In this thesis we study aspects of plane wave spacetimes in the hope of shedding light of the nature of holography for plane waves. In particular, we would like to understand better the space of asymptotically plane wave solutions. We first review the necessary background on plane waves, variational principles for gravity and black holes in higher dimensions. We then propose a definition of asymptotically plane wave spacetimes in vacuum gravity in terms of the asymptotic fall-off of the metric and discuss the relation to previously constructed exact solutions. We construct a well-behaved action principle for such spacetimes, using the formalism developed by Mann and Marolf. We show that the action is finite on-shell and that the variational principle is well-defined for solutions of vacuum gravity satisfying our asymptotically plane wave fall-off conditions.\ud Next we investigate the construction of black holes and black strings in vacuum plane wave spacetimes using the method of matched asymptotic expansions. We find solutions of the linearised equations of motion in the asymptotic region for a general source on a plane wave background. We observe that these solutions have some unusual propeties and do not satisfy our previously defined conditions for being asymptotically plane wave. Hence, the space of asymptotically plane solutions is restricted. We consider the solution in the near horizon region, treating the plane wave as a perturbation of a black object, and find that there is a regular black string solution. We find that no regular black hole solution exists, which is a counter-example to the conjecture of Emparan et. al. We end with a discussion of our results and suggest possible directions for future work
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