A family of explicit exact solutions of Einstein's equations in four and higher dimensions is studied which describes the gravitational field of an object accelerating due to an anisotropic emission of photons. It is possible to prescribe an arbitrary motion, so that the acceleration of such photon rocket need not be uniform both its magnitude and direction may vary with time. Except at the location of the rocket the spacetimes have no curvature singularities, and topological defects like cosmic strings are also absent. Any value of a cosmological constant is allowed. We investigate some particular examples of motion, namely a straight flight and a circular trajectory, and we derive the corresponding radiation patterns and the mass loss of the rockets. We also demonstrate the absence of "gravitational aberration" in such spacetimes. This interesting member of the higher-dimensional RobinsonTrautman class of pure radiation spacetimes of algebraic Type D generalizes the class of Kinnersley's solutions that has long been known in four-dimensional general relativity.
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