Schwarzschild Black Hole Spacetime Research Articles

Radiation Fields on Schwarzschild Spacetime

In this paper we define the radiation field for the wave equation on the Schwarzschild black hole spacetime. In this context it has two components: the rescaled restriction of the time derivative of a solution to null infinity and to the event horizon. In the process, we establish some regularity properties of solutions of the wave equation on the spacetime. In particular, we prove that the regularity of the solution across the event horizon and across null infinity is determined by the regularity and decay rate of the initial data at the event horizon and at infinity. We also show that the radiation field is unitary with respect to the conserved energy and prove support theorems for each piece of the radiation field.

Absorption cross section and Hawking radiation of the electromagnetic field with Weyl corrections

We have investigated the absorption cross section and the Hawking radiation of electromagnetic field with Weyl correction in the background of a four-dimensional Schwarzschild black hole spacetime. Our results show that the properties of the absorption cross section and the Hawking radiation depend not only on the Weyl correction parameter, but also on the parity of the electromagnetic field, which is quite different from those of the usual electromagnetic field without Weyl correction in the four-dimensional spacetime. With increase of Weyl correction parameter, the absorption probability, the absorption cross section, the power emission spectra and the luminosity of Hawking radiation decreases with Weyl correction parameter for the odd-parity electromagnetic field and increases with the even-parity electromagnetic field.

Asymptotic late-time tails of massive spin-2 fields

The late-time dynamics of massive spin-2 fields in flat and curved spacetimes are studied analytically. We find that the time evolutions of the massive fields are characterized by oscillatory power-law decaying tails at asymptotically late times. In a flat spacetime the decaying exponent depends on the multipole number and the parity of the mode. In the curved Schwarzschild black-hole spacetime the decaying exponent is found to be universal.

Upper bound on the radii of black-hole photonspheres

One of the most remarkable predictions of the general theory of relativity is the existence of black-hole “photonspheres”, compact null hypersurfaces on which massless particles can orbit the central black hole. We prove that every spherically-symmetric asymptotically flat black-hole spacetime is characterized by a photonsphere whose radius is bounded from above by rγ⩽3M, where M is the total ADM mass of the black-hole spacetime. It is shown that hairy black-hole configurations conform to this upper bound. In particular, the null circular geodesic of the (bald) Schwarzschild black-hole spacetime saturates the bound.

Dynamical evolution of the electromagnetic perturbation with Weyl corrections

We present firstly the master equation of an electromagnetic perturbation with Weyl correction in the four-dimensional black hole spacetime, which depends not only on the Weyl correction parameter, but also on the parity of the electromagnetic field. It is quite different from that of the usual electromagnetic perturbation without Weyl correction in the four-dimensional spacetime. And then we have investigated numerically the dynamical evolution of the electromagnetic perturbation with Weyl correction in the background of a four-dimensional Schwarzschild black hole spacetime. Our results show that the Weyl correction parameter $\alpha$ and the parities imprint in the wave dynamics of the electromagnetic perturbation. For the odd parity electromagnetic perturbation, we find it grows with exponential rate if the value of $\alpha$ is below the negative critical value $\alpha_c$. However, for the electromagnetic perturbation with even parity, we find that there does not exist such a critical threshold value and the electromagnetic field always decays in the allowed range of $\alpha$.

Decay of linear waves on higher-dimensional Schwarzschild black holes

In this paper we consider solutions to the linear wave equation on higher dimensional Schwarzschild black hole spacetimes and prove robust nondegenerate energy decay estimates that are in principle required in a nonlinear stability problem. More precisely, it is shown that for solutions to the wave equation \Box_g\phi=0 on the domain of outer communications of the Schwarzschild spacetime manifold (M^n_m, g) (where n >= 3 is the spatial dimension, and m > 0 is the mass of the black hole) the associated energy flux E[\phi](\Sigma_\tau) through a foliation of hypersurfaces (\Sigma_\tau) (terminating at future null infinity and to the future of the bifurcation sphere) decays, E[\phi](\Sigma_\tau) <= CD/\tau^2, where C is a constant only depending on n and m, and D < \infty is a suitable higher order initial energy on \Sigma_0; moreover we improve the decay rate for the first order energy to E[\partial_t\phi](\Sigma_\tau^R) <= CD/\tau^(4-2\delta) for any \delta > 0 where \Sigma_\tau^R denotes the hypersurface (\Sigma_\tau) truncated at an arbitrarily large fixed radius R < \infty provided the higher order energy D_\delta on \Sigma_0 is finite. We conclude our paper by interpolating between these two results to obtain the pointwise estimate |\phi|_{\Sigma_\tau^R} <= (C D'_\delta) / \tau^(3/2-\delta). In this work we follow the new physical-space approach to decay for the wave equation of Dafermos and Rodnianski.

Analytic investigation of the branch cut of the Green function in Schwarzschild space-time

The retarded Green function for linear field perturbations in Schwarzschild black hole space-time possesses a branch cut in the complex-frequency plane. This branch cut has remained largely unexplored: only asymptotic analyses either for small-frequency (yielding the known tail decay at late times of an initial perturbation of the black hole) or for large-frequency (quasinormal modes close to the branch cut in this regime have been linked to quantum properties of black holes) have been carried out in the literature. The regime along the cut inaccessible to these asymptotic analyses has so far remained essentially unreachable. We present a new method for the analytic calculation of the branch cut directly on the cut for general-spin fields in Schwarzschild space-time. This method is valid for any values of the frequency on the cut and so it provides analytic access to the whole branch cut for the first time. We calculate the modes along the cut and investigate their properties and connection with quasinormal modes. We also investigate the contribution from these branch cut modes to the self-force acting on a point particle on a Schwarzschild background space-time.

Conformal scalar propagation and Hawking radiation

The construction of the conformal scalar propagator which has been obtained in the preceding two projects as an analytic function of the Schwarzschild black-hole space-time is completed with a boundary condition imposed by the physical context through contour integration in the exterior vicinity of the event horizon. It is shown that, as a consequence of the semi-classical character which the emitted quanta have in that exterior vicinity, the particle production by the Schwarzschild black hole which was formally established in the preceding project is identical to thermal Hawking radiation. By extension, it is established that such a particle production corresponds to a spectrum which detracts from thermality by the amount predicted by Parikh and Wilczek if energy conservation is properly imposed as a constraint on scalar propagation. The results obtained herein support the case made by Hawking on the relation between quantum propagation and observation of particles produced by a black hole.

Branch cut and quasinormal modes at large imaginary frequency in Schwarzschild space-time

The 'retarded' Green function for fields propagating on a Schwarzschild black hole spacetime possesses a branch cut on the complex frequency plane. Classically, the branch cut is important, for example, in order to fully determine the response of the black hole to a linear field perturbation. The branch cut is also useful for the calculation of the self-force on a point particle moving in the Schwarzschild background. In this paper we use techniques of analytic-continuation to the complex plane of the radial coordinate in order to calculate the branch cut contribution to the Green function in the limit of large imaginary frequency. It is expected that the contribution of this frequency regime to the perturbation response and to the self-force will be mostly for short time intervals. We also determine the highly-damped quasinormal mode frequencies for electromagnetic perturbations in Schwarzschild for the first time (previously only the leading imaginary part was known), which seem to have a 'deep connection' with the branch cut. We find that these frequencies behave like $\omega_{\ell,n}= -\dfrac{in}{2}-\dfrac{i[\ell(\ell+1)]^2}{2n}+\dfrac{\pi^{1/2}(1-i)[\ell(\ell+1)]^3}{2^{3/2}n^{3/2}}+O(n^{-2})$. The highly-damped quasinormal modes are particularly interesting for theories of quantum gravity in that they are believed to probe the small scale structure of the spacetime.

Spherical nonlinear absorption of cosmological scalar fields onto a black hole

In this paper we track the nonlinear spherical evolution of a massless scalar field onto a Schwarzschild black hole space-time as a first approximation to the accretion of cosmologically motivated classical scalar fields. We perform an analysis related to wave packets described by wave number and width. We study various values of the wave number $k$ and find that for $k=0$ and width packets bigger than the Schwarzschild radius, the absorption is not total. In the cases we study for $k&gt;0$, the black hole absorbs the total amount of energy density of the scalar field moving toward the horizon. Our results indicate that assuming spherical symmetry in the nonlinear regime, there are cases for which scalar fields are allowed to survive outside black holes and may eventually have lifetimes consistent with cosmological time scales.

Correction to the Dirac field entropy of a Schwarzschild black hole by a modified dispersion relation

Taking the WKB approximation to solve the Dirac field equation in a Schwarzschild black hole spacetime, we can get the classical momenta. Employing the classical momenta and state density equation corrected by a modified dispersion relation, we will obtain the number of quantum states with energy less than $\ensuremath{\omega}$. Then, it is used to calculate the statistical-mechanical entropy of the Dirac field. Solving the integral for $r$ exactly, we obtain the leading term of entropy that is proportional to the event horizon area, and correction terms take the form of ${A}^{\ensuremath{-}1}$, but the logarithmic correction term by this approach is not given.

Correction to Entropy of Schwarzschild Black Hole by Modified Dispersion Relation

Taking WKB approximation to solve the scalar field equation in the Schwarzschild black hole spacetime, we can get the classical momenta. Substituting the classical momenta into state density equation corrected by the modified dispersion relation, we will obtain the number of quantum states with energy less than ω. Then, it is used to calculate the statistical-mechanical entropy of the scalar field in the Schwarzschild black hole spacetime. By taking exact method, we obtained the leader term of entropy which is proportional to the event horizon area and correction terms take the forms of ln A, A−1ln A, A−1 and so on.

Strong gravitational lensing in a noncommutative black-hole spacetime

Noncommutative geometry may be a starting point to a quantum gravity. We study the influence of the spacetime noncommutative parameter on the strong field gravitational lensing in the noncommutative Schwarzschild black-hole spacetime and obtain the angular position and magnification of the relativistic images. Supposing that the gravitational field of the supermassive central object of the galaxy described by this metric, we estimate the numerical values of the coefficients and observables for strong gravitational lensing. Comparing to the Reissner-Norstr\"{om} black hole, we find that the influences of the spacetime noncommutative parameter is similar to those of the charge, just these influences are much smaller. This may offer a way to distinguish a noncommutative black hole from a Reissner-Norstr\"{om} black hole, and may probe the spacetime noncommutative constant $\vartheta$ \cite{foot} by the astronomical instruments in the future.

A causal set black hole

We explicitly compute the causal structure of the Schwarzschild black hole spacetime, by providing an algorithm to decide if any pair of events is causally related. The primary motivation for this study comes from discrete quantum gravity, in particular the causal set approach, in which the fundamental variables can be thought of as the causal ordering of randomly selected events in spacetime. This work opens the way to simulating non-conformally flat spacetimes within the causal set approach, which may allow one to study important questions such as black hole entropy and Hawking radiation on a full four-dimensional causal set black hole.

ABSORPTION CROSS-SECTION AND EMISSION SPECTRA OF SCHWARZSCHILD BLACK HOLE IN DIRAC FIELD

The behavior of a Dirac field in a Schwarzschild black hole spacetime is studied. In this work the Hawking temperature and the absorption cross-section for Schwarzschild black hole placed in Dirac field are calculated, taking into consideration the matter waves reflected from the event horizon. The absorption cross-section σ abs in Dirac field is found to be ⅛ of absorption cross-section in scalar field. The emission spectra of Schwarzschild black hole placed in an environment of Dirac field is also obtained.

Influence of Dark Energy on Gravitational Time Delay

We investigate the gravitational time delay of light in the Schwarzschild black hole space-time surrounded by quintessence. With the analysis and numerical methods, we find that the gravitational time delay of light in the Schwarzschild black hole space-time surrounded by quintessence increases when the normalization factor c increases, and that the gravitational time delay also decreases when the quintessential state parameter ωq increases.

Gravitational frequency-shift and deflection of light by quintessence

In this paper we investigate the gravitational frequency-shift and deflection of light in the Schwarzschild black hole space-time surrounded by quintessence. With the analytical and numerical methods, we find that the gravitational frequency-shift of light in the Schwarzschild black hole space-time surrounded by the quintessence increases as the values of the normalization factor c increases, but the gravitational frequency-shift of light decreases with the quintessential state parameter ωq increasing. We also calculate the deflection of light by quintessence and find that the deflection rate decreases as the values of the quintessential parameters c and ωq increase.

Approximation in Thin Film Brick-Wall Model

The thin film brick-wall model is an important method on calculating the entropy of the non-static black hole. But we can see in this paper that the approximation used in the thin film brick-wall model is a rate of two infinitesimal parameters in fact. in this paper, we make an exact calculation of the approximation term and give the solution in the Schwarzschild black hole space-time.

Influence of Lorentz violation on Dirac quasinormal modes in the Schwarzschild black hole spacetime

Using the third-order WKB approximation and monodromy methods, we investigate the influence of the Lorentz violating coefficient b (associated with a special axial-vector bμ field ) on Dirac quasinormal modes in the Schwarzschild black hole spacetime. At fundamental overtone, the real part decreases linearly as the parameter b increases. But the variation of the imaginary part with b becomes more complex. For the larger multiple moment k, the magnitude of the imaginary part increases with the increase of b, which means that the presence of Lorentz violation makes Dirac field damps more rapidly. At high overtones, it is found that the real part of highly-damped quasinormal frequency does not tend to zero, which is quite different from the asymptotic Dirac quasinormal modes without Lorentz violation.

WKB analysis of the scattering of massive Dirac fields in Schwarzschild black-hole spacetimes

We analyse the radial equations for massive Dirac fields in Schwarzschild black-hole spacetimes. Different approximation formulae under the WKB scheme are developed for the transmission probability of the radial wavefunction with E2 ≫ Vm, E2 ≈ Vm and E2 ≪ Vm, where E is the energy of the field and Vm is the maximum value of the effective potential. Explicit results of in these approximations are given for various values of E, the mass m, and the angular momentum parameter κ of the fields. We also discuss the dependence of on these parameters.