Spacelike Geodesics Research Articles

Excursions beyond the horizon: black hole singularities in Yang-Mills theories (I)

We study black hole singularities in the AdS/CFT correspondence. These singularities show up in CFT in the behavior of finite-temperature correlation functions. We first establish a direct relation between space-like geodesics in the bulk and momentum space Wightman functions of CFT operators of large dimensions. This allows us to probe the regions inside the horizon and near the singularity using the CFT. Information about the black hole singularity is encoded in the exponential falloff of finite-temperature correlators at large imaginary frequency. We construct new gauge invariant observables whose divergences reflect the presence of the singularity. We also find a UV/UV connection that governs physics inside the horizon. Additionally, we comment on the possible resolution of the singularity.

On charged black holes in anti-de Sitter space

We study the region inside the event horizon of charged black holes in five dimensional asymptotically anti-de Sitter space, using as a probe two-sided correlators which are dominated by spacelike geodesics penetrating the horizon. The spacetimes we investigate include the Reissner-Nordstrom black hole and perturbations thereof. The perturbed spacetimes can be found exactly, enabling us to perform a local scan of the region between the inner and outer horizons. Surprisingly, the two-sided correlators we calculate seem to be geometrically protected from the instability of the inner horizon.

The Black Hole Singularity in AdS/CFT

We explore physics behind the horizon in eternal AdS Schwarzschild black holes. In dimension d >3, where the curvature grows large near the singularity, we find distinct but subtle signals of this singularity in the boundary CFT correlators. Building on previous work, we study correlation functions of operators on the two disjoint asymptotic boundaries of the spacetime by investigating the spacelike geodesics that join the boundaries. These dominate the correlators for large mass bulk fields. We show that the Penrose diagram for d>3 is not square. As a result, the real geodesic connecting the two boundary points becomes almost null and bounces off the singularity at a finite boundary time t_c \neq 0. If this geodesic were to dominate the correlator there would be a "light cone" singularity at t_c. However, general properties of the boundary theory rule this out. In fact, we argue that the correlator is actually dominated by a complexified geodesic, whose properties yield the large mass quasinormal mode frequencies previously found for this black hole. We find a branch cut in the correlator at small time (in the limit of large mass), arising from coincidence of three geodesics. The t_c singularity, a signal of the black hole singularity, occurs on a secondary sheet of the analytically continued correlator. Its properties are computationally accessible. The t_c singularity persists to all orders in the 1/m expansion, for finite \alpha', and to all orders in g_s. Certain leading nonperturbative effects can also be studied. The behavior of these boundary theory quantities near t_c gives, in principle, significant information about stringy and quantum behavior in the vicinity of the black hole singularity.

Winding strings andAdS3black holes

We start a systematic study of string theory in AdS_3 black hole backgrounds. Firstly, we analyse in detail the geodesic structure of the BTZ black hole, including spacelike geodesics. Secondly, we study the spectrum for massive and massless scalar fields, paying particular attention to the connection between Sl(2,R) subgroups, the theory of special functions and global properties of the BTZ black holes. We construct classical strings that wind the black holes. Finally, we apply the general formalism to the vacuum black hole background, and formulate the boundary spacetime Virasoro algebra in terms of worldsheet operators. We moreover establish the link between a proposal for a ghost free spectrum for Sl(2,R) string propagation and the massless black hole background, thereby claryfing aspects of the AdS3/CFT correspondence.

INTEGRABILITY OF THE N-BODY PROBLEM IN (2+1)-AdS GRAVITY

We derive a first order formalism for solving the scattering of point sources in (2+1) gravity with negative cosmological constant. We show that their physical motion can be mapped, with a polydromic coordinate transformation, to a trivial motion, in such a way that the point sources move as timelike geodesics (in the case of particles) or as spacelike geodesics (in the case of BTZ black holes) of a three-dimensional hypersurface immersed in a four-dimensional Minkowskian space–time, and that the two-body dynamics is solved by two invariant masses, whose difference is simply related to the total angular momentum of the system.

Geodesics with prescribed energy on static Lorentzian manifolds with convex boundary

By using some variational principles and the Ljusternik-Schnirelmann critical point theory, we extend some previous results dealing with the existence and multiplicity of geodesics having prescribed energy joining a point with a line on static Lorentzian manifolds with convex boundary. Our techniques work also in the case of timelike and spacelike geodesics on manifolds with boundary and for a physically relevant class of spacetimes with non-smooth boundary.

Chaos inpp-wave spacetimes

We demonstrate the chaotic behavior of timelike, null, and spacelike geodesics in nonhomogeneous vacuum $\mathrm{pp}$-wave solutions by analytic and fractal methods. This seems to be the first known example of a chaotic motion in exact radiative spacetime.

Bianchi-II Rotating World

Dynamical and kinematic properties of Bianchi-II cosmological models with rotation and expansion are investigated. Exact solutions of Einstein field equations are obtained which describe the evolution of a rotating Universe. Exact solutions of null, timelike and spacelike geodesics are constructed. Two new cosmological tests for rotating universes are discussed: cosmological lens effect and cosmological mirror effect.

Geodesic Connectedness in Generalized Reissner-Nordström Type Lorentz Manifolds

A detailed study of the existence, causal character and multiplicity of geodesics joining two points is carried out for a wide family of non-static Lorentz manifolds (including intermediate Reissner-Nordstrom, inner Schwarzschild and Generalized Robertson-Walker spacetimes). Results relating causality and connectedness by timelike or lightlike geodesics are obtained, in the spirit of the well-known Avez-Seifert result. The existence of closed spacelike geodesics is also characterized.

How should we measure spatial distances?

The use of time-like geodesics to measure temporal distances is better justified than the use of space-like geodesics for a measurement of spatial distances. We give examples where a “spatial distance” cannot be appropriately determined by the length of a space-like geodesic.

Hawking's chronology protection conjecture: singularity structure of the quantum stress-energy tensor

The recent renaissance of wormhole physics has led to a very disturbing observation: If traversable wormholes exist then it appears to be rather easy to transform such wormholes into time machines. This extremely disturbing state of affairs has lead Hawking to promulgate his chronology protection conjecture. This paper continues a program begun in an earlier paper. An explicit calculation of the vacuum expectation value of the renormalized stress-energy tensor in wormhole spacetimes is presented. Point-splitting techniques are utilized. Particular attention is paid to computation of the Green function (in its Hadamard form), and the structural form of the stress-energy tensor near short self-intersecting spacelike geodesics. Detailed comparisons with previous calculations are presented, leading to a pleasingly unified overview of the situation.

Geodesics in Kasner universes.

Recently, Kasner model universes of various dimensionalities have been explored to see if they could be used in the manner of Kaluza-Klein models to provide the additional unobservable dimensions called for in current elementary-particle models. These discussions have focused narrowly on the number of dimensions and the manner in which those dimensions in excess of the observable four of spacetime behave. This paper discusses some features of Kasner models which ultimately should be considered if these models are to provide a useful arena for studying elementary particles. In particular, geodesics are examined in detail. It is found that there may be a limit to the distance traveled by timelike particles, that spacelike geodesics are limited in their temporal extent, and that for null geodesics in expanding directions the coordinate speed approaches zero.

Spacetime coordinates in the geocentric reference frame

A geocentric relativistic reference frame is established which is close to the conventional non-relativistic equatorial frame of reference. Within post-Newtonian approximation the worldline of the geocentre is used to connect points by spacelike geodesics on the equal proper time hypersurface and to establish a properly chosen tetrad reference frame. Points on the earth surface and near the earth-space are coordinated making use of the Frobenius matrix of integrating factors which connects the geocentric orthonormal tetrad with the tangent spacetime of relativistic pseudo-Riemann geometry. The gravity field of the earth and its relative velocity with respect to the solar system barycentre cause coordinate effects of the order of 10 cm for topocentric point positioning.

Gradient-like and integrable vector fields on ?2

Recently, the authors have obtained criteria for the integral curves of a nonsingular smooth vector field X on a smooth manifold M to be timelike, null or spacelike geodesics for some Lorentzian metric g for M. In this paper, we show that for smoothly contractible subsets S of ℝ2 null geodesibility of a vector field X is equivalent to X being preHamiltonian on S and timelike, spacelike or Riemannian pregeodesibility of X are all equivalent to X being gradient-like. It turns out that null geodesibility is quite rare as we prove that even among real analytic vector fields on S there are many open sets of vector fields which fail to be preHamiltonian.

Spacelike geodesics in the Nordström geometry

We study here the spacelike geodesics in the Nordström geometry. We show that (i) a radially falling tachyon can hit the singularity r=0 if either it is very energetic or the source is highly charged, (ii) nonradial orbits never end into r=0, and (iii) unstable circular orbits exist in the region which is bounded from below as well as from above. In the case of e2/m2=1, scattering cross sections are calculated.

Nonequatorial tachyon trajectories in Kerr space-time and the second law of black-hole physics

The behavior of tachyon trajectories (spacelike geodesics) in Kerr space-time is discussed. It is seen that the trajectories may be broadly classified into three types according to the magnitude of the angular momentum of the tachyon. When the magnitude of angular momentum is large [|h|≤ a(1+Γ 2 ) 1/2 , where h and Γ are the angular momentum and energy at infinity and a=S/M the angular momentum and mass of a rotating black hole] the incoming tachyon from infinity necessarily has a bounce for r>0. In the other cases, a negative value for Carter's constant of motion Q is permitted, which happens to be a necessary condition for the tachyon to fall into the singularity. Next, the second law of black-hole physics is investigated in the general case of nonequatorial trajectories. It is shown that nonequatorial tachyons can decrease the area of the Kerr black hole only if it is rotating sufficiently rapidly [a>(4/3√ 3)M].

Classical null strings

Free Nambu strings in space-time are the two-dimensional sheet analog of one-dimensional curves which are timelike or spacelike geodesics. It is pointed out that there is also a sheet analog of one-dimensional curves which are null geodesics.

On Space-like Geodesics in Two Special Gravitational Fields

Studies are presented of space-like geodesics in the fields of a charged mass point and an uncharged static cylinder. For the first case, it is shown that circular space-like trajectories are unstable in character under radial perturbations for a physically plausible situation (82 < m2). For the second case, it is shown that no trajectories in the r-f/J plane satisfy the conditions of stability.

Causality and Multiply Connected Space-Time

With the introduction of multiply connected topologies into physics, a question of causality arises. There are alternative routes between two points in a multiply connected space. Therefore, one may ask if a signal traveling at the speed of light along one route could be outpaced by a signal which has traveled a much shorter path through a handle or wormhole. This paper examines one such situation and shows that in this example causality is preserved. It proves essential in the analysis to distinguish between those regions of space-time which are catastrophic and those which are not. A catastrophic region is composed of catastrophic points. A catastrophic point in space-time is so located with respect to eventual singularities in the intrinsic geometry that every time-like geodesic through it necessarily runs into a region of infinite curvature at some time in the future---or was born out of a region of infinite curvature at some time in the past---or both. If a classical analysis of nature were possible---which it is not---then it would be natural to postulate that laboratory physics is carried out in noncatastrophic regions of space-time. Two such regions are shown to exist in the example considered in the paper. It is shown that no signal can ever be sent from one to the other. The key point in preventing any violation of causality is simple: The (Schwarzschild) throat of the wormhole pinches off in a finite time and traps the signal in a region of infinite curvature. This investigation also displays some of the unusual geometric features of the Schwarzschild solution of Einstein's equations for a spherically symmetrical center of attraction. Radial spacelike geodesics passing through the throat are calculated and it is shown that there exist regions of space-time unreachable by any radial geodesics that issue from a given point. Also, there exist points in space-time from which light signals can never be received no matter how long one waits.