Dissertation committee:
Θεοδόσιος Χριστοδουλάκης, Καθηγητής, Φυσικής, ΕΚΠΑ
Θεοχάρης Αποστολάτος, Αναπληρωτής Καθηγητής, Φυσικής, ΕΚΠΑ
Βασίλειος Σπανός, Αναπληρωτής Καθηγητής, Φυσικής, ΕΚΠΑ
Νικόλαος Τετράδης, Καθηγητής, Φυσικής, ΕΚΠΑ
Γεώργιος Διαμάντης, Αναπληρωτής Καθηγητής, Φυσικής, ΕΚΠΑ
Θεοφάνης Γραμμένος, Επίκουρος Καθηγητής, Πολιτικών Μηχανικών, Πανεπιστήμιο Θεσσαλίας
Γεώργιος Κοφινάς, Αναπληρωτής Καθηγητής, Μηχανικών Πληροφοριακών και Επικοινωνιακών Συστημάτων, Πανεπιστήμιο Αιγαίου
Summary:
The Bianchi Types were originally studied in the context of cosmology. These Types are the simplest generalization of FLRW geometries since, the spatial homogeneity exists, but the three-dimensional surfaces are generally anisotropic. More precisely, instead of the existence of a unique scale factor as in the case of FLRW geometry, there are many, in some cases up to nine. However, as we will see, this number is not real, since with mathematical arguments can be drastically decrease. The search for solutions and
their uniqueness, for each Bianchi Type, is not a trivial problem due to the complexity of Einstein’s field equations. The first attempts of the scientific community to solve them were based on intuitive simplifications of the spatial metric, leading eventually to solutions of which the generality and uniqueness were not well defined.
The attempts were converted to more mathematically rigorous and complete since the use of the notion of symmetries: symmetries of the spacetime itself and of the Einstein’s differential equations. As we are going to see, the theory of Lie regarding the integration of differential equations by use of their symmetries, eventually leads to the whole solution space for some of the Types, and finally answers the questions of generality and uniqueness. The fundamental role is possessed by the Automorphisms group
which indicates the generators of the Lie symmetries. One of the purposes of this thesis is to reveal the importance of the Bianchi Types both in modern
cosmology (study of four and five-dimensional spacetimes) as well as in other kind of spacetimes such as pp-wave. At the same time, we wish to make clear the necessity of using the method of Lie point symmetries in order to achieve mathematical completeness and clarity in the integration of a system of differential equations.
In particular, this thesis consists of three parts: Part Ι: Theory, Part ΙΙ: Applications, Part ΙΙΙ: Applications in process.
In the first part we present the basic theory that we are going to use in the next two parts: I) Lie point symmetries of differential equations, II) (d+1) Analysis, III) Homogeneous hyper-surfaces (“Bianchi” Types), IV) Symmetries of Einstein+Maxwell+fluids equations on homogeneous spacetimes.
When it comes to the second part, we present the publications:
1) The solution space of the Einstein’s vacuum field equations for the case of five-dimensional Bianchi Type I (Type 4A1).
The solution space of Einstein’s vacuum field equations for the case of five-dimensional
Bianchi Type I (Type 4A1) July 26, 2018 Classical and Quantum Gravity 35(14), 145003.
We consider the 4+1 Einstein’s field equations (EFE’s) in vacuum, simplified by the assumption that there is a 4D 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 submanifold’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 corresponding to 21 distinct 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.
2)Classical and quantum analysis of 3D electromagnetic pp-wave spacetime.
Classical and quantum analysis of 3D electromagnetic pp-wave spacetime T Pailas, N
Dimakis, A Karagiorgos, Petros A Terzis, G O Papadopoulos and T Christodoulakis, 2019
Class.Quantum Grav. 36 135010.
The general classical solution of the 3D electromagnetic pp-wave spacetime has been obtained. The relevant line element contains an arbitrary essential function providing an infinite number of in-equivalent geometries as solutions. A classification is presented based on the symmetry group. To the best of our knowledge, the solution corresponding to only one of the Classes is known. The dynamics of some of the
Classes was also derived from a minisuperspace Lagrangian which has been constructed. This Lagrangian contains a degree of freedom (the lapse) which can be considered either as dynamical or nondynamical (indicating a singular or a regular Lagrangian correspondingly). Surprisingly enough, on the space of
classical solutions, an equivalence of these two points of view can be established. The canonical quantization is then used in order to quantize the system for both the singular and regular Hamiltonian. A subsequent interpretation of quantum states is based on a Bohm-like analysis. The semi-classical trajectories deviate from the classical only for the regular Hamiltonian and in particular for a superposition of eigenstates (a Gaussian initial state has been used). Thus, the above mentioned equivalence is broken
at the quantum level. It is noteworthy that the semiclassical trajectories tend to the classical ones in the limit where the initial wavepacket is widely spread. Hence, even with this simple superposition state, the classical solutions are acquired as a limit of the semi-classical.
3)Dynamically equivalent ΛCDM equations with underlying Bianchi Type geometry.
Dynamically equivalent ΛCDM equations with underlying Bianchi Type geometry. T.
Pailas, T. Christodoulakis, JCAP 07 (2019) 029
Solutions have been found for gravity coupled to electromagnetic field and a set of charged and uncharged perfect fluids for Bianchi Types VI(-1), VIII, IX. It has been assumed that the anisotropy is “frozen”, γ_{μν}= α(t)^{2}m_{μν}, where
γ_{μν} and m_{μν} are the spatial metric and some constant matrix respectively.
This, according to previous works, results in the existence of a conformal Killing vector field proportional to the fluid velocity of the comoving matter, which guarantees the absence of parallax effects and the independence of the temperature (assuming black body spectrum) from the direction of observation.
The electromagnetic field “absorbs” the “frozen” anisotropy and the remaining equations are dynamically equivalent with the equations of CDM. There are solutions with at, negative and positive effective spatial curvature corresponding to the three FLRW classes. Three equations of state for the charged perfect fluid were studied: non-relativistic w = 0, relativistic w = 1/3 and dark energy-like w = -1. For the rest two cases, maximum values exist for the scale factor, in order for the weak energy conditions to be respected, which depend upon the geometric and charged fluid parameters. A minimum
value for the scale factor exists (for the solutions to be valid) in all the cases and Types, indicating the absence of initial spacetime singularity (big bang). This minimum value depends upon the geometric and electromagnetic parameters. The number of essential constants in the final form of each metric is the minimum without loss of generality due to the use of the constant Automorphism’s group. A known solution,
with the anisotropy absorbed via one free scalar field is reproduced with our method and contains the minimum possible number of parameters.
4)Infinite dimensional symmetry groups of the Friedmann equations
Infinite dimensional symmetry groups of the Friedmann equations. T. Pailas, N. Dimakis,
Andronikos Paliathanasis, Petros A. Terzis, T. Christodoulakis, Phys. Rev. D 102,
063524 (2020)
We find the symmetry generators for the Friedman equations emanating from a perfect fluid source in the presence of a cosmological constant term. The relevant dynamics are shown to be governed by two coupled, first order ordinary differential equations, the continuity and the quadratic constraint equation.
Arbitrary functions appear in the components of the symmetry vector, indicating the infinity of the group. When the equation of state is considered as arbitrary but ab initio given, previously known results are recovered and/or generalized. When the pressure is considered among the dynamical variables, solutions for models with different equations of state are mapped to each other, thus enabling the presentation of
solutions to models with complicated equations of state starting from simple known cases.
5)“Time”-Covariant Schrödinger Equation and the Canonical Quantization of the Reissner–
Nordström Black Hole
“Time”-Covariant Schrödinger Equation and the Canonical Quantization of the Reissner–
Nordström Black Hole. Theodoros Pailas, Quantum Reports 2020, 2(3), 414-441.
A “time”-covariant Schrödinger equation is defined for the minisuperspace model of the Reissner–Nordström (RN) black hole, as a “hybrid” between the “intrinsic time” Schrödinger and Wheeler–DeWitt (WDW) equations. To do so, a reduced, regular, and “time(r)”-dependent Hamiltonian density was constructed, without “breaking” the re-parametrization covariance r = f(~r). As a result, the evolution of
states with respect to the parameter r and the probabilistic interpretation of the resulting quantum description is possible, while quantum schemes for different gauge choices are equivalent by construction. The solutions are found for Dirac’s delta and Gaussian initial states. A geometrical interpretation of the wavefunctions is presented via Bohm analysis. Alongside this, a criterion is presented to adjudicate which, between two singular spacetimes, is “more” or “less” singular. Two ways to adjudicate the existence
of singularities are compared (vanishing of the probability density at the classical singularity and semi-classical spacetime singularity). Finally, an equivalence of the reduced equations with those of a 3D electromagnetic pp-wave spacetime is revealed.
Finally, when it comes to the third part, we present the analysies carried out so far, related to the possibility of comparing the results of the publication number 3, with observations.
