Boost-rotation Symmetric Spacetimes Research Articles

The Newtonian limit of spacetimes for accelerated particles and black holes

Solutions of vacuum Einstein’s field equations describing uniformly accelerated particles or black holes belong to the class of boost-rotation symmetric spacetimes. They are the only explicit solutions known which represent moving finite objects. Their Newtonian limit is analyzed using the Ehlers frame theory. Generic spacetimes with axial and boost symmetries are first studied from the Newtonian perspective. The results are then illustrated by specific examples such as C-metric, Bonnor–Swaminarayan solutions, self-accelerating “dipole particles”, and generalized boost-rotation symmetric solutions describing freely falling particles in an external field. In contrast to some previous discussions, our results are physically plausible in the sense that the Newtonian limit corresponds to the fields of classical point masses accelerated uniformly in classical mechanics. This corroborates the physical significance of the boost-rotation symmetric spacetimes.

Characterization of Schwarzschildean initial data

A theorem providing a characterization of Schwarzschildean initial data sets on slices with an asymptotically Euclidean end is proved. This characterization is based on the proportionality of the Weyl tensor and its D'Alambertian that holds for some vacuum Petrov type D spacetimes (e.g. the Schwarzschild spacetime, the C-metric, but not the Kerr solution). The $3+1$ decomposition of this proportionality condition renders necessary conditions for an initial data set to be a Schwarzschildean initial set. These conditions can be written as quadratic expressions of the electric and magnetic parts of the Weyl tensor, and thus involve only the freely specifiable data. In order to complete our characterization, a study of which vacuum static Petrov type D spacetimes admit asymptotically Euclidean slices is undertaken. Furthermore, a discussion of the Arnowitt-Deser-Misner (ADM) 4-momentum for boost-rotation symmetric spacetimes is given. As a by-product of our analysis a certain characterization of the Schwarzschild spacetime is obtained. Finally, a generalization of our characterization, valid for Schwarzschildean hyperboloidal initial data sets is put forward.

The Newtonian limit of space-times describing uniformly accelerated particles

We discuss the Newtonian limit of boost-rotation symmetric spacetimes by means of the Ehler's frame theory. Conditions for the existence of such a limit are given and, in particular, we show that asymptotic flatness is an essential requirement for the existence of such a limit. Consequently, generalized boost-rotation symmetric spacetimes describing particles moving in uniform fields will not possess a Newtonian limit. In the cases where the boost-rotation symmetric spacetime is asymptotically flat and its Newtonian limit exists, then it is non-zero only for the instant of time symmetry and its value is given by a Poisson integral. The relation of this result with the (Newtonian) gravitational potential suggested by the weak field approximation is discussed. We illustrate our analysis through some examples: the two monopoles solution, the Curzon-Chazy particle solution, the generalized Bonnor-Swaminarayan solution, and the C metric.

ON BOOST-ROTATION SYMMETRIC SPACE–TIMES WITH (NON)SPINNING SOURCES

Boost-rotation symmetric spacetimes are the only axially symmetric vacuum spacetimes with an additional symmetry that are radiative and admit global null infinity1,2. They are the only radiative asymptotically flat exact solutions representing the gravitational field of moving objects – uniformly accelerated "particles" of various kinds3. The general theory of boost-rotation symmetric spacetimes was first developed for nonspinning sources4 (i.e. for hypersurface orthogonal Killing vectors) and later partially generalized for spinning sources5. The C-metric3 and the spinning C-metric6,7,8 belonging to this class were analyzed. With appropriately chosen parameters they represent the gravitational field of uniformly accelerated Schwarzschild and Kerr black or white holes. Stability of geodesics describing co-accelerated particles was also studied using an effective potential9 and a Lyapunov function8.

Null Limits of Generalised Bonnor-Swaminarayan Solutions

The Bonnor-Swaminarayan solutions are boost-rotation symmetric space-times which describe the motion of pairs of accelerating particles which are possibly connected to strings (struts). In an explicit and unified form we present a generalised class of such solutions with a few new observations. We then investigate the possible limits in which the accelerations become unbounded. The resulting space-times represent spherical impulsive gravitational waves with snapping or expanding cosmic strings. We also obtain an exact solution for a snapping string of finite length.

Exact radiative spacetimes: some recent developments

Five classes of radiative solutions of Einstein's field equations are discussed in the light of some new developments. These are plane waves and their collisions, cylindrical waves, Robinson-Trautman and type N spacetimes, boost-rotation symmetric spacetimes and generali ed Gowdy-type cosmological models.

Symmetries of asymptotically flat electrovacuum space–times 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 space–times 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 space–times with, in general, a straight “cosmic string” along the axis of symmetry are nonradiative although they can have a nonvanishing news function. The boost-rotation symmetric space–times 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 space–times at null infinity are obtained. The expressions for the mass are new even in the case of vacuum boost-rotation symmetric space–times with hypersurface orthogonal Killing vectors. In the Appendices some errors appearing in previous works are corrected.

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-definedℐ+.