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The tetrad formalism is used to study a broad class of spherically symmetric kink space–times whose Einstein tensor is computed and then diagonalized. A simple example of such a space–time is shown to satisfy the weak energy condition and to be extendible to a space–time that is geodesically complete.

It is well known that a complex four-component Dirac spinor defines an orthonormal tetrad on flat Minkowski spacetime. For example, a spinor solution to the Dirac equation determines the four-velocity and Pauli-Lubanski spin vector, comprising the timelike and first spacelike members of the particle’s tetrad, as well as two other spacelike members that are rarely discussed. Also, of particular note is the complex null tetrad formalism that provides a map from a two-component SL(2, C) spinor and its complex conjugate to a tetrad. The inverse problem is studied here. Given a tetrad, a complex Dirac spinor valued function of the tetrad is explicitly defined in such a way that certain sums of bilinear products of the components of this spinor exactly reproduce the tetrad. This spinor may be used in the Feynman propagator representation of the solution to the Dirac equation to generate a Dirac wave spinor with desired initial properties. The mappings studied here represent a new class of nonlinear mappings SO(...

The null tetrad formalism is used to investigate the structure of the Einstein field equations for Kerr–Schild gravitational fields in the presence of an elastic solid source. It is shown that such equations may be reduced to five nonlinear partial differential equations for five variables. It turns out that, when the interior solutions admit the same preferred null congruence of the vacuum ones and some compatibility conditions hold, it is possible to reduce them to a linear system and to develop a method of solution which closely resembles the ‘‘variation of the arbitrary constants’’ for ordinary differential equations. In the present paper, the above technical framework is developed in general and applied to two simple examples, deferring to future work the approach to the Kerr–interior problem.

The Newman–Penrose equations are solved as an illustrative and pedagogical example of an algebraic computing application. Specifically, a REDUCE program is presented to compute the mathematical objects needed to study the physical and mathematical properties of a given space‐time. This paper is directed toward different audiences: first, to readers interested in learning the capabilities and features of a computer algebra system. Also, to users who could benefit by implementing the developed algorithms to their particular research needs. A brief outline of the Newman–Penrose formalism is also given.

The Ernst metric is analyzed in terms of the null tetrad formalism of Newman and Penrose. Some applications of the formalism are also given.

Recent results concerning globally isometric mappings for arbitrary observers in flat space‐time are generalized to space‐times admitting a time orientation. Critical to the method is the use of an orthonormal tetrad which, when it is defined globally, allows the construction of a global isometry which generalizes the pointwise boost on flat space‐time. Connection coefficients are obtained, thereby defining acceleration covariant differentiation for both particle and tensor field equations. An application to orbiting observers in exterior Schwarzschild geometries is presented.

Quantum field theories on the surface of a four‐dimensional sphere are considered. The Hamiltonian is rotation invariant and its eigenvalues are discrete. Scalar, vector, and spinorial functions on S3 are discussed. The most general Lagrangians for Dirac, Weyl, and Majorana fermions are derived. They are different from the ones in existing literature. The wave functions and propagator are obtained and formulas for matrix elements involving spinors are presented. The discrete symmetries—parity, charge conjugation, and time reversal—are described. The Lagrangian in R×S3 transforms in a nontrivial way under these. Finally, the fermionic Lagrangian is rederived using the tetrad formalism, and conformal transformations are discussed. This leads to a generalization of the formalism to a time‐dependent radius of curvature. As a particular case, a new Lagrangian for de Sitter space is obtained, which, however, is not invariant under the full de Sitter group.

The Kerr−Newman metric is analyzed according to the null tetrad formalism. The components of the Weyl and the Ricci tensors are calculated and these tensors are then projected on a suitable null−tetrad basis. The spin coefficients of Newman and Penrose are also calculated. These results are applied to obtain the equations of gravitational and neutrino perturbations in the Kerr−Newman metric.

The null tetrad formalism of Newman and Penrose is used to investigate empty space‐times which are algebraically special on a given world line. It is found that, when the world line is timelike, the space‐time admits a congruence which is geodesic and shear‐free on the world line. A similar result is obtained for those empty space‐times which are algebraically special on a given space like hypersurface.

The integrability conditions of conformal motions are written in the null tetrad formalism of Newman and Penrose. The maximum order of the group of conformal motions admitted by nonflat empty space‐times of given Petrov type is shown to be at most one greater than the maximum order of the group of Killing motions. The symmetries of those empty space‐times which possess hypersurface orthogonal geodesic rays with nonvanishing divergence are determined. Among these space‐times is one of type III which admits a group of Killing motions of order three. This provides a counter example to a result of Petrov which states that the maximum order of the group of Killing motions for such space‐times is two.

Tetrad formalism is used to derive a set of 36 scalar field equations which correspond to the ordinary field equations Rμν=0. The scalar equations are obtained by beginning with a given Petrov type of empty‐space Riemann tensor and applying the Ricci identity to each of the tetrad vectors. The unknowns or field variables become the 24 Ricci rotation coefficients, the number of which can always be reduced by the Bianchi identities and occasionally by tetrad transformations which leave the form of the Riemann tensor invariant. The use of these scalar field equations is illustrated by their application to a degenerate case of Petrov type I. It is believed that by this method all possible solutions of this particular case have been found.

A new approach to general relativity by means of a tetrad or spinor formalism is presented. The essential feature of this approach is the consistent use of certain complex linear combinations of Ricci rotation coefficients which give, in effect, the spinor affine connection. It is applied to two problems in radiationtheory; a concise proof of a theorem of Goldberg and Sachs and a description of the asymptotic behavior of the Riemann tensor and metric tensor, for outgoing gravitational radiation.

Using a recently developed global isometry method for treating accelerating observers, the induced tangent space transformation on flat Lorentzian R4 is mapped homomorphically onto a time‐dependent D(1/2,0) ⊕ D(0,1/2) representation of SL (2,C). The Dirac equation is shown to take on pseudoterms via this mapping. Eliminating the pseudoterms by identifying an affine connection, an exact analytic expression for the covariant derivative is found for general cases of arbitrary C2 timelike observers. The transformation properties of the connection are shown to satisfy the conditions imposed by a general tetrad formalism. The specific case of the rotating observer is considered wherein the exact expression for the boosted Dirac equation is found.