the small ratios . For example, in the classic experiment in which, at a time when, as viewed from Earth, Venus is just about to pass behind the Sun, a radar signal emitted from Earth which grazes the limb of the Sun, bounces off Venus, and returns to Earth (once again grazing the limb of the Sun), the relativistic time delay is about 20 microseconds according to Nordström's theory and about 240 microseconds according to general relativity. Summary of results We can summarize the results we found above in the following table, in which the given expressions represent appropriate approximations: The last four lines in this table list the so-called four classic solar system tests of relativistic theories of gravitation. Of the three theories appearing in the table, only general relativity is in agreement with the results of experiments and observations in the solar system. Nordström's theory gives the correct result only for the Pound–Rebka experiment; not surprisingly, Newton's theory flunks all four relativistic tests. Vacuum gravitational plane wave In the double null chart for Minkowski spacetime, a simple solution of the wave equation is , where f is an arbitrary smooth function. This represents a plane wave traveling in the z direction. Therefore, Nordström's theory admits the exact vacuum solution which we can interpret in terms of the propagation of a gravitational plane wave. This Lorentzian manifold admits a six-dimensional Lie group of isometries, or equivalently, a six-dimensional Lie algebra of Killing vector fields: (a null translation, "opposing" the wave vector field ) (spatial translation orthogonal to the wavefronts) (rotation about axis parallel to direction of propagation) For example, the Killing vector field integrates to give the one parameter family of isometries Just as in special relativity (and general relativity), it is always possible to change coordinates, without disturbing the form of the solution, so that the wave propagates in any direction transverse to . Note that our isometry group is transitive on the hypersurfaces . In contrast, the generic gravitational plane wave in general relativity has only a five-dimensional Lie group of isometries. (In both theories, special plane waves may have extra symmetries.) We'll say a bit more about why this is so in a moment. Adopting the frame field we find that the corresponding family of test particles are inertial (freely falling), since the acceleration vector vanishes Notice that if f vanishes, this family becomes a family