Special relativity and Kepler's first law

Abstract

Consider a two-body system consisting of a very massive sun and a planet in a circular orbit of radius R around it, as seen by an inertial observer S at rest with respect to the sun. Obviously this sun–planet system obeys Kepler’s first law. For concreteness let the planet revolve on the plane z = 0 and the sun be located at x = y = z = 0. Now consider another inertial observer Sl moving with speed v" = cβˆ x, say, on the plane z = zs relative to the sun.