T. Christodoulakis

The general solution of Bianchi type III vacuum cosmology

The theory of symmetries of systems of coupled, ordinary differential equations (ODE) is used to develop a concise algorithm in order to obtain the entire space of solutions to vacuum Bianchi Einstein’s field equations (EFEs). The symmetries used are the well known automorphisms of the Lie algebra for the corresponding isometry group of each Bianchi Type, as well as the scaling and the time re-parametrization symmetry. The application of the method to Type V I I h results in (a) obtaining the general solution of Type V I I 0 with the aid of the third Painleve transcendental P I I I ; (b) obtaining the general solution of Type V I I h with the aid of the sixth Painleve transcendental P V I ; (c) the recovery of all known solutions (six in total) without a prior assumption of any extra symmetry; (d) The discovery of a new solution (the line element given in closed form) with a G 3 isometry group acting on T 3, i.e., on time-like hyper-surfaces, along with the emergence of the line element describing the flat vacuum Type V I I 0 Bianchi Cosmology.

Lie point and variational symmetries in minisuperspace Einstein gravity

We consider the application of the theory of symmetries of coupled ordinary differential equations to the case of reparametrization invariant Lagrangians quadratic in the velocities; such Lagrangians encompass all minisuperspace models. We find that, in order to acquire the maximum number of possibly existing 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 reparametrization 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 parametrization of the lapse for which the potential becomes constant, the generators of the aforementioned symmetries become the Killing fields of the scaled supermetric and its homothetic field, respectively.

Noether analysis of Scalar-Tensor Cosmology

A scalar--tensor theory of gravity, containing an arbitrary coupling function

Conditional symmetries and the canonical quantization of constrained minisuperspace actions: The Schwarzschild case

F(\phi)

Automorphisms and a cartography of the solution space for vacuum Bianchi cosmologies: The Type III case

and a general potential

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

V(\phi)

Minisuperspace canonical quantization of the Reissner-Nordström black hole via conditional symmetries

, 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

The general solution of Bianchi type V I Ih vacuum cosmology

F(\phi)

Canonical quantization of the BTZ black hole using Noether symmetries

and

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

V(\phi)