On the stability of scalar-vacuum space-times

The European Physical Journal C - A recently formulated framework of highly anisotropic and strongly dissipative hydrodynamics (ADHYDRO) is used to describe the evolution of matter created in...     

Event horizons in static scalar-vacuum space-times

The following theorem is established. Every zero-mass scalar field which is gravitationally coupled, static and asymptotically flat, becomes singular at a simply-connected event horizon. In the special case where the gravitational coupling of the scalar energy density is neglected, the solutions are computed explicitly. Some properties of the singular event horizons are discussed, and a brief mention of related work with non-static scalar fields is given.     

On asymptotically flat space-times

The allowed asymptotic behavior of the Ricci tensor is determined for asymptotically flat space-times. With the aid of Penrose's conformai technique the asymptotic behavior of the components of the metric tensor, Weyl tensor, and spin coefficients in a suitable frame is calculated for such a space-time. For Einstein-Maxwell space-times these results reduce to those of Exton, Newman, Penrose, Unti, and Kozarzewski.     

On purely radiative space-times

The existence of space-times representing pure gravitational radiation which comes in from infinity and interacts with itself is discussed. They are characterized as solutions of Einstein's vacuum field equations possessing a smooth structure at past null infinity which forms the future null cone at past timelike infinity with complete generators. The pure radiation problem is analysed where free initial data for Einstein's field equations are prescribed on the null cone at past time-like infinity. It is demonstrated how the pure radiation problem can be formulated as a local initial value problem for the symmetric hyperbolic system of reduced conformal vacuum field equations. Its solutions are uniquely determined by the free data.Work supported by a Heisenberg-fellowship of the Deutsche Forschungsgemeinschaft     

On purely gravito-magnetic vacuum space-times

It is shown that there are no purely magnetic, vacuum, spacetime metrics where any one of the following conditions holds: (a) the ratio of any two eigenvalues of the Weyl tensor is constant, (b) both of the Riemann principal null directions, defining the time-like blade, are nonrotating, (c) the shear tensor possesses an eigenvector v ^a which is defined by one of the space-like Riemann principal directions, (d) the vorticity is parallel to v ^a , where v ^a is defined by one of the space-like Riemann principal directions.This revised version was published online in April 2005. The publishing date was inserted.     

On static and radiative space-times

The conformal constraint equations on space-like hypersurfaces are discussed near points which represent either time-like or spatial infinity for an asymptotically flat solution of Einstein's vacuum field equations. In the case of time-like infinity a certain radiativity condition, is derived which must be satisfied by the data at that point. The case of space-like infinity is analysed in detail for static space-times with non-vanishing mass. It is shown that the conformal structure implied here on a slice of constant Killing time, which extends analytically through infinity, satisfies at spatial infinity the radiativity condition. Thus to any static solution exists a certain radiative solution which has a smooth structure at past null infinity and is regular at past time-like infinity. A characterization of these solutions by their free data is given and non-symmetry properties are discussed.     

On event horizons in static space-times

A proof of the (vacuum) Israel theorem on event horizons in static space-times is given employing the Newman-Penrose formalism. The theorem is extended to include the case of a static, massive, complex, scalar field.Supported in part by the National Research Council of Canada.     

On the fundamental groups of space-times in general relativity

It is shown that there are upper bounds on the first and second betti numbers of compact space-times or space-times with Cauchy surfaces whose fundamental groups are abelian. Homological classifications of compact space-times and space-times with compact Cauchy surfaces are given.     

On the sources of static plane symmetric vacuum space-times

The sources of static vacuum plane space-times with reflection symmetry are considered. It is argued that the counterpart of the Einstein theory to the gravitational field of a massive Newtonian plane should be described by the Rindler solution, which represents also a uniform gravitational field, and the test particles moving in which have constant accelerations.     

On maximal surfaces in asymptotically flat space-times

Existenc of maximal and almost maximal hypersurfaces in asymptotically flat space-times is established under boundary conditions weaker than those considered previously. We show in particular that every vacuum evolution of asymptotically flat data for the Einstein equations can be foliated by slices maximal outside a spatially compact set and that every (strictly) stationary asymptotically flat space-time can be foliated by maximal hypersurfaces. Amongst other uniqueness results, we show that maximal hypersurfaces can be used to partially fix an asymptotic Poincaré group.Supported in part by the NSF grant PHY 8503072 to Yale University     

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.     

On algebraically special space-times with twisting rays

The reduced gravitational field equations are derived for algebraically special space-times with twisting geodesic and shear-free rays for a large class of Ricci tensors. These equations coincide with those derived by Trim and Wainwright under more restrictive assumptions on the Ricci tensor. Penrose's conformal technique is used to facilitate computation and interpretation of the results. The remaining coordinate freedom and freedom in the choice of tetrad is discussed.     

On quantization of free fields in stationary space-times

In Section 1 we analyse the structure of the infinite-dimensional Hamiltonian system described by the Klein-Gordon equation (free real scalar field) in stationary space-times with closed space sections; we give an existence and uniqueness theorem for the Lichnerowicz distribution kernelG ₁ together with its proper Fourier expansion, and we construct the Hilbert spaces of frequency-part solutions defined by means ofG ₁.In Section 2 an analysis, a theorem and a construction similar to the above are formulated for thefree real field spin 1, massm>0, in one kind of static space-times.In this letter, only results are given. For detailed proofs and further results, see reference [9], [10] and [11].     

On positivity of mass for black hole space-times

For any time-symmetric asymptotically flat initial slice with an apparent horizon, the associated four-momentum measured at spatial infinity is shown to be future-directed and time-like.     

On the rescaling problem of space-times admitting groups of conformal motions

The necessary and sufficient conditions are given that a space-time admitting a group of conformal motions can be mapped conformally on a space-time admitting the same group but of Killing symmetries.     

Linearization stability of coupled gravitational and scalar fields in asymptotically flat space-times

Using the methods of Choquet-Bruhat, Fischer and Marsden and using weighted Sobolev spaces developed recently by Christodoulou and Choquet-Bruhat, it is proved that the Einstein field equations coupled with self-gravitating scalar fields are linearization stable in asymptotically flat space-times.     

The stability of Killing–Cauchy horizons in colliding plane wave space-times

It is confirmed rigorously that the Killing–Cauchy horizons, which sometimes occur in space-times representing the collision and subsequent interaction of plane gravitational waves in a Minkowski background, are unstable with respect to bounded perturbations of the initial waves, at least for the case in which the initial waves have constant aligned polarizations.     

On the algebraic type of space-times possessing a nonsingular Killing horizon

The characteristic initial-value problem is discussed for the Einstein and Einstein-Maxwell equations when initial data are specified on two branches of a bifurcate Killing horizon. In the latter case it is assumed that the principal null directions of the Maxwell tensor coincide with those of a Killing bivector. A theorem specifying necessary and sufficient conditions for such space-times to be Petrov typeD is derived and consequences of this with respect to Israel's theorem and Robinson's theorem are discussed.Part of this work was initiated while the author attended the summer school on global analysis held at the International Center for Theoretical Physics, Trieste, Italy (1972).