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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
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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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 obeys
C=0,S =0,Q =0.
The Levi-Civita connection to a flat metric has zero characteristics:
C=0,S =0,Q =0.
There are so-called teleparallelism theories such that
C=0,S =0,Q =0.
The step from geometry to physics is done by postulating for (g,Γ), or g alone, or Γ alone, a varia-
tional principle. That means, the field equations shall follow from the requirement that some integral
expression becomes stationary:

1 2 n
δ Ldx dx ···dx =0,
M
where the Lagrange density L depends on g and Γ, or g alone, or Γ alone and δ denotes first variation.
There is a well-known one-to-one correspondence between Lagrange densities L and Lagrange scalar
functions L given by
1
2
L = L|detg| , detg := det(g ). (1.1)
αβ
Letussketchherethemainresultsofourwork.
• For a Lagrangian of the form
−1 αβ ν
L = F(g ,C)=F(g ,C )
αβμ
3
we find that
δL ∂F
= ,
αβ αβ
δg ∂g
δL
λαβ λαβ λαβ α λρβ=2∇ X −Q X +2(2S X −S X ),
λ μ λ μ λ μ λρ μ
μ
δΓ
αβ
∂L
αβμ μ μwhere X = are the components of some (3,1)-tensor and S := S ,Q := Q are
ν ν α αμ α αμ
∂Cαβμ
components of traces of S, Q respectivily.
• If L like above as a function of the curvature C is a polynomial then
δL
λρσ λρ σ λρσ2 =2X C +X C −X C .
(α β)λρσ (α |λρ|β)σ (α |λρσ|β)
αβ
δg
Such a polynomial can be decomposed into d-homogeneous parts (d=0,1,2,···). A d-homogeneous
invariant polynomial reads
α α ···α
1 2 4d
L = θ C C ···C ,
α α α α α α α α α α α α
1 2 3 4 5 6 7 8 4d−3 4d−2 4d−1 4d
α α ···α
1 2 4dwhere θ is a linear combination with constant coefficients of expressions
α α α α α απ(1) π(2) π(3) π(4) π(4d−1) π(4d)
g g ···g ,
π being a permutation of (1,2,···,4d).
• Every quadratic (i.e. 2-homogeneous polynomial) L is a linear combination of 16 expressions: L
1
2equals R , L ,···,L are scalar products (with respect to g) of the Ricci tensors (i.e. traces of C)
2 10

T T
Ric, Ric,Ric or their transposes Ric ,Ric and L ,···,L have the form
11 16
α α α απ(1) π(2) π(3) π(4)
C C ,
α α α α
1 2 3 4
δL δL
π being a permutation of (1,2,3,4). For the quadratic case we present , in fully explicit andαβ μ
δg δΓαβ
manifestly covariant form. We specify the q Lagrangian to the subcases where the connection
Γ is symmetric (i.e. S = 0) or metrical (i.e. Q=0)orboth.
• Further, we study the case where L is a smooth function of the scalar curvature R: L = f(R).
• We derive the Second Noether Theorem with respect to diffeomorphism-invariance in full generality
for metric-affine field theories, as follows. Let a Lagrange function L leading to a diffeomorphism-
−1invariant action integral, be a differential expression in the inverse g of a metric g, an affine connec-
αβ μ ition Γ, and a matter field u, with components g ,Γ ,u respectively. Let the matter field equations
αβ
δL
= 0 be satisfied and denotei
δu
1 1
δ δL
− αβ
2 2
E := |detg| (|detg| L),E := .
μ
αβ
αβ μ
δg δΓ
αβ
Then the identity holds:
α α αβ αβ αβ (αβ) μ αβ2∇ E +(4S −Q )E −(4S −Q )E +∇ ∇ E +(4S −Q )∇ E −2S ∇ E
α ρ α α ρ ρ ρ αβ β α ρ β β α ρ ρβ α μ
1 1
(αβ) αβ μ μ αβ
−2Q S E +[ ∇ (4S −Q )+(4S S + Q Q )]E −[(4S −Q )S +C ]E =0,
α β ρ β α α α β α β ρ α α ρβ ραβ μ
2 4
4ν μwhere C ,S ,Q are the components of the curvature C,thetorsionS, and the nonmetricity
αβμ αβ αβμ
Q of Γ respectively.
ToourknowledgethisgeneralNoetherTheoremisnew. Clearly, inapurelymetricaltheoryitsimplifies
αto the well-known divergence-free condition ∇ E =0.
α ρ
• A Lagrangian in a purely metrical theory of the form
−1 αβ ν −1 αβ˜ ˜
L = F(g ,Riem)=F(g ,R )orL = F(g ,Riem)=F(g ,R )
αβμ αβμν
leads to
1 1
δ 1

λρσ ρ λ
2 2
G := |detg| (|detg| L)=Y R − g L+2∇ ∇ Y ,
αβ α βλρσ αβ λαρβ
αβ
δg 2
where
˜
∂F ∂F
αβμ
Y := = g .
ν νλ
ν
∂R ∂R
αβμ αβμλ
• We analyse quadratic Lagrangians L in purely metrical theories. In particular, we show that L is
a second degree Lovelock Lagrangian up to constant factor iff the tensor Y is divergence-free in the
αβμνsense: ∇ Y = 0 identically in g.
α
• The so-called Palatini procedure is applied to Lagrangians of the form
d

−1 αβ ν α β μ α ···α β μ ν
1 1 1 2 d d d k
L = F(g ,C)=F(g ,C )= c L,L= θ C .
αβμ d d d ν ν α β μ
1 d k k k
d≥0 k=1
That means, g and Γ are varied independently from each other, and after variation the Levi-Civita
αβconnectiontog isinsertedforΓ. WeindicatethisinsertionbyanindexLC. TheexpressionsF ,F
μ
αβ
in metric-affine theories by this procedure turn to
1
λρσ λρσ αβ λαβ
F | = Z R −W R − g L| ,F | =2(∇ X )| ,
αβ LC (α β)λρσ (α |λρσ|β) αβ LC μ LC λ μ LC
2
where
αβμν αβ[μν] αβμν αβ(μν)
Z := X | ,W := X | .
LC LC
Also, we derive some relation between purely metrical theories and the Palatini procedure, namely
1
ρ
G = F | + ∇ [F +F −F ] .
αβ αβ LC LC
ρ(αβ) (α|ρ|β) (αβ)ρ
2
• We suggest a new definition for a Lovelock Lagrangian in metric-affine theories. In particular, we
give a list of examples of second degree Lovelock Lagrangians according to this generalized definition.
• We use the Einstein, Bergmann-Thomson, Landau-Lifshitz and Papapetrou energy-momentum com-
plexes to calculate the energy and momentum distributions of a Kantowski-Sachs-spacetime. We
show that the Einstein and Bergmann-Thomson definitions furnish a consistent result for the energy
distribution, but the definitions of Landau-Lifshitz and Papapetrou do not so.
5

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