Schwarzschild Black Hole Spacetime Research Articles

Across-horizon propagation and infinite time-dilation: An independent approach to Hawking radiation

The conformal scalar propagator as an explicit function of the Schwarzschild black-hole space-time has been established in the preceding three projects. The present project examines the precise relation which that propagator has to the amplitude the Schwarzschild black hole emit a scalar particle in a particular mode of positive energy. It is thereby established that the effect of infinite time-dilation on the event horizon results in both, the thermal radiation in the background of a static exterior space-time geometry and the detraction from thermality if energy-conservation is properly imposed as a constraint on scalar propagation. The derivation of the latter signifies an equivalent approach to the result of Parikh and Wilczek from the semi-classical perspective of the distant observer. From that perspective for that matter, Hawking radiation emerges in both contexts as the exclusive consequence of infinite time-dilation on the event horizon. These results are consistent with Hawking radiation as the exclusive consequence of the causal structure of the Schwarzschild black-hole space-time both, in a static exterior geometry and in such dynamical exterior geometry as energy-conservation signifies. The extension of the thermal case to other spin-fields and black-hole geometries is discussed.

Tidal forces in the Simpson–Visser black-bounce and wormhole spacetimes

The concept of regular black holes has gained attention in recent years, especially in the context of quantum gravity theories. In these theories, the existence of singularities is paradoxical as they represent a breakdown of the laws of physics. Motivated by the recent developments in this area, we study the tidal force effects in one such family of regular geometries described by the Simpson–Visser metric. We find the radial and angular force profiles for a radially in-falling particle in this spacetime and calculate the variation of the geodesic separation vector with the radial coordinate using two different initial conditions. These results are then compared with that of Schwarzschild black hole spacetime. We show that for a regular black hole, both radial and angular tidal forces show a peak outside the horizon and then fall to ultimately switch their behavior from stretching to compression and vice-versa. Also, they are finite at r=0 unlike the Schwarzschild spacetime. It is also seen that the angular deviation profile shows an oscillating behavior for a particular initial condition. Our analysis can be used to distinguish between regular black hole, one-way and two-way wormholes and a singular black hole spacetimes.

Relativistic orbits of S2 star in the presence of scalar field

The general theory of relativity predicts the relativistic effect in the orbital motions of S-stars which are orbiting around our Milky-way Galactic Center. The post-Newtonian or higher-order approximated Schwarzschild black hole models have been used by GRAVITY and UCLA Galactic Center groups to carefully investigate the S2 star’s periastron precession. In this paper, we investigate the scalar field effect on the orbital dynamics of S2 star. Hence, we consider a spacetime, namely Janis-Newman-Winicour (JNW) spacetime which is seeded by a minimally coupled, mass-less scalar field. The novel feature of this spacetime is that one can retain the Schwarzschild spacetime from JNW spacetime considering zero scalar charge. We constrain the scalar charge of JNW spacetime by best fitting the astrometric data of S2 star using the Monte-Carlo–Markov-Chain (MCMC) technique assuming the charge to be positive. Our best-fitted result implies that similar to the Schwarzschild black hole spacetime, the JNW naked singularity spacetime with an appropriate scalar charge also offers a satisfactory fitting to the observed data for S2 star. Therefore, the JNW naked singularity could be a contender for explaining the nature of Sgr A* through the orbital motions of the S2 star.

The greybody factor for the monopole and odd-parity modes of the Proca field in the Schwarzschild black hole spacetimes

The greybody factor is one of the famous results of the black hole perturbation theory, which describes the transmission probability of a particle radiated by a black hole into spatial infinity. In this work, we separated the angular parts of the equations of motion and derived the radial equations for the Proca field in the Schwarzschild black hole spacetime. The radial equations for the monopole and odd-parity modes are fully decoupled in the Schrödinger-like form. We study the greybody factor by determining the rigorous bound.

Exploring tidal force effects and shadow constraints for Schwarzschild-like black hole in Starobinsky–Bel-Robinson gravity

The current manuscript deals with the tidal force effects, geodesic deviation, and shadow constraints of the Schwarzschild-like black hole theorised in Starobinsky–Bel-Robinson gravity exhibiting M-theory compactification. In the current analysis, we explore the radial and angular tidal force effects on a radially in-falling particle by the central black hole, which is located in this spacetime. We also numerically solve the geodesic deviation equation and study the variation of the geodesic separation vector with the radial coordinate for two nearby geodesics using suitable initial conditions. All the obtained results are tested for Sag A* and M87* by constraining the value of the stringy gravity parameter β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta $$\\end{document} using the shadow data from the event horizon telescope observations. All the results are compared with Schwarzschild black hole spacetime. In our study, we found that both the radial and angular tidal forces experienced by a particle switch their initial behaviour and turn compressive and stretching, respectively, before reaching the event horizon. The geodesic deviation shows an oscillating trend as well for the chosen initial condition. For the constrained value of β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta $$\\end{document}, we see that the spacetime geometry generated by Sag A* and M87* is effectively same for both Schwarzschild and Starobinsky–Bel-Robinson black hole. Furthermore, we also calculated the angular diameter of the shadow in Starobinsky–Bel-Robinson black hole and compared with the Schwarzschild black hole. It is observed that the angular diameter of shadow for M87* and Sgr A* in Starobinsky–Bel-Robinson black hole is smaller than the Schwarzschild black hole. The calculated results satisfy the event horizon telescope observational constraints.

Black holes and nilmanifolds: quasinormal modes as the fingerprints of extra dimensions?

We investigate whether quasinormal modes (QNMs) can be used in the search for signatures of extra dimensions. To address a gap in the Beyond the Standard Model (BSM) literature, we focus here on higher dimensions characterised by negative Ricci curvature. As a first step, we consider a product space comprised of a four-dimensional Schwarzschild black hole space-time and a three-dimensional nilmanifold (twisted torus); we model the black hole perturbations as a scalar test field. We suggest that the extra-dimensional geometry can be stylised in the QNM effective potential as a squared mass-like term representing the Kaluza–Klein (KK) spectrum. We then compute the corresponding QNM spectrum using three different numerical methods, and determine a possible “detectability bound” beyond which KK masses cannot be detected using QNMs.

Linearised Perturbation of Constant Mass Aspect Function Foliation in Schwarzschild Black Hole Spacetime

We study the linearised perturbation of the constant mass aspect function foliation in a Schwarzschild black hole spacetime. In particular, we investigate the linearised perturbation of the asymptotic geometry of the foliation at null infinity. We show that there is a 4-dimensional linear space for the linearised perturbation of the initial leaf inside the event horizon, corresponding to which the linearised perturbation of the asymptotic geometry of the foliation at null infinity is preserved to be round. For such a linearised perturbation of the initial leaf in this 4-dimensional linear space, we calculate the corresponding linearised perturbation of the energy-momentum vector and the Bondi mass at null infinity. We show that the linearised perturbations of the Bondi energy and the Bondi mass both vanish, and every possible linearised perturbation of the linear momentum can be achieved by a linearised perturbation of the initial leaf in the 4-dimensional linear space.

Polarized images of synchrotron radiations in curved spacetime

In this work, we derive two formulas encoding the polarization direction and luminosity of synchrotron radiations from the moving electrons in curved spacetime under the geometric optics approximation. As an application, we further study the polarized images of synchrotron radiations from electron sources in Schwarzschild black hole spacetime with a vertical and uniform magnetic field. In particular, by focusing on the circular orbits of electrons on the equatorial plane, we show the polarized images of the synchrotron radiations from these orbits for different observational angles and discuss the variations of the polarization directions concerning the angles.

Revisiting the quasinormal modes of the Schwarzschild black hole: Numerical analysis

We revisit the problem of calculating the quasinormal modes of spin 0, 1/2, 1, 3/2, 2, and spin 5/2 fields in the asymptotically flat Schwarzschild black hole spacetime. Our aim is to investigate the problem from the numerical point of view, by comparing some numerical methods available in the literature and still not applied for solving the eigenvalue problems arising from the perturbation equations in the Schwarzschild black hole spacetime. We focus on the pseudo-spectral and the asymptotic iteration methods. These numerical methods are tested against the available results in the literature, and confronting the precision between each other. Besides testing the different numerical methods, we calculate higher overtones quasinormal frequencies for all the investigated perturbation fields in comparison with the known results. Additionally, we obtain purely imaginary frequencies for spin 1/2 and 3/2 fields that are in agreement with analytic results reported previously in the literature. The purely imaginary frequencies for the spin 1/2 perturbation field are exactly the same as the frequencies obtained for the spin 3/2 perturbation field. In turn, the quasinormal frequencies for the spin 5/2 perturbation field are calculated for the very first time, and purely imaginary frequencies are found also in this case. We conclude that both methods provide accurate results and they complement each other.

Mode-sum prescription for the renormalized stress energy tensor on black hole spacetimes

In this paper, we describe an extremely efficient method for computing the renormalized stress-energy tensor of a quantum scalar field in spherically-symmetric black hole spacetimes. The method applies to a scalar field with arbitrary field parameters. We demonstrate the utility of the method by computing the renormalized stress-energy tensor for a scalar field in the Schwarzschild black hole spacetime, applying our results to discuss the null energy condition and the semi-classical backreaction.

Geodesic stability and quasinormal modes of non-commutative Schwarzschild black hole employing Lyapunov exponent

We study the dynamics of test particle and stability of circular geodesics in the gravitational field of a non-commutative geometry-inspired Schwarzschild black hole spacetime. The coordinate time Lyapunov exponent ( $$\lambda _{c}$$ ) is crucial to investigate the stability of equatorial circular geodesics of massive and massless test particles. The stability or instability of circular orbits is discussed by analyzing the variation of Lyapunov exponent with radius of these orbits for different values of non-commutative parameter ( $$\alpha $$ ). In the case of null circular orbits, the instability exponent is calculated and presented to discuss the instability of null circular orbits. Further, by relating parameters corresponding to null circular geodesics (i.e., angular frequency and Lyapunov exponent), the quasinormal modes for a massless scalar field perturbation in the eikonal approximation are evaluated and also visualized by relating the real and imaginary parts. The nature of scalar field potential, by varying the non-commutative parameter ( $$\alpha $$ ) and angular momentum of perturbation (l), is also observed and discussed.

Harvesting entanglement with detectors freely falling into a black hole

We carry out the first investigation of the entanglement and mutual information harvesting protocols for detectors freely falling into a black hole. Working in $(1+1)$-dimensional Schwarzschild black hole spacetime, we consider two pointlike Unruh-DeWitt (UDW) detectors in different combinations of free-falling and static trajectories. Employing a generalization of relative velocity suitable for curved spacetimes, we find that the amount of correlations extracted from the black hole vacuum, at least outside the near-horizon regime, is largely kinematic in origin (i.e. it is mostly due to the relative velocities of the detectors). Second, correlations can be harvested purely from the black hole vacuum even when the detectors are causally disconnected by the event horizon. Finally, we show that the previously known `entanglement shadow' near the horizon is indeed absent for the case of two free-falling-detectors, since their relative gravitational redshift remains finite as the horizon is crossed, in accordance with the equivalence principle.

Diffeomorphism symmetries near a timelike surface in black hole spacetime

Recently symmetries of gravity and gauge fields in the asymptotic regions of spacetime have been shown to play vital role in their low energy scattering phenomena. Further, for the black hole spacetime, near horizon symmetry has been observed to play possible role in understanding the underlying degrees of freedom for thermodynamic behaviour of horizon. Following the similar idea, in this paper, we analysed the symmetry and associated algebra near a timelike surface which is situated at any arbitrary radial position and is embedded in black hole spacetime. In this paper we considered both Schwarzschild and Kerr black hole spacetimes. The families of hypersurfaces with constant radial coordinate (outside the horizon) in these spacetimes is timelike in nature and divide the space into two distinct regions. The symmetry algebra turned out to be reminiscent of Bondi–Metzner–Sach symmetry, found in the asymptotic null infinity.

On pointwise decay of waves

This paper introduces some of the basic mechanisms relating the behavior of the spectral measure of Schrödinger operators near zero energy to the long-term decay and dispersion of the associated Schrödinger and wave evolutions. These principles are illustrated by means of the author’s work on decay of Schrödinger and wave equations under various types of perturbations, including those of the underlying metric. In particular, we consider local decay of solutions to the linear Schrödinger and wave equations on curved backgrounds that exhibit trapping. A particular application is waves on a Schwarzschild black hole spacetime. We elaborate on Price’s law of local decay that accelerates with the angular momentum, which has recently been settled by Hintz, also in the much more difficult Kerr black hole setting. While the author’s work on the same topic was conducted ten years ago, the global semiclassical representation techniques developed there have recently been applied by Krieger, Miao, and the author [“A stability theory beyond the co-rotational setting for critical wave maps blow up,” arXiv:2009.08843 (2020)] to the nonlinear problem of stability of blowup solutions to critical wave maps under non-equivariant perturbations.

Shadows and negative precession in non-Kerr spacetime

It is now known that the shadow is not only the property of a black hole, it can also be cast by other compact objects like naked singularities. However, there exist some novel features of the shadow of the naked singularities which are elaborately discussed in some recent articles. In the earlier literature, it is also shown that a naked singularity may admit negative precession of bound timelike orbits which cannot be seen in Schwarzschild and Kerr black hole spacetimes. This distinguishable behavior of timelike bound orbit in the presence of the naked singularity along with the novel features of the shadow may be useful to distinguish between a black hole and a naked singularity observationally. However, in this paper, it is shown that deformed Kerr spacetime can allow negative precession of bound timelike orbits when the central singularity of that spacetime is naked. We also show that negative precession and shadow both can exist simultaneously in deformed Kerr naked singularity spacetime. Therefore, any observational evidence of negative precession of bound orbits, along with the central shadow may indicate the presence of a deformed Kerr naked singularity.

Harvesting correlations in Schwarzschild and collapsing shell spacetimes

We study the harvesting of correlations by two Unruh-DeWitt static detectors from the vacuum state of a massless scalar field in a background Vaidya spacetime consisting of a collapsing null shell that forms a Schwarzschild black hole (hereafter Vaidya spacetime for brevity), and we compare the results with those associated with the three preferred vacua (Boulware, Unruh, Hartle-Hawking-Israel vacua) of the eternal Schwarzschild black hole spacetime. To do this we make use of the explicit Wightman functions for a massless scalar field available in (1+1)-dimensional models of the collapsing spacetime and Schwarzschild spacetimes, and the detectors couple to the proper time derivative of the field. First we find that, with respect to the harvesting protocol, the Unruh vacuum agrees very well with the Vaidya vacuum near the horizon even for finite-time interactions. Second, all four vacua have different capacities for creating correlations between the detectors, with the Vaidya vacuum interpolating between the Unruh vacuum near the horizon and the Boulware vacuum far from the horizon. Third, we show that the black hole horizon inhibits any correlations, not just entanglement. Finally, we show that the efficiency of the harvesting protocol depend strongly on the signalling ability of the detectors, which is highly non-trivial in presence of curvature. We provide an asymptotic analysis of the Vaidya vacuum to clarify the relationship between the Boulware/Unruh interpolation and the near/far from horizon and early/late-time limits. We demonstrate a straightforward implementation of numerical contour integration to perform all the calculations.

The Thermodynamics of the Perturbed Schwarzschild Black Hole

The phenomena like gravitational waves, Hawking radiation, and growth of black holes not only change the curvature of spacetime continuously but also affect the thermodynamical properties of the sources over time. The present study explores the effect of time in the thermodynamical properties of the corresponding spacetime. We define a general time conformal factor e𝜖ζ(t) in the Schwarzschild black hole spacetime, where 𝜖 is a small parameter that brings perturbation in the said spacetime. This technique re-defines the energy content of spacetime. Moreover, the effect of time is explored in the thermodynamical properties of the perturbed Schwarzschild black hole. We calculate the event horizon of the spacetime as a function of t. Then we study the effect of time in the surface gravity, pressure, volume, and Hawking temperature of the perturbed Schwarzschild black hole and analyze them graphically.

Reducing triangular systems of ODEs with rational coefficients, with applications to coupled Regge-Wheeler equations

We concisely summarize a method of finding all rational solutions to an inhomogeneous rational ODE system of arbitrary order (but solvable for its highest order terms) by converting it into a finite dimensional linear algebra problem. This method is then used to solve the problem of conclusively deciding when certain rational ODE systems in upper triangular form can or cannot be reduced to diagonal form by differential operators with rational coefficients. As specific examples, we consider systems of coupled Regge-Wheeler equations, which have naturally appeared in previous work on vector and tensor perturbations on the Schwarzschild black hole spacetime. Our systematic approach reproduces and complements identities that have been previously found by trial and error methods.

Rotating Disc around a Schwarzschild Black Hole

A stationary and axisymmetric (in fact circular) metric is reviewed which describes the first-order perturbation of a Schwarzschild black-hole space-time due to a rotating finite thin disc encircling the hole symmetrically. The key Green functions of the problem (corresponding to an infinitesimally thin ring)—the one for the gravitational potential and the one for the dragging angular velocity—were already derived, in terms of infinite series, by Will in 1974, but we have now put them into closed forms using elliptic integrals. Such forms are more practical for numerical evaluation and for integration in problems involving extended sources. This last point mostly remains difficult, but we illustrate that it may be workable by using the simple case of a finite thin disc with constant Newtonian surface density.

Stable bound orbits around a supersymmetric black lens

In higher-dimensional Schwarzschild black hole spacetimes, there are no stable bound orbits of particles. In contrast to this, it is shown that there are stable bound orbits in a five-dimensional black lens spacetime.