Om Spacetime Research Articles

Analytic solutions of the geodesic equation in higher dimensional static spherically symmetric spacetimes

The complete analytical solutions of the geodesic equation of massive test particles in higher dimensional Schwarzschild, Schwarzschild--(anti)de Sitter, Reissner-Nordstr\"om and Reissner--Nordstr\"om--(anti)de Sitter spacetimes are presented. Using the Jacobi inversion problem restricted to the theta divisor the explicit solution is given in terms of Kleinian sigma functions. The derived orbits depend on the structure of the roots of the characteristic polynomials which depend on the particle's energy and angular momentum, on the mass and the charge of the gravitational source, and the cosmological constant. We discuss the general structure of the orbits and show that due to the specific dimension-independent form of the angular momentum and the cosmological force a rich variety of orbits can emerge only in four and five dimensions. We present explicit analytical solutions for orbits up to 11 dimensions. A particular feature of Reissner-Nordstr\"om spacetimes is that bound and escape orbits traverse through different universes.

Black hole formation in the head-on collision of ultrarelativistic charges

We study black hole formation in the head-on collision of ultrarelativistic charges. The metric of charged particles is obtained by boosting the Reissner-Nordstr\om spacetime to the speed of light. Using the slice at the instant of collision, we study formation of the apparent horizon (AH) and derive a condition indicating that a critical value of the electric charge is necessary for formation to take place. Evaluating this condition for characteristic values at the LHC, we find that the presence of charge decreases the black hole production rate in accelerators. We comment on possible limitations of our approach.

Avoidance of singularities in spherically symmetric charged dust

In spherically symmetric charged dust, just like in neutral dust, two kinds of singularity may be present: the big bang/big crunch (BB/BC) singularity, and shell crossings. Quite unlike in neutral dust, the BB/BC singularity may be avoided. When the charge density ${\ensuremath{\rho}}_{e}$ and the mass-energy density $ϵ$ obey $|{\ensuremath{\rho}}_{e}|<\phantom{\rule{0ex}{0ex}}{D}_{e}\stackrel{\mathrm{def}}{=}\sqrt{G}ϵ/{c}^{2}$, the conditions that allow the model to avoid the BB/BC singularity necessarily lead to shell crossings. However, when $|{\ensuremath{\rho}}_{e}|\ensuremath{\rightarrow}{D}_{e}$ at the center of symmetry while $|{\ensuremath{\rho}}_{e}|<{D}_{e}$ elsewhere, both kinds of singularity may be avoided for a sufficiently long period that a body of charged dust may go through the tunnel between the singularities in the maximally extended Reissner-Nordstr\"om spacetime, to emerge into another asymptotically flat region. An explicit example of such a configuration is presented and discussed. It does not contradict any astrophysical constraints.

Hawking radiation as tunneling from the Kerr and Kerr-Newman black holes

Recent work, which treats the Hawking radiation as a semiclassical tunneling process at the horizon of the Schwarzschild and Reissner-Nordstr\om spacetimes, indicates that the exact radiant spectrum is no longer pure thermal after considering the black hole background as dynamical and the conservation of energy. In this paper, we extend the method to investigate Hawking radiation as massless particles tunneling across the event horizon of the Kerr black hole and that of charged particles from the Kerr-Newman black hole by taking into account the energy conservation, the angular momentum conservation, and the electric charge conservation. Our results show that when self-gravitation is considered, the tunneling rate is related to the change of Bekenstein-Hawking entropy and the derived emission spectrum deviates from the pure thermal spectrum, but is consistent with an underlying unitary theory.

Letter: Embedding Diagrams for the Reissner-Nordström Spacetime

We consider embedding diagrams for the Reissner-Nordstr\"om spacetime. We embed the $(r-t)$ and $(r-\phi)$ planes into 3-Minkowski/Euclidean space and discuss the relation between the diagrams and the corresponding curvature scalar of the 2-metrics

Gravitating magnetic monopole in the global monopole spacetime

In this paper we study the regular self-gravitating 't Hooft--Polyakov magnetic monopole in a global monopole spacetime. We show that, for the large distance, the structure of the manifold corresponds to the Reissner-Nordstr\"om spacetime with a solid angle deficit factor. Although we analyze static and spherically symmetric solutions, it is not possible to solve analytically the system of coupled differential equations and only numerical evaluations can provide detailed information about the behavior of this system at the neighborhood of the defect's core. So, for this reason we solve numerically the set of differential equations for the metric tensor and for the matter fields for different values of the Higgs field vacuum expectation value \ensuremath{\eta} and the self-coupling constant \ensuremath{\lambda}.

Stress-energy tensor in soluble models of spherically symmetric charged black hole evaporation

We study the decay of a near-extremal black hole in AdS$_2$, related to the near-horizon region of 3+1-dimensional Reissner-Nordstr\"om spacetime, following Fabbri, Navarro, and Navarro-Salas. Back-reaction is included in a semiclassical approximation. Calculations of the stress-energy tensor of matter coupled to the physical spacetime for an affine null observer demonstrate that the black hole evaporation proceeds smoothly and the near-extremal black hole evolves back to an extremal ground state, until this approximation breaks down.

Late-time Evolution of the Yang-Mills Field in the Spherically Symmetric Gravitational Collapse

We investigate the late-time evolution of the Yang-Mills field in the self-gravitating backgrounds: Schwarzschild and Reissner-Nordstr\"om spacetimes. The late-time power-law tails develop in the three asymptotic regions: the future timelike infinity, the future null infinity and the black hole horizon. In these two backgrounds, however, the late-time evolution has quantitative and qualitative differences. In the Schwarzschild black hole background, the late-time tails of the Yang-Mills field are the same as those of the neutral massless scalar field with multipole moment l=1. The late-time evolution is dominated by the spacetime curvature. When the background is the Reissner-Nordstr\"om black hole, the late-time tails have not only a smaller power-law exponent, but also an oscillatory factor. The late-time evolution is dominated by the self-interacting term of the Yang-Mills field. The cause responsible for the differences is revealed.

Late-time tails in gravitational collapse of a self-interacting (massive) scalar-field and decay of a self-interacting scalar hair

We study analytically the initial value problem for a self-interacting (massive) scalar field on a Reissner-Nordstr\"om spacetime. Following the no-hair theorem we examine the dynamical physical mechanism by which the self-interacting(SI) hair decays. We show analytically that the intermediate asymptotic behavior of SI perturbations is dominated by an oscillatory inverse power-law decaying tail. We confirm numerically this result. However, the numerical examination reveals that at late times the decay of SI hair is slower than any power law.

Late-time evolution of charged gravitational collapse and decay of charged scalar hair. II

We study analytically the initial value problem for a charged massless scalar-field on a Reissner-Nordstr\"om spacetime. Using the technique of spectral decomposition we extend recent results on this problem. Following the no-hair theorem we reveal the dynamical physical mechanism by which the charged hair is radiated away. We show that the charged perturbations decay according to an inverse power-law behaviour at future timelike infinity and along future null infinity. Along the future outer horizon we find an oscillatory inverse power-law relaxation of the charged fields. We find that a charged black hole becomes ``bald'' slower than a neutral one, due to the existence of charged perturbations. Our results are also important to the study of mass-inflation and the stability of Cauchy horizons during a dynamical gravitational collapse of charged matter in which a charged black-hole is formed.

Constant mean curvature slices and trapped surfaces in asymptotically flat spherical spacetimes.

We investigate trapped surfaces in asymptotically flat spherical spacetimes using constant mean curvature slicing. Precise necessary and sufficient conditions for the formation of such surfaces are derived. We write down an explicit expression for the constant mean curvature foliation of the Reissner-Nordstr\"om spacetime. A set of criteria describing the formation of horizons in arbitrary slicings of asymptotically flat and spherically symmetric spacetimes is given in the Appendix.

Inevitability of shell crossing in the gravitational collapse of weakly charged dust spheres.

It is shown here that shell crossing is inevitable in the gravitational collapse of weakly charged (\ensuremath{\Vert}charge density\ensuremath{\Vert}) dust spheres. That is, all the dust shells in the central part of the dust cloud are subject to shell crossing before they reach the stage of maximal contraction. This shell crossing, which is accompanied by a density singularity, indicates the breakdown of the dust model there. Most previous analyses of the gravitational collapse and gravitational bounce of charged dust spheres were based on the ``free-surface approach''---the assumption that the surface of the dust sphere moves like a test particle in Reissner-Nordstr\om spacetime. Because of the occurrence of shell crossing, the free-surface approach cannot be applied to the late stages of evolution. This makes it difficult to analyze, in general, what will be the final stage in the gravitational collapse of charged dust. However, for idealized models, in which the dust interior is considered to be self-similar and the shells are considered to collide inelastically, the shell crossing is shown to have a very significant effect on the causal structure: It completely prevents the gravitational bounce. This result, the inevitability of shell crossing (and its potential to affect drastically the causal structure), is in remarkable contrast to common ideas about charged-dust collapse.

Quantum wormholes.

This paper presents an application of quantum-mechanical principles to a microscopic variant of the traversable wormholes recently introduced by Morris and Thorne. The analysis, based on the surgical grafting of two Reissner-Nordstr\"om spacetimes, proceeds by using a minisuperspace model to approximate the geometry of these wormholes. The "thin shell" formalism is applied to this minisuperspace model to extract the effective Lagrangian appropriate to this one-degree-of-freedom system. This effective Lagrangian is then quantized and the wave function for the wormhole is explicitly exhibited. A slightly more general class of wormholes---corresponding to the addition of some "dust" to the wormhole throat---is analyzed by recourse to WKB techniques. In all cases discussed in this paper, the expectation value of the wormhole radius is calculated to be of the order of the Planck length. Accordingly, though these quantum wormholes are of considerable theoretical interest they do not appear to be useful as a means for interstellar travel. The results of this paper may also have a bearing on the question of topological fluctuations in quantum gravity. These calculations serve to suggest that topology-changing effects might in fact be suppressed by quantum-gravity effects.

A method to compute < phi2> in asymptotically flat, static, spherically symmetric spacetimes.

A method is presented which allows for the computation of $〈{\ensuremath{\varphi}}^{2}〉$ for scalar fields at zero and nonzero temperatures in asymptotically flat, static, spherically symmetric spacetimes. If a spacetime has event horizons, the method works in the region exterior to the outermost event horizon. The method is used to compute $〈{\ensuremath{\varphi}}^{2}〉$ for massless and massive scalar fields in Reissner-Nordstr\om and extreme Reissner-Nordstr\om spacetimes.

Maximally slicing a black hole with minimal distortion.

Equations are derived which determine the maximal hypersurfaces of the analytically extended Kerr-Newman spacetime. An analytic solution is obtained in the charged, nonrotating case for the asymptotically flat maximal hypersurfaces of the Reissner-Nordstr\"om spacetime using spatial coordinates which minimize the coordinate distortion. The slices tend asymptotically in time to a limiting hypersurface lying between the inner and outer horizons, while covering the domain of outer communication of the black hole. The coordinate lines are drawn down the black hole if coordinate symmetry is maintained across the throat. The equation for the limiting hypersurface in the Kerr geometry is solved numerically. An apparently unique solution exists for all rotating black holes with specific angular momentum \ensuremath{\Vert}a\ensuremath{\Vert}&lt;M, where M is the black-hole mass. An excellent analytic approximation is derived for the value of the Boyer-Lindquist coordinate r on the hypersurface. Implications for gravitational-collapse calculations are discussed.

Gravitational collapse of a charged fluid sphere

A class of solutions of Einstein's gravitational field equations is discussed which describes the collapse or expansion of a charged, perfect-fluid spherical distribution of matter. These solutions reduce in the appropriate limits to certain charged Newtonian polytropes. A physical interpretation of the collapsing configurations is given, and it is shown that these solutions can describe the gravitational collapse of a bounded, charged fluid around a charged black hole. In all these configurations the singular region is either a spacelike or a null hypersurface. Therefore, the final state of collapse cannot be described by the complete analytic extension of the Reissner-Nordstr\om spacetime. The special case of uniform density models is investigated in detail and it is shown that they describe the accretion of neutral matter by a charged black hole. On the basis of this analysis, it is suggested that for the realistic collapse of charged matter the singular region formed within the matter is either spacelike or null.

Horizon stability in asymptotically flat spacetimes

It is shown that, provided the ideal point structure of spacetime is stable, generic infinitesimal perturbations cause singularities to appear on the event horizons in both the maximally extended Schwarzschild spacetime and the maximally extended Reissner-Nordstr\om spacetime. Furthermore, singularities will also appear on the inner event of the maximally extended Reissner-Nordstr\om solution when this spacetime is perturbed generically. The Cauchy horizon appearing in the Reissner-Nordstr\om solution is shown to be unstable to spherically symmetric infinitesimal perturbations. The barrier that prevents travel through a wormhole from one asymptotically flat region to another is shown in the Schwarzschild case to be stable to spherically symmetric perturbations.