Transformation properties and invariants in scalar-tensor theories of gravity
Failid
Kuupäev
2019-01-30
Autorid
Ajakirja pealkiri
Ajakirja ISSN
Köite pealkiri
Kirjastaja
Abstrakt
Einsteini üldrelatiivsusteooria, tänane gravitatsiooni standardteooria on tensorteooria. Keskseks objektiks selles on meetriline tensor ehk sisuliselt juhend lõikude pikkuste ja nendevaheliste nurkade arvutamiseks. Termin tensor rõhutab, et eelmainitud juhend, mida võib ette kujutada sisendiga ja väljundiga kastina, seab täpselt kahele lõigule (pikkuse korral sama lõik kaks korda) vastavusse ühe arvu, mille väärtuses erinevad vaatlejad nõustuvad. Einsteini teooria ideeks on uurida situatsiooni, kus lisaks välisele informatsioonile ehk lõikude omadustele sõltub väljund, arv, ka juhendi enda sisemisest informatsioonist. Just eelnev, pikkuste ja nurkade arvutamise juhendi sisemine informatsioon, on üldrelatiivsusteoorias dünaamiline ja aegruumi punktist sõltuv, lubades seega kirjeldada erinevaid kõveraid aegruume.
Skalaar-tensortüüpi gravitatsiooniteooriates üldistatakse Einsteini teooriat, lisades sellele skalaarse suuruse, tensori sugulase, mis lõikudest sõltumatuna ehk puhtalt sisemise informatsiooni najal väljastab vaatlejast sõltumatu, kuid aegruumi punktist sõltuva arvu. Väitekirjas uurin teisendusi, mis segavad meetrilise tensori ja skalaari sisemist informatsiooni, jättes sealjuures aegruumi põhjusliku struktuuri muutumatuks. Täpsemalt uurin erinevate avaldiste teisenemist sellise segamise tagajärjel ning kirjutan välja muutumatud kombinatsioonid ehk niinimetatud invariandid. Viimaseid kasutan sama teooriate pere erinevate formuleeringute ja erinevate abstraktsuse tasemete vaheliste suhete selgitamiseks. Töö tulemusena järeldan, et mugavaim ja sirgjooneliseim formuleering teisenemiseeskirjade uurimiseks on kõige abstraktsem, sest siis igal avaldisel on kindel ja ühene teisenemiseeskiri.
Einstein’s general relativity, current standard theory of gravity is a tensor theory. Its central object is the metric tensor, i.e., the prescription for calculating the length of line segments and angles between two of them. The term ‘tensor’ emphasizes that the prescription, pictured as a box with input and output, takes exactly two of those line segments (or twice the same in case of length) and produces one number which is the same for different observers. The key idea behind Einstein’s theory is to consider the situation where in addition to extrinsic information, i.e., the properties of the line segments, the output depends also on the intrinsic information of the prescription. It is the latter, the intrinsic information of the prescription for calculating lengths and angles that is dynamical and spacetime-point-dependent in general relativity, thus permitting to describe different curved spacetime geometries. In scalar-tensor theories of gravity Einstein’s theory is extended by adding a scalar, a sibling of a tensor that receives no line segments and thus produces an observer-independent number solely based on spacetime-point-dependent intrinsic information. In the thesis I study transformations that mix the intrinsic information of the metric tensor and the scalar in a manner that preserves the causal structure. In particular I consider how different expressions change under the transformations, and identify those which remain unchanged, i.e., the so-called invariants. The latter are used to clarify relations between different formulations of the same family of theories on different levels of abstraction. I conclude that the most convenient level for studying the transformation properties under such transformations is the most abstract one as in that case each expression has an explicit and unique transformation rule.
Einstein’s general relativity, current standard theory of gravity is a tensor theory. Its central object is the metric tensor, i.e., the prescription for calculating the length of line segments and angles between two of them. The term ‘tensor’ emphasizes that the prescription, pictured as a box with input and output, takes exactly two of those line segments (or twice the same in case of length) and produces one number which is the same for different observers. The key idea behind Einstein’s theory is to consider the situation where in addition to extrinsic information, i.e., the properties of the line segments, the output depends also on the intrinsic information of the prescription. It is the latter, the intrinsic information of the prescription for calculating lengths and angles that is dynamical and spacetime-point-dependent in general relativity, thus permitting to describe different curved spacetime geometries. In scalar-tensor theories of gravity Einstein’s theory is extended by adding a scalar, a sibling of a tensor that receives no line segments and thus produces an observer-independent number solely based on spacetime-point-dependent intrinsic information. In the thesis I study transformations that mix the intrinsic information of the metric tensor and the scalar in a manner that preserves the causal structure. In particular I consider how different expressions change under the transformations, and identify those which remain unchanged, i.e., the so-called invariants. The latter are used to clarify relations between different formulations of the same family of theories on different levels of abstraction. I conclude that the most convenient level for studying the transformation properties under such transformations is the most abstract one as in that case each expression has an explicit and unique transformation rule.
Kirjeldus
Märksõnad
transformations, invariance, relativistic gravitation theory, gravitation