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