Newman-Penrose Formalism - University of Florida Formalism... · Newman-Penrose Formalism-Tetrad...

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!