Symmetries of asymptotically flat electrovacuum spacetimes and radiation

Symmetries compatible with asymptotic flatness and admitting gravitational and electromagnetic radiation are studied by using the Bondi-Sachs-van der Burg formalism. It is shown that in axially symmetric electrovacuum spacetimes in which at least locally a smooth null infinity in the sense of Penrose exists, the only second allowable symmetry is either the translational symmetry or the boost symmetry. Translationally invariant spacetimes with in general a straight "cosmic string" along the axis of symmetry are non-radiative although they can have a non-vanishing news function. The boost-rotation symmetric spacetimes are radiative. They describe "uniformly accelerated charged particles" or black holes which in general may also be rotating - the axial and an additional Killing vector are not assumed to be hypersurface orthogonal. The general functional forms of both gravitational and electromagnetic news functions, and of the mass aspect and total mass of asymptotically flat boost-rotation symmetric spacetimes at null infinity are obtained. The expressions for the mass are new even in the case of vacuum boost-rotation symmetric spacetimes with hypersurface orthogonal Killing vectors. In Appendices some errors appearing in previous works are corrected.Comment: 23 pages, RevTeX, to appear in JM

Bianchi identities in higher dimensions

A higher dimensional frame formalism is developed in order to study implications of the Bianchi identities for the Weyl tensor in vacuum spacetimes of the algebraic types III and N in arbitrary dimension $n$. It follows that the principal null congruence is geodesic and expands isotropically in two dimensions and does not expand in $n-4$ spacelike dimensions or does not expand at all. It is shown that the existence of such principal geodesic null congruence in vacuum (together with an additional condition on twist) implies an algebraically special spacetime. We also use the Myers-Perry metric as an explicit example of a vacuum type D spacetime to show that principal geodesic null congruences in vacuum type D spacetimes do not share this property.Comment: 25 pages, v3: Corrections to Appendix B as given in Erratum-ibid.24:1691,2007 are now incorporated (A factor of 2 was missing in certain Bianchi equations.

A note on the peeling theorem in higher dimensions

We demonstrate the ``peeling property'' of the Weyl tensor in higher dimensions in the case of even dimensions (and with some additional assumptions), thereby providing a first step towards understanding of the general peeling behaviour of the Weyl tensor, and the asymptotic structure at null infinity, in higher dimensions.Comment: 5 pages, to appear in Class. Quantum Gra

Ricci identities in higher dimensions

We explore connections between geometrical properties of null congruences and the algebraic structure of the Weyl tensor in n>4 spacetime dimensions. First, we present the full set of Ricci identities on a suitable "null" frame, thus completing the extension of the Newman-Penrose formalism to higher dimensions. Then we specialize to geodetic null congruences and study specific consequences of the Sachs equations. These imply, for example, that Kundt spacetimes are of type II or more special (like for n=4) and that for odd n a twisting geodetic WAND must also be shearing (in contrast to the case n=4).Comment: 8 pages. v2: typo corrected between Propositions 2 and 3. v3: typo in the last term in the first line of (11f) corrected, missing term on the r.h.s. of (11p) added, first sentence between Propositions 2 and 3 slightly change

Axisymmetric electrovacuum spacetimes with a translational Killing vector at null infinity

By using the Bondi-Sachs-van der Burg formalism we analyze the asymptotic properties at null infinity of axisymmetric electrovacuum spacetimes with a translational Killing vector and, in general, an infinite ``cosmic string'' (represented by a conical singularity) along the axis. Such spacetimes admit only a local null infinity. There is a non-vanishing news function due to the existence of the string even though there is no radiation. We prove that if null infinity has a smooth compact cross section and the spacetime is not flat in a neighbourhood of null infinity, then the translational Killing vector must be timelike and the spacetime is stationary. The other case in which an additional symmetry of axisymmetric spacetimes admits compact cross sections of null infinity is the boost symmetry, which leads to radiative spacetimes representing ``uniformly accelerated objects''. These cases were analyzed in detail in our previous works. If the translational Killing vector is spacelike or null, corresponding to cylindrical or plane waves, some complete generators of null infinity are ``singular'' but null infinity itself can be smooth apart from these generators. As two explicit examples of local null infinity, Schwarzschild spacetime with a string and a class of cylindrical waves with a string are discussed in detail in the Appendix.Comment: 15 pages, RevTeX, submitted to Class. Quantum Gra

Asymptotic structure of radiation in higher dimensions

We characterize a general gravitational field near conformal infinity (null, spacelike, or timelike) in spacetimes of any dimension. This is based on an explicit evaluation of the dependence of the radiative component of the Weyl tensor on the null direction from which infinity is approached. The behaviour similar to peeling property is recovered, and it is shown that the directional structure of radiation has a universal character that is determined by the algebraic type of the spacetime. This is a natural generalization of analogous results obtained previously in the four-dimensional case.Comment: 14 pages, no figures (two references added

Newman-Penrose formalism in higher dimensions: vacuum spacetimes with a non-twisting geodetic multiple Weyl aligned null direction

Vacuum spacetimes admitting a non-twisting geodetic multiple Weyl aligned null direction (WAND) are analyzed in arbitrary dimension using recently developed higher-dimensional Newman-Penrose (NP) formalism. We determine dependence of the metric and of the Weyl tensor on the affine parameter r along null geodesics generated by the WAND for type III and N spacetimes and for a special class of type II and D spacetimes, containing e.g. Schwarzschild-Tangherlini black holes and black strings and branes. For types III and N, all metric components are at most quadratic polynomials in r while for types II and D the r-dependence of the metric as well as of the Weyl tensor is determined by an integer m corresponding to the rank of the expansion matrix S_{ij}. It is shown that for non-vanishing expansion, all these spacetimes contain a curvature singularity. As an illustrative example, a shearing expanding type N five-dimensional vacuum solution is also re-derived using higher-dimensional NP formalism. This solution can be, however, identified with a direct product of a known four-dimensional type N metric with an extra dimension.Comment: 25 pages, version to be published in Class. Quantum Grav. (expanded -background material included, 3 references added, small change in notation

Boost-rotation symmetric vacuum spacetimes with spinning sources

Boost-rotation symmetric vacuum spacetimes with spinning sources which correspond to gravitational field of uniformly accelerated spinning "particles" are studied. Regularity conditions and asymptotic properties are analyzed. News functions are derived by transforming the general spinning boost-rotation symmetric vacuum metric to Bondi-Sachs coordinates.Comment: REVTeX 4, 9 page

Alignment and algebraically special tensors in Lorentzian geometry

We develop a dimension-independent theory of alignment in Lorentzian geometry, and apply it to the tensor classification problem for the Weyl and Ricci tensors. First, we show that the alignment condition is equivalent to the PND equation. In 4D, this recovers the usual Petrov types. For higher dimensions, we prove that, in general, a Weyl tensor does not possess aligned directions. We then go on to describe a number of additional algebraic types for the various alignment configurations. For the case of second-order symmetric (Ricci) tensors, we perform the classification by considering the geometric properties of the corresponding alignment variety.Comment: 19 pages. Revised presentatio

Co-accelerated particles in the C-metric

With appropriately chosen parameters, the C-metric represents two uniformly accelerated black holes moving in the opposite directions on the axis of the axial symmetry (the z-axis). The acceleration is caused by nodal singularities located on the z-axis. In the~present paper, geodesics in the~C-metric are examined. In general there exist three types of timelike or null geodesics in the C-metric: geodesics describing particles 1) falling under the black hole horizon; 2)crossing the acceleration horizon; and 3) orbiting around the z-axis and co-accelerating with the black holes. Using an effective potential, it can be shown that there exist stable timelike geodesics of the third type if the product of the parameters of the C-metric, mA, is smaller than a certain critical value. Null geodesics of the third type are always unstable. Special timelike and null geodesics of the third type are also found in an analytical form.Comment: 10 pages, 12 EPS figures, changes mainly in abstract & introductio