We solve the equivalence problem for vacuum pp-wave spacetimes by employing the Karlhede algorithm. Our main result is a suite of Cartan invariants that allows for the complete invariant classification of the vacuum pp-waves. In particular, we derive the invariant characterization of the G2 and G3 sub-classes in terms of these invariants. It is known [J. M. Collins, “The Karlhede classification of type N vacuum spacetimes,” Class. Quantum Grav.8, 1859–1869 (Year: 1991)10.1088/0264-9381/8/10/011] that the invariant classification of vacuum pp-waves requires at most the fourth order covariant derivative of the curvature tensor, but no specific examples requiring the fourth order were known. Using our comprehensive classification, we prove that the q ⩽ 4 bound is sharp and explicitly describe all such maximal order solutions.
Received 16 October 2012Accepted 25 January 2013Published online 20 February 2013
The authors would like to thank Georgios Papadopoulos for useful discussions. The research of R.M. and A.C. is supported, in part, by Natural Sciences and Engineering Research Council (Canada) (NSERC) discovery grants.
Article outline: I. INTRODUCTION II. VACUUM PP-WAVE SPACETIMES III. THE G1SOLUTIONS IV. THE (0, 1) CLASS V. THE G2 PRECURSORS VI. THE G2SOLUTIONS VII. THE MAXIMAL IC ORDER CLASS VIII. THE G3SOLUTIONS IX. THE G5 AND G6SOLUTIONS X. CONCLUSIONS