String dynamics in cosmological and black hole backgrounds: The null string expansion.

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 X0(σ, τ) = q(σ)τ 1 1+2β + c2B0(σ, τ) + · · ·, B0(σ, τ) = ∑k bk(σ)τ where bk(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 ∼ ηβ . 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 r = ∑ nA 1 n(σ)(−τ) , φ = ∑ nA 3 n(σ)(−τ) , and t = ∑ n A 0 n(σ)(−τ) . 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. 11.25.Db, 04.70.Bw, 98.80.Hw Typeset using REVTEX 1