GHP formalism

The GHP formalism (or Geroch–Held–Penrose formalism), also known as the compacted spin-coefficient formalism, is a technique used in the mathematics of general relativity that involves singling out a pair of null directions at each point of spacetime. It is a rewriting of the Newman–Penrose formalism which respects the covariance of Lorentz transformations preserving two null directions. This is desirable for Petrov Type D spacetimes, including black holes in general relativity, where there is a preferred pair of degenerate principal null directions but no natural additional structure to fully fix a preferred Newman–Penrose (NP) frame.

Covariance

The GHP formalism notices that given a spin-frame $(o^{A},\iota ^{A})$ with $o_{A}\iota ^{A}=1,$ the complex rescaling $(o^{A},\iota ^{A})\rightarrow (\lambda o^{A},\lambda ^{-1}\iota ^{A})$ does not change normalization. The magnitude of this transformation is a boost, and the phase tells one how much to rotate. A quantity of weight $(p,q)$ is one that transforms like $\eta \rightarrow \lambda ^{p}{\bar {\lambda }}^{q}\eta .$ One then defines derivative operators which take tensors under these transformations to tensors. This simplifies many NP equations, and allows one to define scalars on 2-surfaces in a natural way.

References

Geroch, Robert, Held, A. and Penrose, Roger (1973). "A space-time calculus based on pairs of null directions". Journal of Mathematical Physics. 14 (7): 874–881. Bibcode:1973JMP....14..874G. doi:10.1063/1.1666410.{{cite journal}}: CS1 maint: multiple names: authors list (link)

Penrose, Roger; Rindler, Wolfgang (1987). Spinors and Space-Time: Volume 1, Two-Spinor Calculus and Relativistic Fields. Cambridge: Cambridge University Press. ISBN 0-521-33707-0.

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GHP formalism

Covariance

See also

References