Shear-free Null Congruences Research Articles

Algebrodynamics: Shear-Free Null Congruences and New Types of Electromagnetic Fields

We briefly present our version of noncommutative analysis over matrix algebras, the algebra of biquaternions (B) in particular. We demonstrate that any B-differentiable function gives rise to a null shear-free congruence (NSFC) on the B-vector space CM and on its Minkowski subspace M. Making use of the Kerr–Penrose correspondence between NSFC and twistor functions, we obtain the general solution to the equations of B-differentiability and demonstrate that the source of an NSFC is, generically, a world sheet of a string in CM. Any singular point, caustic of an NSFC, is located on the complex null cone of a point on the generating string. Further we describe symmetries and associated gauge and spinor fields, with two electromagnetic types among them. A number of familiar and novel examples of NSFC and their singular loci are described. Finally, we describe a conservative algebraic dynamics of a set of identical particles on the “Unique Worldline” and discuss the connections of the theory with the Feynman–Wheeler concept of “One-Electron Universe”.

О некоторых свойствах бессдвиговых изотропных конгруэнций

О некоторых свойствах бессдвиговых изотропных конгруэнций

A nonstationary generalization of the Kerr congruence

Making use of the Kerr theorem for shear-free null congruences and of Newman's representation for a virtual charge ``moving'' in complex space-time, we obtain an axisymmetric time-dependent generalization of the Kerr congruence, with a singular ring uniformly contracting to a point and expanding then to infinity. Electromagnetic and complex eikonal field distributions are naturally associated with the obtained congruence, with electric charge being necesssarily unit (``elementary''). We conjecture that the corresponding solution to the Einstein-Maxwell equations could describe the process of continious transition of the naked ringlike singularitiy into a rotating black hole and vice versa, under a particular current radius of the singular ring.

Asymptotic twistor theory and the Kerr theorem

We first review asymptotic twistor theory with its real subspace of null asymptotic twistors: a five-dimensional CR manifold. This is followed by a description of the Kerr theorem (the identification of shear-free null congruences, in Minkowski space, with the zeros of holomorphic functions of three variables) and an asymptotic version of the Kerr theorem that produces regular asymptotically shear-free null geodesic congruences in arbitrary asymptotically flat Einstein or Einstein–Maxwell spacetimes. A surprising aspect of this work is the role played by analytic curves in H-space, each curve generating an asymptotically flat null geodesic congruence. Also there is a discussion of the physical space realizations of the two associated five- and three-dimensional CR manifolds.

Twisting null geodesic congruences, scri, H-space and spin-angular momentum

The purpose of this work is to return, with a new observation and rather unconventional point of view, to the study of asymptotically flat solutions of Einstein equations. The essential observation is that from a given asymptotically flat spacetime with a given Bondi shear, one can find (by integrating a partial differential equation) a class of asymptotically shear-free (but, in general, twisting) null geodesic congruences. The class is uniquely given up to the arbitrary choice of a complex analytic world line in a four-parameter complex space. Surprisingly, this parameter space turns out to be the H-space that is associated with the real physical spacetime under consideration. The main development in this work is the demonstration of how this complex world line can be made both unique and also given a physical meaning. More specifically, by forcing or requiring a certain term in the asymptotic Weyl tensor to vanish, the world line is uniquely determined and becomes (by several arguments) identified as the ‘complex centre of mass’. Roughly, its imaginary part becomes identified with the intrinsic spin-angular momentum while the real part yields the orbital angular momentum. One should think of this work as developing a generalization of the properties of the algebraically special spacetimes in the sense that the term that is forced here to vanish is automatically vanishing (among many other terms) for all the algebraically special metrics. This is demonstrated in the several given examples. It was, in fact, an understanding of the algebraically special metrics and their associated shear-free null congruence that led us to this construction of the asymptotically shear-free congruences and the unique complex world line. The Robinson–Trautman metrics and the Kerr and charged Kerr metrics with their properties are explicit examples of the construction given here.

Effective Connections and Fields Associated with Shear-Free Null Congruences

A special subclass of shear-free null congruences (SFC) is studied, with tangent vector field being a repeated principal null direction of the Weyl tensor. We demonstrate that this field is parallel with respect to an effective affine connection which contains the Weyl nonmetricity and the skew symmetric torsion. On the other hand, a Maxwell-like field can be directly associated with any special SFC, and the electric charge for bounded singularities of this field turns to be ``self-quantized''. Two invariant differential operators are introduced which can be thought of as spinor analogues of the Beltrami operators and both nullify the principal spinor of any special SFC.

Stringy Robinson-Trautman solutions.

A class of solutions of the low-energy string theory in four dimensions is studied. This class admits a geodesic, shear-free null congruence which is nontwisting but in general diverging and the corresponding solutions in Einstein's theory form the Robinson-Trautman family together with a subset of Kundt's class. The Robinson-Trautman conditions are found to be frame invariant in string theory. The Lorentz Chern-Simons three-form of the stringy Robinson-Trautman solutions is shown to always be closed. The stringy generalizations of the vacuum Robinson-Trautman equation are obtained and three subclasses of solutions are identified. One of these subclasses exists, among all the dilatonic theories, only in Einstein's theory and in string theory. Several known solutions including the dilatonic black holes, the $\mathrm{pp}$ waves, the stringy $C$ metric, and certain solutions which correspond to exact conformal field theories are shown to be particular members of the stringy Robinson-Trautman family. Some new solutions which are static or asymptotically flat and radiating are also presented. The radiating solutions have a positive Bondi mass. One of these radiating solutions has the property that it settles down smoothly to a black hole state at late retarded times.

TypeN vacuum space-times as special functions in ?2

The members of one explicit class of functions in ℂ2 are identified with the geodetic shear-free null congruences in Minkowski's space-time. Members of a second explicit class are identified with the type-N vacuum space-times with twist-free rays. These two classes are special subclasses from a larger class of functions associated with the type-N space-times. This larger class is characterized in the following way: Ifϕ andζ are holomorphic variables in ℂ2, thenu (ϕ, ζ, $$\bar \zeta $$ ), a function holomorphic inϕ, belongs to the class provided the function $$\partial _{\zeta } \partial _{\zeta } $$ ∂ ϕ u/∂ ϕ u satisfies the tangential Cauchy-Riemann equation for an antiholomorphic function on the 3-surface whereu (ϕ, ζ, $$\bar \zeta $$ ) has real values.

A conventional proof of Kerr's Theorem

A proof of Kerr's Theorem for generating geodesic and shearfree null congruences in Minkowski space is given in the Newman-Penrose null tetrad formalism.

An integrable structure for type D spacetimes

A structure analogous to an almost complex structure on a manifold is presented. It is integrable for spacetime manifolds admitting two geodesic and shear-free null congruences. This fact sheds light on Newman's “complex coordinate transformations”.

Twistor Algebra

A new type of algebra for Minkowski space-time is described, in terms of which it is possible to express any conformally covariant or Poincaré covariant operation. The elements of the algebra (twistors) are combined according to tensor-type rules, but they differ from tensors or spinors in that they describe locational properties in addition to directional ones. The representation of a null line by a pair of two-component spinors, one of which defines the direction of the line and the other, its moment about the origin, gives the simplest type of twistor, with four complex components. The rules for generating other types of twistor are then determined by the geometry. One-index twistors define a four-dimensional, four-valued (``spinor'') representation of the (restricted) conformal group. For the Poincaré group a skew-symmetric metric twistor is introduced. Twistor space defines a complex projective three-space C, which gives an alternative picture equivalent to the Minkowski space-time M (which must be completed by a null cone at infinity). Points in C represent null lines or ``complexified'' null lines in M; lines in C represent real or complex points in M (so M, when complexified, is the Klein representation of C. Conformal transformations of M, including space and time reversals (and complex conjugation) are discussed in detail in twistor terms. A theorem of Kerr is described which shows that the complex analytic surfaces in C define the shear-free null congruences in the real space M. Twistors are used to derive new theorems about the real geometry of M. The general twistor description of physical fields is left to a later paper.

�u�erer Differentialkalk�l f�r Spinorformen und Anwendung auf das allgemeine reine Gravitationsstrahlungsfeld

An exterior differential calculus for spinor forms quite analogous to the well known tensorial Cartan calculus is developed which renders possible the direct spinorial calculation of the Weyl- and Ricci-spinors. As an application, the metric and curvature spinors of the most general space-time admitting a normal shearfree null congruence are derived; the rather modest amount of calculation illustrates the advantages of the new method as compared to the techniques used thus far.