Η βαρύτητα ως θεωρία βαθμίδας και κάποιες τροποποιήσεις αυτής

Παλληκάρης, Κωνσταντίνος; Pallikaris, Konstantinos

dc.contributor.author	Παλληκάρης, Κωνσταντίνος	el
dc.contributor.author	Pallikaris, Konstantinos	en
dc.date.accessioned	2018-07-02T11:14:07Z
dc.date.available	2018-07-02T11:14:07Z
dc.date.issued	2018-07-02
dc.identifier.uri	https://dspace.lib.ntua.gr/xmlui/handle/123456789/47178
dc.identifier.uri	http://dx.doi.org/10.26240/heal.ntua.15339
dc.rights	Default License
dc.subject	Βαρύτητα	el
dc.subject	Διαφορική γεωμετρία	el
dc.subject	Θεωρίες βαθμίδας	el
dc.subject	Μαθηματική φυσική	el
dc.subject	Gravitation	en
dc.subject	Τροποποιημένη βαρύτητα	el
dc.subject	Modified gravity	en
dc.subject	Gauge theories	en
dc.subject	Differential geometry	en
dc.subject	Mathematical physics	en
dc.title	Η βαρύτητα ως θεωρία βαθμίδας και κάποιες τροποποιήσεις αυτής	el
heal.type	bachelorThesis
heal.classification	Βαρύτητα	el
heal.classification	Γενική σχετικότητα και βαρύτητα	el
heal.classification	Gauge fields (Physics)	en
heal.classification	Διαφορική γεωμετρία	el
heal.classificationURI	http://data.seab.gr/concepts/9bab51b354b67876c6284de28df0e549940fb873
heal.classificationURI	http://data.seab.gr/concepts/c94fc33eba1cd953b0edc5227f0628025f2fd1e0
heal.classificationURI	http://id.loc.gov/authorities/subjects/sh85053534
heal.classificationURI	http://data.seab.gr/concepts/b23b2a188e64d48ed369e7951509209b8ae6150f
heal.language	el
heal.access	free
heal.recordProvider	ntua	el
heal.publicationDate	2018-02-23
heal.abstract	The main object of this thesis is to present a well-established strategy for the gauging of gravity, i.e the description of gravity as a gauge theory. It is well known that the usual geometric background of General Relativity (and thus, for most of its modifications) differs greatly from that of the fundamental non-gravitational interactions (Standard Model), mostly in the geometrical objects acting as dynamical variables of each theory. One can argue that gauge theories (for example, Yang-Mills theories) are built upon the foundations of Ehresmann geometry, i.e their geometric configuration roughly consists of a (smooth) principal fiber bundle with a Lie group acting (freely, properly and transi- tively) from the right on the total space (namely the structure group of the bundle) and an Ehresmann connection, which lies in the total space and is valued in the Lie algebra of the structure group. The latter, specifically the frame-dependent connection 1-form, plays the role of the gauge potential that in a way produces the degrees of freedom for each model. Since a gauge is just a choice of a local section of the bundle, gauge transfor- mations are simply transformations between sections. Roughly speaking, gauge theories may be described in terms of connective geometry over spacetime with the gauge group, i.e the vertical automorphisms of the total space, being the symmetry group of each theory. Due to the fact that gauge transformations do not relate different fibers, but rather take place in a fiber at a point of the base manifold, this whole process may be viewed as an internal feature, if we consider spacetime as an external object. On the other hand, General Relativity approaches gravitational interaction in a com- pletely different manner, describing it in terms of the metric geometry of spacetime. Here, the symmetry group is simply the group of diffeomorphisms of spacetime and there is no dynamical connection gauging a local symmetry. One may argue that in the mainstream description of gravity, the geometry is in a way “external”. Perhaps the first attempt to a unified geometrical description was the a la Palatini approach, where the spin connec- tion was introduced, acting as the associated gauge field for the local Lorentz symmetry. However, ad hoc introduction of a new field, the soldering form, was necessary, in order to compensate for the difference in degrees of freedom between the two approaches. As there is no analog of this additional field in gauge theories, this early attempt (as it was) fails to be a complete and consistent mechanism for gauging gravity. We use the work of Cartan to show that the fundamental variables of the (first order) Palatini formula- tion, namely the spin connection and the soldering form, can be treated as parts of a single field, known as the Cartan connection. In this manner, we successfully establish a gauge-theoretical description of gravity. We do so in both ways, either by defining a Cartan geometry from scratch or by reducing the Ehresmann bundle. The physical intuition of the latter way is far better, since the reduction may be thought as a “partial” symmetry-breaking process, where the gauge associated to the local translational invari- ance is fixed, thus leaving only the local Lorentz symmetry unbroken. We then argue that the (external) diffeomorphism invariance of spacetime is the reason that the initially broken translational symmetry is after all preserved. In order to achieve all of the above, we thoroughly study the mathematical framework that will allow us to construct these neat geometries. The most part of this thesis is mathematically rigorous, meaning that we give proofs for most of the theorems, claims, propositions etc. We give great emphasis at differential forms and exterior algebra in general, firstly because they are necessary for the geometri- sation process and finally, because they form the, closest to the underlying geometry, language one can use in a practical way, i.e for doing chart-independent calculations. We try to cover a decent part of old and modern differential geometry (at least those subjects that are crucial to the understanding of the last chapters of this thesis), in order to give a consistent bottom to top (as from structurally poorest to richest notions) image of the geometries we present. Finally, we give some detailed examples of torsion gravity, specif- ically two theories of its teleparallel class, as a special case of Poincaré gauge gravity, while we also examine some well known examples of modified gravity, namely the f(R) class and Horndeski’s generalized scalar-tensor theory, where the latter covers a great i ii part of the scalar-tensor type spectrum. Our engagement with the above mentioned toy models remains at the level of eom (equations of motion) extraction in a detailed manner. Apart from the f(R) case, the rest is chosen due to the technical challenge it presents, i.e the amount of not so straightforward calculus one must do to obtain results. In other words, they act as a nice playground for many of the identities we introduce in former chapters. In the case of the teleparallel theories, we try to handle the eom extraction process in a dense and short way (by avoiding, as much as possible, the “debauchery of indices”, as Cartan once said), thus proving that the use of exterior calculus is opti- mal for extended calculations, once appropriate familiarity with the latter formulation is achieved. The transition to the usual “tensorial” index notation is always a rather trivial task, that can be done afterwards (as we also do after the extraction of the eom). Finally, we propose smarter ways of dealing with index notation by using antisymmetry brackets, symmetry parentheses, generalised Kronecker deltas and generalised permanent symbols, thus avoiding unnecessary index gymnastics. The effectiveness of the above may be seen in the f(R) cases.	en
heal.advisorName	Αναγνωστόπουλος, Κωνσταντίνος	el
heal.committeeMemberName	Ήργες, Νίκος	el
heal.committeeMemberName	Κουτσούμπας, Γεώργιος	el
heal.committeeMemberName	Αναγνωστόπουλος, Κωνσταντίνος	el
heal.academicPublisher	Εθνικό Μετσόβιο Πολυτεχνείο. Σχολή Εφαρμοσμένων Μαθηματικών και Φυσικών Επιστημών. Τομέας Φυσικής	el
heal.academicPublisherID	ntua
heal.numberOfPages	219 σ.
heal.fullTextAvailability	true