arXiv subjects

A. L. Larsen

Publications and source records attributed to A. L. Larsen.

At least 37 records · Page 2Linked to original sources

arXiv1998

From the WZWN Model to the Liouville Equation: Exact String Dynamics in Conformally Invariant AdS Background

It has been known for some time that the SL(2,R) WZWN model reduces to Liouville theory. Here we give a direct and physical derivation of this result based on the classical string equations of motion and the proper string size. This allows us to extract precisely the physical effects of the metric and antisymmetric tensor, respectively, on the {\it exact} string dynamics in the SL(2,R) background. The general solution to the proper string size is also found. We show that the antisymmetric tensor (corresponding to conformal invariance) generally gives rise to repulsion, and it precisely cancels the dominant attractive term arising from the metric. Both the sinh-Gordon and the cosh-Gordon sectors of the string dynamics in non-conformally invariant AdS spacetime reduce here to the Liouville equation (with different signs of the potential), while the original Liouville sector reduces to the free wave equation. Only the very large classical string size is affected by the torsion. Medium and small size string behaviours are unchanged. We also find illustrative classes of string solutions in the SL(2,R) background: dynamical closed as well as stationary open spiralling strings, for which the effect of torsion is somewhat like the effect of rotation in the metric. Similarly, the string solutions in the 2+1 BH-AdS background with torsion and angular momentum are fully analyzed.

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arXiv1998

Domain wall interacting with a black hole: A new example of critical phenomena

We study a simple system that comprises all main features of critical gravitational collapse, originally discovered by Choptuik and discussed in many subsequent publications. These features include universality of phenomena, mass-scaling relations, self-similarity, symmetry between super-critical and sub-critical solutions, etc. The system we consider is a stationary membrane (representing a domain wall) in a static gravitational field of a black hole. For a membrane that spreads to infinity, the induced 2+1 geometry is asymptotically flat. Besides solutions with Minkowski topology there exists also solutions with the induced metric and topology of a 2+1 dimensional black hole. By changing boundary conditions at infinity, one finds that there is a transition between these two families. This transition is critical and it possesses all the above-mentioned properties of critical gravitational collapse. It is remarkable that characteristics of this transition can be obtained analytically. In particular, we find exact analytical expressions for scaling exponents and wiggle-periods. Our results imply that black hole formation as a critical phenomenon is far more general than one might expect.

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arXiv1998

Soap Bubbles in Outer Space: Interaction of a Domain Wall with a Black Hole

We discuss the generalized Plateau problem in the 3+1 dimensional Schwarzschild background. This represents the physical situation, which could for instance have appeared in the early universe, where a cosmic membrane (thin domain wall) is located near a black hole. Considering stationary axially symmetric membranes, three different membrane-topologies are possible depending on the boundary conditions at infinity: 2+1 Minkowski topology, 2+1 wormhole topology and 2+1 black hole topology. Interestingly, we find that the different membrane-topologies are connected via phase transitions of the form first discussed by Choptuik in investigations of scalar field collapse. More precisely, we find a first order phase transition (finite mass gap) between wormhole topology and black hole topology; the intermediate membrane being an unstable wormhole collapsing to a black hole. Moreover, we find a second order phase transition (no mass gap) between Minkowski topology and black hole topology; the intermediate membrane being a naked singularity. For the membranes of black hole topology, we find a mass scaling relation analogous to that originally found by Choptuik. However, in our case the parameter $p$ is replaced by a 2-vector $\vec{p}$ parametrizing the solutions. We find that $Mass\propto|\vec{p}-\vec{p}_*|^γ$ where $γ\approx 0.66$. We also find a periodic wiggle in the scaling relation. Our results show that black hole formation as a critical phenomenon is far more general than expected.

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arXiv1998

Quantum String Dynamics in the conformal invariant SL(2,R) WZWN Background: Anti-de Sitter Space with Torsion

We consider classical and quantum strings in the conformally invariant background corresponding to the SL(2,R) WZWN model. This background is locally anti-de Sitter spacetime with non-vanishing torsion. Conformal invariance is expressed as the torsion being parallelized. The precise effect of the conformal invariance on the dynamics of both circular and generic classical strings is extracted. In particular, the conformal invariance gives rise to a repulsive interaction of the string with the background which precisely cancels the dominant attractive term arising from gravity. We perform both semi-classical and canonical string-quantization, in order to see the effect of the conformal invariance of the background on the string mass spectrum. Both approaches yield that the high-mass states are governed by m sim HN (N,`large integer'), where m is the string mass and H is the Hubble constant. It follows that the level spacing grows proportionally to N: d(m^2 alpha')/dN sim N, while the entropy goes like: S sim sqrt{m}. Moreover, it follows that there is no Hagedorn temperature,so that the partition function is well defined at any positive temperature. All results are compared with the analogue results in Anti- de Sitter spacetime, which is a non conformal invariant background. Conformal invariance simplifies the mathematics of the problem but the physics remains mainly unchanged. Differences between conformal and non-conformal backgrounds only appear in the intermediate region of the string spectrum, but these differences are minor. For low and high masses, the string mass spectra in conformal and non-conformal backgrounds are identical. Interestingly enough, conformal invariance fixes the value of the spacetime curvature to be -69/(26 alpha').

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arXiv1996

Are Higher Order Membranes stable in Black Hole Spacetimes?

We continue the study of the existence and stability of static spherical membrane configurations in curved spacetimes. We first consider higher order membranes described by a Lagrangian which, besides the Dirac term, includes a term proportional to the scalar curvature of the world--volume ${}^{(3)}R$. Notably, in this case, the equations of motion can be reduced to second order ones and an effective potential analysis can be made. The conditions for stability are then explicitly derived. We find a self--consistent static spherical membrane, determining the spacetime generated by the membrane itself. In this case we find, however, that the total energy of the membrane has to be negative, and no {\it stable} equilibrium can be achieved. We then generalize the discussion to a membrane described by a Lagrangian including all possible second derivative terms. We conclude the paper with some discussion on the generality of the results obtained.

gr-qc↗

arXiv1996

Cosmic Strings and Black Holes

In the first part of this talk, I consider some exact string solutions in curved spacetimes. In curved spacetimes with a Killing vector (timelike or spacelike), the string equations of motion and constraints are reduced to the Hamilton equations of a relativistic point-particle in a scalar potential, by imposing a particular ansatz. As special examples I consider circular strings in axially symmetric spacetimes, as well as stationary strings in stationary spacetimes. In the second part of the talk, I then consider in more detail the stationary strings in the Kerr -Newman geometry. It is shown that the world-sheet of a stationary string, that passes the static limit of the 4-D Kerr-Newman black hole, describes a 2-D black hole. Mathematical results for 2-D black holes can therefore be applied to physical objects; (say) cosmic strings in the vicinity of Kerr black holes. As an immediate general result, it follows that the string modes are thermally excited.

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arXiv1996

Sinh-Gordon, Cosh-Gordon and Liouville Equations for Strings and Multi-Strings in Constant Curvature Spacetimes

We find that the fundamental quadratic form of classical string propagation in $2+1$ dimensional constant curvature spacetimes solves the Sinh-Gordon equation, the Cosh-Gordon equation or the Liouville equation. We show that in both de Sitter and anti de Sitter spacetimes (as well as in the $2+1$ black hole anti de Sitter spacetime), {\it all} three equations must be included to cover the generic string dynamics. The generic properties of the string dynamics are directly extracted from the properties of these three equations and their associated potentials (irrespective of any solution). These results complete and generalize earlier discussions on this topic (until now, only the Sinh-Gordon sector in de Sitter spacetime was known). We also construct new classes of multi-string solutions, in terms of elliptic functions, to all three equations in both de Sitter and anti de Sitter spacetimes. Our results can be straightforwardly generalized to constant curvature spacetimes of arbitrary dimension, by replacing the Sinh-Gordon equation, the Cosh-Gordon equation and the Liouville equation by higher dimensional generalizations.

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arXiv1996

Supermembranes and Superstrings with Extrinsic Curvature

In a recent paper Townsend suggested to associate to the D=11 solitonic membrane of N=1 supergravity a certain thickness, and then to identify this membrane with the fundamental supermembrane. By integrating out the 8 transverse dimensions of the "thick" solitonic membrane, we show that the resulting world-volume action indeed contains all the usual supermembrane terms, as well as background curvature terms and extrinsic curvature terms, which are believed to render the membrane spectrum discrete. We also outline the analog derivation for the "thick" D=10 solitonic string solution of N=IIA supergravity. The resulting world-sheet action contains the usual type IIA superstring terms, as well as extra terms whose presence can be interpreted as a rescaling of the background metric, thus preserving kappa-symmetry and conformal invariance.

hep-th↗

arXiv1996

Stationary Strings and Principal Killing Triads in 2+1 Gravity

A new tool for the investigation of 2+1 dimensional gravity is proposed. It is shown that in a stationary 2+1 dimensional spacetime, the eigenvectors of the covariant derivative of the timelike Killing vector form a rigid structure, the {\it principal Killing triad}. Two of the triad vectors are null, and in many respects they play the role similar to the principal null directions in the algebraically special 4-D spacetimes. It is demonstrated that the principal Killing triad can be efficiently used for classification and study of stationary 2+1 spacetimes. One of the most interesting applications is a study of minimal surfaces in a stationary spacetime. A {\it principal Killing surface} is defined as a surface formed by Killing trajectories passing through a null ray, which is tangent to one of the null vectors of the principal Killing triad. We prove that a principal Killing surface is minimal if and only if the corresponding null vector is geodesic. Furthermore, we prove that if the 2+1 dimensional spacetime contains a static limit, then the only regular stationary timelike minimal 2-surfaces that cross the static limit, are the minimal principal Killing surfaces. A timelike minimal surface is a solution to the Nambu-Goto equations of motion and hence it describes a cosmic string configuration. A stationary string interacting with a 2+1 dimensional rotating black hole is discussed in detail.

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arXiv1996

On the Stability of Spherical Membranes in Curved Spacetimes

We study the existence and stability of spherical membranes in curved spacetimes. For Dirac membranes in the Schwarzschild--de Sitter background we find that there exists an equilibrium solution. By fine--tuning the dimensionless parameter $ΛM^2,$ the static membrane can be at any position outside the black hole event horizon, even at the stretched horizon, but the solution is unstable. We show that modes having $l=0$ (and for $ΛM^2<16/243$ also $l=1$) are responsible for the instability. We also find that spherical higher order membranes (membranes with extrinsic curvature corrections), contrary to what happens in flat Minkowski space, {\it do} have equilibrium solutions in a general curved background and, in particular, also in the ``plain'' Schwarzschild geometry (while Dirac membranes do not have equilibrium solutions there). These solutions, however, are also unstable. We shall discuss a way of by--passing these instability problems, and we also relate our results to the recent ideas of representing the black hole event horizon as a relativistic bosonic membrane.

gr-qc↗

arXiv1995

How to Create a 2-D Black Hole

The interaction of a cosmic string with a four-dimensional stationary black hole is considered. If a part of an infinitely long string passes close to a black hole it can be captured. The final stationary configurations of such captured strings are investigated. A uniqueness theorem is proved, namely it is shown that the minimal 2-D surface $Σ$ describing a captured stationary string coincides with a {\it principal Killing surface}, i.e. a surface formed by Killing trajectories passing through a principal null ray of the Kerr-Newman geometry. Geometrical properties of principal Killing surfaces are investigated and it is shown that the internal geometry of $Σ$ coincides with the geometry of a 2-D black or white hole ({\it string hole}). The equations for propagation of string perturbations are shown to be identical with the equations for a coupled pair of scalar fields 'living' in the spacetime of a 2-D string hole. Some interesting features of physics of 2-D string holes are described. In particular, it is shown that the existence of the extra dimensions of the surrounding spacetime makes interaction possible between the interior and exterior of a string black hole; from the point of view of the 2-D geometry this interaction is acausal. Possible application of this result to the information loss puzzle is briefly discussed.

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arXiv1995

How to Create a 2-D Black Hole

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arXiv1995

Quantum Tunneling Effect in Oscillating Friedmann Cosmology

It is shown that the tunneling effect in quantum cosmology is possible not only at the very beginning or the very end of the evolution, but also at the moment of maximum expansion of the universe. A positive curvature expanding Friedmann universe changes its state of evolution spontaneously and completely, {\it without} any changes in the matter content, avoiding recollapse, and falling into oscillations between the nonzero values of the scale factor. On the other hand, an oscillating nonsingular universe can tunnel spontaneously to a recollapsing regime. The probability of such kind of tunneling is given explicitly. It is inversely related to the amount of nonrelativistic matter (dust), and grows from a certain fixed value to unity if the negative cosmological constant approaches zero.

gr-qc↗

arXiv1995

Circular Strings and Multi-Strings in de Sitter and Anti de Sitter Spacetimes

The exact general solution of circular strings in $2+1$ dimensional de Sitter spacetime is described completely in terms of elliptic functions. The novel feature here is that one single world-sheet generically describes {\it infinitely many} (different and independent) strings. This has no analogue in flat spacetime. The circular strings are either oscillating ("stable") or indefinitely expanding ("unstable"). We then compute the {\it exact} equation of state of circular strings in the $2+1$ dimensional de Sitter (dS) and anti de Sitter (AdS) spacetimes, and analyze its properties for the different (oscillating, contracting and expanding) strings. We finally perform a semi-classical quantization of the oscillating circular strings. We find the mass formula $α'm^2_{\mbox{dS}}\approx 4n-5H^2α'n^2,\;(n\in N_0),$ and a {\it finite} number of states $N_{\mbox{dS}}\approx 0.34/(H^2α')$ in de Sitter spacetime; $m^2_{\mbox{AdS}}\approx H^2n^2$ (large $n\in N$) and $N_{\mbox{AdS}}=\infty$ in anti de Sitter spacetime. The level spacing grows with $n$ in AdS spacetime, while is approximately constant (although smaller than in Minkowski spacetime and slightly decreasing) in dS spacetime.

hep-th↗

arXiv1995

Strings and Multi-Strings in Black Hole and Cosmological Spacetimes

Recent results on classical and quantum strings in a variety of black hole and cosmological spacetimes, in various dimensions, are presented. The curved backgrounds under consideration include the $2+1$ black hole anti de Sitter spacetime and its dual, the black string, the ordinary $D\geq 4$ black holes with or without a cosmological constant, the de Sitter and anti de Sitter spacetimes and static Robertson-Walker spacetimes. Exact solutions to the string equations of motion and constraints are obtained in these backgrounds and their physical properties (length, energy, pressure) are described. The existence of {\it multi-string} solutions, describing finitely or infinitely many strings, is shown to be a general feature of spacetimes with a positive or negative cosmological constant. Furthermore, using a canonical quantization procedure, we find the string mass spectrum in de Sitter and anti de Sitter spacetimes. New features as compared to the string spectrum in flat Minkowski spacetime appear, for instance the {\it fine-structure effect} at all levels beyond the graviton in both de Sitter and anti de Sitter spacetimes, and the {\it non-existence} of a Hagedorn temperature in anti de Sitter spacetime. Finally, we consider the effect of spatial curvature on the string dynamics in Robertson-Walker spacetimes.

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arXiv1995

STATIONARY STRINGS AND 2-D BLACK HOLES

A general description of string excitations in stationary spacetimes is developed. If a stationary string passes through the ergosphere of a 4-dimensional black hole, its world-sheet describes a 2-dimensional black (or white) hole with horizon coinciding with the static limit of the 4-dimensional black hole. Mathematical results for 2-dimensional black holes can therefore be applied to physical objects (say) cosmic strings in the vicinity of Kerr black holes. An immediate general result is that the string modes are thermally excited. The string excitations are determined by a coupled system of scalar field equations in the world-sheet metric. In the special case of excitations propagating along a stationary string in the equatorial plane of the Kerr-Newman black hole, they reduce to the $s$-wave scalar field equations in the 4-dimensional Reissner-Nordström black hole. We briefly discuss possible applications of our results to the black hole information puzzle.

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arXiv1995

New Classes of Exact Multi-String Solutions in Curved Spacetimes

We find new classes of {\it exact} string solutions in a variety of curved backgrounds. They include stationary and dynamical (open, closed, straight, finitely and infinitely long) strings as well as {\it multi-string} solutions, in terms of elliptic functions. The physical properties, string length, energy and pressure are computed and analyzed. In anti de Sitter spacetime, the solutions describe an {\it infinite} number of infinitely long stationary strings of equal energy but different pressures. In de Sitter spacetime, outside the horizon, they describe infinitely many {\it dynamical} strings infalling non-radially, scattering at the horizon and going back to spatial infinity in different directions. For special values of the constants of motion, there are families of solutions with {\it selected finite} numbers of different and independent strings. In black hole spacetimes (without cosmological constant), {\it no} multi-string solutions are found. In the Schwarzschild black hole, inside the horizon, we find one straight string infalling non-radially, with {\it indefinetely} growing size, into the $r=0$ singularity. In the $2+1$ black hole anti de Sitter background, the string stops at $r=0$ with {\it finite} length.

hep-th↗

arXiv1995

The Effect of Spatial Curvature on the Classical and Quantum Strings

We study the effects of the spatial curvature on the classical and quantum string dynamics. We find the general solution of the circular string motion in static Robertson-Walker spacetimes with closed or open sections. This is given closely and completely in terms of elliptic functions. The physical properties, string length, energy and pressure are computed and analyzed. We find the {\it back-reaction} effect of these strings on the spacetime: the self-consistent solution to the Einstein equations is a spatially closed $(K>0)$ spacetime with a selected value of the curvature index $K$ (the scale f* is normalized to unity). No self-consistent solutions with $K\leq 0$ exist. We semi-classically quantize the circular strings and find the mass $m$ in each case. For $K>0,$ the very massive strings, oscillating on the full hypersphere, have $m^2\sim K n^2\;\;(n\in N_0)$ {\it independent} of $α'$ and the level spacing {\it grows} with $n,$ while the strings oscillating on one hemisphere (without crossing the equator) have $m^2α'\sim n$ and a {\it finite} number of states $N\sim 1/(Kα').$ For $K<0,$ there are infinitely many string states with masses $m\log m\sim n,$ that is, the level spacing grows {\it slower} than $n.$ The stationary string solutions as well as the generic string fluctuations around the center of mass are also found and analyzed in closed form.

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