In a cosmological context, the electric and magnetic parts of the Weyl tensor, E ab and H ab , represent the locally free curvature -i.e. they are not pointwise determined by the matter fields. By performing a complete covariant decomposition of ∇ c E ab and ∇ c H ab , we show that the parts of the derivative of the curvature which are locally free (i.e. not pointwise determined by the matter via the Bianchi identities) are exactly the symmetrised trace-free spatial derivatives of E ab and H ab together with their spatial curls. These parts of the derivatives are shown to be crucial for the existence of gravitational waves. The terms divergence and curl as applied to rank-2 tensors are defined by equation (1). ‡ Angle brackets · · · enclosing indices represent their totally symmetric trace-free and spatially projected part.Round and square brackets denote, respectively, symmetrisation and antisymmetrisation.
We consider one-dimensional classical time-dependent Hamiltonian systems with quasi-periodic orbits. It is well-known that such systems possess an adiabatic invariant which coincides with the action variable of the Hamiltonian formalism. We present a new proof of the adiabatic invariance of this quantity and illustrate our arguments by means of explicit calculations for the harmonic oscillator.The new proof makes essential use of the Hamiltonian formalism. The key step is the introduction of a slowly-varying quantity closely related to the action variable. This new quantity arises naturally within the Hamiltonian framework as follows: a canonical transformation is first performed to convert the system to action-angle coordinates; then the new quantity is constructed as an action integral (effectively a new action variable) using the new coordinates. The integration required for this construction provides, in a natural way, the averaging procedure introduced in other proofs, though here it is an average in phase space rather than over time. : 45.05.+x, 45.20.Jj.
PACS
The problem of ascertaining the stability of a given spatially homogeneous solution of Einstein's equations to small metric perturbations was examined by Barrow and Sonoda (1986), and their method was discussed by Siklos (1988). In this paper, one of the points mentioned by Siklos is investigated further: the role of the constraint equation. It is shown that the constraint equation can lead to problems aside from those (e.g. linearization stability) usually considered. A particularly simple example is used to illustrate these points, namely the homogeneous vacuum plane wave metric, which for a certain range of values of its parameters admits spatially homogeneous hypersurfaces. One way of avoiding the problems associated with the constraint equation is simply to ignore it. In most cases, this will not affect the outcome.
Homogeneous space-times (i.e. those admitting a three-parameter group of isometries) are studied using the Newman Penrose formalism. It is found that solutions containing horizons depend on two fewer parameters than the most general solution, so that horizons and the associated whimper singularities are not stable features of homogeneous space-times. In the vacuum case, there are just three two-parameter families with horizons, two of which are the NUT solutions and certain plane waves.
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