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Newman-Penrose Formalism
-Tetrad formalism
-Special cases
-NP formalism
-GHP method
-Application Summary
Tetrad formalism
-Introduction
At every point in space set up four linearly independent vectors
where
Tetrad formalism
- Intrinsic Derivative & Ricci Rotation Coefficients
Define Ricci rotation coefficients
Intrinsic derivative
Tetrad formalism
- Commutation relations
Tetrad formalism
- Ricci & Bianchi Identities
-Generalization
-Coordinate and Tetrad transformation
Special Tetrad system
-Four vectors at each point are in the direction of the
coordinate axes; that is, parallel to the four coordinate
differentials :
-base vectors of a Cartesian coordinate system in the local
Minkowski system of the point concerned:
-null vectors as tetrad vectors
using this system, complex tetrad components can arise
• Special cases
NP tetrad
Null tetrad approach to NP Formalism
-Introduction
- Spin Coefficient in terms of Ricci Rotation
Coefficients
Weyl, Ricci and Riemann Tensors in NP
formalism
NP set of equations:
• commutation relations,
• Ricci Identities,
• eliminant relations
• Bianchi Identities
Ricci
Identities
Bianchi Identities:
Spinor calculus
• Spinors in minkowskian space,
Isomorphism between Unimodular T. and L.T.s
Spinors in minkowskian space
define
is invariant
Spinors in minkowskian space
General connection between Tensors and
Spinors and Spinor Affine Connection
2-Spinor approach to NP Formalism
Dyad Formalism
Spin Coefficients in terms of Spinor Affine
Connection
GHP
New and appropriate operations
δ’=
Application summary
-In finding & analyzing new solutions of Einstein
field equations
-In studying asymptotic properties of radiation
fields
-In particular GHP method turn out to be very
effective in 2-surfaces calculations
-Developing approaches to quantization through
the study of complexified space times!