On Gravitational Shock Waves in Curved Spacetimes

Some years ago Dray and 't Hooft found the necessary and sufficient conditions to introduce a gravitational shock wave in a particular class of vacuum solutions to Einstein's equations. We extend this work to cover cases where non-vanishing matter fields and cosmological constant are present. The sources of gravitational waves are massless particles moving along a null surface such as a horizon in the case of black holes. After we discuss the general case we give many explicit examples. Among them are the $d$-dimensional charged black hole (that includes the 4-dimensional Reissner-Nordstr\"om and the $d$-dimensional Schwarzschild solution as subcases), the 4-dimensional De-Sitter and Anti-De-Sitter spaces (and the Schwarzschild-De-Sitter black hole), the 3-dimensional Anti-De-Sitter black hole, as well as backgrounds with a covariantly constant null Killing vector. We also address the analogous problem for string inspired gravitational solutions and give a few examples.