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”.     

Singularities at real points of ℋ space

In the presence of gravitational radiation, there are ordinarily no shear-free slices of null infinity. A four-complex-dimensional set of shear-free slices of complexified null infinity do exist. They comprise the manifold space. In general, there are no preferred real subspaces of space associated with slices of real null infinity. However, for radiation fields possessing a twist-free axial symmetry, a two-parameter family of shear-free slices of real null infinity exist and therefore pick out a preferred two-dimensional real subspace of space. In this paper, we study the geometry of these 2-spaces for the particular case of quadrupole radiation fields for which determination of the shear-free slices reduces to the standard problem of determining orbits of a particle moving in a potential. Our principal interest is the investigation of possible singularities caused by sufficiently intense radiation fields. We find that such singularities do occur for radiation fields having the characteristic powerc ⁵/G.     

A radiating Kerr-Newman solution

A non-static exact solution of Einstein’s equations corresponding to a field of flowing null radiation plus an electromagnetic field is presented. The geometry of the solution is described by the Kerr-Schild metric. The solution admits a shear-free, geodetic null congruence. It has the symmetry of the Kerr-Newman solution and when a certain parameter is put equal to zero the solution becomes static and reduces to the Kerr-Newman solution.     

Integrable optical geometry

It is shown that a Lorentzian 4-manifold admitting a congruence of optical (null) geodesics without shear and twist defines an optical geometry which is integrable (locally flat) in the sense of the theory of G-structures. The existence of a symmetric linear connection compatible with the optical geometry is another condition equivalent to the integrability of the optical G-structure.     

Light-cones,almost light-cones and almost-complex light-cones

We point out (and then apply to a general situation) an unusual relationship among a variety of null geodesic congruences; (a) the generators of ordinary light-cones and (b) certain (related) shear-free but twisting congruences in Minkowski space–time as well as (c) asymptotically shear-free null geodesic congruences that exist in the neighborhood of Penrose’s \(I^{ +}\) in Einstein or Einstein–Maxwell asymptotically flat-space–times. We refer to these geodesic congruences respectively as: Lignt-Cones (LCs), as “Almost-Complex”-Light-Cones (ACLCs), [though they are real they resemble complex light-cones in complex Minkowski space] and finally to a family of congruences in asymptotically flat-spaces as ‘Almost Light-Cones’ (ALC). The two essential points of resemblance among the three families are: (1) they are all either shear-free or asymptotically shear-free and (2) in each family the individual members of the family can be labeled by the points in a real or complex four-dimensional manifold. As an example, the Minkowski space LCs are labeled by the (real) coordinate value of their apex. In the case of (ACLCs) (complex coordinate values), the congruences will have non-vanishing twist whose magnitude is determined by the imaginary part of the complex coordinate values. In studies of gravitational radiation, Bondi-type of null surfaces and their associated Bondi coordinates have been almost exclusively used for calculations. It turns out that some surprising relations arise if, instead of the Bondi coordinates, one uses ALCs and their associated coordinate systems in the analysis of the Einstein–Maxwell equations in the neighborhood of \(I^{+}\). More explicitly and surprisingly, the asymptotic Bianchi Identities (arising directly from the Einstein equations), expressed in the coordinates of the ALCs, turn directly into many of the standard definitions and equations and relations of classical mechanics coupled with Maxwell’s equations. These results extend and generalize the beautiful results of Bondi and Sachs with their expressions for, and loss of, mass and linear momentum.     

Thermodynamical laws and spacetime geometry

We examine the conditions imposed on spacetime geometry by linear and extended thermodynamics. In this analysis we confine ourselves on shear-free spacetimes with divergence-free Weyl tensor. This results in a variety of well-known spacetimes which have to have simple kinematic properties as well as very restricted source structure. In all cases the thermodynamical considerations show the privileged role of the equation p = – which can be interpreted as cosmological constant. Moreover, it is interesting to observe that the restrictions imposed on the spacetime geometry in the case of extended thermodynamics (for vanishing anisotropic pressure) are much stronger than in the linear case.     

On the space-times admitting two shear-free geodesic null congruences

We analyze the space-times admitting two shear-free geodesic null congruences. The integrability conditions are presented in a plain tensorial way as equations on the volume element U of the time-like 2-plane that these directions define. From these we easily deduce significant consequences. We obtain explicit expressions for the Ricci and Weyl tensors in terms of U and its first and second order covariant derivatives. We study the different compatible Petrov-Bel types and give the necessary and sufficient conditions that characterize every type in terms of U. The type D case is analyzed in detail and we show that every type D space-time admitting a 2 + 2 conformal Killing tensor also admits a conformal Killing-Yano tensor.     

The solution of Dirac's equation in algebraically special space-times

A procedure for obtaining solutions to Dirac's equation in algebraically special space-times which admit a shear-free congruence of null geodesies along the repeated principal null direction of the Weyl tensor, is presented. By aligning one of the Dirac spinors to the congruence the problem is reduced to solving one second-order linear partial differential equation for a scalar potential. The solution of the massless field equations for null fields of arbitrary spin s>1/2 aligned to the congruence is also given.     

Cosmological Horizons and Reconstruction of Quantum Field Theories

As a starting point, we state some relevant geometrical properties enjoyed by the cosmological horizon of a certain class of Friedmann-Robertson-Walker backgrounds. Those properties are generalised to a larger class of expanding spacetimes M admitting a geodesically complete cosmological horizon common to all co-moving observers. This structure is later exploited in order to recast, in a cosmological background, some recent results for a linear scalar quantum field theory in spacetimes asymptotically flat at null infinity. Under suitable hypotheses on M, encompassing both the cosmological de Sitter background and a large class of other FRW spacetimes, the algebra of observables for a Klein-Gordon field is mapped into a subalgebra of the algebra of observables constructed on the cosmological horizon. There is exactly one pure quasifree state λ on which fulfills a suitable energy-positivity condition with respect to a generator related with the cosmological time displacements. Furthermore λ induces a preferred physically meaningful quantum state λ _M for the quantum theory in the bulk. If M admits a timelike Killing generator preserving , then the associated self-adjoint generator in the GNS representation of λ _M has positive spectrum (i.e., energy). Moreover λ _M turns out to be invariant under every symmetry of the bulk metric which preserves the cosmological horizon. In the case of an expanding de Sitter spacetime, λ _M coincides with the Euclidean (Bunch-Davies) vacuum state, hence being Hadamard in this case. Remarks on the validity of the Hadamard property for λ _M in more general spacetimes are presented. Dedicated to Professor Klaus Fredenhagen on the occasion of his 60th birthday.     

Weyl type N solutions with null electromagnetic fields in the Einstein–Maxwell <Emphasis Type="Italic">p</Emphasis>-form theory

We consider d-dimensional solutions to the electrovacuum Einstein–Maxwell equations with the Weyl tensor of type N and a null Maxwell \((p+1)\)-form field. We prove that such spacetimes are necessarily aligned, i.e. the Weyl tensor of the corresponding spacetime and the electromagnetic field share the same aligned null direction (AND). Moreover, this AND is geodetic, shear-free, non-expanding and non-twisting and hence Einstein–Maxwell equations imply that Weyl type N spacetimes with a null Maxwell \((p+1)\)-form field belong to the Kundt class. Moreover, these Kundt spacetimes are necessarily \({ CSI}\) and the \((p+1)\)-form is \({ VSI}\). Finally, a general coordinate form of solutions and a reduction of the field equations are discussed.     

A few properties of a certain class of degenerate space-times

The properties are studied of a class of space-times determined by assuming the shape of the metric formds ² including disposable coordinate functions. It has been found that this class includes degenerate space-times with geodetic, null, shear-free congruences with nonvanishing expansion. The theorem has been proved that this class of solutions of the Einstein equations can easily be expanded to solutions of Einstein-Maxwell equations with a fairly general electromagnetic field. For a selected subclass relations are given between the functions determining the metric form, and two new explicit solutions with arbitrary functions of the Einstein-Maxwell equations with a cosmological constant are found.On leave from the Institute of Theoretical Physics, Warsaw University, Warsaw, Poland.     

The shear-free condition in Robinson's theorem

A geometrical interpretation of the shear-free condition, required by Robinson's theorem, is given. In particular it is proved that the shear-free condition for a (geodesic) null congruence is necessary and sufficient in order that the null conditions be preserved along the rays.Lavoro eseguito nell'ambito dell'attività del Gruppo Nazionale per la Fisica Matematica del C.N.R.     

Plane symmetric metrics associated with semi-plane symmetric electromagnetic fields in higher dimensions

Electromagnetic fields yielding plane symmetric metrics in higher-dimensional spacetimes are exhausted and classified. It is shown that these EM fields must fall into one of the following two cases: (i)F _it =F _iz =0,i=1,...,n; (ii)F_tz=0. We give the general solution to the Einstein-Maxwell equations in higher dimensions corresponding to electromagnetic fields of case (ii) withF _it =F _iz , which covers all even-dimensional spacetimes as well as a subcase of odd-dimensional spacetimes.     

On the superposition of electromagnetic waves in general relativity

It is shown that, in a region of space-time containing two independent electromagnetic waves propagating in different directions, it is not possible for the two waves to follow simultaneously affinely parameterised shear-free and twist-free null geodesic congruences.     

An equation satisfied by the tangent to a shear-free,geodesic, null congruence

A tensorial equation satisfied by the tangent to a shear-free geodesic, null congruence is presented. If the congruence is neither twist-free nor expansion-free then the equation defines a second, unique, null direction previously obtained, using the spinor formalism, by Somers. Some further properties of the equation are discussed.     

A class of algebraically general solutions of the Einstein-Maxwell equations for non-null electromagnetic fields

The Einstein-Maxwell field equations for non-null electromagnetic fields are studied under the conditions that the null tetrad is parallelly propagated along both principal null congruences. It is shown that the resulting spacetime solutions are necessarily algebraically general. The twist-free solution found in a previous article is shown to be the most general twist-free solution. An expansionfree solution with twist and shear is also found.     

Gravitational field of a radiating rotating body

A nonstationary solution of the Einstein field equations, corresponding to the field of a radiating rotating body, is presented. The solution is algebraically special of Petrov type II with a twisting, shear-free, null congruence identical to that of the Kerr metric. The new metric bears the same relation to the Kerr metric as does Vaidya's metric to the Schwarzschild metric, in the sense that in both cases the radiating solution is generated from the nonradiating one by replacing the mass parameter by an arbitrary function of a retarded time coordinate. The energy-momentum tensor in the present case, however, has two terms, a Vaidya type radiative one and an additional nonradiative residual term. Due to the presence of the nonradiative term in this case, however, the energy-momentum tensor becomes Vaidya-like asymptotically only, thus allowing for a geometrical optics interpretation. Asymptotically, part of the radiation field is purely electromagnetic with a Maxwell tensor which admits only one principal null direction corresponding to the undirectional flow of radiation.     

Metric and Ricci tensors for a certain class of space-times ofD type

The class of space-times has been determined at the connection level, assuming the existence of some symmetrical relations between the Ricci rotation coefficients. It has been assumed, for instance, that at least two shear-free congruences of null geodesics exist. We have shown that onlyD type or conformally flat space-times can belong to this class. The theorem has been proved that a system of coordinates exists in which the metric tensor can depend on two coordinates, only. The metric tensor has been determined with an accuracy to two functions, each of which is a function of only one coordinate. Linear, second-order differential expressions have been found for these two functions. They determine the Ricci tensor. Several solutions of the Einstein-Maxwell equations with a cosmological constant are given.On leave from the Institute of Theoretical Physics, Warsaw University, Warsaw, Poland.     

Boost-rotation symmetric gravitational null cone data

The potential role of boost-rotation symmetric vacuum spacetimes as test beds for numerical studies of gravitational radiation is discussed. For application to null cone evolution codes, these spacetimes are analyzed in terms of their data on the preferred null cone left invariant by the symmetry group. On this cone, an explicit solution of the Bondi hypersurface and evolution equations is found. This solution has a smooth vertex, a smooth interior, and, except for polar singularities, admits a well-defined ⁺.This work was supported in part by NSF grant No. PHY800823 to the University of Pittsburgh.