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 $\tau$ coordinate is given by $X^0(\sigma , \tau )=q\left(\sigma \right){\tau }^{\frac{1}{(1+2\mathrm{}\beta )}}+c^2{B}^0(\sigma , \tau )+\dots$, $B^0(\sigma , \tau )={\Sigma }_k{b}_k\left(\sigma \right){\tau }^k$ where $q\left(\sigma \right)$ is a function of the momentum component $P_0\left(\sigma \right)$, $b_k\left(\sigma \right)$ are obtained from the equation for the first order fluctuations, 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 such as the conformal factor $R\left(\eta \right)$ 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 ($r$, $\phi$, $t$) are $r={\Sigma }_n{A}_n^1\left(\sigma \right){(-\tau )}^{\frac{2n}{\mathrm{}(D+1\mathrm{})\mathrm{}}}$, $\phi ={\Sigma }_n{A}_n^3\left(\sigma \right){(-\tau )}^{\frac{\mathrm{}(D-5+2n)\mathrm{}}{\mathrm{}(D+1\mathrm{})\mathrm{}}}$, and $t={\Sigma }_n{A}_n^0\left(\sigma \right){(-\tau )}^{1+\frac{2n(D-3\mathrm{})\mathrm{}}{\mathrm{}(D+1\mathrm{})\mathrm{}}}$. The first terms of the series represent a generic approach to the Schwarzschild singularity at $r=0\mathrm{}$. 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 $D-1\mathrm{}$ dimensions. As the string approaches the $r=0\mathrm{}$ singularity its proper size grows indefinitely such as $\sim {(-\tau )}^{-\frac{\mathrm{}(D-3\mathrm{})\mathrm{}}{\mathrm{}(D+1\mathrm{})\mathrm{}}}$. 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.