Null Tetrad Research Articles

GRAVITATIONAL PERTURBATIONS OF THE KERR BLACK HOLE DUE TO ARBITRARY SOURCES

We describe the Kerr black hole in the ingoing and outgoing Kerr–Schild horizon penetrating coordinates. Starting from the null vector naturally defined in these coordinates, we construct the null tetrad for each case, as well as the corresponding geometrical quantities allowing us to explicitly derive the field equations for the perturbed scalar projections Ψ0(1) and Ψ4(1) of the Weyl tensor, including arbitrary source terms. This perturbative description, including arbitrary sources, described in horizon penetrating coordinates is desirable in several lines of research on black holes, and contributes to the implementation of a formalism aimed to study the evolution of the spacetime in the region where two black holes are close together.

Quantum entropy of the Kerr black hole arising from gravitational perturbation

The quantum entropy of the Kerr black hole arising from gravitational perturbation is investigated by using Null tetrad and \'t Hooft\'s brick-wall model. It is shown that effect of the graviton\'s spins on the subleading correction is dependent of the square of the spins and the angular momentum per unit mass of the black hole, and contribution of the logarithmic term to the entropy will be positive, zero, and negative for different value of $a/r_+$.

Notes on isolated horizons

A general analysis for characterizing and classifying `isolated horizons'is presented in terms of null tetrads and spin coefficients. The freelyspecifiable spin coefficients corresponding to isolated horizons areidentified and specific symmetry classes are enumerated. For isolatedhorizons admitting at least one spatial isometry, a standard set ofspherical coordinates are introduced and an associated metric is obtained. Anangular momentum is also defined.

Newman-Penrose Formalism and Gravitational Impulsive and Shock Waves**The project supported by the Scientific Foundation of the Chinese Academy of Sciences

Vacuum spacetimes endowed with two commuting spacelike Killing vector fields are considered. Subject to the hypothesis that there exists a shearfree null geodesic congruence orthogonal to the two-surface generated by the two commuting spacelike Killing vector fields,it is shown that, with a specific choice of null tetrad, the Newman-Penrose equations are reduced to an ordinary differential equation of Riccati type. fiom the consideration of this differential equation, exact solutions of the vacuum Einstein field equations with distribution valued Weyl curvature describing the propagation of gravitational impulsive and shock wave of variable polarization are then constructed.

A Canonical Null Gauge for the Einstein Equations

Associated with any choice of outgoing null (characteristic) coordinate, we construct a null tetrad which is determined uniquely by purely local information. Unlike other well-known characteristic gauge conditions [1–4], this “canonical” tetrad is determined at any point solely by the metric and its first derivatives at that point. The tetrad leads to a radial coordinate which is also uniquely determined by purely local information. These properties greatly simplify the structure of the hypersurface Einstein equations and may be used to directly compare metrics with compatible null foliations.

New form of the Kerr solution

A new form of the Kerr solution is presented. The solution involves a time coordinate which represents the local proper time for free-falling observers on a set of simple trajectories. Many physical phenomena are particularly clear when related to this time coordinate. The chosen coordinates also ensure that the solution is well behaved at the horizon. The solution is well suited to the tetrad formalism and a convenient null tetrad is presented. The Dirac Hamiltonian in a Kerr background is also given and, for one choice of tetrad, it takes on a simple, Hermitian form.

Einstein-Maxwell Spacetime with Two Commuting Spacelike Killing Vector Fields and Newman-Penrose Formalism

Einstein-Maxwell spacetimes endowed with twocommuting spacelike Killing vector fields areconsidered. Subject to the hypotheses that one of thetwo null geodesic congruence orthogonal to thetwo-surface generated by the two commuting spacelikeKilling vector fields is shearfree and theelectromagnetic field is non null, it is shown that,with a specific choice of null tetrad, theNewman-Penrose equations together with the Maxwell equations for theclass of spacetime considered may be reduced to asecond-order ode of Sturm-Liouville type, from whichexact solutions of the class of spacetimes consideredmay be constructed. Examples of exact solutions arethen given. Exact solutions with distribution-valuedWeyl curvature describing the scattering ofelectromagnetic shock wave with gravitational impulsiveor shock wave of variable polarisation are also constructed.

The maple package NPTOOLS; a symbolic algebra package for tetrad formalisms in general relativity

In this paper, we present and discuss the NPTOOLS package written for the computer algebra system Maple. The package contains the main routines “nulltran”, “inv”, “nullor”, “ornull” and “pptype”. The routine “nulltran” is an implementation of the 6-parameter Lorentz group in the Newman-Penrose (null tetrad) formalism, and “inv” checks for invariance of equations under subgroups of the Lorentz group. The routines “nullor” and “ornull” provide a translation between the Newman-Penrose quantities associated with the null tetrad and the orthonormal tetrad quantities associated with the Ellis-MacCallum formalism. Finally, “pptype” determines the Petrov-Plebański type of the Ricci tensor.

The Lanczos potential for vacuum space–times with an Ernst potential

By a particular choice of tetrad and coordinate systems, we find the Lanczos potential for general vacuum stationary axisymmetric space–times and for general vacuum cylindrically symmetric solutions. This result is particularly elegant and unifies the treatment of the Lanczos potential and the Ernst potential for vacuum stationary axially symmetric and cylindrically symmetric space–times. The Lanczos potential for such large classes of vacuum space–times is expressed very simply in terms of spin-coefficients for a null tetrad.

Symmetry operators and separation of variables for spin-wave equations in oblate spheroidal coordinates

A family of second-order differential operators that characterize the solution of the massless spin s field equations, obtained via separation of variables in oblate spheroidal coordinates and using a null tetrad is found. The first two members of the family also characterize the separable solutions in the Kerr space-time. It is also shown that these operators are symmetry operators of the field equations in empty space-times whenever the space-time admits a second-order Killing–Yano tensor.

Numerical relativity - III. The Bondi mass revisited

Expressions for the Bondi mass and mass flux far from an isolated system are derived by using an arbitrary Newman–Penrose null tetrad in the conformally related space-time. A numerical tool for relating the near field of an isolated gravitationally radiating system to the far field is developed, so that the mass of such a system can be computed and some simple examples are described.

On the Kerr–Newman metric in cosmological background

Using the method of null tetrad, the Kerr–Newman metric in the background of the Robertson–Walker universe is derived in terms of a metric that is conformal to a generalized Kerr–Schild metric. A new two–fluid interpretation of this metric is presented. The Kerr–Newman metric in the background of the Einstein–de Sitter universe is discussed in detail.

On the electromagnetic field and the Teukolsky–Press relations in arbitrary space-times

The relations on the electromagnetic field obtained by Teukolsky and Press for type D vacuum space-times are considered; these are four second-order equations in two complex components of the field with respect to a principal null tetrad. A rigorous geometric interpretation of these relations is given, showing the essential role played by the Maxwellian character of the basic null tetrad. It appears that, generically, the Teukolsky–Press relations are incomplete. Once completed, their generalizations to the general Maxwell equations (with source term) with respect to non-necessarily Maxwellian tetrads on arbitrary space-times are given.

Characterization of locally rotationally symmetric space-times

We give a simple characterization of locally rotationally symmetric space-times in terms of the existence of a canonical null tetrad or canonical orthonormal tetrad. The result is applied to space-times which satisfy the Einstein field equations with a perfect fluid or electromagnetic field as source.

Dirac brackets for general relativity on a null cone

The Hamiltonian for the Einstein equations is constructed on a outgoing null cone with the help of the usual null tetrad. The resulting null surface constraints are shown to be second class in the terminology of Dirac. These second class constraints are eliminated by use of the “starring” procedure of Bergmann and Komar.

The Hamiltonian of general relativity on a null surface

The Hamiltonian for the Einstein equations is constructed on an outgoing null cone with the help of the usual null tetrad used in the study of the asymptotical gravitational radiation field.

Local black holes are typeD on the Horizon

For the most general static and axisymmetric local black hole with (topologically) spherical or toroidal horizon we calculate the Riemann tensor, we construct a Newman-Penrose null tetrad, we evaluate Weyl scalars and we investigate the algebraic type of the solutions. We find that the spherical black holes are of Petrov typeD on the horizon and on the axis of azimuthal symmetry, while the toroidal ones are of typeD on the horizon and on the plane of the reflectional symmetry.

Pure radiation fields admitting nontrivial null symmetries

The sixteen types of geometrical symmetries corresponding to the continuous groups of collineations and motions generated by a null vector n are considered. The common propagation vector of a pure electromagnetic radiation field and a pure gravitational radiation field is chosen to be n. For such radiation fields all the sixteen symmetries are expressed in terms of the Newman–Penrose (NP) spin coefficients and then it is shown that when n is a gradient field there are only five independent symmetries. The existence of these five nontrivial null symmetries is established by finding exact solutions of Einstein–Maxwell field equations when n satisfies freedom conditions and when l of the NP null tetrad (l, m, m̄, n) is shear-free. Thus a class of space-times of pure radiation fields that admit (i) a Ricci collineation which is not a curvature collineation (CC), (ii) a CC which is not a special curvature collineation (SCC), (iii) a SCC which is not an affine collineation (AC), (iv) an AC which is not a motion, and (v) a motion is determined.

Separation of variables and symmetry operators for the neutrino and Dirac equations in the space-times admitting a two-parameter abelian orthogonally transitive isometry group and a pair of shearfree geodesic null congruences

We show that there exist a coordinate system and null tetrad for the space-times admitting a two-parameter abelian orthogonally transitive isometry group and a pair of shearfree geodesic null congruences in which the neutrino equation is solvable by separation of variables if and only if the Weyl tensor is Petrov type D. The massive Dirac equation is separable if in addition the conformal factor satisfies a certain functional equation. As a corollary, we deduce that the neutrino equation is separable in a canonical system of coordinates and tetrad for the solution of Einstein’s type D vacuum or electrovac field equations with cosmological constant admitting a nonsingular aligned Maxwell field and that the Dirac equation is separable only in the subclass of Carter’s [Ã] solutions and the Debever–McLenaghan null orbit solution A0. We also compute the symmetry operators which arise from the above separability properties.

Orthogonal transitivity, invertibility and null geodesic separability in type D vacuum solutions of Einstein’s field equations with cosmological constant

It is shown without explicit integration that all Petrov type-D vacuum solutions with cosmological constant admit at least a two-parameter, abelian, orthogonally transitive group of local isometries. In the case when the orbits are non-null the group is invertible and a symmetric null tetrad is shown to exist in which it is manifest that the principal null congruences defined by the type-D Weyl tensor are indistinguishable. General forms for the metrics are given for both the case of non-null and null group orbits. It is also demonstrated that there always exists a system of coordinates for these solutions in which the Hamilton–Jacobi equation for the null geodesics is solvable by separation of variables, a fact that explains the existence of a conformal Killing tensor therein. A partial invariant characterization of Kinnersley’s type-D vacuum solutions is given from which it follows that all his metrics except those of Case III admit a (0,2) Yano–Killing tensor and hence a full (0,2) Killing tensor.