Classical and quantum strings near spacetime singularities: gravitational plane waves with arbitrary polarization

Abstract

We study the classical and quantum dynamics of strings near spacetime singularities. We deal with singular gravitational plane waves of arbitrary polarization and arbitrary profile functions W1(U) and W2(U), whose behaviour for U to 0 is W1(U)= alpha 1/ mod U mod ( beta 1), W2(U)= alpha 2/ mod U mod ( beta 2) (U being a null variable). In these spacetimes, the string dynamics is exactly (and explicitly) solvable even at the spacetime singularities. The string behaviour depends crucially on whether both parameters beta 1 and beta 2 are smaller or larger than two. When beta 1>or=2 and/or beta 2>or=2, the string does not cross the singularity U = 0 but goes off to infinity in a given direction alpha depending on the polarization of the gravitational wave. The string time evolution is fully determined by the spacetime geometry, whereas the overall sigma -dependence is fixed by the initial string state. The proper length at fixed tau to 0 (U to 0) stretches infinitely. For beta 1<2 and beta 2<2, the string passes smoothly through the spacetime singularity and reaches the U>0 region. In this case, outgoing operators make sense and we find the explicit transformation relating in and out operators. For the quantum string states, this implies spin polarization rotations and particle transmutations. The expectation values of the outgoing mass and mode number operators are computed and the excitation of the string modes after the crossing of the spacetime singularity is analysed.