An improved algorithm for quartic equation classification and Petrov classification

The determination of multiplicities of the roots of quartic equations with (in general) nonconstant coefficients is studied in the context of the Petrov classification of the Weyl conformal curvature tensor. A history of existing algorithms for this determination is given. An alternative algorithm is described and a qualitative comparison to the above-mentioned algorithms given. Following some notes on the actual computer implementation, a quantitative comparison is made between three of the algorithms, using the symbolic computer language Maple. The algorithm is also implemented in the symbolic language MuSimp.     

Algebraic classification of the gravitational field

We rewrite the tensorial method of Peres [1] for the Petrov classification in terms of the Newman-Penrose formalism [2]. It results an algorithm computationally simpler that the one proposed by DInverno-Russell Clark [3].This revised version was published online in April 2005. The publishing date was inserted.     

A New Algorithm for the Petrov Classification of the Weyl Tensor

A new algorithm for the Petrov classification of the Weyl tensor is introduced. It is similar to the Letniowski-McLenaghan algorithm [1] when some of the 's are zero, but offers a completely new approach when all of the 's are nonzero. In all cases, new code in Maple has been implemented.     

Late-time behavior of primordial gravitational waves in expanding universe

The results of two different approaches to the late-time behavior of primordial gravitational radiation are compared. Using asymptotic expansion in time of the tetrad components of the Riemann tensor one finds that initially chaotic behavior transfers itself into a radiative gravitational field of Petrov typeN ast. On the other hand, to accomplish the physical picture, we study the high-frequency behavior of the field variables in the same formalism. We show that the background spacetime is of general Petrov type I, and then calculate the tetrad components of the stress-energy tensor induced by the disappeared radiation with the help of Newman-Penrose equations.     

Normal-dominated singularities in static space-times

We define normal-dominated singularities of static solutions of the Einstein equations and show that a uniquely and invariantly defined structure can be assigned to these singularities. We find for the general solution that the dominant term of the Riemann tensor near the singularity is of Petrov Type N. Except for one special class of solutions, it seems that in general the shear of the null geodesics blows up at the same rate as their convergence near the singularity, in contradistinction to the elementary singularity of Newman and Posadas. We compute the structure for a variety of known static solutions as well as the stationary Kerr-Newman metrics.Supported in part by NSF Grant GP 34639 X.     

Necessary and sufficient conditions for trivial solutions in supergravity

We determine the conditions necessary for a solution of the supergravity field equations with infinitesimal spin-3/2 field to be a pure gauge transformation of an Einstein vacuum field. The analysis depends on the Petrov classification of the curvature tensor and uses two-component spinor calculus. For general type I, the type II, and typeD, the necessary conditions found are also shown to be sufficient, and the explicit form of the gauge transformation can be given.Work supported in part by the Einstein Memorial Foundation.     

Conformally flat Einstein-Maxwell spaces and some generalizations of them

It has been noted that the family of plane electromagnetic waves and the electromagnetic universe of Bertotti-Robinson exhaust the entire class of conformally flat Einstein-Maxwell spaces. In the formalism of Newman-Penrose a family of exact solutions of the Einstein-Maxwell equations of the type of Bertotti-Robinson is obtained with a cosmological term belonging to the degenerate type D in the algebraic classification of Petrov and describing the space-time generated by a covariantly constant, nonisotropic electromagnetic field.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 50–55, June, 1979.     

General form of the maxwell equations and group of external transformations of electromagnetic quantities

The Maxwell equations in their general form, corresponding to the presence of electric and magnetic charge and currents, are shown to be invariant with respect to a previously unknown group, the group of external transformations of electromagnetic quantities. The introduction of this group is based essentially on the use of an algebraic method of writing the Maxwell equations in terms of a real-spinor basis algebra (one of the real forms of Clifford algebra). The algebraic notation is also used to determine the transformation properties of electromagnetic quantities with respect to spatial and temporal reflections.Translated from Izvestiya VUZ. Fizika, No. 12, pp. 19–23, December, 1969.The author thanks A. Z. Petrov, D. D. Ivanenko, A. E. Levashov, the participants in the seminars under their guidance, and participants of the Ivanovo Interinstitute Seminar on Mathematical Physics for discussion of these results.     

Qualitative analysis of ten-dimensional lovelock cosmological models

The Einstein-like field equations obtained from the variation of a Lagrangian containing linear, quadratic (Gauss-Bonnet) and quartic terms for a ten-dimensional cosmological model cannot be solved analytically. However, we can reduce them to a system of dynamical equations for the Hubble parameters. The study of the mathematical properties of the fixed points of this system gives a qualitative picture of the behaviour of a class of possible solutions. The inclusion of a quartic term generates an extremely rich structure for the corresponding dynamical system. Some solutions are shown to exhibit the interesting property of dimensional reduction, which has been proposed as a possible explanation of the three-dimensional character of our universe.     

Algebraic classification of gravitational fields in five-dimensional space-time

We consider the algebraic classification of five-dimensional empty space-time (Kalutsa type) with one time-like direction as a generalization of the Petrov algebraic classification of gravitational fields in four-dimensional space-time. We study two special cases: a) zero electromagnetic field and zero scalar field; b) nonzero electromagnetic field and zero scalar field. For the (1+4) separated Kalutsa five-metric we introduce the pentad metric of a tangent five-space, which is mapped together with the curvature tensor into a ten-dimensional real flat vector space. The classification is constructed in local geodesic coordinates for the above two cases. In both cases the characteristic equation can be reduced to a sixth-order equation that can be simplified when certain requirements are satisfied. Our results demonstrate the nontrivial nature of algebraic classification in five dimensions.This work was performed within the framework of the State Science and Technology Astronomy program.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 73–78, March, 1995.     

Petrov Types of Slowly Rotating Fluid Balls

Circularly rotating axisymmetric perfect fluid space-times are investigated to second order in the small angular velocity. The conditions of various special Petrov types are solved in a comoving tetrad formalism. A number of theorems are stated on the possible Petrov types of various fluid models. It is shown that Petrov type II solutions must reduce to the de Sitter spacetime in the static limit. Two space-times with a physically satisfactory energy-momentum tensor are investigated in detail. For the rotating incompressible fluid, it is proven that the Petrov type cannot be D. The equation of the rotation function can be solved for the Tolman type IV fluid in terms of quadratures. It is also shown that the rotating version of the Tolman IV space-time cannot be Petrov type D.     

Perfect Fluid Spacetimes with a Purely Magnetic Weyl Tensor

We present a solution of the gravitational fieldequations which is similar in form to that given byWainwright. Several cases are considered, in particularwe find a general algebraic perfect fluid solution with equation of state p = whose Weyl tensor is of the purely magnetic type within a finiteregion of the spacetime. It is shown, for an observerwith four-velocity, u_mag say, that themetric's Weyl tensor is purely magnetic within the finiteregion while it is purely electric, as read by anotherobserver with four-velocity u_ele, elsewhere.Another observer, independent of the observers whomeasure the Weyl tensor to be purely electric ormagnetic, interprets the perfect fluid to have anequation of state p = . The Petrov type of themetric, in this case, is I(M) by theArianrhod-McIntosh classification and therefore there exists noconformally related metric which is vacuum. The vacuumseed metrics are derived for the perfect fluidsolutions.     

Self-gravitating general-relativistic cosmic strings

We examine the coupled Einstem-Euler-Lagrange equations for nonstationary cosmic strings. Self-consistent solutions to all the equations are found under the assumption that the energy-momentum tensor is of the formT _t ^t =T _z ^z while all other components vanish. It is shown that the strings are necessarily static in this case and that the scalar field potential must be of the usual quartic form with the coupling constants satisfying e²=8.     

Uniqueness of the Newman–Janis Algorithm in Generating the Kerr–Newman Metric

After the original discovery of the Kerr metric, Newman and Janis showed that this solution could be derived by making an elementary complex transformation to the Schwarzschild solution. The same method was then used to obtain a new stationary axisymmetric solution to Einstein's field equations now known as the Kerr–Newman metric, representing a rotating massive charged black hole. However no clear reason has ever been given as to why the Newman–Janis algorithm works, many physicist considering it to be an ad hoc procedure or fluke and not worthy of further investigation. Contrary to this belief this paper shows why the Newman–Janis algorithm is successful in obtaining the Kerr–Newman metric by removing some of the ambiguities present in the original derivation. Finally we show that the only perfect fluid generated by the Newman–Janis algorithm is the (vacuum) Kerr metric and that the only Petrov typed D solution to the Einstein–Maxwell equations is the Kerr–Newman metric.     

Stationary axisymmetric solutions of the Einstein equations with rigidly rotating perfect fluid and nonlinear charged sources

A class of stationary, rigidly rotating perfect fluids coupled with nonlinear electromagnetic fields was investigated. An exact solution of the Einstein equations with sources for the Carter B(+) branch was found for the equation of state 3p+=const. We use a structure function for the Born-Infeld nonlinear electrodynamics which is invariant under duality rotations and a metric possessing a four-parameter group of motions. The solution is of Petrov type D and the eigenvectors of the electromagnetic field are aligned to the Debever-Penrose vectors.     

Infinitesimal holonomy groups of Einstein-Maxwell space-times

Some general results concerning the Petrov classification of the Weyl tensor of space-times whose infinitesimal holonomy group belongs to a given conjugacy class of the Lorentz group ₊ are established using the null tetrad notation. Those conjugacy classes, corresponding to proper subgroups of ₊ which contain the infinitesimal holonomy groups of Einstein-Maxwell space-times, are determined, and explicit examples of such space-times are given for each conjugacy class.     

Microscopic and macroscopic entropy of extremal black holes in string theory

It has been observed on a number of occasions that complex transformations, of real solutions of the field equations to other real solutions, often preserve certain properties of the Weyl tensor. That is, the Petrov type and/or gravito-electromagnetic (GEM) properties of the Weyl tensor are preserved. In this context, we present an outstanding example of a complex windmill transformation of a static (non-physical) anisotropic fluid spacetime of Petrov type $I(M^+)$ that maps to a purely magnetic (PM) spacetime of Petrov type $I(M^{\infty })$ . The PM spacetime is analyzed and compared to the Arianrhod–Lun–McIntosh–Perjés spacetime. It is shown that these spacetimes, although similar in some aspects, are distinct solutions. The main distinction is that the generated PM spacetime satisfies all the standard energy-conditions. This intriguing but purely mathematical scenario may have implications in the area of GEM duality.     

Renormalization of the neutrino mass operators in the multi-Higgs-doublet standard model

We derive the renormalization group equations (RGE) for the flavor coupling matrices of the effective dimension-five operators which yield Majorana neutrino masses in the multi-Higgs-doublet standard model; in particular, we consider the case where two different scalar doublets occur in those operators. We also write down the RGE for the scalar-potential quartic couplings and for the Yukawa couplings of that model, in the absence of quarks. As an application of the RGE, we consider two models which, based on a - interchange symmetry, predict maximal atmospheric neutrino mixing, together with U _e3 = 0, at the seesaw scale. We estimate the change of those predictions due to the evolution of the coupling matrices of the effective mass operators from the seesaw scale down to the electroweak scale. We derive an upper bound on that change, thereby finding that the radiative corrections to those predictions are in general negligible.Received: 30 September 2004, Published online: 11 January 2005PACS: 11.10.Hi, 14.60.Pq, 12.60.Fr, 11.30.Hv     

Transverse Frames for Petrov Type I Spacetimes: A General Algebraic Procedure

We develop an algebraic procedure to rotate a general Newman-Penrose tetrad in a Petrov type I spacetime into a frame with Weyl scalars ₁ and ₃ equal to zero, assuming that initially all the Weyl scalars are non vanishing. The new frame highlights the physical properties of the spacetime. In particular, in a Petrov type I spacetime, setting ₁ and ₃ to zero makes apparent the superposition of a Coulomb-type effect ₂ with transverse degrees of freedom ₀ and ₄.     

Asymptotic directional structure of radiation for fields of algebraic type D

The directional behavior of dominant components of algebraically special spin-s fields near a spacelike, timelike or null conformal infinity is studied. By extending our previous general investigations, we concentrate on fields which admit a pair of equivalent algebraically special null directions, such as the Petrov type-D gravitational fields or algebraically general electromagnetic fields. We introduce and discuss a canonical choice of the reference tetrad near infinity in all possible situations, and we present the corresponding asymptotic directional structures using the most natural parametrizations.Dedicated to Prof. Jií Horáek on the occasion of his 60th birthday