String Theory in Curved Space Times

The following sections are included:

• PREFACE

• CONTENTS

Recent progress on string theory in curved spacetimes is reviewed. The string dynamics in cosmological and black hole spacetimes is investigated. The different methods available to solve the string equations of motion and constraints in curved spacetimes are described. That is, the string perturbation approach, the null string approach, the τ-expansion, and the construction of global solutions (for instance by inverse scattering methods).

The classical behaviour of strings in FRW and inflationary spacetimes is now understood in a large extent from the various types of explicit string solutions. Three different types of behaviour appear: unstable, dual to unstable and stable. For the unstable strings, the energy and size grow for large scale factors R → ∞, proportional to R. For the dual to unstable strings, the energy and size blow up for R → 0 as 1/R. For stable strings, the energy and proper size are bounded. (In Minkowski spacetime, all string solutions are of the stable type).

Recent progress on self-consistent solutions to the Einstein equations for string dominated universes is reviewed. The energy-momentum tensor for a gas of strings is then considered as source of the spacetime geometry and from the above string behaviours the string equation of state is derived. The self-consistent string solution exhibits the realistic matter dominated behaviour R ≃ T^2/3 for large times and the radiation dominated behaviour R ≃ T^1/2 for early times (T being the cosmic time).

We report on the exact integrability of the string equations plus the constraints in de Sitter spacetime that allows to systematically find exact string solutions by soliton methods and the multistring solutions. Multistring solutions are a new feature in curved spacetimes. That is, a single world-sheet simultaneously describes many different and independent strings. This phenomenon has no analogue in flat spacetime and follows to the coupling of the strings with the geometry.

We discuss the temporal variation of the equation of state of a classical string network, evolving in a background in which the Hubble radius H⁻¹ shrinks to a minimum and then re-expands to infinity. We also present a method to look for self-consistent non-vacuum string backgrounds, corresponding to the simultaneous solution of the gravi-dilaton background equations and of the string equations of motion.

We study bosonic and fermionic first quantized strings that propagate in a background. New phenomena appear called particle transmutations. Fermion number is violated in supergravity backgrounds.

The ultrarelativistic limit of the Kerr - Newman geometry is studied in detail. We find the corresponding gravitational shock wave background associated with this limit. Interestingly, this allows us to find the source of the Kerr - Newman geometry in the ultrarelativistic regime. We study the scattering of scalar fields in the gravitational shock wave geometries, and discuss the presence of the poles iGs = n = 0,1,2, ... , already present in the Aichelburg - Sexl metric. We compare this with the scattering by ultrarelativistic extended sources, for which such poles do not appear and with the scattering of fundamental strings.

We also study planckian energy string collisions in flat spacetime as the scattering of a string in the effective curved background produced by the others as the impact parameter b decreases. We find the effective energy density distribution σ(ρ) ∼ exp{−ρ²/Δ²}, generated by these collisions. Two different regimes can be studied: intermediate impact parameters , (x^d characterizing the string fluctuations) and large impact parameters, . The effective metric generated by these collisions is a gravitational shock wave of profile f(ρ) ∼ pρ^4−D , i.e. the Aichelburg - Sexl geometry for a point-like particle of momentum p for large b. For intermediate b, f(ρ) ∼ qρ², corresponding to an extended source of momentum q. The scattering matrix in this geometry and its implications for the string collision process are analysed. We show that the poles iGs = n , n = 0,1, ..., characteristic of the scattering by the A - S geometry are absent here, due to the extended nature of the effective source.

We finally study the emergence of string instabilities in D - dimensional black hole spacetimes (Schwarzschild and Reissner - Nordstrom), and De Sitter space (in static coordinates to allow a better comparison with the black hole case). We solve the first order string fluctuations around the center of mass motion at spatial infinity, near the horizon and at the spacetime singularity. We find that the time components are always well behaved in the three regions and in the three backgrounds. The radial components are unstable: imaginary frequencies develop in the oscillatory modes near the horizon, and the evolution is like (τ − τ₀)^−P, (P > 0), near the spacetime singularity, r → 0, where the world - sheet time (τ − τ₀) → 0, and the proper string length grows infinitely. In the Schwarzschild black hole, the angular components are always well - behaved, while in the Reissner - Nordström case they develop instabilities inside the horizon, near r → 0 where the repulsive effects of the charge dominate over those of the mass. In general, whenever large enough repulsive effects in the gravitational background are present, string instabilities develop. In De Sitter space, all the spatial components exhibit instability. The infalling of the string to the black hole singularity is like the motion of a particle in a potential γ(τ − τ₀)⁻² where γ depends on the D spacetime dimensions and string angular momentum, with γ > 0 for Schwarzschild and γ < 0 for Reissner - Nordström black holes. For (τ − τ₀) → 0 the string ends trapped by the black hole singularity.

The string propagation in curved spacetimes is investigated in the null string approach. In this approach the string equations of motion and constraints are systematically expanded in powers of c (the speed of light in the world-sheet). We obtain to zeroth order the null string (string with zero tension), while the first order correction incorporates the string dynamics.

We apply our formalism to null strings in de Sitter spacetime. After a reparametrization of the world-sheet coordinates, the equations of motion are solved. The quantum algebra generated by the constraints is considered, ordering the momentum operators to the right of the coordinate operators. No critical dimension appears. It is anticipated however that the conformal anomaly will appear when the first order corrections proportional to T₀, are introduced.

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 ≥ 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, representing circular strings, stationary open strings and dynamical straight strings, are obtained in these backgrounds and their physical properties (length, energy, pressure) are described. The existence of 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. Generic approximative solutions are obtained using the string perturbation series approach, and the question of the stability of the solutions is addressed.

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 fine-structure effect at all levels beyond the graviton in both de Sitter and anti de Sitter spacetimes, and the non-existence of a Hagedorn temperature in anti de Sitter spacetime. We discuss the physical implications of these results. Finally, we consider the effect of spatial curvature on the string dynamics in Robertson-Walker spacetimes.

Recent developments on integrable field theories and statistical models in two dimensions are reviewed.

The best way to investigate the continuum limit of lattice models is through the light-cone approach. This method allows to find the underlying quantum field theory for any integrable lattice model in its gapless regime. The relativistic spectrum and S-matrix follows straightforwardly in this way through the Bethe Ansatz. We show here how to derive the infinite number of local commuting and non-local and non-commuting conserved charges in integrable QFT, taking the massive Thirring model (sine-Gordon) as an example. They are generated by quantum monodromy operators and provide a representation of q–deformed affine Lie algebras . Therefore, these models enjoy infinite dimensional non-abelian symmetries.

A new lattice local formulation of integrable field models is presented and applied to the massive Thirring model (sine-Gordon) as an example.

Finally, the new thermodynamic Bethe Ansatz (DDV equations) allowing an unified treatment of finite size and finite temperature integrable models is reviewed.

We show that the effective coarse graining of a two-mode squeezed density matrix, implicit in the Wehrl approaches to a semiclassical phase-space distribution, leads to results in agreement with previous different definitions of entropy for the process of pair production from the vacuum. We also present, in this context, a possible interpretation of the entropy growth as an amplification (due to the squeezing) of our lack of knowledge about the initial conditions, which gives rise to an effective decoherence of the squeezed density matrix.

Recent progress in the preheating phenomena for inflationary cosmology is reviewed. We first discuss estimates of the preheating time scale and particle production at the early stages of parametric amplification within the Mathieu and Lamé approximations and we analyze their precision and limitations. The necessity of self-consistent calculations including the non-linearity of the field theory equations in an energy conserving scheme is stressed. The large N calculations including the field back-reaction are reviewed. For spontaneously broken theories the issue of symmetry restoration is analyzed. A discussion of the possibility and criterion for symmetry restoration is presented. (To appear in the Proceedings of the Paris Euronetwork Meeting ‘String Gravity’, Observatoire de Paris, June 1996).

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 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 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. The string mass is , C being the Casimir operator, C = −L_μνL^μν, of the O(3, 1)-dS [O(2, 2)-AdS] group, and H is the Hubble constant. We find the mass formula , (n ∈ N₀), and a finite number of states N_dS ≈ 0.34/(H²α′) in de Sitter spacetime; (large n ∈ N) and N_AdS = ∞ 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.

The string propagation in the two-dimensional stringy black-hole is investigated from a new approach . We completely solve the classical and quantum string dynamics in the lorentzian and euclidean regimes. In the lorentzian case all the physics reduces to a massless scalar particle described by a Klein-Gordon type equation with a singular effective potential. The scattering matrix is found and it reproduces the results obtained by coset CFT techniques. It factorizes into two pieces : an elastic coulombian amplitude and an absorption part. In both parts, an infinite sequence of imaginary poles in the energy appear. The generic features of string propagation in curved D-dimensional backgrounds (string stretching, fall into spacetime singularities) are analyzed in the present case. A new physical phenomenon specific to the present black-hole is found : the quantum renormalization of the speed of light. We find , where k is the integer in front of the WZW action. Only for k → ∞ this new effect disappears (although the conformal anomaly is present). We analyze all the classical euclidean string solutions and exactly compute the quantum partition function. No critical Hagedorn temperature appears here.

I discuss the status of present knowledge about a possible background of relic gravitons left by an early, pre-big bang cosmological epoch, whose existence in the past of our Universe is suggested by the duality symmetries of string theory.

We find exact solutions of the string equations of motion and constraints describing the classical splitting of a string into two. We show that for the same Cauchy data, the strings that split have smaller action than the string without splitting. This phenomenon is already present in flat space-time. The mass, energy and momentum carried out by the strings are computed. We show that the splitting solution describes a natural decay process of one string of mass M into two strings with a smaller total mass and some kinetic energy. The standard non-splitting solution is contained as a particular case. We also describe the splitting of a closed string in the background of a singular gravitational plane wave, and show how the presence of the strong gravitational field increases (and amplifies by an overall factor) the negative difference between the action of the splitting and non-splitting solutions.

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, and I review some recent work done in collaboration with N. Sànchez and H.J. de Vega.

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. This second part of the talk is a review of some recent work done in collaboration with V. Frolov and S. Hendy.

The classical string equations of motion and constraints are solved near the horizon and near the singularity of a Schwarzschild black hole.

In a conformal gauge such that τ = 0 (τ = worldsheet time coordinate) corresponds to the horizon (r = 1) or to the black hole singularity (r = 0), the string coordinates express in power series in τ near the horizon and in power series in τ^1/5 around r = 0.

We compute the string invariant size and the string energy-momentum tensor. Near the horizon both are finite and analytic. Near the black hole singularity, the string size, the string energy and the transverse pressures (in the angular directions) tend to infinity as r⁻¹. To leading order near r = 0, the string behaves as two dimensional radiation. This two spatial dimensions are describing the S² sphere in the Schwarzschild manifold.

We study the classical dynamics of a bosonic string in the D-dimensional flat Friedmann–Robertson–Walker and Schwarzschild backgrounds. We make a perturbative development in the string coordinates around a null string configuration; the background geometry is taken into account exactly. In the cosmological case we uncouple and solve the first order fluctuations; the string time evolution with the conformal gauge world-sheet τ–coordinate is given by , B⁰(σ,τ) = Σ_kb_k(σ)τ^k where b_k(σ) are given by Eqs. (3.15), and β is the exponent of the conformal factor in the Friedmann–Robertson–Walker metric, i.e. R ∼ η^β. The string proper size, at first order in the fluctuations, grows like the conformal factor R(η) and the string energy–momentum tensor corresponds to that of a null fluid. For a string in the black hole background, we study the planar case, but keep the dimensionality of the spacetime D generic. In the null string expansion, the radial, azimuthal, and time coordinates (r, ϕ, t) are , and . The first terms of the series represent a generic approach to the Schwarzschild singularity at r = 0. First and higher order string perturbations contribute with higher powers of τ. The integrated string energy-momentum tensor corresponds to that of a null fluid in D – 1 dimensions. As the string approaches the r = 0 singularity its proper size grows indefinitely like ∼ (−τ)^{−(D−3)/(D+1)}. We end the paper giving three particular exact string solutions inside the black hole. They represent respectively straight strings across the origin, twisted, and rigidly rotating strings.

The following sections are included:

• INTRODUCTION AND RESULTS

• PARTIAL WAVE ANALYSIS
  • Absorption cross sections
  • Elastic Scattering

• DIFFERENTIAL ELASTIC CROSS SECTION

• HAWKING EMISSION RATES

• REMARKS ON APPROXIMATIONS

• CONCLUSIONS

• References

A l’occasion du “Second Paris Cosmology Colloquium” tenu à l’Observatoire de Paris en 1994 (2-4 juin), une visite de l’Observatoire de Paris a fait connaître aux participants les richesses de l’Etablissement dont la première pierre fut posée le 21 juin 1667, et où une activité de recherche n’a pas cessé d’y être menée. Le descriptif de la visite conduite par l’un d’entre nous (S. Débarbat) est paru aux pages 528 à 531 des Actes du Colloque (World Scientific, 1995). Plusieurs des photographies reproduites aux pages 533 à 543 permettent d’apprécier “l’état des lieux”, tandis que la page IV de couverture représente la pendule “astronomique” de 1710 qui orne l’une des salles du Bâtiment Perrault⋯