to Brinkmann coordinates, in the form where is any smooth function. This was the original definition of Brinkmann, and it has the virtue of being easy to understand. The definition which is now standard in the literature is more sophisticated. It makes no reference to any coordinate chart, so it is a coordinate-free definition. It states that any Lorentzian manifold which admits a covariantly constant null vector field is called a pp-wave spacetime. That is, the covariant derivative of must vanish identically: This definition was introduced by Ehlers and Kundt in 1962. To relate Brinkmann's definition to this one, take , the coordinate vector orthogonal to the hypersurfaces . In the index-gymnastics notation for tensor equations, the condition on can be written . Neither of these definitions make any mention of any field equation; in fact, they are entirely independent of physics. The vacuum Einstein equations are very simple for pp waves, and in fact linear: the metric obeys these equations if and only if . But the definition of a pp-wave spacetime does not impose this equation, so it is entirely mathematical and belongs to the study of pseudo-Riemannian geometry. In the next section we turn to physical interpretations of pp-wave spacetimes. Ehlers and Kundt gave several more coordinate-free characterizations, including: A Lorentzian manifold is a pp-wave if and only if it admits a one-parameter subgroup of isometries having null orbits, and whose curvature tensor has vanishing eigenvalues. A Lorentzian manifold with nonvanishing curvature is a (nontrivial) pp-wave if and only if it admits a covariantly constant bivector. (If so, this bivector is a null bivector.) Physical interpretation It is a purely mathematical fact that the characteristic polynomial of the Einstein tensor of any pp-wave spacetime vanishes identically. Equivalently, we can find a Newman–Penrose complex null tetrad such that the Ricci-NP scalars (describing any matter or nongravitational fields which may be present in a spacetime) and the Weyl-NP scalars (describing any gravitational field which may be present) each have only one nonvanishing component. Specifically, with respect to the NP tetrad the only nonvanishing component of the Ricci spinor is and the only nonvanishing component of the Weyl spinor is This means that any pp-wave spacetime can be interpreted, in the context of general relativity, as a null dust solution. Also, the Weyl tensor always has Petrov type N as may be verified by using the Bel criteria. In other