Schwarzschild Black String Research Articles

The Gregory–Laflamme instability of the Schwarzschild black string exterior

In this paper, a direct rigorous mathematical proof of the Gregory–Laflamme instability for the five-dimensional Schwarzschild black string is presented. Under a choice of ansatz for the perturbation and a gauge choice, the linearized vacuum Einstein equation reduces to an ordinary differential equation (ODE) problem for a single function. In this work, a suitable rescaling and change of variables is applied, which casts the ODE into a Schrödinger eigenvalue equation to which an energy functional is assigned. It is then shown by direct variational methods that the lowest eigenfunction gives rise to an exponentially growing mode solution, which has admissible behavior at the future event horizon and spacelike infinity. After the addition of a pure gauge solution, this gives rise to a regular exponentially growing mode solution of the linearized vacuum Einstein equation in harmonic/transverse-traceless gauge.

Strongly coupled quarks and colorful black holes in AdS/CFT correspondence

Small black holes potentially created by the Large Hadron Collider could typically carry color charges inherited from their parton progenitors. The dynamics of quarks near such a black hole depends on the curved spacetime geometry as well as the strong interaction due to the color charge of the black hole. We use the AdS/CFT Correspondence to study the behavior of strongly-coupled quarks near a color-charged black hole. The supergravity background consists of a six-dimensional Schwarzschild-black string AdS soliton, for which the bulk horizon extends from the AdS boundary down to an infra-red floor. By going to higher energy scales, the regime of validity of the classical supergravity background can be extended closer to the singularity than might be expected from the four-dimensional perspective. We study the resulting behavior of quarks and compute the rate at which a quark rotating around the black hole loses energy as a function of its mass parameter and angular velocity. We also analyze the behavior of quark-antiquark string configurations and discuss features such as the quark-antiquark screening length.

HYPERCYLINDRICAL VACUUM SPACETIME SOLUTIONS

A brief report is given on hypercylindrical vacuum spacetime solutions and their geometrical properties. The five-dimensional Schwarzschild black string solution characterized by the mass density, which is known to reveal the so-called Gregory-Laflamme instability, turns out to be a special case of a wider class of hypercylindrical vacuum spacetime solutions characterized by two parameters, i.e., mass density and tension. Recent work on the solution space even including a momentum flow or a cosmological constant, higher dimensional extensions than five and some stability analysis under small perturbations are briefly summarized.

Dynamical evolution of phantom scalar perturbation in the Schwarzschild black string spacetime

Using Leaver's continue fraction and time domain method, we study the wave dynamics of phantom scalar perturbation in a Schwarzschild black string spacetime. We find that the quasinormal modes contain the imprint from the wavenumber k of the fifth dimension. The late-time behaviors are dominated by the difference between the wavenumber k and the mass μ of the phantom scalar perturbation. For k μ, the late-time evolution of phantom scalar perturbation is dominated by a decaying tail with an oscillation which is consistent with that of the usual massive scalar field. Thus, the Schwarzschild black string is unstable only against the phantom scalar perturbations which satisfy the wavelength λ > 2π/μ. These information can help us know more about the wave dynamics of phantom scalar perturbation and the properties of black string.

Spacetime Structure of 5D Hypercylindrical Vacuum Solutions with Tension

We investigate geometrical properties of 5D cylindrical vacuum solutions with a transverse spherical symmetry. The metric is uniform along the fifth direction and characterized by tension and mass densities. The solutions are classified by the tension-to-mass ratio. One particular example is the well-known Schwarzschild black string which has a curvature singularity enclosed by a horizon. We focus mainly on geometry of other solutions which possess a naked singularity. The light signal emitted by an object approaching the singularity reaches a distant observer with finite time, but is infinitely red-shifted.

BLACK STRING AND VELOCITY FRAME DRAGGING

We investigate velocity frame dragging with the boosted Schwarzschild black string solution and the boosted Kaluza–Klein bubble solution, in which a translational symmetry along the boosted z-coordinate is implemented. The velocity frame dragging effect can be nullified by the motion of an observer using the boost symmetry along the z-coordinate if it is not compact. However, in spacetime with the compact z-coordinate, we show that the effect cannot be removed since the compactification breaks the global Lorentz boost symmetry. As a result, the comoving velocity depends on r and the momentum parameter along the z-coordinate becomes an observer independent characteristic quantity of the black string and bubble solutions. The dragging induces a spherical ergo-region around the black string.

KALUZA–KLEIN SOLITONS RE-EXAMINED

In 4 + 1 gravity the assumption that the five-dimensional metric is independent of the fifth coordinate permits the extra dimension to be either spacelike or timelike. As a consequence of this, the time coordinate and the extra coordinate are interchangeable, which in turn allows the conception of different scenarios in 4D from a single solution in 5D. In this paper, we make a thorough investigation of all possible 4D scenarios, associated with this interchange, for the well-known Kramer–Gross–Perry–Davidson–Owen set of solutions. We show that there are three families of solutions with very distinct geometrical and physical properties. They correspond to different sets of values of the parameters which characterize the solutions in 5D. The solutions of physical interest are identified on the basis of physical requirements on the induced matter in 4D. We find that only one family satisfies these requirements; the other two violate the positivity of mass-energy density. The "physical" solutions possess a lightlike singularity which coincides with the horizon. The Schwarzschild black string solution as well as the zero moment dipole solution of Gross and Perry are obtained in different limits. These are analyzed in the context of Lake's geometrical approach. We demonstrate that the parameters of the solutions in 5D are not free, as previously considered. Instead, they are totally determined by measurements in 4D — namely, by the surface gravitational potential of the astrophysical phenomena, like the Sun or other stars, modeled in Kaluza–Klein theory. This is an important result which may help in observations for an experimental/observational test of the theory.

Quantum effects in black holes from the Schwarzschild black string?

The holographic conjecture for black holes localized on a 3-brane in Randall–Sundrum braneworld models RS2 predicts the existence of a classical 5D time-dependent solution dual to a 4D evaporating black hole. After briefly reviewing recent criticism and presenting some difficulties in the holographic interpretation of the Gregory–Laflamme instability, we simulate some basic features of such a solution by studying null geodesics of the Schwarzschild black string, in particular those propagating nontrivially in the bulk, and using holographic arguments.

Fate of black string instabilities: An observation

Gregory and Laflamme (hep-th/9301052) have argued that an instability causes the Schwarzschild black string to break up into disjoint black holes. On the other hand, Horowitz and Maeda (arXiv:hep-th/0105111) derived bounds on the rate at which the smallest sphere can pinch off, showing that, if it happens at all, such a pinch-off can occur only at infinite affine parameter along the horizon. An interesting point is that, if a singularity forms, such an infinite affine parameter may correspond to a finite advanced time -- which is in fact a more appropriate notion of time at infinity. We argue below that pinch-off at a finite advanced time is in fact a natural expectation under the bounds derived by Horowitz and Maeda.

Statistical entropy of Schwarzschild black strings and black holes

The statistical entropy of a Schwarzschild black string in five dimensions is obtained by counting the black string states which form a representation of the near-horizon conformal symmetry with a central charge. The statistical entropy of the string agrees with its Bekenstein-Hawking entropy as well as that of the Schwarzschild black hole in four dimensions. The choice of the string length which gives the Virasoro algebra also reproduces the precise value of the Bekenstein-Hawking entropy and lies inside the stability bound of the string.