192 research outputs found
Lie and Noether symmetries of geodesic equations and collineations
The Lie symmetries of the geodesic equations in a Riemannian space are
computed in terms of the special projective group and its degenerates (affine
vectors, homothetic vector and Killing vectors) of the metric. The Noether
symmetries of the same equations are given in terms of the homothetic and the
Killing vectors of the metric. It is shown that the geodesic equations in a
Riemannian space admit three linear first integrals and two quadratic first
integrals. We apply the results in the case of Einstein spaces, the
Schwarzschild spacetime and the Friedman Robertson Walker spacetime. In each
case the Lie and the Noether symmetries are computed explicitly together with
the corresponding linear and quadratic first integrals.Comment: 19 page
Ricci Collineations for Non-Degenerate, Diagonal and Spherically Symmetric Ricci Tensors
The expression of the vector field generator of a Ricci Collineation for
diagonal, spherically symmetric and non-degenerate Ricci tensors is obtained.
The resulting expressions show that the time and radial first derivatives of
the components of the Ricci tensor can be used to classify the collineation,
leading to 64 families.
Some examples illustrate how to obtain the collineation vector
Hypersurface homogeneous locally rotationally symmetric spacetimes admitting conformal symmetries
All hypersurface homogeneous locally rotationally symmetric spacetimes which
admit conformal symmetries are determined and the symmetry vectors are given
explicitly. It is shown that these spacetimes must be considered in two sets.
One set containing Ellis Class II and the other containing Ellis Class I, III
LRS spacetimes. The determination of the conformal algebra in the first set is
achieved by systematizing and completing results on the determination of CKVs
in 2+2 decomposable spacetimes. In the second set new methods are developed.
The results are applied to obtain the classification of the conformal algebra
of all static LRS spacetimes in terms of geometrical variables. Furthermore all
perfect fluid nontilted LRS spacetimes which admit proper conformal symmetries
are determined and the physical properties some of them are discussed.Comment: 15 pages; to appear in Classical Quantum Gravity; some misprints in
Tables 3,5 and in section 4 correcte
Classification of Spherically Symmetric Static Spacetimes according to their Matter Collineations
The spherically symmetric static spacetimes are classified according to their
matter collineations. These are studied when the energy-momentum tensor is
degenerate and also when it is non-degenerate. We have found a case where the
energy-momentum tensor is degenerate but the group of matter collineations is
finite. For the non-degenerate case, we obtain either {\it four}, {\it five},
{\it six} or {\it ten} independent matter collineations in which four are
isometries and the rest are proper. We conclude that the matter collineations
coincide with the Ricci collineations but the constraint equations are
different which on solving can provide physically interesting cosmological
solutions.Comment: 15 pages, no figure, Late
Ricci Collineations of the Bianchi Type II, VIII, and IX Space-times
Ricci and contracted Ricci collineations of the Bianchi type II, VIII, and IX
space-times, associated with the vector fields of the form (i) one component of
is different from zero and (ii) two components of are
different from zero, for , are presented. In subcase (i.b), which
is , some known solutions are found, and in subcase
(i.d), which is , choosing ,
the Bianchi type II, VIII, and IX space-times is reduced to the
Robertson-Walker metric.Comment: 12 Pages, LaTeX, 1 Table, no figure
Killing tensors in pp-wave spacetimes
The formal solution of the second order Killing tensor equations for the
general pp-wave spacetime is given. The Killing tensor equations are integrated
fully for some specific pp-wave spacetimes. In particular, the complete
solution is given for the conformally flat plane wave spacetimes and we find
that irreducible Killing tensors arise for specific classes. The maximum number
of independent irreducible Killing tensors admitted by a conformally flat plane
wave spacetime is shown to be six. It is shown that every pp-wave spacetime
that admits an homothety will admit a Killing tensor of Koutras type and, with
the exception of the singular scale-invariant plane wave spacetimes, this
Killing tensor is irreducible.Comment: 18 page
Matter collineations of Spacetime Homogeneous G\"odel-type Metrics
The spacetime homogeneous G\"odel-type spacetimes which have four classes of
metrics are studied according to their matter collineations. The obtained
results are compared with Killing vectors and Ricci collineations. It is found
that these spacetimes have infinite number of matter collineations in
degenerate case, i.e. det, and do not admit proper matter
collineations in non-degenerate case, i.e. det. The degenerate
case has the new constraints on the parameters and which characterize
the causality features of the G\"odel-type spacetimes.Comment: 12 pages, LaTex, no figures, Class. Quantum.Grav.20 (2003) 216
Ricci Collineations for type B warped space-times
We present the general structure of proper Ricci Collineations (RC) for type
B warped space-times. Within this framework, we give a detailed description of
the most general proper RC for spherically symmetric metrics. As examples,
static spherically symmetric and Friedmann-Robertson-Walker space-times are
considered.Comment: 18 pages, Latex, To appear in GR
Kerr-Schild Symmetries
We study continuous groups of generalized Kerr-Schild transformations and the
vector fields that generate them in any n-dimensional manifold with a
Lorentzian metric. We prove that all these vector fields can be intrinsically
characterized and that they constitute a Lie algebra if the null deformation
direction is fixed. The properties of these Lie algebras are briefly analyzed
and we show that they are generically finite-dimensional but that they may have
infinite dimension in some relevant situations. The most general vector fields
of the above type are explicitly constructed for the following cases: any
two-dimensional metric, the general spherically symmetric metric and
deformation direction, and the flat metric with parallel or cylindrical
deformation directions.Comment: 15 pages, no figures, LaTe
- âŠ