arXiv subjects

T. Christodoulakis

Publications and source records attributed to T. Christodoulakis.

At least 37 records · Page 2Linked to original sources

arXiv2018

The solution space to the Einstein's vacuum field equations for the case of five-dimensional Bianchi Type I (Type 4A1)

We consider the 4+1 Einstein's field equations (EFE's) in vacuum, simplified by the assumption that there is a four-dimensional sub-manifold on which an isometry group of dimension four acts simply transitive. In particular we consider the Abelian group Type 4A1; and thus the emerging homogeneous sub-space is flat. Through the use of coordinate transformations that preserve the sub-manifold's manifest homogeneity, a coordinate system is chosen in which the shift vector is zero. The resulting equations remain form invariant under the action of the constant Automorphisms group. This group is used in order to simplify the equations and obtain their complete solution space which consists of seven families of solutions. Apart form the Kasner type all the other solutions found are, to the best of our knowledge, new. Some of them correspond to cosmological solutions, others seem to depend on some spatial coordinate and there are also pp-wave solutions.

gr-qc↗

arXiv2018

Quantum cosmology of a Bianchi III LRS geometry coupled to a source free electromagnetic field

We consider a Bianchi type III axisymmetric geometry in the presence of an electromagnetic field. A first result at the classical level is that the symmetry of the geometry need not be applied on the electromagnetic tensor $F_{μν}$; the algebraic restrictions, implied by the Einstein field equations to the stress energy tensor $T_{μν}$, suffice to reduce the general $F_{μν}$ to the appropriate form. The classical solution thus found contains a time dependent electric and a constant magnetic charge. The solution is also reachable from the corresponding mini-superspace action, which is strikingly similar to the Reissner-Nordstr{ö}m one. This points to a connection between the black hole geometry and the cosmological solution here found, which is the analog of the known correlation between the Schwarzschild and the Kantowski-Sachs metrics. The configuration space is drastically modified by the presence of the magnetic charge from a 3D flat to a 3D pp wave geometry. We map the emerging linear and quadratic classical integrals of motion, to quantum observables. Along with the Wheeler-DeWitt equation these observables provide unique, up to constants, wave functions. The employment of a Bohmian interpretation of these quantum states results in deterministic (semi-classical) geometries most of which are singularity free.

gr-qc↗

arXiv2017

Discrete spectrum of the quantum Reissner - Nordström geometry

We start from a static, spherically symmetric space-time in the presence of an electrostatic field and construct the mini-superspace Lagrangian that reproduces the well known Reissner - Nordström solution. We identify the classical integrals of motion that are to be mapped to quantum observables and which are associated with the mass and charge. Their eigenvalue equations are used as supplementary conditions to the Wheeler-DeWitt equation and a link is provided between the existence of an horizon and to whether the spectrum of the observables is fully discrete or not. For each case we provide an orthonormal basis of states as emerges through the process of canonical quantization.

gr-qc↗

arXiv2016

Decoupling of the re-parametrization degree of freedom and a generalized probability in quantum cosmology

The high degree of symmetry renders the dynamics of cosmological as well as some black hole spacetimes describable by a system of finite degrees of freedom. These systems are generally known as minisuperspace models. One of their important key features is the invariance of the corresponding reduced actions under reparametrizations of the independent variable, a fact that can be seen as the remnant of the general covariance of the full theory. In the case of a system of $n$ degrees of freedom, described by a Lagrangian quadratic in velocities, one can use the lapse by either gauge fixing it or letting it be defined by the constraint and subsequently substitute into the rest of the equations. In the first case, the system is solvable for $n$ accelerations and the constraint becomes a restriction among constants. In the second case, the system can only be solved for $n-1$ accelerations and the "gauge" freedom is transferred to the choice of one of the scalar degrees of freedom. In this paper, we take the second path and express all $n-1$ scalar degrees of freedom in terms of the remaining one, say $q$. By considering these $n-1$ degrees of freedom as arbitrary but given functions of $q$, we manage to extract a two dimensional pure gauge system consisting of the lapse $N$ and the arbitrary $q$: in a way, we decouple the reparametrization invariance from the rest of the equations of motion. The solution of the corresponding quantum two dimensional system is used for the definition of a generalized probability for every configuration $f^i (q)$, be it classical or not. The main result is that, interestingly enough, this probability attains its extrema on the classical solution of the initial $n$-dimensional system.

gr-qc↗

arXiv2016

General Analytic Solutions of Scalar Field Cosmology with Arbitrary Potential

We present the solution space for the case of a minimally coupled scalar field with arbitrary potential in a FLRW metric. This is made possible due to the existence of a nonlocal integral of motion corresponding to the conformal Killing field of the two-dimensional minisuperspace metric. The case for both spatially flat and non flat are studied first in the presence of only the scalar field and subsequently with the addition of non interacting perfect fluids. It is verified that this addition does not change the general form of the solution, but only the particular expressions of the scalar field and the potential. The results are applied in the case of parametric dark energy models where we derive the scalar field equivalence solution for some proposed models in the literature.

gr-qc↗

arXiv2016

Conditional symmetries in axisymmetric quantum cosmologies with scalar fields and the fate of the classical singularities

In this paper, the classical and quantum solutions of some axisymmetric cosmologies coupled to a massless scalar field are studied in the context of minisuperspace approximation. In these models, the singular nature of the Lagrangians entails a search for possible conditional symmetries. These have been proven to be the simultaneous conformal symmetries of the supermetric and the superpotential. The quantization is performed by adopting the Dirac proposal for constrained systems, i.e. promoting the first-class constraints to operators annihilating the wave function. To further enrich the approach, we follow \cite{Christodoulakis:2012eg} and impose the operators related to the classical conditional symmetries on the wave function. These additional equations select particular solutions of the Wheeler-DeWitt equation. In order to gain some physical insight from the quantization of these cosmological systems, we perform a semiclassical analysis following the Bohmian approach to quantum theory. The generic result is that, in all but one model, one can find appropriate ranges of the parameters, so that the emerging semiclassical geometries are non-singular. An attempt for physical interpretation involves the study of the effective energy-momentum tensor which corresponds to an imperfect fluid.

gr-qc↗

arXiv2015

Variational Contact Symmetries of Constraint Lagrangians

The investigation of contact symmetries of re--parametrization invariant Lagrangians of finite degrees of freedom and quadratic in the velocities is presented. The main concern of the paper is those symmetry generators which depend linearly in the velocities. A natural extension of the symmetry generator along the lapse function $N(t)$, with the appropriate extension of the dependence in $\dot{N}(t)$ of the gauge function, is assumed; this action yields new results. The central finding is that the integrals of motion are either linear or quadratic in velocities and are generated, respectively by the conformal Killing vector fields and the conformal Killing tensors of the configuration space metric deduced from the kinetic part of the Lagrangian (with appropriate conformal factors). The freedom of re--parametrization allows one to appropriately scale $N(t)$, so that the potential becomes constant; in this case the integrals of motion can be constructed from the Killing fields and Killing tensors of the scaled metric. A rather interesting result is the non--necessity of the gauge function in Noether's theorem due to the presence of the Hamiltonian constraint

gr-qc↗

arXiv2014

Noether analysis of Scalar-Tensor Cosmology

A scalar--tensor theory of gravity, containing an arbitrary coupling function $F(ϕ)$ and a general potential $V(ϕ)$, is considered in the context of a spatially flat FLRW model. The use of reparametrization invariance enables a particular lapse parametrization in which the mini--superspace metric completely specifies the dynamics of the system. A requirement of existence of the maximal possible number of autonomous integrals of motion is imposed. This leads to a flat mini--superspace metric realized by a particular relation between the coupling function and the potential. The space of solutions is completely described in terms of the three autonomous integrals of motion constructed by the Killing fields of the mini--supermetric and an additional rheonomous emanating from the homothetic field. The solutions contain the arbitrary function which remains after the imposition of the relation between $F(ϕ)$ and $V(ϕ)$. To exemplify the use of the general results, we select some particular cases and study their physical implications through an effective energy--momentum tensor, which tends out to be that of a perfect fluid.

gr-qc↗

arXiv2014

Canonical Quantization of the BTZ Black Hole using Noether Symmetries

The well-known BTZ black hole solution of (2+1) Einstein's gravity, in the presence of a cosmological constant, is treated both at the classical and quantum level. Classically, the imposition of the two manifest local Killing fields of the BTZ geometry at the level of the full action results in a mini-superspace constraint action with the radial coordinate playing the role of the independent dynamical variable. The Noether symmetries of this reduced action are then shown to completely determine the classical solution space, without any further need to solve the dynamical equations of motion. At a quantum mechanical level, all the admissible sets of the quantum counterparts of the generators of the above mentioned symmetries are utilized as supplementary conditions acting on the wave-function. These additional restrictions, in conjunction with the Wheeler-DeWitt equation, help to determine (up to constants) the wave-function which is then treated semiclassically, in the sense of Bohm. The ensuing space-times are, either identical to the classical geometry, thus exhibiting a good correlation of the corresponding quantization to the classical theory, or are less symmetric but exhibit no Killing or event horizon and no curvature singularity, thus indicating a softening of the classical conical singularity of the BTZ geometry.

gr-qc↗

arXiv2013

FLRW metric $f(R)$ cosmology with a perfect fluid by generating integrals of motion

In the context of metric f(R) gravity, we consider a FLRW space-time, filled with a perfect fluid described by a barotropic equation of state (p = γρ). We give the equivalent mini-superspace description and use the reparametrization invariance of the resulting Lagrangian to work in the equivalent constant potential description. At that point, we restrict our analysis to those models for which the ensuing scaled mini-superspace is maximally symmetric. Those models exhibit the maximum number of autonomous integrals of motion linear in the momenta, which are constructed by the Killing vectors of the respective mini-supermetric. The integrals of motion are used to analytically solve the equations of the corresponding models. Finally, a brief description of the properties of the resulting Hubble parameters is given.

gr-qc↗

arXiv2013

Minisuperspace Canonical Quantization of the Reissner-Nordstrom Black Hole via Conditional Symmetries

We use the conditional symmetry approach to study the $r$-evolution of a minisuperspace spherically symmetric model both at the classical and quantum level. After integration of the coordinates $t$, $θ$ and $ϕ$ in the gravitational plus electromagnetic action the configuration space dependent dynamical variables turn out to correspond to the $r$-dependent metric functions and the electrostatic field. In the context of the formalism for constrained systems (Dirac - Bergmann, ADM) with respect to the radial coordinate $r$, we set up a point-like reparameterization invariant Lagrangian. It is seen that, in the constant potential parametrization of the lapse, the corresponding minisuperspace is a Lorentzian three-dimensional flat manifold which obviously admits six Killing vector fields plus a homothetic one. The weakly vanishing $r$-Hamiltonian guarantees that the phase space quantities associated to the six Killing fields are linear holonomic integrals of motion. The homothetic field provides one more rheonomic integral of motion. These seven integrals are shown to comprise the entire classical solution space, i.e. the space-time of a Reissner-Nordström black hole, the $r$-reparametrization invariance since one dependent variable remains unfixed, and the two quadratic relations satisfied by the integration constants. We then quantize the model using the quantum analogues of the classical conditional symmetries, and show that the existence of such symmetries yields solutions to the Wheeler-DeWitt equation which, as a semiclassical analysis shows, exhibit a good correlation with the classical regime. We use the resulting wave functions to investigate the possibility of removing the classical singularities.

gr-qc↗

arXiv2013

Lie - point and variational symmetries in minisuperspace Einstein's gravity

We consider the application of the theory of symmetries of coupled ordinary differential equations to the case of reparametrisation invariant Lagrangians quadratic in the velocities; such Lagrangians encompass all minisuperspace models. We find that, in order to acquire the maximum number of symmetry generators, one must (a) consider the lapse $N(t)$ among the degrees of freedom and (b) allow the action of the generator on the Lagrangian and/or the equations of motion to produce a multiple of the constraint, rather than strictly zero. The result of this necessary modification of the standard theory (concerning regular systems) is that the Lie - point symmetries of the equations of motion are exactly the variational symmetries (containing the time reparametrisation symmetry) plus the well known scaling symmetry. These variational symmetries are seen to be the simultaneous conformal Killing fields of both the metric and the potential, thus coinciding with the conditional symmetries defined in phase space. In a parametrisation of the lapse for which the potential becomes constant, the generators of the aforementioned symmetries become the Killing fields of the scaled supermetric and the homothetic field respectively.

gr-qc↗

arXiv2012

Lie algebra automorphisms as Lie point symmetries and the solution space for Bianchi Type I, II, IV, V vacuum geometries

Lie group symmetry analysis for systems of coupled, nonlinear ordinary differential equations is performed in order to obtain the entire solution space to Einstein's field equations for vacuum Bianchi spacetime geometries. The symmetries used are the automorphisms of the Lie algebra of the corresponding three-dimensional isometry group acting on the hyper-surfaces of simultaneity for each Bianchi Type, as well as the scaling and the time reparametrization symmetry. A detailed application of the method is presented for Bianchi Type IV. The result is the acquisition of the general solution of Type IV in terms of sixth Painleve transcendent PVI, along with the known pp-wave solution. For Bianchi Types I, II, V the known entire solution space is attained and very briefly listed, along with two new Type V solutions of Euclidean and neutral signature and a Type I pp-wave metric.

gr-qc↗

arXiv2012

Conditional Symmetries and the Canonical Quantization of Constrained Minisuperspace Actions: the Schwarzschild case

A conditional symmetry is defined, in the phase-space of a quadratic in velocities constrained action, as a simultaneous conformal symmetry of the supermetric and the superpotential. It is proven that such a symmetry corresponds to a variational (Noether) symmetry.The use of these symmetries as quantum conditions on the wave-function entails a kind of selection rule. As an example, the minisuperspace model ensuing from a reduction of the Einstein - Hilbert action by considering static, spherically symmetric configurations and r as the independent dynamical variable, is canonically quantized. The conditional symmetries of this reduced action are used as supplementary conditions on the wave function. Their integrability conditions dictate, at a first stage, that only one of the three existing symmetries can be consistently imposed. At a second stage one is led to the unique Casimir invariant, which is the product of the remaining two, as the only possible second condition on $Ψ$. The uniqueness of the dynamical evolution implies the need to identify this quadratic integral of motion to the reparametrisation generator. This can be achieved by fixing a suitable parametrization of the r-lapse function, exploiting the freedom to arbitrarily rescale it. In this particular parametrization the measure is chosen to be the determinant of the supermetric. The solutions to the combined Wheeler - DeWitt and linear conditional symmetry equations are found and seen to depend on the product of the two "scale factors"

gr-qc↗

arXiv2011

Classical and Quantum Bianchi Type III vacuum Horava - Lifshitz Cosmology

A diagonal Bianchi Type III space-time is treated, both at the classical and quantum level, in the context of Horava - Lifshitz gravity. The system of the classical equations of motion is reduced to one independent Abel's equation of the first kind. Closed form solution are presented for various values of the coupling constants appearing in the action. Due to the method used, solutions of Euclidean, Lorentzian and neutral signature are attained. The solutions corresponding to \lamda <> 1 are seen to develop curvature singularities as the other constants approach their Einsteinian values, in contrast to those with \lamda = 1 which tend to the known Einstein gravity solutions. At the quantum level, the resulting Wheeler-DeWitt equation is explicitly solved for \lamda = 1, σ= 0 and \lamda = 1/3 . The ensuing wave-functions diverge in the Einsteinian limit.

gr-qc↗

arXiv2009

Towards Canonical Quantum Gravity for Geometries Admitting Maximally Symmetric Two-dimensional Surfaces

The 3+1 (canonical) decomposition of all geometries admitting two-dimensional space-like surfaces is exhibited. A proposal consisting of a specific re-normalization {\bf Assumption} and an accompanying {\bf Requirement} is put forward, which enables the canonical quantization of these geometries. The resulting Wheeler-deWitt equation is based on a re-normalized manifold parameterized by three smooth scalar functionals. The entire space of solutions to this equation is analytically given, exploiting the freedom left by the imposition of the {\bf Requirement} and contained in the third functional.

gr-qc↗

arXiv2009

Towards Canonical Quantum Gravity for G1 Geometries in 2+1 Dimensions with a Lambda--Term

The canonical analysis and subsequent quantization of the (2+1)-dimensional action of pure gravity plus a cosmological constant term is considered, under the assumption of the existence of one spacelike Killing vector field. The proper imposition of the quantum analogues of the two linear (momentum) constraints reduces an initial collection of state vectors, consisting of all smooth functionals of the components (and/or their derivatives) of the spatial metric, to particular scalar smooth functionals. The demand that the midi-superspace metric (inferred from the kinetic part of the quadratic (Hamiltonian) constraint) must define on the space of these states an induced metric whose components are given in terms of the same states, which is made possible through an appropriate re-normalization assumption, severely reduces the possible state vectors to three unique (up to general coordinate transformations) smooth scalar functionals. The quantum analogue of the Hamiltonian constraint produces a Wheeler-DeWitt equation based on this reduced manifold of states, which is completely integrated.

gr-qc↗

arXiv2007

From Feshbach-Resonance Managed Bose-Einstein Condensates to Anisotropic Universes: Some Applications of the Ermakov-Pinney equation with Time-Dependent Nonlinearity

In this work we revisit the topic of two-dimensional Bose-Einstein condensates under the influence of time-dependent magnetic confinement and time-dependent scattering length. A moment approach reduces the examination of moments of the wavefunction (in particular, of its width) to an Ermakov-Pinney (EP) ordinary differential equation (ODE). We use the well-known structure of the solutions of this nonlinear ODE to ``engineer'' trapping and interatomic interaction conditions that lead to condensates dispersing, breathing or even collapsing. The advantage of the approach is that it is fully tractable analytically, in excellent agreement with our numerical observations. As an aside, we also discuss how similar time-dependent EP equations may arise in the description of anisotropic scalar field cosmologies.

cond-mat.other↗