Static Black String Research Articles

Black strings from dark matter

In this paper, we obtain two static black string solutions by considering as sources axisymmetric dark matter (DM) distributions in 3+1 dimensions, in two different scenarios, namely, purely isotropic and radial orbits, which are regulated by two DM parameters. Both solutions tend asymptotically to the usual static and uncharged black string vacuum solution predicted by General Relativity (GR), and each of them presents an event horizon, which is larger than the one corresponding to the vacuum solution, as it is shown. We also show that the presence of DM turns these solutions singular at their symmetry axis. Then, we calculate the Hawking temperatures of these black string horizons and discuss some of their consequences. Unlike what occurs with the static black string vacuum solution, we find that there exists a linear density of mass (or tension) remnant associated with a vanishing Hawking temperature for the solutions under consideration. Thus, we analyze how the presence of DM affects the occurrence of the remnants. Further, we calculate other thermodynamic quantities, namely entropy, heat capacity, and free energy per unit length, showing that thermal phase transitions can occur in the presence of DM. We also analyze the weak (and null) energy conditions and conclude that DM does not behave like an exotic fluid. Finally, the corresponding stationary solutions are obtained, as well as their new tensions as functions of both the mass and angular momentum of the black strings.

R2 corrections to the black string instability and the boosted black string

We consider a perturbative Gauss-Bonnet term supplementing the Einstein-Hilbert action, and evaluate its effect on the spectrum of the scalar mode that triggers the Gregory-Laflamme instability of black strings in five dimensional General Relativity. After studying some properties of the static black string, we provide the correction to the Lichnerowicz operator up to $\mathcal{O}(\alpha^2)$. For the scalar mode of the gravitational perturbation, we find a master variable and study its spectrum, providing an analysis of the regime of validity of our scheme. We show that the instability persists under the inclusion of the $R^2$-correction, and that the critical wavelength increases with the value of $\alpha/r_+^2<<1$, providing new evidence of the phenomenon which was already observed in other approaches. We also construct the boosted black strings and compute the correction to the mass, the momentum and the tension due to the higher curvature term. The presence of the dimensionful coupling $\alpha$ spoils the validity of the Smarr relation, which is the gravitational version of the Euler relation which must hold for every homogeneous, thermodynamic system. We show that this identity can be restored by working in an extended thermodynamic setup that includes variations of the Gauss-Bonnet coupling.

Four-dimensional regular black strings in bilocal gravity

In this paper, we obtain a static black string solution for a bilocal gravitational source in 3+1 dimensions. The solution is regular at the origin and tends asymptotically to the ordinary static uncharged black string solution of general relativity. It allows an event horizon and an internal horizon depending on the value of the mass density. A mass remnant associated with a vanishing Hawking temperature is also found. In order to stabilize the solution, a perfect cosmological fluid with state parameter ω>−1 should be present throughout space. However, energy conditions do not exclude an exotic substance nearby the black string. Finally, we find the stationary counterpart of the solution and analyze the behavior of the horizons according to the mass and angular momentum of the spinning object.

Particle dynamics around the black string

In this paper, some dynamical properties of neutral and charged particles around a weakly magnetized five-dimensional static black string have been studied. The perturbation method was also used to calculate the innermost stable circular orbit (ISCO) of this metric in the presence of a magnetic field. The escape velocity of neutral and charged particles around the black string was derived. In the next step, the analytical solutions of the equations of motion were discussed and some possible orbits for particles in the black string space–time were plotted. Interestingly, it was found that adding an extra dimension has a slight influence on the effective potential and one term of the effective force. The magnitude of the new constant of motion (J) affects both the shape of the potential and existence of the stable circular orbits. In conclusion, by comparing a black hole and a black string, it is realized that the value of a new constant of motion causes slight but interesting differences.

From quasinormal modes of rotating black strings to hydrodynamics of a moving CFT plasma

A certain identification of points in a planar Schwarzschild-anti de Sitter (AdS) black hole generates a four-dimensional static black string. In turn, a rotating black string can be obtained from a static one by means of an `illegitimate coordinate transformation', a local boost in the compact direction. On the basis of the gauge/gravity duality, these black strings are dual to rotating thermal states of a strongly interacting conformal field theory (CFT) that lives on a cylinder. In this work, we obtain the complete quasinormal mode (QNM) spectrum of the gravitational perturbations of rotating black strings. Analytic solutions for the dispersion relations are found in the hydrodynamic limit, characterized by fluctuations with wavenumber and frequency much smaller than the Hawking temperature of the string (or the temperature in the CFT dual description). We obtain these dispersion relations both by studying the gravitational perturbations of rotating black strings and by investigating the hydrodynamic fluctuations of a moving fluid living on the boundary of the AdS spacetime. Relativistic effects like the Doppler shift of the frequencies, wavelength contraction, and dilation of the thermalization time are shown explicitly in such a regime. We also investigate the behavior of a sound wave propagating in a viscous fluid for several values of the rotation parameter. The numerical solutions for the fundamental QNMs show a crossover (a transition) from a hydrodynamic-like behavior to a linear relativistic scaling for large wavenumbers. Additionally, we find a new family of QNMs which are purely damped in the zero wavenumber limit and that does not follow as a continuation of QNMs of the static black string, but that appears to be closely related to the algebraically special perturbation modes.

Acceleration of particles in spacetimes of black string

The particle acceleration mechanism in $S^2 \times\mathbb{R}^1$ topology, namely in the spacetime of the five-dimensional compact black string has been studied. The expression of center-of-mass energy of the colliding neutral particles near static black string has been found. The collision of charged particle moving at the innermost stable circular orbit with neutral particle coming from infinity have been considered when black string is immersed in external uniform magnetic field. It has been shown that the unlimited center-of-mass energy can be approached in case of the extremal rotation of the black string which is similar to the analogous effect in the Kerr spacetime. We have also obtained that the scattering energy of particles in the center-of-mass system can take arbitrary large values not only for extremal black string but also for nonextremal one. It has been derived that the presence of the extra dimension can, in principle, increase the upper limit of efficiency of energy extraction from the extremely rotating black string up to $203\%$ versus to $143\%$ which can be extracted from the extreme Kerr black hole.

Black holes and black strings of N = 2, d = 5 supergravity in the H-FGK formalism

We study general classes and properties of extremal and non-extremal static black-hole solutions of N = 2, d = 5 supergravity coupled to vector multiplets using the recently proposed H-FGK formalism, which we also extend to static black strings. We explain how to determine the integration constants and physical parameters of the blackhole and black-string solutions. We derive some model-independent statements, including the transformation of non-extremal flow equations to the form of those for the extremal flow. We apply our methods to the construction of example solutions (among others a new extremal string solution of heterotic string theory on K 3 × S 1). In the cases where we have calculated it explicitly, the product of areas of the inner and outer horizon of a non-extremal solution coincides with the square of the moduli-independent area of the horizon of the extremal solution with the same charges.

Black strings in Hořava-Lifshitz gravity

We examine a class of cylindrically symmetric solutions in Horava-Lifshitz gravity. For the relativistic value of the coupling constant, $ \lambda=1 $, we find the "hedgehog" type static black string solution with the nonvanishing radial shift in the ADM-type decomposition of the spacetime metric. With zero radial shift, this solution corresponds to the usual BTZ black string in general relativity. However, unlike the general relativity case, the BTZ type black strings do naturally exist in HL gravity, without the need for any specific source term. We also find a rotating BTZ type black string solution which requires the nonvanishing radial shift for its very existence. We calculate the mass and the angular momentum of this solution, using the canonical Hamiltonian approach. Next, we discuss the Lemos type black string, which is inherent in general relativity with a negative cosmological constant, and present the static metric for any value of $ \lambda > 1/3 $. Finally, we show that while, for $\lambda=1 $, the entropy of the Lemos type black string is given by one quarter of the horizon area, the entropy of the static BTZ type black string is one half of its horizon area.

Brane holes

The aim of this paper is to demonstrate that in models with large extra dimensions under special conditions one can extract information from the interior of 4D black holes. For this purpose we study an induced geometry on a test brane in the background of a higher-dimensional static black string or a black brane. We show that, at the intersection surface of the test brane and the bulk black string or brane, the induced metric has an event horizon, so that the test brane contains a black hole. We call it a brane hole. When the test brane moves with a constant velocity $V$ with respect to the bulk black object, it also has a brane hole, but its gravitational radius ${r}_{e}$ is greater than the size of the bulk black string or brane ${r}_{0}$ by the factor $(1\ensuremath{-}{V}^{2}{)}^{\ensuremath{-}1}$. We show that bulk ``photon'' emitted in the region between ${r}_{0}$ and ${r}_{e}$ can meet the test brane again at a point outside ${r}_{e}$. From the point of view of observers on the test brane, the events of emission and capture of the bulk photon are connected by a spacelike curve in the induced geometry. This shows an example in which extra dimensions can be used to extract information from the interior of a lower-dimensional black object. Instead of the bulk black string or brane, one can also consider a bulk geometry without a horizon. We show that nevertheless the induced geometry on the moving test brane can include a brane hole. In such a case the extra dimensions can be used to extract information from the complete region of the brane-hole interior. We discuss thermodynamic properties of brane holes and interesting questions which arise when such an extra-dimensional channel for the information mining exists.

Black strings with a negative cosmological constant: inclusion of electric charge and rotation

We generalize the vacuum static black string solutions of Einstein's equations with negative cosmological constant recently discussed in literature, by including an electromagnetic field. These higher-dimensional configurations have no dependence on the ‘compact’ extra dimension, while their boundary topology is the product of time and Sd−3 × S1 or Hd−3 × S1. Rotating generalizations of the even dimensional black string configurations are considered as well. Different from the static, neutral case, no regular limit is found for a vanishing event horizon radius. We explore numerically the general properties of such classical solutions and, using a counterterm prescription, we compute their conserved charges and discuss their thermodynamics. We find that the thermodynamics of the black strings follows the pattern of the corresponding black hole solutions in AdS backgrounds.

Black rings, boosted strings, and Gregory-Laflamme instability

We investigate the Gregory-Laflamme instability for black strings carrying KK momentum along the internal direction. We demonstrate a simple kinematical relation between the thresholds of the classical instability for the boosted and static black strings. We also find that Sorkin's critical dimension depends on the internal velocity and in fact disappears for sufficiently large boosts. Our analysis implies the existence of an analogous instability for the five-dimensional black ring of Emparan and Reall. We also use our results for boosted black strings to construct a simple model of the black ring and argue that such rings exist in any number of space-time dimensions.

Fate of the black string instability.

Gregory and Laflamme showed that certain nonextremal black strings (and p-branes) are unstable to linearized perturbations. It is widely believed that this instability will cause the black string horizon to classically pinch off and then quantum mechanically separate, resulting in higher dimensional black holes. We argue that this cannot happen. Under very mild assumptions, classical event horizons cannot pinch off. Instead, they settle down to new static black string solutions which are not translationally invariant along the string.