Tensor[AdaptedNullTetrad] - find a null tetrad which transforms the Newman-Penrose Weyl scalars to a standard form
AdaptedNullTetrad(NT, PT, options )
AdaptedNullTetrad(NT, PT, W, options )
AdaptedNullTetrad(NT, PT, NP , options )
NT - a null tetrad for the spacetime metric g
PT - the Petrov type of g
W - (optional) the Weyl tensor of g
NP - (optional) the Newman-Penrose Weyl scalars
options - one or more of the keyword arguments method and output
The Newman-Penrose Weyl scalars are a set of 5 complex scalars, labeled Ψ0, Ψ1, Ψ2, Ψ3, Ψ4 , and defined by certain components of the Weyl tensor with respect to a given null tetrad in a four dimensional spacetime of signature [1, -1, -1, -1]. Under local Lorentz transformations, the Newman-Penrose Weyl scalars transform among themselves in a natural way. Depending upon the Petrov type of the spacetime it is possible to transform the Newman-Penrose Weyl scalars to one of following normal forms. Below, η and χ are complex scalars. See NPCurvatureScalars, NullTetradTransformation.
Type I. Ψ0= 32 η χ , Ψ1 = 0, Ψ2=12η2 − χ, Ψ3 =0, Ψ4 = 32η χ .
Type II. Ψ0= 0, Ψ1= 0, Ψ2 =η, Ψ3=0,Ψ4 = 6 η.
Type III. Ψ0 = 0, Ψ1=0, Ψ2 =0, Ψ3=1, Ψ4 = 0.
Type D. Ψ0= 0, Ψ1 = 0, Ψ2 =η, Ψ3 = 0, Ψ4= 0.
Type N. Ψ0= 0, Ψ1 =0, Ψ2 = 0, Ψ3 = 0, Ψ4 = 1.
Type O. Ψ0= 0, Ψ1 = 0, Ψ2 =0, Ψ3 =0, Ψ4 = 0.
See Penrose and Rindle Vol. 2, Section 8.3.
Null tetrads for which the Newman-Penrose Weyl scalars are in the above normal form are called adapted null tetrads. Calculations are often simplified by using an adapted null tetrad.
The command AdaptedNullTetrad returns a null tetrad which will put the Newman-Penrose Weyl scalars in the above normal form.