and strong limits. In both cases, the corrections to the deflection angle are of the order ϵ/E Pl , where E Pl is the Planck energy and ϵ, the energy of the photon at spatial infinite. The corrections to the angular separation image is also of order ϵ/E Pl , for the weak limit, while in the strong case there is a corrective term with the shape ϵ/E Pl n e2nπ for the relativistic image of order n. The amplification of the image is also discussed. Even if corrections are tiny, in both limits, we discuss the qualitative effects for different types of Lorentz invariance deformations.

Evolution of massive fields around a black hole in Hořava gravity

We study the evolution of massive scalar field in the spacetime geometry of Kehagias-Sfetsos(KS) black hole in deformed Horava-Lifshitz(HL) gravity by numerical analysis. We find that the signature of HL theory is encoded in the quasinormal mode(QNM) phase of the evolution of field. The QNM phase in the evolution process lasts for a longer time in HL theory. QNMs involved in the evolution of massive field are calculated and find that they have a higher oscillation frequency and a lower damping rate than the Schwarzschild spacetime case. We also study the relaxation of field in the intermediate and asymptotic range and verified that behaviors of field in these phases are independent of the HL parameter and is identical to the Schwarzschild case.

Fermion analysis of IR modified Hořava–Lifshitz gravity: tunneling and perturbation perspectives

In this paper, we investigate the fermion Hawking radiation and quasinormal (QN) modes in infrared (IR) modified Hořava–Lifshitz (HL) gravity under tunneling and perturbation perspectives. Firstly, through the fermion tunneling in IR modified HL gravity, we obtain the Hawking radiation emission rate, tunneling temperature and entropy for the Kehagias–Sfetsos black hole. It is found that the results of fermion tunneling are consistent with the thermodynamics results obtained by calculating surface gravity. Secondly, we numerically calculate the low-lying QN mode frequencies of fermion perturbations by using WKB formulas including the third-order and sixth-order approximations simultaneously. It turns out that the actual frequency of fermion perturbation is larger than that in the Schwarzschild case, and the damping rate is smaller than that for the pure Schwarzschild. The results of fermion perturbation suggest the QN modes could live longer in HL gravity.

Comment on “Quasinormal modes in Schwarzschild–de Sitter spacetime: A simple derivation of the level spacing of the frequencies”

It is shown here that the extraction of quasinormal modes within the first Born approximation of the scattering amplitude is mathematically not well-founded. Indeed, the constraints on the existence of the scattering amplitude integral lead to inequalities for the imaginary parts of the quasinormal mode frequencies. For instance, in the Schwarzschild case, $0\ensuremath{\le}{\ensuremath{\omega}}_{I}<\ensuremath{\kappa}$ (where $\ensuremath{\kappa}$ is the surface gravity at the horizon) invalidates the poles deduced from the first Born approximation method, namely, ${\ensuremath{\omega}}_{n}=in\ensuremath{\kappa}$.

Linearized gravity and gauge conditions

In this paper we consider the field equations for linearized gravity and other integer spin fields on the Kerr spacetime, and more generally on spacetimes of Petrov type D. We give a derivation, using the Geroch–Held–Penrose formalism, of decoupled field equations for the linearized Weyl scalars for all spin weights and identify the gauge source functions occurring in these. For the spin weight 0 Weyl scalar, imposing a generalized harmonic coordinate gauge yields a generalization of the Regge–Wheeler equation. Specializing to the Schwarzschild case, we derive the gauge invariant Regge–Wheeler and Zerilli equation directly from the equation for the spin 0 scalar.

Emergent flux from particle collisions near a Kerr black hole

The escape fraction at infinity is evaluated for massless particles produced in collisions of weakly interacting particles accreted into a density spike near the particle horizon of an extremal Kerr black hole, for the case of equatorial orbits. We compare with the Schwarzschild case, and argue that in the case of extremal black holes, redshifted signatures can be produced that could potentially explore the physics of particle collisions at centre of mass energies that extend beyond those of any feasible terrestrial accelerator.

Linearized perturbations of the Kerr spacetime and outer boundary conditions in numerical relativity

This paper discusses linearized vacuum gravitational perturbations of the Kerr spacetime in a neighbourhood of future null infinity . Unlike earlier discussion of perturbations of the Kerr spacetime we avoid the use of spheroidal harmonics and harmonic time dependence. Instead we develop the theory in terms of Hertz potentials and spherical harmonics with coupling between modes. The ‘master equation’ is a single complex scalar wave equation which, in the Minkowski limit, reduces to the Euler–Poisson–Darboux equation. We solve this by Picard iteration making extensive use of the flat spacetime Riemann–Green function. As an application we consider the problem of outer boundary conditions for numerical relativity and generalize earlier results of Buchman and Sarbach (2006 Class. Quantum Grav. 23 6709–44, 2007 Class. Quantum Grav. 24 S307–26) for the Schwarzschild case.

Solar system constraints on asymptotically flat IR modified Hořava gravity through light deflection

In this paper, we study the motion of photons around a Kehagias-Sfetsos (KS) black hole and obtain constraints on IR modified Ho$\check{r}$ava gravity without cosmological constant ($\sim \Lambda_{W}$). An analytic formula for the light deflection angle is obtained. For a propagating photon, the deflection angle $\delta \phi$ increases with large values of the Ho$\check{r}$ava gravity parameter $\omega$. Under the UV limit $\omega \longrightarrow \infty$, deflection angle reduces to the result of usual Schwarzschild case, $4GM/R$. It is also found that with increasing scale of astronomical observation system the Ho$\check{r}$ava-Lifshitz gravity should satisfy $|\omega M^2|>1.1725 \times10^{-16}$ with 12% precision for Earth system, $|\omega M^2| > 8.27649 \times 10^{-17}$ with 17% precision for Jupiter system and $|\omega M^2| > 8.27650\times 10^{-15}$ with 0.17% precision for solar system.

Regularity Results for the Spherically Symmetric Einstein–Vlasov System

The spherically symmetric Einstein-Vlasov system is considered in Schwarzschild coordinates and in maximal-isotropic coordinates. An open problem is the issue of global existence for initial data without size restrictions. The main purpose of the present work is to propose a method of approach for general initial data, which improves the regularity of the terms that need to be estimated compared to previous methods. We prove that global existence holds outside the centre in both these coordinate systems. In the Schwarzschild case we improve the bound on the momentum support obtained in \cite{RRS} for compact initial data. The improvement implies that we can admit non-compact data with both ingoing and outgoing matter. This extends one of the results in \cite{AR1}. In particular our method avoids the difficult task of treating the pointwise matter terms. Furthermore, we show that singularities never form in Schwarzschild time for ingoing matter as long as $3m\leq r.$ This removes an additional assumption made in \cite{A1}. Our result in maximal-isotropic coordinates is analogous to the result in \cite{R1}, but our method is different and it improves the regularity of the terms that need to be estimated for proving global existence in general.