On metric-affine gravitational theories with a Lagrangian quadratic in the curvature and the energy-momentum problem [Elektronische Ressource] / Ahmad Fouad Abdalwahab Abdellatif
122 pages
English

On metric-affine gravitational theories with a Lagrangian quadratic in the curvature and the energy-momentum problem [Elektronische Ressource] / Ahmad Fouad Abdalwahab Abdellatif

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122 pages
English
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On metric-affine gravitational theories with a Lagrangian quadratic in the curvature and the energy-momentum problemI n a u g u r a l d i s s e r t a t i o n zur Erlangung des akademischen Grades doctor rerum naturalium (Dr. rer. nat.) an der Mathematisch-Naturwissenschaftlichen Fakultät der Ernst-Moritz-Arndt-Universität Greifswald vorgelegt von Ahmad Fouad Abdalwahab Abdellatif geboren am 10. 8. 1973in El-Minia, Egypt Greifswald, Oktober 2011 Dekan: Prof. Dr. Klaus Fesser 1. Gutachter: Prof. Dr. Rainer Schimming 2. Gutachter: Prof. Dr. Felix Finster Tag der Promotion: 17. Januar 2012 On metric-affine gravitational theories with a Lagrangian quadratic in the curvatureand the energy-momentum problemThesis submitted for the degree Dr. rer. nat.byAhmad Fouad Abdalwahab AbdellatifMathematics Department, Faculty of Science,Minia UniversityEl-Minia, Egypt.SupervisorsProf. Dr. Rainer SchimmingInstitute for Mathematics and Computer ScienceErnst-Moritz-Arndt UniversityGreifswald, Germany.Prof. Dr. Ragab M. M. Gad andProf. Dr. Abdel Rahman H. EssawayMathematics Department Faculty of Science, Minia University El-Minia, Egypt. On metric-affine gravitational theories with a Lagrangian quadratic in the curvatureand the energy-momentum problemContents1 Introduction 12 Preliminaries.

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Publié le 01 janvier 2012
Nombre de lectures 24
Langue English

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On metric-affine gravitational theories with a Lagrangian quadratic in the curvature
and the energy-momentum problem
I n a u g u r a l d i s s e r t a t i o n
zur
Erlangung des akademischen Grades
doctor rerum naturalium (Dr. rer. nat.)
an der Mathematisch-Naturwissenschaftlichen Fakultät
der
Ernst-Moritz-Arndt-Universität Greifswald
vorgelegt von
Ahmad Fouad Abdalwahab Abdellatif
geboren am 10. 8. 1973
in El-Minia, Egypt
Greifswald, Oktober 2011





































Dekan: Prof. Dr. Klaus Fesser


1. Gutachter: Prof. Dr. Rainer Schimming

2. Gutachter: Prof. Dr. Felix Finster

Tag der Promotion: 17. Januar 2012 On metric-affine gravitational theories with a Lagrangian quadratic in the curvature
and the energy-momentum problem
Thesis submitted for the degree Dr. rer. nat.
by
Ahmad Fouad Abdalwahab Abdellatif
Mathematics Department, Faculty of Science,
Minia University
El-Minia, Egypt.
Supervisors
Prof. Dr. Rainer Schimming
Institute for Mathematics and Computer Science
Ernst-Moritz-Arndt University
Greifswald, Germany.
Prof. Dr. Ragab M. M. Gad
and
Prof. Dr. Abdel Rahman H. Essaway
Mathematics Department
Faculty of Science, Minia University
El-Minia, Egypt. On metric-affine gravitational theories with a Lagrangian quadratic in the curvature
and the energy-momentum problem
Contents
1 Introduction 1
2 Preliminaries. Geometric objects 6
3 Manifolds with an affine connection 9
n4 Variational calculus on R 17
5 Variational calculus on a manifoldM 21
6 Metric-affine field theories 27
7 Purely metrical field theories 53
8 The Palatini case 72
9 On energy-momentum complexes 94
References 100












Chapter 1













1 Introduction
Inthisworkwesystematicallystudyso-calledmetric-affine theories, i.e. fieldtheoriesforbothametric
g and an affine connection Γ on a smooth n-dimensional manifold M. We also study purely metrical
theories for g only as a special case. We assume that the field equations follow from a variational
principle with a Lagrange function L or a Lagrange density L. More precisely, we assume that L or L
is built from g and from the curvature C of Γ.
Several scientific disciplines meet in such field theories, as follows.
• A smooth manifold M, a Riemannian or pseudo-Riemannian metric g, and an affine connection Γ
are fundamental concepts of higher differential geometry.
• Variational calculus with a Lagrangian L or L is an important part of mathematical analysis.
• The Euler-Lagrange equations, which follow from the variational principle, are interpreted as phys-
ical field equations. In particular, if the metric g has Lorentzian signature, then (M,g) is called a
spacetime manifold and g, Γ are interpreted as descriptions of gravitation. According to the Kaluza-
Klein principle the spacetime may have any dimension n.
Let us shortly, following the survey papers [70,71], recall some prehistory of geometrized field theories.
For about 2000 years there was only one kind of geometry, namely what we call today Euclidean
geometry in two or three dimensions, and there was no clear distinction of the mathematical, physical,
and philosophical aspects of geometry. This simple view was disturbed in the 19th century by the
discovery of hyperbolic (also called Lobachevsky) geometry and of spherical geometry as consistent
mathematical theories. Moreover, the concepts of a vector space and other abstract spaces introduced
the idea of a dimension n into geometry. An important landmark is B. Riemann’s proposal in 1854 of
a very general kind of geometry, which was later named after him. Thereby he introduced the notion
of an n-dimensional manifold M. The curvature of a Riemannian metric g over M generally varies
from point to point and depends, in a sense, from the direction. Euclidean, hyperbolic, and spherical
geometries appear as very special cases of Riemannian geometry. Naturally, the question arose: which
mathematical geometry is the best description of the physical space? Moreover, W.K. Clifford already
in 1876 initiated the idea of geometrization of physics, i.e. identification of physical fields with geo-
metrical quantities. The idea was realized in 1915: Einstein’s general relativity theory (GRT) merges
space and time to a four-dimensional spacetime manifold M and equips M with a Lorentzian metric g
which is identified with gravitation. All fields other than g, particles, and media form physical matter
in GRT. The dynamics in GRT is characterized by the impressive slogan: ”Matter tells spacetime how
to curve; spacetime tells matter how to move.” [51].
Einstein’s GRT is the standard theory of space, time, and gravitation until today. It stands well all
experimental tests in the solar system and on earth. Moreover, astronomy and GRT together give a
fairly consistent picture of the world at large. Nevertheless, there is a strong tendency to search for
alternatives of GRT. The production of alternative theories began already soon after 1915. Einstein
himself was one of the greatest theory-makers.
The motives of the quest for alternative theories are theoretical imperfections of GRT:
–Only one field, gravitation, is geometrized.
1–There are solutions of the field equations with unwanted features, namely with singularities and / or
causality violations.
–Spinors enter the field equations only through tensors, while in quantum theory spinors are primary
and tensors are secondary quantities.
–No general definition of a localizable gravitational energy independent of an observer is available.
–Quantization of GRT leads to non-renormalizable expressions.
There is, additionally, a list of wishes towards a better gravitational theory:
–Unification of fundamental interactions, i.e. gravitational, electromagnetic, weak and strong interac-
tions. As a first step, gravity and electromagnetism shall be unified.
–Replacement of the field-particle dualism by a field monism (Einstein’s particle program).
–Realization of Mach’s principle (It claims that inertia is induced by the masses in the cosmos).
–Explanation of the hypothetical dark energy.
–More wishes not specified here.
Let us classify geometrized theories of gravitation and those of gravitation unified with electromag-
netism or with another physical field. There are purely metrical theories which have a metric g as
the only primary object and extended theories which rely on a richer geometric structure. GRT itself
belongs to the first class. Any alternative purely metrical theory differs from GRT in the dimension
of the spacetime manifold, the order of the field equations, or in some other essential feature. An
extended theory of gravitation either relies on a mixed geometry, where there another geometrical
object is added to the metric g, or there is one geometrical ”superobject”, which induces a metric g.
The following types of mixed geometries are met in alternative theories:
metric + scalar,
metric + vector,
metric + torsion,
metric + affine connection,
metric + another metric,
and more configurations metric + geometrical object.
Otherwise, the following superobjects are met in alternative theories:
non-symmetric fundamental tensor, the symmetric part of which is a metric g,
complex fundamental tensor, the real part of which is a metric g,
Hermitean metric on a complex manifold,
teleparallelism (i.e. existence of a global frame of vector fields),
Finsler metric,
and more.
Let us sketch the historical and conceptual background of the notion of an affine connection.Itiswel
known that a metric g in a natural way defines covariant differentiation of vector and tensor fields
or, equivalently, a notion of parallel propagation of such fields along curves. Several authors observed
that covariant differentiation or parallel p can be defined by simple axioms without use of a
metric. Thus, the concept of a general affine connection Γ was introduced. Soon alternative theories
based on a mixed geometry g +Γ were proposed. Such geometries can be classified according to the
three characteristics: curvature C, torsion S,andnon-metricity Q. Note that curvature and torsion
2depend only on Γ, while Q is built from both g and Γ. The following list of theories tells whether the
characteristic quantity generically is = 0 or is = 0 or has a special form.
For a general metric-affine theory one expects
C=0,S =0,Q =0.
A theory for (g,Γ) is called metrical iff Q = 0. Well-studied field theories such that
C=0,S =0,Q =0
are Einstein-Cartan theory, also called ECSK theory after Einstein [27], Cartan [15–18], Sciama [75],
Kibble [42], and Poincar´e gauge field theory [33–38].
A connection Γ is called symmetric iff S = 0. Eddington [22,23] and Einstein [26] proposed theories
with
C=0,S =0,Q =0.
H. Weyl’s conformal relativity theory [80–82] assumes
C=0,S =0,Q = φ⊗g.
where φ denotes Weyl’s one-form.
In a purely metrical theory, Γ is set equal to the Levi-Civita connection, which generically

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