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

Abstract

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 {\it 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 $\tau$--coordinate is given by $X^0(\sigma ,\tau )=q(\sigma ){\tau }^{\frac{1}{1+2\beta }}+c^2{B}^0(\sigma ,\tau )+\cdots$, $B^0(\sigma ,\tau )={\sum }_k{b}_k(\sigma ){\tau }^k$ where $b_k(\sigma )$ are given by Eqs.\ (3.15), and $\beta$ is the exponent of the conformal factor in the Friedmann--Robertson--Walker metric, i.e. $R\sim {\eta }^{\beta }$. The string proper size, at first order in the fluctuations, grows like the conformal factor $R(\eta )$ 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 $(r,\varphi ,t)$ are $r={\sum }_n{A}_n^1(\sigma )(-\tau {)}^{2n/(D+1)},$ $\varphi ={\sum }_n{A}_n^3(\sigma )(-\tau {)}^{(D-5+2n)/(D+1)},$ and $t={\sum }_n{A}_n^0(\sigma )(-\tau {)}^{1+2n(D-3)/(D+1)}.$ The first terms of the series represent a {\it generic} approach to the Schwarzschild singularity at $r=0$. First and higher order string perturbations contribute with higher powers of $\tau$. 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 $\sim (-\tau {)}^{-(D-3)/(D+1)}$. We end the paper giving three particular exact string solutions inside the black hole.

17 pages, REVTEX, no figures

References in corpus (1)

• Strings Next To and Inside Black Holes

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