89 research outputs found
Geometrization of Lie and Noether symmetries with applications in Cosmology
We derive the Lie and the Noether conditions for the equations of motion of a
dynamical system in a dimensional Riemannian space. We solve these
conditions in the sense that we express the symmetry generating vectors in
terms of the special projective and the homothetic vectors of the space.
Therefore the Lie and the Noether symmetries for these equations are geometric
symmetries or, equivalently, the geometry of the space is modulating the motion
of dynamical systems in that space. We give two theorems which contain all the
necessary conditions which allow one to determine the Lie and the Noether
symmetries of a specific dynamical system in a given Riemannian space. We apply
the theorems to various interesting situations covering Newtonian 2d and 3d
systems as well as dynamical systems in cosmology.Comment: 15 pages, no figures, 11 tables, Talk given at the 15th Conference on
Recent Developments in Gravity (NEB XV), 20-23 June 2012, Chania, Greec
Two scalar field cosmology: Conservation laws and exact solutions
We consider the two scalar field cosmology in a FRW spatially flat spacetime
where the scalar fields interact both in the kinetic part and the potential. We
apply the Noether point symmetries in order to define the interaction of the
scalar fields. We use the point symmetries in order to write the field
equations in the normal coordinates and we find that the Lagrangian of the
field equations which admits at least three Noether point symmetries describes
linear Newtonian systems. Furthermore, by using the corresponding conservation
laws we find exact solutions of the field equations. Finally, we generalize our
results to the case of N scalar fields interacting both in their potential and
their kinematic part in a flat FRW background.Comment: 17 pages, to be published in Phys. Rev.
Symmetries of Differential Equations in Cosmology
The purpose of the current article is to present a brief albeit accurate
presentation of the main tools used in the study of symmetries of Lagrange
equations for holonomic systems and subsequently to show how these tools are
applied in the major models of modern cosmology in order to derive exact
solutions and deal with the problem of dark matter/energy. The key role in this
approach are the first integrals of the field equations. We start with the Lie
point symmetries and the first integrals defined by them, that is the Hojman
integrals. Subsequently we discuss the Noether point symmetries and the well
known method for deriving the Noether integrals. By means of the Inverse
Noether Theorem we show that to every Hojman quadratic first integral one is
possible to associate a Noether symmetry whose Noether integral is the original
Hojman integral. It is emphasized that the point transformation generating this
Noether symmetry need not coincide with the point transformation defining the
Lie symmetry which produces the Hojman integral. We discuss the close
connection between the Lie point and the Noether point symmetries with the
collineations of the metric defined by the kinetic energy of the Lagrangian. In
particular the generators of Noether point symmetries are elements of the
homothetic algebra of that metric. The key point in the current study of
cosmological models is the introduction of the mini superspace, that is the
space which is defined by the physical variables of the model, which is not the
spacetime where the model evolves. The metric in the mini superspace is found
from the kinematic part of the Lagrangian and we call it the kinetic metric.
The rest part of the Lagrangian is the effective potential.Comment: 44 pages, review article to appear in Symmetr
Lie and Noether point symmetries of a class of quasilinear systems of second-order differential equations
We study the Lie and Noether point symmetries of a class of systems of
second-order differential equations with independent and dependent
variables ( systems). We solve the symmetry conditions in a
geometric way and determine the general form of the symmetry vector and of the
Noetherian conservation laws. We prove that the point symmetries are generated
by the collineations of two (pseudo)metrics, which are defined in the spaces of
independent and dependent variables. We demonstrate the general results in two
special cases (a) a system of coupled Laplace equations and (b) the
Klein-Gordon equation of a particle in the context of Generalized Uncertainty
Principle. In the second case we determine the complete invariant group of
point transformations, and we apply the Lie invariants in order to find
invariant solutions of the wave function for a spin- particle in the two
dimensional hyperbolic space.Comment: 20 pages, to be published in Journal of Geometry and Physic
Type II hidden symmetries for the homogeneous heat equation in some general classes of Riemannian spaces
We study the reduction of the heat equation in Riemannian spaces which admit
a gradient Killing vector, a gradient homothetic vector and in Petrov Type
D,N,II and Type III space-times. In each reduction we identify the source of
the Type II hidden symmetries. More specifically we find that a) If we reduce
the heat equation by the symmetries generated by the gradient KV the reduced
equation is a linear heat equation in the nondecomposable space. b) If we
reduce the heat equation via the symmetries generated by the gradient HV the
reduced equation is a Laplace equation for an appropriate metric. In this case
the Type II hidden symmetries are generated from the proper CKVs. c) In the
Petrov spacetimes the reduction of the heat equation by the symmetry generated
from the nongradient HV gives PDEs which inherit the Lie symmetries hence no
Type II hidden symmetries appear. We apply the general results to cases in
which the initial metric is specified. We consider the case that the
irreducible part of the decomposed space is a space of constant nonvanishing
curvature and the case of the spatially flat Friedmann-Robertson-Walker space
time used in Cosmology. In each case we give explicitly the Type II hidden
symmetries provided they exist.Comment: 18 pages, Accepted for publication in Journal of Geometry and Physic
Three fluid cosmological model using Lie and Noether symmetries
We employ a three fluid model in order to construct a cosmological model in
the Friedmann Robertson Walker flat spacetime, which contains three types of
matter dark energy, dark matter and a perfect fluid with a linear equation of
state. Dark matter is described by dust and dark energy with a scalar field
with potential V({\phi}). In order to fix the scalar field potential we demand
Lie symmetry invariance of the field equations, which is a model-independent
assumption. The requirement of an extra Lie symmetry selects the exponential
scalar field potential. The further requirement that the analytic solution is
invariant under the point transformation generated by the Lie symmetry
eliminates dark matter and leads to a quintessence and a phantom cosmological
model containing a perfect fluid and a scalar field. Next we assume that the
Lagrangian of the system admits an extra Noether symmetry. This new assumption
selects the scalar field potential to be exponential and forces the perfect
fluid to be stiff. Furthermore the existence of the Noether integral allows for
the integration of the dynamical equations. We find new analytic solutions to
quintessence and phantom cosmologies which contain all three fluids. Using
these solutions one is able to compute analytically all main cosmological
functions, such as the scale factor, the scalar field, the Hubble expansion
rate, the deceleration parameter etc.Comment: accepted for publication in Class. Quantum Grav. (12 pages
The reduction of Laplace equation in certain Riemannian spaces and the resulting Type II hidden symmetries
We prove a general theorem which allows the determination of Lie symmetries
of Laplace equation in a general Riemannian space using the conformal group of
the space. Algebraic computing is not necessary. We apply the theorem in the
study of the reduction of Laplace equation in certain classes of Riemannian
spaces which admit a gradient Killing vector, a gradient Homothetic vector and
a special Conformal Killing vector. In each reduction we identify the source of
Type II hidden symmetries. We find that in general the Type II hidden
symmetries of Laplace equation are directly related to the transition of the
CKVs from the space where the original equation is defined to the space where
the reduced equation resides. In particular we consider the reduction of
Laplace equation (i.e. the wave equation) in Minkowski space and obtain the
results of all previous studies in a straightforward manner. We consider the
reduction of Laplace equation in spaces which admit Lie point symmetries
generated from a non gradient HV and a proper CKV and we show that the
reduction with these vectors does not produce Type II hidden symmetries. We
apply the results to General Relativity and consider the reduction of Laplace
equation in locally rotational symmetric space times (LRS) and in algebraically
special vacuum solutions of Einstein's equations which admit a homothetic
algebra acting simply transitively. In each case we determine the Type II
hidden symmetries.Comment: Accepted for publication in Journal of Geometry and Physics (22
pages
The geometric origin of Lie point symmetries of the Schr\"{o}dinger and the Klein Gordon equations
We determine the Lie point symmetries of the Schr\"{o}dinger and the Klein
Gordon equations in a general Riemannian space. It is shown that these
symmetries are related with the homothetic and the conformal algebra of the
metric of the space respectively. We consider the kinematic metric defined by
the classical Lagrangian and show how the Lie point symmetries of the
Schr\"{o}dinger equation and the Klein Gordon equation are related with the
Noether point symmetries of this Lagrangian. The general results are applied to
two practical problems a. The classification of all two and three dimensional
potentials in a Euclidian space for which the Schr\"{o}dinger equation and the
Klein Gordon equation admit Lie point symmetries and b. The application of Lie
point symmetries of the Klein Gordon equation in the exterior Schwarzschild
spacetime and the determination of the metric by means of conformally related
Lagrangians.Comment: to be published in IJGMMP, 18 page
The generic model of General Relativity
We develop a generic spacetime model in General Relativity which can be used
to build any gravitational model within General Relativity. The generic model
uses two types of assumptions: (a) Geometric assumptions additional to the
inherent geometric identities of the Riemannian geometry of spacetime and (b)
Assumptions defining a class of observers by means of their 4-velocity
which is a unit timelike vector field. The geometric assumptions as a rule
concern symmetry assumptions (the so called collineations). The latter
introduces the 1+3 decomposition of tensor fields in spacetime. The 1+3
decomposition results in two major results. The 1+3 decomposition of
defines the kinematic variables of the model (expansion, rotation, shear and
4-acceleration) and defines the kinematics of the gravitational model. The 1+3
decomposition of the energy momentum tensor representing all gravitating matter
introduces the dynamic variables of the model (energy density, the isotropic
pressure, the momentum transfer or heat flux vector and the traceless tensor of
the anisotropic pressure) as measured by the defined observers and define the
dynamics of he model. The symmetries assumed by the model act as constraints on
both the kinematical and the dynamical variables of the model. As a second
further development of the generic model we assume that in addition to the
4-velocity of the observers there exists a second universal vector
field in spacetime so that one has a so called double congruence
which can be used to define the 1+1+2 decomposition of tensor
fields. The 1+1+2 decomposition leads to an extended kinematics concerning both
fields building the double congruence and to a finer dynamics involving more
physical variables.Comment: 55 pages, no figure
Symmetries of second-order PDEs and conformal Killing vectors
We study the Lie point symmetries of a general class of partial differential
equations (PDE) of second order. An equation from this class naturally defines
a second-order symmetric tensor (metric). In the case the PDE is linear on the
first derivatives we show that the Lie point symmetries are given by the
conformal algebra of the metric modulo a constraint involving the linear part
of the PDE. Important elements in this class are the Klein--Gordon equation and
the Laplace equation. We apply the general results and determine the Lie point
symmetries of these equations in various general classes of Riemannian spaces.
Finally we study the type II\ hidden symmetries of the wave equation in a
Riemannian space with a Lorenzian metric.Comment: 16 pages; to be published in J.Phys.Conf.Ser.: Proceedings of GADEIS
VII, June 2014, Larnaca Cypru
- β¦