The Einstein–Cartan theory, Studia, Pracownie
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TheEinstein–Cartantheory:
themeaningandconsequencesoftorsion
(TeoriaEinsteina–Cartana:znaczenieikonsekwencjetorsji)
PawełLasko±-Grabowski
Master’sthesis
Advisor:
ProfessorJerzyKowalski-Glikman
UniversityofWrocław
FacultyofPhysicsandAstronomy
Theoreticalphysics
August11,2009
Abstract
In the following paper we focus on one of the oldest extensions to general rela-
tivity, the Einstein–Cartan theory. We explain the physical sense of involved quan-
tities, stressing the meaning of torsion. We discuss the possibilities of measuring
torsion-related phenomena. Finally, we derive and solve the Mathisson–Papapetrou
equations in few simple cases and provide explanations of the results.
W poni»szej pracy skupimy si¦ na jednym z najstarszych rozszerze« ogólnej
teorii wzgl¦dno±ci — teorii Einsteina–Cartana. Omówimy i wyja±nimy znaczenie
podstawowych wielko±ci, podkre±laj¡c znaczenie poj¦cia torsji. Rozwa»ymy mo»-
liwo±ci pomiaru zjawisk zwi¡zanych z torsj¡. Na koniec wyprowadzimy równanie
Mathissona–Papapetrou, rozwi¡»emy je w pewnych prostych przypadkach i zinter-
pretujemy otrzymane wyniki.
Contents
1Introduction
5
2TheEinstein–Cartantheory
7
2.1
Preliminaries
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.2
From the ECT to traditional general relativity
. . . . . . . . . . . .
8
2.3
The geometrical meaning of curvature and torsion
. . . . . . . . . .
10
2.4
Measuring curvature and torsion
. . . . . . . . . . . . . . . . . . . .
12
2.5
Torsion and the principle of equivalence
. . . . . . . . . . . . . . . .
15
3TheMathisson–Papapetrouequation
17
3.1
Derivation of the Mathisson–Papapetrou equation
. . . . . . . . . .
17
3.2
Flat, torsioned spacetime
. . . . . . . . . . . . . . . . . . . . . . . .
19
3.3
Nonspinning particle . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
3.4
Derivation of the equations of motion (a simple case) . . . . . . . . .
21
3.5
Graphical presentation of the results . . . . . . . . . . . . . . . . . .
24
4Summary
31
Bibliography
33
3
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