Jiri Bicak

Asymptotic structure of symmetry reduced general relativity

Gravitational waves with a space-translation Killing field are considered. Because of the symmetry, the four-dimensional Einstein vacuum equations are equivalent to the three-dimensional Einstein equations with certain matter sources. This interplay between four- and three-dimensional general relativity can be exploited effectively to analyze issues pertaining to four dimensions in terms of the three-dimensional structures. An example is provided by the asymptotic structure at null infinity: While these space-times fail to be asymptotically flat in four dimensions, they can admit a regular completion at null infinity in three dimensions. This completion is used to analyze the asymptotic symmetries, introduce the analogue of the four-dimensional Bondi energy momentum, and write down a flux formula. The analysis is also of interest from a purely three-dimensional perspective because it pertains to a diffeomorphism-invariant three-dimensional field theory with local degrees of freedom, i.e., to a midisuperspace. Furthermore, because of certain peculiarities of three dimensions, the description of null infinity has a number of features that are quite surprising because they do not arise in the Bondi-Penrose description in four dimensions.

Selected Solutions of Einstein’s Field Equations: Their Role in General Relativity and Astrophysics

The primary purpose of all physical theory is rooted in reality, and most relativists pretend to be physicists. We may often be members of departments of mathematics and our work oriented towards the mathematical aspects of Einstein’s theory, but even those of us who hold a permanent position on “scri”, are primarily looking there for gravitational waves. Of course, the builder of this theory and its field equations was the physicist. Jurgen Ehlers has always been very much interested in the conceptual and axiomatic foundations of physical theories and their rigorous, mathematically elegant formulation; but he has also developed and emphasized the importance of such areas of relativity as kinetic theory, the mechanics of continuous media, thermodynamics and, more recently, gravitational lensing. Feynman expressed his view on the relation of physics to mathematics as follows [1]:

Gravitational Radiation from Uniformly Accelerated Particles in General Relativity

An exact solution of Einstein’s vacuum equations representing uniformly accelerated particles was given by Bonnor & Swaminarayan. The radiative properties of this solution are here investigated by Bondi’s method. After the introduction of the metric due to Bondi, the news function is found and the asymptotic behaviour of the Riemann tensor is analysed. The non-vanishing news function and the radiative character of the Riemann tensor at light infinity indicate that the solution is of radiative type; in particular, the solution is radiative in the case of freely gravitating masses. By comparing the solution of Bonnor & Swaminarayan with Born’s solution in electrodynamics the results of Bondi’s method are modified for the system which is not permanently isolated. The angular distribution of the radiated energy and the total rate of radiation of energy and momentum are found in the case of freely moving particles with sufficiently small masses. Gravitational radiation of the particles in question is very similar to electromagnetic radiation of an analogous system of charges. At the moment when the masses or the charges are at rest, the resulting expressions are corroborated on the basis of the approximation method of Bonnor & Rotenberg and of the standard multipole expansion method. The radiation has octupole character. In addition to outgoing radiation, incoming radiation with similar properties is present. This incoming radiation can exist in spite of the radiation condition; an additional condition which excludes the incoming radiation is suggested.

Isometries compatible with gravitational radiation

Isometries compatible with asymptotic flatness and admitting radiation are studied by using Bondi’s formalism. In axially symmetric space‐times, the only second allowable symmetry that does not exclude radiation is boost symmetry. The boost‐rotation symmetric solutions describe ‘‘uniformly accelerated particles’’ of various kinds. The news function is restricted by a differential equation; however, it need not vanish, as has been claimed in the literature. If two Killing fields corresponding to null rotations at null infinity are present, then it is shown that the vacuum field equations imply a further isometry. The resulting space‐time is a plane wave.

Behavior of Einstein-Rosen waves at null infinity

The asymptotic behavior of Einstein-Rosen waves at null infinity in 4 dimensions is investigated in {\it all} directions by exploiting the relation between the 4-dimensional space-time and the 3-dimensional symmetry reduction thereof. Somewhat surprisingly, the behavior in a generic direction is {\it better} than that in directions orthogonal to the symmetry axis. The geometric origin of this difference can be understood most clearly from the 3-dimensional perspective.

Einstein Equations: Exact Solutions

In Einsteins general relativity, with its nonlinear field equations, the discoveries and analyzes of various specific explicit solutions made a great impact on understanding many of the unforeseen features of the theory. Some solutions found fundamental applications in astrophysics, cosmology and, more recently, in the developments inspired by string theory. In this short article we survey the invariant characterization and classification of the solutions and describe the properties and role of the most relevant classes: Minkowski, (anti-)de Sitter spacetimes, spherical Schwarzschild and Reissner-Nordstroem metrics, stationary axisymmetric solutions, radiative metrics describing plane and cylindrical waves, radiative fields of uniformly accelerated sources and Robinson-Trautman solutions. Metrics representing regions of spacetimes filled with matter are also discussed and cosmological models are very briefly mentioned. Some parts of the text are based on a detailed survey which appeared in gr-qc/0004016 (see Ref. 2).

Curvature invariants in type-N spacetimes

The results of Bicak and Pravda (1998 Class. Quantum Grav. 15 1539) are generalized for vacuum type-III solutions with, in general, a non-vanishing cosmological constant . It is shown that all curvature invariants containing derivatives of the Weyl tensor vanish if a type-III spacetime admits a non-expanding and non-twisting null geodesic congruence. A non-vanishing curvature invariant containing first derivatives of the Weyl tensor is found in the case of type-III spacetime with expansion or twist.

Global structure of Robinson-Trautman radiative space-times with cosmological constant

Robinson-Trautman radiative space-times of Petrov type II with a nonvanishing cosmological constant {Lambda} and mass parameter m{gt}0 are studied using analytical methods. They are shown to approach the corresponding spherically symmetric Schwarzschild{endash}de Sitter or Schwarzschild{endash}anti-de Sitter solution at large retarded times. Their global structure is analyzed, and it is demonstrated that the smoothness of the extension of the metrics across the horizon, as compared with the case {Lambda}=0, can increase for {Lambda}{gt}0 and decreases for {Lambda}{lt}0. For the extreme value 9{Lambda}m{sup 2}=1, the extension is smooth but nonanalytic. This case appears to be the first example of a smooth but nonanalytic horizon. The models with {Lambda}{gt}0 exhibit explicitly the cosmic no-hair conjecture under the presence of gravitational waves. {copyright} {ital 1997} {ital The American Physical Society}

Accelerated sources in de Sitter spacetime and the insufficiency of retarded fields

The scalar and electromagnetic fields produced by the geodesic and uniformly accelerated discrete charges in de Sitter spacetime are constructed by employing the conformal relation between de Sitter and Minkowski space. Special attention is paid to new effects arising in spacetimes which, like de Sitter space, have spacelike conformal infinities. Under the presence of particle and event horizons, purely retarded fields (appropriately defined) become necessarily singular or even cannot be constructed at the “creation light cones” — future light cones of the “points” at which the sources “enter” the universe. We construct smooth (outside the sources) fields involving both retarded and advanced effects, and analyze the fields in detail in case of (i) scalar monopoles, (ii) electromagnetic monopoles, and (iii) electromagnetic rigid and geodesic dipoles.

Null dust in canonical gravity

We present the Lagrangian and Hamiltonian framework which incorporates null dust as a source into canonical gravity. Null dust is a generalized Lagrangian system which is described by six Clebsch potentials of its four-velocity Pfaff form. The Dirac--ADM decomposition splits these into three canonical coordinates (the comoving coordinates of the dust) and their conjugate momenta (appropriate projections of four-velocity). Unlike ordinary dust of massive particles, null dust therefore has three rather than four degrees of freedom per space point. These are evolved by a Hamiltonian which is a linear combination of energy and momentum densities of the dust. The energy density is the norm of the momentum density with respect to the spatial metric. The coupling to geometry is achieved by adding these densities to the gravitational super-Hamiltonian and supermomentum. This leads to appropriate Hamiltonian and momentum constraints in the phase space of the system. The constraints can be rewritten in two alternative forms in which they generate a true Lie algebra. The Dirac constraint quantization of the system is formally accomplished by imposing the new constraints as quantum operator restrictions on state functionals. We compare the canonical schemes for null and ordinary dust and emhasize their differences.