Invariant differential operators and the equivalence problem of algebraically special spacetimes

Abstract

Many calculations in general relativity are simplified when using a tetrad formalism. As an important example we have the Newman-Penrose (NP) formalism which uses a complex null tetrad as basis for writing all information corresponding to Einstein's equations. However, certain physical problems are best described when the formalism is adapted to the geometry of such physical situations, i.e, when the basis vectors (or spinors) are not completely arbitrary but related to the geometry or physics in some natural way. A well known example is the Geroch-Held-Penrose (GHP) formalism which best describes the geometry of a null 2-surface and which is invariant under the group of spin and boost transformations.

The GHP formalism is ideally suited to situations where two null directions are naturally singled out, but in many physical cases one is faced with only one preferred null direction. As important examples we have null congruences, null hypersurfaces or wave fronts and type N spacetimes.

A formalism which is invariant under null rotations is presented. The fundamental objects are totally symmetric spinors. From this notation we develop a formalism based on a single null direction which is covariant under both spin and boost transformation and null rotations.

Although both formalisms, which we refer to in this thesis as the generalized NP formalism and the generalized GHP formalism, have many other applications mainly to do with null congruences and null hypersurfaces they are used in here as an application to the equivalence problem of type N spacetimes.

The problem of determining whether two given metrics expressed in different coordinate systems are actually the same metric, i.e., can be mapped into each other by a coordinate transformation is the well known equivalence problem of metrics. The theoretical resolution of this problem was originally provided by Cartan and later refined by Karlhede who provided the useful Karlhede algorithm of classifying different Petrov types of spacetimes.

More information

Identifiers

Catalogue record

Export record

ASCII CitationAtomBibTeXData Cite XMLDublin CoreDublin CoreEP3 XMLEndNoteHTML CitationHTML CitationHTML ListJSONMETSMODSMPEG-21 DIDLOpenURL ContextObjectOpenURL ContextObject in SpanRDF+N-TriplesRDF+N3RDF+XMLRIOXX2 XMLReferReference ManagerSimple Metadata