How to obtain the transformation of Null Tetrad ?

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    Tetrad Transformation
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SUMMARY

The discussion focuses on the null tetrad transformations within the Newman-Penrose formalism, specifically addressing the mathematical structure of null rotations. Participants seek clarity on the derivation of the transformation coefficients that maintain orthonormality among the tetrad vectors k, ℓ, and m. The general transformation form is outlined, emphasizing the conditions for the coefficients: α must equal 0, and relationships between β, γ, and ε are established. This foundational understanding is crucial for further exploration of null transformations in general relativity.

PREREQUISITES
  • Familiarity with the Newman-Penrose formalism
  • Understanding of tetrad formalism in general relativity
  • Knowledge of orthonormality conditions in vector spaces
  • Basic grasp of complex numbers and their conjugates
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  • Study the derivation of null tetrads in the context of general relativity
  • Explore the implications of null rotations on spacetime geometry
  • Learn about the role of complex coefficients in tetrad transformations
  • Investigate applications of the Newman-Penrose formalism in gravitational wave analysis
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This discussion is beneficial for physicists, mathematicians, and students specializing in general relativity, particularly those interested in the mathematical foundations of tetrad formalism and null transformations.

Karmerlo
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Currently, I meet with the so-called null rotation in my study. I cannot understand why it has a mathematical form like that? Is there anyone familiar with this? Can anyone give a lucid explanation of it or provide steps to derive it.

See the image above on the null transformation (in Newman-Penrose Formalism)

Thanks.
 
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These are all the transformations that preserve the orthonormality relations among k, ℓ, m. For example, ask yourself, what tetrad rotation preserves ℓ? The most general form is

ℓ → ℓ
m → m + α k + β ℓ
k → k + γ ℓ + δ m + ε m*

and try to determine the coefficients:

k = k* ⇒ δ = ε*
ℓ·m = 0 ⇒ α = 0
k·m = 0 ⇒ β - ε
k·k = 0 ⇒ γ - ε ε*
 

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