16 research outputs found
Lorentz transformation and vector field flows
The parameter changes resulting from a combination of Lorentz transformation
are shown to form vector field flows. The exact, finite Thomas rotation angle
is determined and interpreted intuitively. Using phase portraits, the
parameters evolution can be clearly visualized. In addition to identifying the
fixed points, we obtain an analytic invariant, which correlates the evolution
of parameters.Comment: 11 pages, 3 figures. Section IV revised and title change
When physics helps mathematics: calculation of the sophisticated multiple integral
There exists a remarkable connection between the quantum mechanical
Landau-Zener problem and purely classical-mechanical problem of a ball rolling
on a Cornu spiral. This correspondence allows us to calculate a complicated
multiple integral, a kind of multi-dimensional generalization of Fresnel
integrals. A direct method of calculation is also considered but found to be
successful only in some low-dimensional cases. As a byproduct of this direct
method, an interesting new integral representation for is obtained.Comment: 13 pages, no figure
Form Geometry and the 'tHooft-Plebanski Action
Riemannian geometry in four dimensions, including Einstein's equations, can
be described by means of a connection that annihilates a triad of two-forms
(rather than a tetrad of vector fields). Our treatment of the conformal factor
of the metric differs from the original presentation of this result, due to
'tHooft. In the action the conformal factor now appears as a field to be
varied.Comment: 12pp, LaTe
so(4) Plebanski Action and Relativistic Spin Foam Model
In this note we study the correspondence between the ``relativistic spin
foam'' model introduced by Barrett, Crane and Baez and the so(4) Plebanski
action. We argue that the Plebanski model is the continuum analog of
the relativistic spin foam model. We prove that the Plebanski action possess
four phases, one of which is gravity and outline the discrepancy between this
model and the model of Euclidean gravity. We also show that the Plebanski model
possess another natural dicretisation and can be associate with another, new,
spin foam model that appear to be the counterpart of the spin foam
model describing the self dual formulation of gravity.Comment: 12 pages, REVTeX using AMS fonts. Some minor corrections and
improvement
Hamiltonian Analysis of Plebanski Theory
We study the Hamiltonian formulation of Plebanski theory in both the
Euclidean and Lorentzian cases. A careful analysis of the constraints shows
that the system is non regular, i.e. the rank of the Dirac matrix is
non-constant on the non-reduced phase space. We identify the gravitational and
topological sectors which are regular sub-spaces of the non-reduced phase
space. The theory can be restricted to the regular subspace which contains the
gravitational sector. We explicitly identify first and second class constraints
in this case. We compute the determinant of the Dirac matrix and the natural
measure for the path integral of the Plebanski theory (restricted to the
gravitational sector). This measure is the analogue of the
Leutwyler-Fradkin-Vilkovisky measure of quantum gravity.Comment: 25 pages, no figures, references adde
Higher dimensional Kerr-Schild spacetimes
We investigate general properties of Kerr-Schild (KS) metrics in n>4
spacetime dimensions. First, we show that the Weyl tensor is of type II or more
special if the null KS vector k is geodetic (or, equivalently, if
T_{ab}k^ak^b=0). We subsequently specialize to vacuum KS solutions, which
naturally split into two families of non-expanding and expanding metrics. After
demonstrating that non-expanding solutions are equivalent to the known class of
vacuum Kundt solutions of type N, we analyze expanding solutions in detail. We
show that they can only be of the type II or D, and we characterize optical
properties of the multiple Weyl aligned null direction (WAND) k. In general, k
has caustics corresponding to curvature singularities. In addition, it is
generically shearing. Nevertheless, we arrive at a possible "weak" n>4
extension of the Goldberg-Sachs theorem, limited to the KS class, which matches
previous conclusions for general type III/N solutions. In passing, properties
of Myers-Perry black holes and black rings related to our results are also
briefly discussed.Comment: 33 pages. v2: minor changes, new reference