Pp-wave Spacetime - Mathematical Definition

Mathematical Definition

A pp-wave spacetime is any Lorentzian manifold whose metric tensor can be described, with respect 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. In this sense, the notion of a pp-wave spacetime is entirely mathematical and belongs to the study of pseudo-Riemannian geometry. In the next section, we will turn to the physical interpretation of pp-waves.

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.)