On Cosmologies with Torsion Nikolaos Chatzarakis

On Cosmologies with Torsion

Nikolaos Chatzarakis Register Number 13603

B.Sc. Thesis

Supervisor: Christos G. Tsagas

Department of Physics Aristotle University of Thessaloniki Greece Friday 19 February 2016 On the Cosmology of the Einstein-Cartan Universe

Nikolaos Chatzarakis

Abstract In this thesis, we will discuss the construction of a cosmological theory in a Riemann-Cartan space-time. This theory presents similarities to the relativistic theory of cosmology, and thus we will present the two theories at the same time, comparing the results. We begin the analysis by introducing the reader to the geometry of a Riemann-Cartan space-time, where torsion is non-zero and appears as the antisymmetric part of the affine connection; this causes several interesting changes to the classical geometric features of the Riemann geometry, most notably in the curves of zero “acceleration” and the form of the curvature tensor and its components. From this geometrical background, we are introduced to the field equations of the Einstein-Cartan theory and we focus on the differences of the “matter-to-geometry” coupling between this theory and the General Relativity; one of the most noticeable results is the immediate coupling of spin to torsion. We also take a look into the conservation laws in this theory of gravity, since they appear to take different forms from their usual relativistic ones. Then, we continue with the kinematics of this theory, presenting the most interesting changes that torsion causes in the kinematic equations (e.g. the Rauchaudhuri equation). Finally, we construct a homogeneous and isotropic cosmological model that bears all the symmetries of the relativistic Friedmann model; this model has the Robertson-Walker metric, but also a scalar fields related to torsion.

1 2 Contents

1 Preface 5

2 Introduction to the Einstein-Cartan Theory 11 2.1 The Structure of a Riemann-Cartan Space-time ...... 11 2.1.1 The Metric Tensor ...... 11 2.1.2 The Affine Connection ...... 12 2.1.3 The Torsion and Contorsion Tensors ...... 13 2.2 The Effects of Torsion ...... 15 2.2.1 The Parallel Transport and the Unclosed Parallelogram ...... 15 2.2.2 The Co-moving Transport and the Ricci Identity ...... 16 2.2.3 Autoparallel and Geodesic Equations ...... 18 2.3 Curvature with Torsion ...... 19 2.3.1 The Curvature Tensor ...... 19 2.3.2 The Ricci Curvature ...... 21 2.3.3 The Weyl Curvature ...... 22 2.3.4 The Weitzenböck Identities ...... 24

3 The Field Equations of the Einstein-Cartan Theory 27 3.1 The Einstein-Hilbert Action ...... 27 3.2 The Field Equations ...... 28 3.2.1 The Variation in respect to the Metric ...... 29 3.2.2 The Variation in respect to the Contorsion ...... 29 3.2.3 The Algebraic Form of the Field Equations ...... 30 3.2.4 The Curvature Tensor as a Field Equation ...... 31 3.2.5 The Eﬀective Field Equations ...... 33 3.3 Matter Fields ...... 34 3.3.1 The Matter Lagrangian ...... 34 3.3.2 The Energy-Momentum density tensor ...... 37 3.3.3 The Spin density tensor ...... 40 3.4 Conservation Laws ...... 42 3.4.1 The Conservation of Energy and Momentum ...... 42 3.4.2 The Conservation of Spin ...... 45 3.4.3 The Conservation of Angular Momentum ...... 46 3.5 The Equations of Motion ...... 47 3.5.1 The Mathisson-Papapetrou-Dixon Equations ...... 48 3.5.2 The Dixon-Souriau Equations ...... 50 3.5.3 Supplementary Conditions ...... 51

4 The Kinematics and the Dynamics of the Einstein-Cartan Theory 52 4.1 The Kinematic Parameters ...... 52 4.1.1 The Raychaudhuri Equation ...... 53 4.1.2 The Shear Propagation Equation ...... 55 4.1.3 The Vorticity Propagation Equation ...... 58

3 4.1.4 The Constraints ...... 59 4.2 Electrodynamics and Magnetohydrodynamics ...... 61 4.2.1 Maxwell Equations and Conservation Laws ...... 61 4.2.2 Propagation of Electromagnetic Waves ...... 67 4.2.3 Kinematics of a Charged Fluid ...... 70 4.2.4 Kinematics of a Magnetized Fluid ...... 71 4.3 The “Electric” and “Magnetic” Parts of the Weyl Tensor ...... 72 4.3.1 The Propagation Equations ...... 73 4.3.2 The Constraints ...... 76 4.4 Comparison to Other Analyses ...... 78 4.4.1 The Relativistic Approach of Spinning Fluid Kinematics ...... 79 4.4.2 The Einstein-Cartan Approach of Spinning Fluid Kinematics ...... 80

5 Constructing a Cosmological Model in the Einstein-Cartan Theory 83 5.1 The Spatial Curvature ...... 83 5.2 The Friedmann-Lemaˆıtre-Robertson-Walker Model ...... 85 5.2.1 The Metric, the Torsion and the Matter Fields ...... 86 5.2.2 The Evolution of the Universe ...... 87 5.2.3 The Friedmann Equations ...... 88 5.3 Solutions to the Friedmann Model ...... 90 5.3.1 The Dust Solution ...... 92 5.3.2 The Radiation Solution ...... 93 5.3.3 The Stiﬀ Matter Solution ...... 94 5.3.4 The Inﬂation Solution ...... 94 5.3.5 The Curvature Solution ...... 95 5.3.6 The Vacuum Solution ...... 96 5.3.7 The Static Solution ...... 97 5.4 Comparison to Other Analyses ...... 97 5.4.1 The Relativistic Approach of Spinning Fluid Cosmologies ...... 97 5.4.2 The Einstein-Cartan Approach of Spinning Fluid Cosmologies ...... 98

6 Conclusions 103 6.1 The Main Results ...... 103 6.2 Cosmological Implications ...... 104

4 Chapter 1

Preface

In 1915, Albert Einstein proposed a theory that could, on certain circumstances, summarize all known physical laws -with the exception of the laws on the microscopic scales that were being constructed at the very time by the first quantum theories. This theory tried to perceive every physical phenomenon as taking place in a four-dimensional space-time, whose geometry was not the typical euclidean one. On the contrary this space-time was curved and its curvature was relative to the physical phenomena that took place. Even deeper, the curvature of the space-time was expressing these physical phenomena, while the physical phenomena were depending on the curvature; this space-time could not exist without any physical phenomena taking place on it -without matter, energy and momentum existing on it- and it was a form of existence of these physical phenomena -of matter, energy and momentum included. This was the General Theory of Relativity. One could say that the General Theory of Relativity was an extension of another theory Einstein himself had proposed ten years before: the Special Theory of Relativity. The latter was a theory for understanding the -microscopic- physical phenomena that developed high velocities -very close to the one of light. In this theory, the four-dimensional space-time is flat, pseudo-euclidean; this did not allow any strong phenomena to occur, neither could include macroscopic phenomena, such as gravity. Gravity until that time was perceived as a force acting between two bodies that have mass -the property of matter related to gravitational interactions. Of course, this idea that was originally posted by Isaac Newton was highly criticised, since the very concept of force was not very stable -it did not seem natural at all. Many physicists and mathematicians have tried to remove this “curse” from the physical theories, but until then this was not made possible. However, the opinion that the light speed is the upper limit for any velocity, and as a result of any transfer, blew a new breath to this old controversy. It seemed impossible that the information of the gravitational field would be transferred immediately throughout any distance via the mechanism of the gravitational force; the electromagnetic field was not acting like this at all. It is remarkable that, a few years after Einstein published his Special Theory, the French mathematician Henri Poincarèstated that the space-time of this theory -known as Minkowski space-time- should be curved in the presence of mass and that the information of the existence of this mass should travel across the space-time with the speed of light, just like a wave. Einstein -with some help from K. Karatheodori and D. Hilbert- managed to find this connection of physics and geometry Poincarèwas trying to point out, by using the Riemannian geometry. The latter is an approach of the geometry of curved spaces; it allows us to study the curvature of these space-times and perform any kind of “measures” on it. Einstein, with this theory, united the physical phenomena to the curved space-time, which he would use in order to describe the gravitational field. The idea is simple: the space-time exists only when physical phenomena occur and is curved proportionally to these phenomena. In simpler words, the presence of mass, energy or momentum not only “creates” space-time on which the previous will “live”, but it also curves this space-time according to the density of them. As a result, the physical systems -eg. a material body or a physical process- do not “feel” any gravitational force; they “live” in a curved space-time whose

5 curvature they understand as gravity. The geometry that Einstein used in order to describe these -the Riemann geometry- is based on one simple fact: the connections of the space-time are non-zero and symmetric. As a result, they can describe the curvature immediately. The metric tensor of the space-time is turned into the main variable for the description of the space-time and, at the same time, the main variable for any physical theory in this space-time. On the other hand, no other geometrical variable appears as necessary for this description; the majority of them are considered to be zero. One of those variables is the torsion of the space-time. The torsion is related to the antisymmetric part of the connection, which we have demanded to be zero, so that we would exclude its effects. If we choose, however, to allow the torsion to be non-zero, we would pass to the Riemann-Cartan geometry, a similar approach of curved spaces that are torsional. In this case, we could also construct a similar theory that would include torsion as a main variable too. This theory was originally proposed by Elie Cartan himself in 1922 so that the angular momentum could be described, although Arthur Eddington had spoken about such a case a little before. In 1940’s the theory was restudied, this time in order to describe the spin of the matter fields -something the General Relativity is not capable of. Dennis W. Sciama and Tom W.B. Kibble were eventually the ones that constructed this theory as a gauge approach to gravitation, in 1964, and Friedrich W. Hehl reconstructed it as a U4-theory. This was the birth of the Einstein-Cartan theory of cosmology. The key concept of this theory is that we wish to treat the spin of the matter fields as another source -after energy and momentum- of curvature, and respectively of gravitation. Spin should also be coupled to a certain geometrical parameter and the torsion of space-time seemed the most appropriate. The curvature and especially the metric tensor of the space-time seemed proper in order to be coupled with matter, energy and momentum; in the same manner, the torsion seemed reasonable as a choice to be coupled with spin. The question that might arise is whether describing the spin of the matter fields is so important, when the General Relativity is recognised as a classical theory of macroscopic perspective; in a such a theory, the spin -a completely microscopic and quantum size- should be of no real importance. The objection is formed of two different arguments: The first is that we do not know whether the spin of the average matter of the universe, although not important when each particle is studied isolated, might interact with the geometry of the space-time, thus effecting the total geometry of the universe in a way we might have detected but not being able to explain. The second is that the ability to express the spin and its effects in a curved space-time -no matter how small they are- in a covariant formulation, might be a step towards the unification of the macroscopic relativistic theory of the gravitational field with the microscopic quantum theory of the electromagnetic field, the weak nuclear interaction field and the strong nuclear interaction field -the other three “force” fields in Physics. As for the experimental and observational results concerning torsion, one must agree we are very poor. The presence of spin interacting with the space-time has not been detected yet, while even the most optimistic theoretical works agree that the interaction is so small that we need at least the tenth radius of the today-observed universe in order to be certain. Propositions that torsion could also be coupled to any rotation of a material field, have led several people -like Adamowicz and Trautman in 1975, Rumpf in 1979, Kleinert in 2000 and Mao in 2007- to believe that we could have an observational result -via the gyroscope of the Gravity Probe B Experiment for example- that would justify the work done in this theory. Yet, until today, no data has proved any of their arguments.

However, the case we will focus in this thesis is quite different. We will remain on the macroscopic level and try to present the foundations and the main equations for a cosmology in the Einstein- Cartan theory; that is for a cosmological theory that will include the spin of the matter fields and the torsion of the space-time. This thesis will survey both the geometry and the physics of a universe described by the Einstein-Cartan theory and will try to present the main differences between such a universe and a universe described by the General Theory of Relativity. Throughout the conduction of this thesis, I found myself to the constant aid of my supervising professor, C.G. Tsagas, whom I would like to thank -officially this once, through this comment. I

6 would also like to thank Proﬀ. D. Papadopoulos for the interesting conversations in the beginning of my search and Damianos Iosiﬁdis for the warm-hearted cooperation near the end of it. Furthermore, I would like to thank my family and friends for their ever-lasting help and support, even to the worse of the days. Finally, I wish to dedicate this work to my late grandmother, Anna, who tried the most and hardest to complete her education, while never having the chance to see her grandson completing his.

7 Preliminaries Before we move further into the the presentation of the Einstein-Cartan theory, I would like to point out several important features of this thesis -especially, of the symbols used. Tensors are denoted with upper and lower indices as

efg... Tabc... (1.1) while scalars have no indices. The metric tensor of any space-time will be denoted as gab, while ηab will be the metric tensor of the ﬂat Minkowski space-time. The Kronecker tensor will be δab and will follow the deﬁnition a δaa = 1 δab = 0 δ a = n (1.2) where a 6= b and n is the number of the space-time dimensions. Notice that the Einstein Sum Notation is used, since 3 a X a 0 1 2 3 δ a = δ a = δ 0 + δ 1 + δ 2 + δ 3 = 4 a=0

The totally antisymmetric Levi-Civita symbol of the three-dimensional space will be abc and will obey the following deﬁnitions

123 = 231 = 312 = 1 132 = 213 = 321 = −1 112 = 121 = 211 = ... = 0 (1.3)

Its counterpart of the four-dimensional space-time is abcd and follows similar identities. The relation between them is d abcd = 2u[ab]cd − 2ab[cud] abc = abcdtensoru (1.4) c a where abcu = 0 -usually, the vector u denotes the four-velocity. Note that both Levi-Civita symbols are traceless; they also follow these two simple multiplication laws

klmn k l m n klm k l m abcd = −4!δ[a δb δc δd] abc = 3!h[a hb hc] (1.5) where hab = gab + uaub -usually denoting the projection tensor. Throughout the text of this thesis, both Greek and Latin letters will be used as indices. The Greek ones will refer to the n-dimensional space-time (with n=4, in our case) and will take numbers from 0 to 3; with 0 we will denote time and with 1,2,3 the space dimensions. The Latin letters will refer to other m-dimensional spaces and will take numbers from 1 to m. Resulting from this, our metric tensor (for four dimensions) will have a [−, +, +, +] signature, with − referring to time and + to space; this means that the Minkowski metric tensor will be

ηαβ = diag[−1, 1, 1, 1] (1.6) But, how can one define a tensor field? The tensor fields have one major thing in common; when translated from one coordinate system to another they maintain their properties. Their form might change, but all the information they have and all the identities they fulfil are conserved throughout the change. In order to ensure this, we have come up to the following law of transformation from 0 the coordinate system xa to the coordinate system x a.

0a 0b 0c l m n 0 ∂x ∂x ∂x ∂x ∂x ∂x T efg... = ... T lmn... (1.7) abc... ∂xi ∂xj ∂xk ∂x0e ∂x0f ∂x0g ijk... and we may also have the it the opposite way -from the coordinate system x,a to the coordinate system xa. If and only if a quantity fulfils this transformation formula, can be considered a tensor. Scalar and vector fields fulfil it and are considered tensor fields of zeroth and of first order respectively. As for differentiating, what is valid in flat Euclidean space-times is not for the curved space-times; in the latter we have the covariant derivative instead of the common partial derivative, that ensures that the derivative of any tensor field will be a tensor field itself -according to the previous statement. The covariant derivative imports the effects of the space-time curvature in the differentiating of any

8 field “living” on that space-time. To give some examples: a) When taking the derivative of a scalar field in respect to the coordinates, we must note that the partial derivative identifies with the covariant one.

∇aΦ = ∂aΦ (1.8) b) When taking the derivative of a vector ﬁeld in respect to the coordinates, we must notice that a new term appears, which is related with the curvature of the space-time.

c a a a c ∇bua = ∂bua − Γ abua ∇bu = ∂bu + Γ cbu (1.9)

a where Γ bc is the connection of the space-time. c) When taking the derivative of a tensor ﬁeld of second order, two new terms are added in the same manner as before.

d d ab ab a db d ad ∇cTab = ∂cTab−Γ Tdb − Γ Tad ∇cT = ∂cT + Γ T + Γ T ac bc dc dc (1.10) b b d b b d ∇cTa = ∂cTa − Γ acTd + Γ dcTa

For any other tensor field of higher order, we add as many new terms -similar to the ones presented before- as the order of this field. Notice that the last index of the connection denotes the differentiation; this will be proved important in defining several things later. Furthermore, any tensor can decompose in symmetric and antisymmetric parts. Symmetric tensors we call those tensors whose components are equal to one another if a certain transposition is held, usually denoted by the shifting of their indices; as a result, a symmetric tensor has the following property: Tabcd... = Tbacd... (1.11) On the contrary, the antisymmetric tensors have components that are exactly opposite if a certain symmetry is applied on them; as a result, an antisymmetric tensor has the following property:

Tabcd... = −Tabdc... (1.12)

Any tensor can decompose in a totally symmetric and a totally antisymmetric part. The symmetric part is depicted as T(abcd...) and is deﬁned as 1 T = (T + T + T + T + ...) (1.13) (abcd...) n! abcd... bacd... bcad... bcda...

The antisymmetric part, on the other hand, is depicted as T[abcd...] and is defined as 1 T = (T − T + T − T + ...) (1.14) [abcd...] n! abcd... bacd... bcad... bcda... Note that the symbol “|”, when used, denotes that the indices between two of it are not affected by any symmetry. And finally, in this work tensor of two different space-times will be used: some will be in respect to the Riemannian space-time -and to the General Theory of Relativity- and some will be in respect to the Riemann-Cartan space-time -and to the Einstein-Cartan Theory. The first of those will be denoted with a circumflex over their symbol. For example, the tensor Rãb is the Ricci tensor in a Riemann space-time, while the tensor Rab is the Ricci tensor in a Riemann-Cartan space-time.

The 3+1 Formalism Let us make another small comment here, since we will meet the following throughout the thesis. The formalism that is used here -known as 3+1 formalism- is based on a certain fact: the “breaking” of the four-dimensional space-time into a three-dimensional space and time. This, as will be understood later on, allows us to study our equations in two diﬀerent levels that are “separated” by nature itself;

9 since time and space do not function in the same way, we can study our variables separately, bringing them also closer to our every-day thought. dxα In order to put this formalism in action, we need the velocity vector, uα = , where τ is dτ the proper time, and the projection tensor, hαβ = gαβ + uαuβ. Using these two quantities, we will have the ability to project our space-time equations on time and space respectively. To give a more speciﬁc example, the covariant derivative of a tensor Xαβ can split into the temporal derivative

˙ µ Xαβ = u ∇µXαβ and the spatial derivative κ λ µ Dγ Xαβ = hα hβ hγ ∇µXκλ In this way, all the following equations will be either temporal or spatial; the ﬁrst of them are recognised as propagation equations, since they are depicting the time evolution of a certain tensor, while the second are called constraints, since they are depicting the space limits of this tensor. The two equations are usually going side by side, since none of them can sustain all initial information for this tensor alone. We should also mention that the temporal and spatial derivatives of the Riemannian quantities will be denoted diﬀerently, so that the discrimination between the equations will be easier. Specifying ˜ for the covariant derivative of a tensor Xαβ , we will have the following temporal derivative

˜ 0 µ ˜ ˜ X αβ = u ∇µXαβ and the following spatial derivative

˜ ˜ κ λ µ ˜ ˜ Dγ Xαβ = hα hβ hγ ∇µXκλ

Finally, it is interesting to show the relations between the Riemann-Cartan and the Riemann derivatives of the projection tensor hαβ; it is also interesting to present the temporal and spatial derivatives of the three-dimensional Levi-Civita symbol. For the ﬁrst case, it is easy to show that the temporal derivatives of the projection tensor are related as follows

˙ 0 µ ν hαβ = h αβ − 4u u S(α|µν u|β) while its spatial derivatives are both equal to zero, so that the metricity of the space-time is preserved. ˜ Dµhαβ = Dµhαβ = 0

As for the Levi-Civita symbol, we will have the following temporal derivative

µ ˙αβγ = 3u[αβγ]µu˙

10 Chapter 2

Introduction to the Einstein-Cartan Theory

2.1 The Structure of a Riemann-Cartan Space-time

As we have explained, the Einstein-Cartan theory is constructed similarly to the General Relativity, by connecting gravity -and physics in general- with the geometry of the space-time. The important difference of two theories is the structure of this space-time. In General Relativity, a Riemann space-time is used; that is a space-time which presents non-zero affine connection and is curved, as a result. In the Einstein-Cartan theory, a Riemann-Cartan space-time is used; that is a space-time that is curved, but also torsional. The Riemann-Cartan space-times could be considered an extension of Riemann space-times, proposing we can include torsion in them. To do so, we will suppose that the affine connection is non-symmetric, with its symmetric part representing the curvature and its antisymmetric presenting the torsion. Beyond that, the analysis of the two space-times is similar, but with a few differences. In the first chapter, we will introduce ourselves to the geometry of an Einstein-Cartan universe, by studying the main characteristics of this space-time.

2.1.1 The Metric Tensor The primary description of an n-dimensional space-time is made possible only by the deﬁnition of the metric tensor gαβ. The metric tensor is a symmetric tensor of second order that is deﬁned from the linear element of the space-time, if we demand that it is a scalar and, as a result, it is independent of the coordinate system

2 α β ds = gαβ dx dx (2.1) where dxα and dxβ are coordinate vectors [1; 2]. It is easy to see that this expression is taking the place of what is known in Euclidean geometry as the Pythagorean Theorem. Consequently, it is this line element that we will use in order to measure distances in this space-time. Furthermore, if we write this element as an indefinite differential form ∂s ds = dxβ ∂xα where s = s(xα) is the proper distance of the space-time -in other words, the affine parameter we use in order to parametrise curves-, we can express the metric tensor as a tensor product of the ∂s unitary tangent vectors ∂xα ∂s ∂s g = ε ε = αβ α β ∂xα ∂xβ

11 M N We can also assume transformation matrices, eα and eβ , that will allow us to derive the metric tensor of any space-time from the Minkowski space-time metric tensor [1].

M N gαβ = eα eβ ηMN (2.2) where M and N are indexes that denote the tetrad identity of the transformation matrices. The metric tensor fulﬁls the following condition

γβ β gαγ g = δα (2.3)

β where δα is the Kronecker tensor. It also allows us to raise and lower the indices of a vector, or in other words to transform the covariant components of vector into contravariants and the opposite. This is also possible with the indices of a tensor

α αβ β u = g uβ uα = gαβ u αβ ακ βµ κλ T = g g Tκλ Tαβ = gακgβµT αβγ ακ βλ γµ κλµ Z = g g g Zκλµ Zαβγ = gακgβλgγµZ Thus, we can also use the metric tensor in order to deﬁne the inner product of two vectors or tensors. In metrical manifolds, the metric tensor must always be covariantly constant, since the space-time is metric; this means that its covariant derivative must always be zero.

ν ν ∇µgαβ = ∂µgαβ − gαν Γ βγ − gνβΓ αµ = 0 (2.4)

α where Γ βγ is the aﬃne connection of the space-time [1].

2.1.2 The Aﬃne Connection α Beginning from equation (2.4), we can discuss the properties of the aﬃne connection Γ βγ . De- manding that this property holds at any space-time that can be described so, no matter what its connection is like, we conclude that the connection of such a space-time must have the form

α α α µα ν µα ν Γ βγ = Γ (βγ) + Γ [βγ] − g gβν Γ [µγ] + g gγν Γ [βµ] (2.5)

α α where Γ (βγ) is the symmetric part and Γ [βγ] the antisymmetric. The symmetric part can be identified as the connection of a Riemann space-time, known as Christoffel Symbol 1 Γα = Γ˜α = gαµ(∂ g + ∂ g − ∂ g ) (2.6) (βγ) βγ 2 γ µβ β µγ µ βγ and it is associated with the curvature of the space-time. The Christoffel Symbol can also be defined by using the unitary coordinate vectors in the following way

∂ε ∂2s Γ˜α ε = γ = βγ α ∂xβ ∂xβ∂xγ That definition implies that the Schwarz-Young theorem holds for all space-times that have a symmetric affine connection [1]. Usually, the Christoffel Symbol is called Levi-Civita connection, since it serves as the only part of the affine connection in a torsion-free space-time, like the Riemann manifolds [3]. The antisymmetric part, on the other side, can be identified as the Torsion tensor [4; 5; 6; 7].

α α Γ [βγ] = Sβγ (2.7) This tensor is associated -as it is declared by its name- with the torsion of the space-time. Using those deﬁnitions, the connection ﬁnally takes the form

α ˜α α Γ βγ = Γ βγ + Cβγ (2.8)

12 α α α α where Cβγ = Sβγ − S γβ + S βγ is the Contorsion tensor. Here, we must notice something rather odd: while the antisymmetric part of the connection is a tensor, the Christoﬀel Symbol does not fulﬁl the tensor criterion ans as a result is not a tensor.

λ µ 0α 0α λ µ 0α 2 µ 0 ∂x ∂x ∂x 0 ∂x ∂x ∂x ∂x ∂ x S α = S κ Γ˜ α = Γ˜κ + βγ ∂x0β ∂x0γ ∂xκ λµ βγ ∂xκ ∂x0β ∂x0γ λµ ∂xµ ∂x0β∂x0γ

This means that neither the aﬃne connection is a tensor.

The Spinor Connection Since later on in the thesis, we are likely to meet the Dirac formulation of spin, it would be nice to see the Affine Connection in terms of spinors. Spinors are elements of a complex vector space that are distinguished from vectors and tensors through their transformation laws -that are also used as criteria. In order to derive the Spinor Connection from the Affine Connection, we need the transformation matrices we have used before in order to derive the general metric tensor form the Minkowski one. The reason is simple: spinors are defined in a co-frame of our space-time that can be approached locally as a Minkowski space-time, and as a result we use the transformation matrices to “jump” from this co-frame to the space-time we will work. The relation is

α α M M N ω βγ = e M (∂γ e β + Γ γN e β) (2.9) As with the Aﬃne Connection, the Spinor Connection is not a tensor.

2.1.3 The Torsion and Contorsion Tensors Having deﬁned the torsion and contorsion tensors, we should refer to the properties and symmetries of them, since those tensors are essential for the understanding of the structure of Riemann-Cartan space-time. Firstly, we will treat the torsion tensor. Being the antisymmetric part of the connection, it follows one simple symmetry: α α Sβγ = −Sγβ (2.10) It is interesting that the torsion tensor is not fully antisymmetric, something that is going to be proved very useful in the next chapters. Therefore, we can move into the contractions of the tensor which are βα αβ β β Sβ = Sβ = 0 Sαβ = −Sβα = Sα (2.11) with Sα being the torsion vector - which, in turn, is nothing more than the only non-zero trace of the torsion tensor. Usually, this vector is replacing the torsion tensor so that the equations will be simpliﬁed. Another way to simplify the torsion tensor is to decompose it as 2 Sα = Sˆα − δα S (2.12) βγ βγ 3 [β γ]

â where S bc is the trace-free component of the torsion tensor - also known as the Modified Torsion tensor [5; 6]. We can also define the torsion pseudo-vector that exists due to the symmetries of the torsion tensor. If we multiply the latter one with the Levi-Civita symbol, we have 1 Sˇ = βµν S (2.13) α 2 α [βµν] This pseudo-vector cannot replace torsion in order to simplify the equations -as the torsion vector can-, but possesses all information that the torsion tensor itself can provide us with. An interesting remark we can make here is the relation of torsion tensor to the translation group. For F.W. Hehl and Y.N. Obukhov (in 2007), the torsion is the translation gauge field strength,

13 associated to the translational displacements and distortions. In other words torsion can function as translational curvature [7].

It is easy to conclude, from its given symmetries, that the torsion tensor has twenty four non-zero independent components. According to S. Capozziello, G. Lambiase and C. Stornaiolo (in 2007), we can split the torsion tensor in three irreducible parts -the V-torsion, the A-torsion and T-torsion- each of whom contains different information and contributes differently to the theory of gravity we will discuss later. The first part is the called the V-torsion tensor, it has four non-zero independent components and is formed as 2 V S α = S δ α (2.14) βγ 3 [β γ] This is the part of the torsion associated to its only trace: the torsion vector. The second part is called the A-torsion tensor, it has four non-zero and independent components too, and is given as A α γδ Sβγ = g S[αβγ] (2.15) but it can also be expressed in terms of the torsion pseudo-vector A ˇδ Sαβγ = αβγδS (2.16) Obviously, this is the part of the torsion associated with the torsion pseudo-vector. Finally, the third part, known as the T-torsion, contains the remaining sixteen components. T α α V α A α ˆ α γδ Sβγ = Sβγ − Sβγ − Sβγ = Sβγ − g S[αβγ] (2.17) B a Usually, the T-torsion is expressed in terms of an arbitrary matrix CA and a vector V ; this is given by the following expression T α κλ A B γ Sβγ = αβV[κ eλ] CA e B (2.18) A γ where eλ and e B are the orthonormal vectors defining a tetrad field -the second index, depicted with a capital letter, denotes whether the vector V a is a time-like, a space-like or a null one. This decomposition of the torsion tensor, along with the geometric classification of its three parts, allow us to demonstrate the different ways torsion contributes in the space-time, is comparison to a Riemann space-time. S. Capozziello, G. Lambiase and C. Stornaiolo are also discussing the contributions of torsion to the Einstein-Cartan theory of gravity through this classification [4].

Concerning the contorsion tensor, which is the non-Riemannian part of the connection, it has several symmetries, the ﬁrst of which is derived right from its deﬁnition

C(α|β|γ) = 0 (2.19) since the “β” and “γ” indices cannot be shifted. Using this, as well as the deﬁnition of the contorsion tensor and the symmetries of the torsion tensor that have been deﬁned before, we derive the other symmetries which follow

Cα(βγ) = 2Sα(βγ) Cα[βγ] = S[βγ]α C(αβ)γ = 2Sγ(αβ) C[αβ]γ = S[αβ]γ (2.20) Again, we see that the contorsion tensor is not fully antisymmetric -something reasonable, considering the symmetries of the torsion tensor; this will also prove important later. Finally, we can easily calculate the contractions of the contorsion tensor, using the contractions of the torsion tensor and the following form of its deﬁnition α α α α Cβγ = Sβγ + S βγ + S γβ (2.21) Thus, the contractions of the contorsion tensor are αβ αβ β Cα = 0 Cβ = −2Sα Cαβ = 2Sα (2.22) The contorsion is also associated with the translation group, although its role is more important in the connection; it is also related to the translational contortions -as its very name denotes [5; 6; 7].

14 2.2 The Eﬀects of Torsion

We have gathered the main parameters of a Riemann-Cartan space-time: the metric tensor, the Christoffel symbol, the torsion tensor and the contorsion tensor. We Have also figured out which one of them is associated with the curvature of the space-time -the Christoffel symbol- and which ones are associated with its torsion -the torsion and the contorsion tensors. It is easy to see that demanding a zero torsion tensor, transforms a Riemann-Cartan space-time into a Riemann one. This means that the first and main difference between the two spaces is -as we have already said- the presence of torsion. This presence of torsion alters several of the widely known geometrical features of a curved space-time; most notably, the parallel and the co-moving transport of vectors and the two difference between the autoparallel and the geodesic curves.

2.2.1 The Parallel Transport and the Unclosed Parallelogram Let us assume that we have a curve, C, and its parametric equation, xα = xα(λ), where λ is the parameter. A corresponding tangent vector of this curve will have the form dxα uα = dλ In a Euclidean space-time, this vector can be transported parallel to itself in every point of the space-time without changing; this is a result of the zero curvature of the space-time. As curvature appears, we must take it in consideration when parallel transporting vectors. The way the curvature eﬀects on the parallel transport of a vector is simply with its “carrier”: the Christoﬀel symbol. So, if we want to transport a vector parallel to itself in a Riemann space-time -that has only curvature and no torsion- we must allow it change through the transport on the following way. 0α α ˜α β γ u = u − Γ βγ u u

Now, we can also assume that we have a second curve, C˘, its parametric equation,x ˘α =x ˘α(λ), dx˘α and a corresponding tangent vector,u ˘α = . If we wish to transport the first vector, uα parallel dλ to the second,u ˘α, and the second parallel to the first, we must also have on mind that the space-time is curved; the parallel transport will now take the form 0α α ˜α β γ 0α α ˜α β γ u = u − Γ βγ u u˘ u˘ =u ˘ − Γ βγ u˘ u respectively. What we must notice here is that the second term on the right part of both relations is similar: this happens because of the symmetry of the Christoffel symbol. As a result, if we try to create a parallelogram with those four vectors -the two new ones, that occurred from the parallel transport, and the two old ones, that we used as generators- this parallelogram will be exactly as in a Euclidean space-time. The reason is simple: is we consider the sums V α = uα +u ˘0α and W α =u ˘α + u0α, that have the form α α α ˜α β γ a α α ˜α β γ V = u +u ˘ − Γ βγ u˘ u W = u +u ˘ − Γ βγ u u˘ , we can clearly see that these two vectors are identified with each other, since those second terms vanish because of their symmetry. These two sums are nothing more but a different approach of the diagonal of the parallelogram we have created.

The image in a Riemann-Cartan space-time is quite different; the presence of torsion along with curvature makes thins a little odder. If we assume the same curve, C, with parametric expression dxα xα = xα(λ) and corresponding tangent vector uα = in a Riemann-Cartan space-time and try dλ to transport it parallel to itself, we notice that torsion also appears effecting this process, since the affine connection is defined as in equation (2.7). The result will be 0α α α β γ α ˜α β γ α β γ u = u − Γ βγ u u = u − Γ βγ u u − 2S (βγ) u u (2.23)

15 since Cα(βγ) = 2Sα(βγ) (from the equation (2.20)). Assuming a second curve, C˘, with parametric expressionx ˘α =x ˘α(λ) and corresponding tangent dx˘α vectoru ˘α = , we can also derive the parallel transport of the one vector in respect to the other dλ -like before. Again, we see that torsion eﬀects this.

0α α α β γ α ˜α β γ α β γ u = u − Γ βγ u u˘ = u − Γ βγ u u˘ − 2S (βγ) u u˘ (2.24) 0α α α β γ α ˜α β γ α β γ u˘ =u ˘ − Γ βγ u˘ u =u ˘ − Γ βγ u˘ u − 2S (βγ) u˘ u

If we try now two create a parallelogram, we will see that the two sums we have identiﬁed as diﬀerent form of the diagonal before, are not the same any longer.

α α α α β γ α α ˜α β γ α β γ V = u +u ˘ − Γ βγ u˘ u = u +u ˘ − Γ βγ u˘ u − 2S (βγ) u˘ u (2.25) α α α α β γ α α ˜α β γ α β γ W = u +u ˘ − Γ βγ u u˘ = u +u ˘ − Γ βγ u u˘ − 2S (βγ) u u˘

Subtracting the above relations, we have

α α α β γ β γ V − W = 2S βγ (u u˘ +u ˘ u ) (2.26)

This simply means that in a Riemann-Cartan space-time, a parallelogram cannot close; torsion is acting like a force that is pulling the parallel transported vectors apart from each other, not allowing the parallelogram to be formed. This is clear from the existence of two diﬀerent “diagonals”, the V α and the W α that are not identical as we would expect.

This result -of the unclosed parallelogram- can be generalised for every closed curve in a Riemann- Cartan space-time. While, in Riemann space-times, the curves are allowed to close, no matter how much has the curvature altered the corresponding tangent vectors, in Riemann-Cartan space-times, torsion is a catalyst that does not allow this to happen. We must also notice that the presence of torsion alone -without any curvature- is suﬃcient for the unclosed parallelogram and any other similar curve [7].

2.2.2 The Co-moving Transport and the Ricci Identity We have discussed the parallel transport of a vector, both in respect of itself and in respect to another vector; we can now move on to the co-moving transport of a field -disregarding whether it is a scalar field, a vector field or a tensor field. The co-moving transport, unlike the parallel one, occurs when a specific field is transported along a curve. We will begin by transporting a scalar field -let us call it Φ = Φ(xµ)- along a curve. Its covariant derivative, as we have introduced it, is ∇αΦ = ∂αΦ (2.27) without any extra terms related to curvature, since it is a scalar field. However, the second covariant derivative, which transports the field along a new curve, is µ ˜µ µ ∇β∇αΦ = ∂β∇αΦ − Γ αβ∂µΦ = ∂β∇αΦ − Γ αβ∂µΦ − Cαβ ∂µΦ (2.28) Now, we can try to apply this to a close path created by four curves, two of whom are parallel to the other two respectively. What we will take as a result is the following relation.

µ ∇α∇βΦ − ∇β∇αΦ = 2Sαβ ∇µΦ (2.29) This relation shows the transport of a scalar field along a closed path; this transport obviously does not leave the scalar field unchanged. If the space-time on which this transport takes place, was a Euclidean flat space-time, then the right hand of this equation would be zero, which would mean that the scalar field would not change at all after this transport along a close path. In a

16 Riemannian space-time, the right hand would also be zero -since the torsion is zero in such a space- time-, which denotes the very same thing. However, in a Riemann-Cartan space-time, the scalar field that “travels” along a closed path suffers a “penalty”: by the end of this journey has changed due to the torsion of the space-time. This relation is very important for a space-time and it is known as Ricci Identity for a scalar field. It can tell us whether the commutation of second-order covariant derivatives is allowed in the space-time that we are. It is obvious, that any torsional space-time does not respect the Schwarz- Young theorem and does not allow us to shift turns in the second-order derivatives [2; 8; 9].

Let us move now to the vector fields; in order to transport a vector field -let us say uα = uα(xµ)- along a curve, we are using again the covariant derivative α α α µ α ˜α µ α µ ∇βu = ∂βu + Γ βµu = ∂βu + Γ βµu + Cβµ u (2.30) and to transport it again along another curve, we will take the second covariant derivative. µ µ µ ν ν µ ν µ µ ν κ µ ν µ κ ν ∇α∇βu = ∂α∂βu + Γ νβ∂αu + u ∂αΓ νβ − Γ βα∂ν u + Γ να∂βu + Γ βαΓ νκu − Γ καΓ νβu (2.31) The same picture holds for the contravariant components of the field. In the same manner as before, we try to transport the vector field along a closed path. What we will take as a result is the following relation µ µ ν µ ν µ ν κ µ ν κ µ ν µ ∇β∇αu − ∇α∇βu = u ∂βΓ να − u ∂αΓ νβ + u Γ ναΓ κβ − u Γ νβΓ κα − 2Sαβ ∇ν u (2.32) or, for the contravariant components of the vector field, ν ν κ µ ν κ µ ν ∇β∇αuµ − ∇α∇βuµ = −uν ∂βΓ µα + uν ∂αΓ µβ − uν Γ µαΓ κβ + u Γ νβΓ κα − 2Sαβ ∇ν uµ (2.33) This relation shows that the transport of a vector field along a closed path is not leaving the vector unchanged; both the curvature and the torsion of the space-time have effected on it. The geometrical meaning of this is simple: by the end of the transport, but the vector field is no longer the same field - like the scalar field, it presents a certain difference with its “past self”, which is encoded on the right hand of this equation. What is interesting to see is that in a Euclidean space-time -as we know- the vector field would not change; this is easy to say so, since the partial derivatives -that would replace the covariant ones- will be following the Schwarz-Young theorem for the symmetry of their indices. However, in a Riemann space-time, the vector would again change and its change would come as a result of the curvature. In a Riemann-Cartan space-time, the vector would change even more, due to the presence of torsion as well as curvature. The relations (2.32) and (2.33) is known as the Ricci identity for a vector field and they are the main proof for the existence of curvature in a space-time. On the following section, we will use this to define the curvature tensor - a parameter immediately connected to the torsion of the space-time, as we have specified it here [2; 8; 9].

One can continue this work for tensors of second or greater order; it is merely the same handling, only a little more diﬃcult in calculating. For example, the Ricci identity for a tensor ﬁeld of second order would take the form µν µν κν µ κν µ κν λ µ κν λ µ ∇β∇αT − ∇α∇βT =T ∂κΓ αβ − T ∂βΓ ακ + T Γ αβΓ λκ − T Γ ακΓ λβ+ µκ ν µκ ν µκ λ ν µκ λ ν κ µν + T ∂κΓ αβ − T ∂βΓ ακ + T Γ αβΓ λκ − T Γ ακΓ λβ − 2Sαβ ∇κT , for its contravariant components κ κ λ κ λ κ ∇β∇αTµν − ∇α∇βTµν = − Tκν ∂µΓ αβ + Tκν ∂βΓ αµ − Tκν Γ αβΓ λµ + Tκν Γ αν Γ λβ− κ κ λ κ λ κ κ − Tµκ∂ν Γ αβ + Tµκ∂βΓ αν − TµκΓ αβΓ λν + TµκΓ αν Γ λβ − 2Sαβ ∇κTµν and for its “mixed” components µ µ κ µ κ µ κ λ µ κ λ µ ∇β∇αT ν − ∇α∇βT ν =T ν ∂κΓ αβ − T ν ∂βΓ ακ + T ν Γ αβΓ λκ − T ν Γ ακΓ λβ− µ κ µ κ µ λ κ µ λ κ κ µ − T κ ∂ν Γ αβ + T κ ∂βΓ αν − T κ Γ αβΓ λν + T κ Γ αν Γ λβ − 2Sαβ ∇κT ν

17 2.2.3 Autoparallel and Geodesic Equations So far, we have referred more than once to the curves of the Riemann-Cartan space-time in order to define our vectors and their transports. We must say though, that it is the existence of certain curves that comes with quite an importance to us: the curves that present the shortest path from one point of the space-time to another and the curves that have a zero “acceleration”. In Euclidean and a Riemannian space-time, those two curves are one and they are known as the geodesic curves. This does not hold true for the Riemann-Cartan space-time as well. So, what we need to see is the definition of those two curves, the differences they present and which one of them fits betters to the work we intend to do. The autoparallel curves are the curves that have zero “acceleration”; in other words, the derivative of the corresponding tangent vector in respect to the associated-at affine parameter - the proper distance or proper time- is zero. So, if we assume a curve with parametric equation xα = xα(s), dxα where s is the proper distance, and the tangent vector is uα = , we can demand that ds

β α u ∇βu = 0 (2.34) and from here we can derive immediately the equation for the autoparallel curves [2; 8; 9].

d2xα dxβ dxγ + Γα = 0 ds2 βγ ds ds If we substitute the connection from the equation (2.7), we have

d2xα dxβ dxγ dxβ dxγ + Γ˜α + Cα = 0 ds2 βγ ds ds βγ ds ds dxβ dxγ But, since the product is symmetric, this equation can be formed as ds ds d2xα dxβ dxγ dxβ dxγ + Γ˜α + 2S α = 0 (2.35) ds2 βγ ds ds (βγ) ds ds This is the full form of the autoparallel equation, which stands for a curve that has zero “acceleration”. Those are the “straight” world-lines of the space-time, yet not the “shortest” ones [7]. The property of “extremum” -either minimum or maximum- length belongs to the geodesic curves; those are usually the “shortest paths in the space-time. The geodesic curves are formed in a similar way with the autoparallel ones. The main difference is that we do not use the covariant derivative in respect to the affine connection, but the covariant derivative in respect only to the Christoffel symbol. As a result, we must have

β ˜ α u ∇βu = 0 (2.36) and from here we have the equation for a geodesic curve

d2xα dxβ dxγ + Γ˜α = 0 (2.37) ds2 βγ ds ds This equation is similar to the geodesic equation of a Riemann space-time [2; 7; 8; 9].

We must notice that the two equations differ by a term relative to the torsion. This causes a small trouble, since in the Riemann-Cartan geometry we cannot have zero “acceleration” without having this extra term, which can take the form of a “force” applied by the space-time due to the torsion. Since the importance of the geodesic equation in relativistic physics is that they give us the equations of motion for test particles that are not affected by external forces (e.g. an electromagnetic field, this trouble seems to become even larger in the Einstein-Cartan theory of gravity: it “breaks” the principles that Einstein himself introduced with General Relativity, that there must

18 be no gravitational force. The Einstein-Cartan theory, by adding this torsion-related term in the autoparallel equation, is bringing the concept of gravitational force back from the dead. It is also making it diﬃcult for us to choose which of the two equations -the autoparallel or the geodesic- should we use in order to describe the motion of a test particle in a Riemann-Cartan space-time. This problem will be discussed later, in Chapter 3.

The case of another parameter An interesting question is what happens when another parameter is used instead of the proper time or proper distance. The treatment of this is simple and reﬂects the one followed in Riemann space-times. Let us assume that we have a curve with the parametric equation xα = xα(λ), where λ is a dxα random parameter, then we will again take the tangent vector ua = and demand that dλ

β α u ∇βu = 0 in order to ﬁnd the autoparallel equation. In this case, we will have

d2xα dxβ dxγ dxβ dxγ dxα + Γ˜α + 2S α = f(λ) (2.38) dλ2 βγ dλ dλ (βγ) dλ dλ dλ where f(λ) is a function of the parameter. In the same manner, for the geodesic equations we demand that β α u ∇˜ βu = 0 and we will have d2xα dxβ dxγ dxα + Γ˜α = f(λ) (2.39) dλ2 βγ dλ dλ dλ 2.3 Curvature with Torsion

We know that the curvature of the space-time is expressed by the connection. However, as we have explained, the connection is not a tensor and as a result comes with several problems - one of which is that it cannot be related to any physical parameter. As a result, we must use a tensor that will stand for all the geometrical properties of the space-time. This tensor can be easily derived by the Ricci identity we have already come across, since this relation expresses in the best way the eﬀects of curvature in the space-time.

2.3.1 The Curvature Tensor Writing the Ricci identities as

ν ν 2∇[α∇β]uµ = R µαβ uν − 2Sαβ ∇ν uµ µ µ ν ν (2.40) 2∇[α∇β]u = −R ναβ u − 2Sαβ ∇ν uµ we have deﬁned the curvature tensor, a fourth order tensor that is created because of the diﬀerence of the two vectors we have discusses before: one is the vector at the starting point before the transport begins, and one is the vector that has completed the transport and returned to the starting point. According to the equation (2.32), this curvature tensor must have the following form

α α α κ α κ α R βµν = −∂ν Γ βµ + ∂µΓ βν − Γ βµΓ κν + Γ βν Γ κµ (2.41)

It is reasonable why the curvature tensor must be expressed in respect to the aﬃne connection, since it is the latter one that embodies all the deviations of a curved space-time with the Euclidean space-time.

19 If, however, we use the expression for the connection of a Riemann-Cartan space-time, from the equation (2.7), then we can write the curvature tensor in another form, with respect to the Christoﬀel symbol and the contorsion tensor. In this form, we can detect in which way the torsion of the space-time aﬀects its curvature.

˜ ˜λ κ λ ˜κ ˜λ κ λ ˜κ Rαβµν = Rαβµν + Qαβµν + gακ(−Γ βµCλν − Cβµ Γ λν + Γ βν Cλµ + Cβν Γ λµ) (2.42)

˜ ˜κ ˜κ ˜λ ˜κ ˜λ ˜κ where Rαβµν = gακ(−∂ν Γ βµ + ∂µΓ βν − Γ βµΓ λν + Γ βν Γ λµ) is the Riemann-Christoffel tensor, κ κ λ κ with respect only to the Christoffel symbols, and Qαβµν = gακ(−∂ν Cβµ + ∂µCβν − Cβµ Cλν + λ κ Cβν Cλν ) is another tensor, similar in form to the Riemann-Christoffel tensor but with respect to the contorsion tensor rather than the Christoffel symbol. The Riemann-Christoffel tensor can also be written in respect to the metric tensor, since the Christoffel symbol can be written as in the equation (2.6) [2; 7; 8]. 1 R˜ = (∂ ∂ g + ∂ ∂ g − ∂ ∂ g − ∂ ∂ g ) αβµν 2 β µ αν α ν βµ β ν αµ α µ βν This tensor serves as curvature tensor in Riemann space-time. We also notice that there is another term appearing in the expression of the curvature tensor, which is a sum of four products of the Christoffel symbols with the contorsion tensor. This part express a “coupling” between the two main geometrical characteristics: the curvature and the torsion. The curvature tensor follows two certain symmetries, which are

Rαβµν = −Rβαµν Rαβµν = −Rαβµν (2.43)

All other symmetries that appear in a Riemann-Christoﬀel tensor, do not appear in this tensor, since they demand a symmetric connection in order to appear [9; 10].

An Alternative Expression for the Curvature Tensor What we can see from the expression we have derived for the curvature tensor is that consists of three parts. The first is the “Riemannian part” - the tensor we would have if the space-time was Riemannian. The second is the “Cartan part” - a tensor similar in form to the Riemannian one, but with respect to the contorsion tensor instead of the Christoffel symbol. The third is the “Coupling part” - the sum of the Christoffel symbol multiplied with the contorsion tensor. These parts govern, as it can be shown by their “names” the different effects of torsion in curvature: the first being exclusively relative to curvature, the second being solely the affection of torsion, and the third being the coupling of torsion with curvature. This order is preserved in all other forms of the curvature tensor or its contractions, as we will see right after. However, this is not the only approach. Using the expression for the covariant derivative of the contorsion tensor in respect to the Christoffel symbols alone

α ˜ α ˜κ α ˜κ α ˜α κ ∂µCβν = ∇µCβν + Γ µβCκν + Γ µν Cβκ − Γ µκCβν we can also write the curvature tensor in the following way

α ˜α ˜ α ˜ α κ α κ α R βµν = R βµν + ∇µCβν − ∇ν Cβµ + Cβν Cκµ − Cβµ Cκν (2.44)

Thus, the curvature tensor is expressed as the Riemann-Christoffel tensor plus some “correction” terms in respect to the contorsion. This approach, which is commonly used by N.J. Poplawski [11] and K. Pasmatsiou [12], differs with the previous one in a very specific place: the first is trying to express the curvature tensor of a Riemann-Cartan space-time in comparison to that space-time, while the second expresses it in comparison to a Riemann space-time. Obviously, the second one is better when it comes to comparisons of the two geometries, but it is explicitly the first that we will use in this work.

20 2.3.2 The Ricci Curvature Having defined and derived the curvature tensor, we can make a small comment: it is obvious that this tensor is a very hard to work with. It is a tensor of fourth order, having n4 components, with n being the dimension of the space-time -in ours case n = 4. The equations that would include such a tensor would obviously be very difficult to be solved. Consequently, we can use its contractions; of those contractions two have the greatest importance. The first is the Ricci tensor, that is defined as µ Rαβ = R αµβ (2.45) If we use the curvature tensor we have just presented, we have the following tensor

˜ ˜λ µ λ ˜µ ˜λ µ αβ λ ˜µ Rαβ = Rαβ + Qαβ + (Γ αµCλβ + Cαµ Γ λβ − Γ αβCλµ − C Γ λµ) (2.46) ˜ ˜µ where Rαβ = R αµβ is the Ricci tensor of a Riemann space-time, with respect only to the Christoﬀel µ µ λ µ λ symbols, and Qαβ = Q αµβ = −∂β Sα + ∂µ Cαβ − Cαµ Cλβ + Cαβ Sλ is the contraction of the other tensor. We must notice here that this tensor has no symmetries, while the Ricci tensor of a Riemann space-time is symmetric[2; 8]; we can however ﬁnd easily its symmetric part. ˜ ˜κ λ ˜κ λ κ ˜λ λ λ µ R(αβ) = Rαβ + Γ αβCκλ − Γ (α|λCκ|β) − C(α|λ Γ κ|β) − ∂(βCα)λ − C(α|µ Cλ|β) (2.47) As for the antisymmetric, it is proved to be κ κ R[αβ] = −∇κSαβ + 2∇[αSβ] − 2Sαβ Sκ (2.48) 1 Finally, if we continue the contractions, we take the trace of the Ricci tensor, that is α R = R α (2.49) This is called Ricci scalar or Scalar Curvature. And again, if we use the curvature tensor we have just showed, this scalar takes the form ˜ ακ ˜λ µ λ ˜µ ˜λ µ λ ˜µ R = R + Q + g (Γ κµCλα + Cκµ Γ λα − Γ καCλµ − Cκα Γ λµ) (2.50) ˜ ˜α α α κα λ where R = R α the scalar curvature of a Riemann space-time, and Q = Q α = −∂ Sα + Cλ Cκα is the second contraction of the other tensor [2; 8].

We notice again, as we have mentioned before, that the division of the curvature tensor in three parts -the “Riemann” one, the “Cartan” one and the “Coupling” one- is preserved throughout these contractions as well. This is very interesting, since one would expect the traces of the curvature tensor to exclude the eﬀects of the torsion, due to the omitting of the antisymmetric parts. However, the presence of the torsion is preserved, since the torsion and the contorsion tensors are not -as many would expect- fully antisymmetric. Even, if we take the fully symmetric part of the Ricci tensor, we will see that torsion is preserved in there too, as it is preserved in the Scalar Curvature. Of course, this would not be obvious, if we choose the alternative expression of the curvature tensor [12]. In that case, the Ricci tensor would take the form ˜ ˜ κ ˜ κ λ κ Rαβ = Rαβ + ∇κC ab − 2∇βSα + 2C αβ Sκ − C ακ C λβ while the Ricci scalar would be ˜ ˜ αβ βκ α α κβ ˜ ˜ α α α κβ R = R − 2∇βC α + C κ C βα − C κβ C α = R − 4∇αS − 4S Sα − C κβ C α

Finally, we must say that there is another contraction of the curvature tensor: the Homothetic Curvature. This contraction is of no real interest to us, since it is identically zero in any metrical manifold. ˇ µ Rαβ = R µαβ = 0 1This relation is also given from the algebraic Weitzenb¨ock identities that we will discuss later.

21 2.3.3 The Weyl Curvature So far, we have seen the curvature tensor, the Ricci tensor and the Ricci scalar; those geometrical quantities express the curvature of the space-time. Especially, the Ricci tensor and the Ricci curvature, as the traces of the curvature tensor, are standing for the curvature locally -in “small distances” from its “source”. Of course, the curvature tensor stands for all curvature, which means that what is left of it when we exclude its traces is this “curvature-in-distance”. In order to describe and measure this, we can define another tensor which will be the conformal traceless part of the Curvature tensor. This tensor is known as Weyl tensor. In Riemann space-times, it is given from the relation 2 2 W = R − (g R − g R ) + Rg g (2.51) αβµν αβµν n − 2 α[µ ν]β β[µ ν]α (n − 1)(n − 2) α[µ ν]β where n is the dimension of the space-time - in our case, n = 4. However, we cannot use this definition in nor-Riemann space-times, like the one we have been analysing so far. in other words, we need a new definition derived from the curvature tensor of that space-time [2]. In a Riemann-Cartan space-time the definition of the Weyl tensor has been given by B.N. Frolov (in 1996) as 1 1 Wαβµν = Rαβµν + (gαµR(νβ) − gαν R(µβ) − gβµR(να) + gβν R(µα)) − R(gα[µgν]β)− 2 3 (2.52) 1 1 − (κ D − κ D ) − D 4 αβ[µ| κ|ν] µν[α| κ|β] 12 αβµν

˜ ˜κ λ ˜κ λ κ ˜λ where Rαβµν is the Riemann curvature tensor, R(αβ) = Rαβ+Γ αβC κλ −Γ (α|λC κ|β) −C (α|λ Γ κ|β)− λ λ µ α ∂(αC β)λ − C (α|µ C λ|β) is the symmetric part of the Ricci tensor, R = R α is the Ricci scalar, 1 Dµ = − ηαβκµR and D = Dλ . These new parameters, D and D, which are formed by the ν 2 αβκν λ µν antisymmetric parts of the curvature tensor, are useful in order to exclude all the effects of those antisymmetric parts in the traces; these are actually the “new” terms that were “added” by Frolov so that we could take the Weyl tensor in this space-time [13]. However, this definition comes with a problem; this Weyl tensor, although invariant under conformal transformations, in not traceless since one of its two contractions is not zero. Analytically, the first contraction of this tensor reflects what we defined as homothetic curvature in the previous section α α ˆ W αµν = R αµν = Rµν , which is identically zero, as we have discussed before. The second contraction refelcts what we have referred to as the Ricci tensor 1 W α = R + R − R µαν µν (µν) 2 [µν]

, which is not zero. But, we can easily see that if the term R(µν) is replaced with the term −R(µν) and the coeﬃcient of R[µν] becomes one, then this trace will be zero as well. Consequently, we can alter slightly the form of the Weyl tensor B.N. Frolov has given us, in order to make it completely traceless. This new form will be 1 1 Wαβµν = Rαβµν − (gαµR(νβ) − gαν R(µβ) − gβµR(να) + gβν R(µα)) + Rgα[µgν]β− 2 3 (2.53) 1 1 − (κ D − κ D ) − D 2 αβ[µ| κ|ν] µν[α| κ|β] 12 αβµν In this new form, the two contractions of the Weyl tensor are totally zero.

κ κ W ακβ = W καβ = 0 (2.54)

22 Furthermore, the Weyl tensor preserves the two main symmetries of the Curvature tensor

Wαβµν = −Wβαµν Wαβµν = −Wαβνµ (2.55)

, as well as its form, since it can split in three parts: a “Riemann”, a “Cartan” and a “Coupling” one.

We can plainly see that this form of the Weyl tensor proposed by B.N. Frolov is divided in two groups of terms. The first is similar to the Riemannian one, with one slight change -the Riemannian Ricci tensors are replaced with the symmetric parts of the Riemann-Cartan counterparts; the second is introduced here for the first time and represents the antisymmetric parts of the Ricci tensor and Ricci scalar in the Riemann-Cartan space-time. However, considering the change we have made here, it is obvious that we have come even closer to the Riemannian form of the Weyl tensor. One would easily believe that we could write it in the same manner, if we added the antisymmetric parts in the symmetric and excluded the new terms that Frolov used 2. Such a work was conducted by S. Jensen (in 2005); the form that he proposes is almost the same to the Riemannian one [10]. 1 1 W = R − (R g − R g − R g + R g ) + R(g g − g g ) (2.56) αβµν αβµν 2 αµ νβ αν µβ βµ να βν µα 6 αµ βν αν µβ It is obvious that this form of the Weyl tensor is traceless as the previous one. The only problem appearing is -as stated by Jensen himself- that this Weyl tensor is no checked for invariance under conformal transformations. for this reason, we will use the corrected form of the Weyl tensor we have just shown for our next steps -however, not much will change in the course of this work if we choose Jensesn’s definition instead of Frolov’s

It is obvious that any remark we have made for the Curvature tensor and its contractions before, is valid for the Weyl tensor as well - again we have the “Riemann” part, the “Cartan” part and the “coupling” part.

Something very important to the nature of the Weyl tensor is the fact that it is traceless; this means that we cannot take any contractions of the Weyl tensor in order to simplify our equations. However, this tensor can be decomposed in two second-order tensors, that are called the “electric” and the “magnetic” part of the Weyl tensor. Their names have been chosen as these, since their evolution equations present a great similarity to the ones of the vectors of the electric and the magnetic ﬁeld - something which could imply a deeper connection between relativistic gravity and classical electrodynamics. The two tensors are deﬁned by the relation

W = [(g g − g g )(g g − g g ) − ]Eκν uλuµ− αβγδ ακ βλ αλ βκ γµ δν γν δµ αβκλ γδµν (2.57) κν λ µ −[αβκλ(gγµgδν − gγν gδµ) − (gαλgβκ − gακgβλ)γδµν ]H u u or, by breaking the latter one, by the two following relations 1 E = W uµuν H = µν W uκ αβ αµβν αβ 2 α µνβκ where Eαβ its “electric” part and Hαβ its “magnetic” part. The full form of the “electric” part will be 1 1 E = W uµuν = R uµuν − (g R uµuν − R uµu − R uν u + R ) + Rh − αβ αµβν αµβν 2 αβ (µν) (βµ) α (αν) β (αβ) 6 αβ 6 − (u[κuλδ γ]R − u[κuλδ γ]R ) + δ [κδ λuγ uδ]R 4 α κλγβ β κλγα α β κλγδ (2.58)

2We must note here that Frolov’s Weyl tensor was calculated for a non-metric aﬃne manifold, while ours is metric; as a result, it carries several information we do not need, since they are identically zero in a Riemann-Cartan space-time.

23 and for the “magnetic” 1 1 1 1 H = µν W uκ = µν R uκ − ( ν R − ν R + µR )uκ + R ν u − αβ 2 α µνβκ 2 α µνβκ 4 αβ (κν) ακ (βν) αβ (κν) 6 αβ ν 6 1 − ( [κλuγ]R − [κλδ γ]R uδ) − [κλδ γ uδ]R 8 α κλγβ α β κλγδ 2 α β κλγδ (2.59)

We must notice that the “electric” and the “magnetic” parts of the Weyl tensor are traceless and orthonormal to the vector uα that was used for their deﬁnition.

α α β β E α = H α = 0 Eαβ u = Hαβ u = 0 (2.60) Both of these properties are easily proved by their very deﬁnitions. The ﬁrst is of course trivial to understand since the Weyl tensor is traceless. As for the second, it becomes more interesting if this vector is the velocity of an observer; in that case, the orthonormality of the “electric” and “magnetic” parts to the velocity vector means that only their spatial components are non-zero.

As we have said before, the Weyl tensor is traceless and as a result it cannot be contracted. The only way we have to simplify the study of curvature in great distances is through this very decomposition; by defining the “Electric” and “Magnetic” parts of the Weyl tensor we can study all effects of curvature in great distances in a much more comfortable way. Among these effects, one can consider the large-scale structure, the tidal forces and the gravitational waves.

2.3.4 The Weitzenböck Identities The curvature tensor is the very essence for the Riemann-Cartan space-time. Thus, we use it in order to describe the structure and the symmetries of it. A great role in this description is played by the Weitzenböck identities, which are expressing some further symmetries of the curvature tensor and of its derivatives; these identities are nothing more the generalised Bianchi identities for non- Riemannian space-times [9; 10]. The first of them is known as the Algebraic Witzenböck identity, although several dismiss it as a Bianchi identity and categorise it as an extra symmetry relation of the curvature tensor. In a Riemann-Cartan space-time, it has the following form

ν ν κ ν R [αβµ] = 2∇[αSβµ] − 4S[αβ Sµ]κ (2.61) This relation also provides the Weitzenböck identities for the torsion tensor as 1 ∇ S ν = Rν + 2S κS ν (2.62) [α βµ] 2 [αβµ] [αβ µ]κ If we want to write this for the Ricci tensor, we can simply contract the ν and β indices and take the relation (2.48) 1 1 ∇ S = R + ∇ S κ + S κS [α β] 2 [αβ] 2 κ αβ αβ κ that we can call Contracted Algebraic Weitzenböck identities, or the Weitzenböck identities for the torsion vector. [9; 10]. The second is known as the Differentiating Weitzenböck identity and is the one widely acknowl- edged as “the” Weitzenböck identity. It summarizes the symmetries of the first-order covariant derivatives of the curvature tensor and allows us to think of a certain evolution equation of the curvature tensor that could be derived from it. Its form in a Riemann-Cartan space-time is

κ λ κ ∇[αR βµ]ν = 2S[αβ R µ]λν (2.63) Furthermore, we can calculate the Diﬀerentiating Weitzenbo¨ock identity for the Ricci tensor, since the only thing we have to do is to contract the curvature tensor. What we have to do is to

24 contract the κ and µ indices in the equation (2.63). If we write it down analytically, we have

κ κ κ κ κ κ ∇αR βµν + ∇βR µαν + ∇µR αβν − ∇αR µβν − ∇βR αµν − ∇µR βαν = λ κ λ κ λ κ λ κ λ κ λ κ = 2(Sαβ R µλν + Sβµ R αλν + Sµα R βλν − Sαµ R βλν − Sβα R µλν − Sµβ R αλν ) and contracting the two indices

κ κ κ κ ∇αRβν + ∇βR καν + ∇κR αβν − ∇αR κβν − ∇βRαν − ∇κR βαν = λ κ λ κ λ κ λ κ λ κ λ κ = 2(Sαβ R κλν + Sβκ R αλν + Sκα R βλν − Sακ R βλν − Sβα R κλν − Sκβ R αλν ) and ﬁnally we have the Contracted Weitzenbock identities in the form

κ λ κ ∇[αRβ]ν + ∇κR [αβ]ν = 4S[β|κ R |α]λν (2.64) We can go even further by a second contraction -this time for the β and ν indices. By taking the analytic form of equation (2.64),

κ κ λ κ λ κ ∇αRβν − ∇βRαν + ∇κR αβν − ∇κR βαν = 4(Sβκ R αλν − Sακ R βλν ) and contract the two indices

ν ν κ ν κν λ κ ν λ κν ∇αRν − ∇ν Rα + ∇κR αν − ∇κR αν = 4(Sνκ R αλ − Sακ R λν ) we can ﬁnally obtain the form of the Twice Contracted Weitzenb¨ock identities.

ν λµ λµν ∇αR − 2∇ R(αν) = 4(Sα Rλµ − S Rαλµν ) (2.65) Finally, we can calculate the Differentiating Weitzenböck identity for the Weyl tensor, since we know it preserves the symmetries of the curvature tensor and can replace it at large scales - where the effects of the trace vanishes. In order to find the form of the Weitzenböck identities for the Weyl tensor, we begin by taking the divergence of the Weyl tensor. 1 1 ∇ν W = ∇ν R − (g ∇ν R − g ∇ν R + ∇ R − ∇ R ) + g ∇ R− αβµν αβµν 2 αµ (νβ) βµ (να) β (µα) α (µβ) 3 α[µ β] 1 1 − (κ ∇ν D − κ ∇ν D − κ ∇ν D + κ ∇ν D ) − ∇ν D 4 αβµ κν αβν κµ µνα κβ µνβ κα 12 αβµν

γδκν γ δ κ γδκλ γ δ κ λ We know the permutation formulas αβµν = 6δ[α δβ δµ] and αβµν = 24δ[α δβ δµ δν] ; this allows us to expand several of the terms of our relation. 1 1 ∇ν W = ∇ν R − (g ∇ν R − g ∇ν R + ∇ R − ∇ R ) − g ∇ R− αβµν αβµν 2 αµ (νβ) βµ (να) β (µα) α (µβ) 3 α[µ β] 6 − (δ γ δ δδ κ∇ν R − δ γ δ δ∇κR − δ γ δ δ∇κR + δ γ δ κ∇δR )− 4 [α β µ] γδκν [α β] γδκµ [µ α] γδκβ [µ β] γδκα γ δ κ λ ν −2δ[α δβ δµ δν] ∇ Rγδκλ It is obvious that the “permutation” terms are zero, due to the symmetries of the curvature tensor. Furthermore, from the Contracted Diﬀerentiating Weitzenb¨ock identity (equation (2.62)), we have

ν κ ν κ ν κ ∇ Rαβµν = ∇αRβµ − ∇βRαµ + 2(Sνα Rβκµ + Sνβ Rακµ + Sαβ Rκµ) If we substitute this into the previous relation, then we will have 1 ∇ν W = ∇ R + g ∇ R + 4S κR ν + 2S κR (2.66) αβµν [α β]µ 6 µ[α β] ν[α β]κµ αβ κµ This is the form the Weitzenbock identities take for the Weyl tensor in a Riemann-Cartan space-time.

25 We must observe -for once more- the great similarity of these relations with the equivalent equations of a Riemann space-time. Their main differences can be found in two parts. The first is the obvious fact that all Riemannian curvature tensors -including the Ricci and the Weyl tensors- are replaced with their Riemann-Cartan counterparts; the second is that the torsion of the Riemann- Cartan space-time seems to be a regulator for those equations, since its presence in the right part of all five formulas is essential in understanding its effect on the space-time. Furthermore, the fact that we have the (Differentiating) Weitzenböck identities for the curvature and the Weyl tensors means that we can calculate propagation equations or, in other words, equations of curvature transfer in great distances. Physically, this is of great importance because it gives us the opportunity to study the large scale structure and the transfer of the gravitational field in great distances -in other words, the gravitational waves- in a Riemann-Cartan space-time.

26 Chapter 3

The Field Equations of the Einstein-Cartan Theory

3.1 The Einstein-Hilbert Action

In Einstein’s General Theory of Relativity, the curvature of space-time is a result of the existence of mass or energy and expresses the gravitation of this mass or energy. To be more precise, matter, energy and momentum can curve the space-time and all phenomena concerning gravitational attraction due to that matter, energy and momentum - such as the acceleration of a body falling on Earth, or the relative motion of galactic super-clusters -, are consequences of this curvature. Or, as it is more simply expressed:“The matter tells space-time how to curve; the space-time tell matter how to move”. The relation between the curvature of space-time, that is described by the Curvature tensor and its contractions, and the density of matter, energy and momentum that curve it, is shown by the Einstein equations 1 R˜ − Rg˜ + Λg = κT (3.1) αβ 2 αβ αβ αβ 4 where Λ is the cosmological constant, κ = 8πG/c the gravitational constant and Tab the energy- momentum-stress tensor -a symmetric second-order tensor. Since, the cosmological constant is present, this form of the equation describes the curvature in local terms as well as greater distances. If, however, we want to study relativity in local terms, we can create a new tensor, that is called Einstein tensor and is deﬁned as 1 G˜ = R˜ − Rg˜ αβ αβ 2 αβ and rewrite the Einstein equations in local terms as

G˜αβ = κTαβ

In order to derive those field equations we use the Einstein-Hilbert Action, which is nothing more than the action of a Lagrangian density that connects the geometry of space-time with any matter, energy and momentum of the system. This Lagrangian is 1 √ L = R˜ −g + L (3.2) 2κ M where g = det|gµν | is the determinant of the metric tensor and LM is the matter Lagrangian, that is defined by the characteristics of the cosmic fluid we choose to study - in our case, a spin fluid. Thus, the Einstein-Hilbert Action is written as Z 1 √ S = R˜ −g + L dΩ 2κ M

27 and, by using the standard lagrangian formulation, we must minimise it. This means that, in respect to all variables of the Lagrangian, the variation of the Action must be zero - this is called the Action principle. δS = 0 (3.3) In the Einstein-Cartan theory, we must follow the same path in order to write the ﬁeld equations. To do so, we assume that the Einstein-Hilbert Action and its lagrangian formulation stand for this theory as well. As a result, we take the same form for the Lagrangian density, with one change: the Ricci scalar is replaced with its counterpart of the Riemann-Cartan space-time. 1 √ L = R −g + L (3.4) 2κ M 1 Here, the matter Lagrangian has the same form with the Einstein universe, since the same physical laws apply in the two diﬀerent universes - the geometry is what really changes. Eventually, the Einstein-Hilbert Action takes the form Z 1 √ S = R −g + L dΩ = 2κ M (3.5) Z h 1 √ i = R˜ + Q + gαν (−Γ˜κ Cµ − Cκ Γ˜µ + Γ˜κ Cµ + Cκ Γ˜µ ) −g + L 2κ νµ κα νµ κα να κµ να κµ M In the same manner as before, in the case of an Einstein universe, we apply the Action principle and demand that the variance of the Action, in respect to all its variables, is zero.

3.2 The Field Equations

The Field Equations are the main body of any field theory. They are the equations that describe the form and structure of the physical field we intend to study. The first example of such equations are none else than the Maxwell Equations in the electromagnetic theory, that combine the vectors of the electric and the magnetic fields with their sources: the electric charge density and the electric current density. This is the typical form of field equations and this is like the one we mean to calculate here. In General Relativity, as we already have shown, Einstein managed to create a relation between the geometry of the space-time and the energy, stress and momentum of the physical system. This relation is shown by the equation (3.1). These equations, as we have explained, are derived by the Einstein-Hilbert action, by demanding that it has a zero variance. In the Einstein-Cartan theory, we use again this action in order to find the field equations of this theory. Combining the equations (3.3) and (3.5), and since the Einstein-Cartan theory has two dynamical variables: the metric tensor and the contorsion tensor (that can also be replaces by the torsion tensor), we come to the conclusion that we have two sets of field equations that arise from the variation of the action in respect to the metric tensor and from the variation of the action in respect to the contorsion tensor [6; 12].

δ 1 √ R −g + L = 0 δgαβ 2κ M (3.6) δ 1 √ R −g + L = 0 δCαβγ 2κ M Λ 1If one wishes to add the cosmological constant in his analysis, he can easily add the term − in the Lagrangian κ density, making the Einstein-Hilbert Action to take the following form: Z 1 √ S = (R − 2Λ) −g + LM dΩ 2κ

28 3.2.1 The Variation in respect to the Metric Taking the variation in respect to the metric tensor, we will end up to the form the first of our field equations must have. Demanding that δL = 0 δgαβ we can prove that √ δR 1 δ −g 2κ δL + √ R = −√ M δgαβ −g δgαβ −g δgαβ √ 1 δ −g 1 δg 1 or, since R = R gαβ and √ = = − g , (αβ) −g δgαβ 2g δgαβ 2 αβ 1 2κ δL R − Rg = −√ M (3.7) (αβ) 2 αβ −g δgαβ where R is the Scalar Curvature. Defining the term on the right hand of the equation as the energy-momentum density tensor, we can turn this into a more familiar equation 1 R − Rg = κT (3.8) (αβ) 2 αβ αβ

It is obvious that we have created the ﬁrst of the ﬁeld equations, the one that is similar to the Einstein equations of the General Relativity. This equation has in its left hand all the information we need for the geometry of the space-time -locally- and in its right hand all the information we need for the physical system. Its importance is that it allows to “translate” the one kind of information into the other. It is remarkable, although not without sense, that this equation is almost completely similar to the Einstein equation of General Relativity [8].

3.2.2 The Variation in respect to the Contorsion Following the same procedure, this time in respect to the contorsion tensor, we can prove the form of the second field equation. This one, alike the other, will be concentrating on the spin of the system and will “translate” it to geometrical characteristics -more accurately, into torsion. We can assume, from before starting the procedure, that the second equation will be similar in structure to the first; on the one side we will have the geometrical factors we are interested in, and on the other a density tensor that will contain all necessary information about the spin. As before, we demand that δL = 0 δCαβγ and we can prove that √ δR 1 δ −g 2κ δL + √ R = −√ M δCαβγ −g δCαβγ −g δgαβ √ 1 δ −g δR or, since R = R gαβ, √ = 0 and = 0, (αβ) −g δCαβγ δCαβγ δC µ δC ν δC ν δCµ 2κ δL gκλ κ µ C ν + gκλ λν C µ − gκλ κ µ C µ − gκλ λν C ν = −√ M δCαβγ λν δCαβγ κ µ δCαβγ λν δCαβγ κ µ −g δCαβγ This very complex expression can be simplified into the following one, by using the rule of derivation δCαβγ = δαδβδγ δCκλµ κ λ µ 2κ δL C + δ C µ = √ M (3.9) α[βγ] α[β γ]µ −g δCαβγ

29 which is one of the forms of the Cartan equations, the second of the Einstein-Cartan field equations. 2 δL As before, we can define the term √ M as the spin density tensor, the equation can be written −g δCαβγ in the following way µ Cγ[αβ] + gγ[αC β]µ = κsαβγ (3.10) or, if we use the symmetries of the contorsion tensor, we can further simplify it in 1 S − S g + S g = − κs (3.11) αβγ β αγ α βγ 2 αβγ which is the most familiar form of them [5; 6; 12]. The Cartan equations is much simpler in form and structure than the firs set of field equations; instead of a complex relation between the curvature of the space-time and the energy and momentum of the matter fields, what we have here is an immediate algebraic coupling of spin to torsion. In the form we have presented it in the equation (3.11), this coupling appears with emphasis to the geometric terms of the space-time. But, due to this simple algebraic form, we can solve this form of the equation to the torsion tensor, and present it in the following form, where we emphasize in the spin density terms. 1 1 1 S = − κ s + g sµ − g sµ (3.12) αβγ 2 αβγ 2 βγ αµ 2 αγ βµ Furthermore, if we contract the β and γ indices, we can take an even simpler equations that µ would couple the torsion vector with the spin density vector sα = sαµ ; this contracted Bianchi equation will be Sα = κsα (3.13)

Having a small glimpse into what we will do a little later, we could say a few words concerning the Weyssenhoff fluid, a fluid consisted of spinning particles -eg. electrons, protons and neutrons. This fluid has a very simple description: As long as its energy-momentum density tensor is concerned, it is nothing more but a slightly modified perfect fluid energy-momentum tensor; as long as it spin density tensor is concerned, it usually takes the form

sαβµ = sαβuµ where uµ the velocity vector and sαβ the canonical spin density tensor -also known as Spin Matrix. This tensor is antisymmetric and has a certain property, know as the Frenkel condition

β sαβu = 0

2 As a result of all these, the Cartan equations take a much simpler form, which is

Sαβγ = κuαsβγ

3.2.3 The Algebraic Form of the Field Equations What one can plainly see from those two sets of field equations is that they are a purely geometrical approach of the field equations, and a very complex one to be used immediately. What we can try now to do is to find an immediate way of coupling geometry with physics. Taking the first of the field equations (equation(3.8)) and contracting its a and b indices, we have 1 R − δα R = κT 2 α 2All these will be discussed in the next section.

30 α α where R is the scalar curvature, T = T α is the trace of the energy-momentum tensor, and δ α = n is the Kronecker symbol -with n = 4 being the number of dimensions of the space-time. As a result, we can connect the scalar curvature to a scalar of the energy-momentum density. 2 R = − κT = −κT n − 2 If we use this in the ﬁled equations, we can take an alternative form of them, which is focuses on the energy-momentum density, rather than on geometry. 1 1 T − T g = R (3.14) αβ 2 αβ κ (αβ) If we take the internal product of this form of the ﬁeld equations with twice the velocity vector 1 R uαuβ = κ T uαuβ − T g uαuβ (αβ) αβ 2 αβ

α β α and use the normalisation gαβ u u = u uα = −1, we have 1 R uαuβ = κ T uαuβ − T (3.15) (αβ) ab 2 Furthermore, we can take the internal product of these ﬁeld equations with the velocity tensor and the projection tensor hαβ = gαβ + uαuβ -a symmetric tensor that allows us to project vertically the velocity vector on the three-dimensional rest space of an observer- 1 h βR uµ = κ h βT uµ − T h βg uµ α (βµ) α βµ 2 α βµ

β γ β and then use the condition hα gβγ u = hα uβ = 0, we have

β µ β µ hα R(βµ)u = κhα Tβµ u (3.16) And ﬁnally, we can perform this internal product one last time, between the ﬁeld equations and twice the projection tensor, 1 h µh ν R = κ h µh ν T − T h µh ν g α β (µν) α β µν 2 α β µν µ ν µ and, by using the relation hα hβ gµν = hα hβµ = hαβ, we have 1 h µh ν R = κ h µh ν T − h T (3.17) α β (µν) α β µν 2 αβ .

Those three equations -the (3.15), the (3.16) and the (3.17)- are depicting the immediate algebraic connection of physics with geometry, the coupling of matter, energy, momentum and spin to curvature and torsion. Their form is much simpler and easier in use than the ﬁeld equations -and this will be proved soon enough.

3.2.4 The Curvature Tensor as a Field Equation We know that the curvature tensor decomposes to the Weyl tensor, the Ricci tensor and the Ricci scalar according to the relation (2.53). Having seen that the Weyl tensor can decompose again to the “electric” and “magnetic” parts of it -according to the relations (2.58) and (2.59)- and the Ricci tensor and scalar can be written in terms of the energy-momentum tensor -by using the ﬁeld equations-, we can prove that the curvature tensor can be written in terms of the main physical and

31 geometrical parameters, as those are described by the energy-momentum tensor and the “electric” and “magnetic” ﬁelds. This is rather simple, though it demands quite a lot of time. First of all, it is easy to prove that the Weyl tensor is written as

κν λ µ Wαβγδ = [(gακgβλ − gαλgβκ)(gγµgδν − gγν gδµ) − αβκλγδµν ]E u u − κν λ µ −[αβκλ(gγµgδν − gγν gδµ) − (gαλgβκ − gακgβλ)γδµν ]H u u which transforms easily into the following form

κ κ κ Wαβµν = 4(h[α Eβ][µ hν]κ − u[αEβ][µ uν]) + 2(αβ u[µHν]κ + µν u[αHβ]κ ) After that, we have to express the Ricci tensor and Ricci scalar in terms of energy and momentum. Of course, we know from the field equations that κ R = κT − T g R = −κT (αβ) αβ 2 αβ , as we have shown right before. If we consider the form of the curvature tensor and try to substitute these wherever possible, we can easily obtain the following relation 1 1 5 1 (g R −g R −g R +g R )− Rg g = −κ(g T −g T )+κT g g − g g 2 αµ (νβ) αν (µβ) βµ (να) βν (µα) 3 α[µ ν]β α[µ ν]β β[µ ν]α 6 α[µ ν]β 2 β[µ ν]α Finally, we know that the decomposition of the curvature tensor into Weyl and Ricci fields according to equation (2.53), includes some antisymmetric parts as well. If we use the relation (2.61) for the symmetry between three indices of the curvature tensor and the definitions of Dµν and D, we will find that 1 (κ D − κ D ) = 2 αβ[µ| κ|ν] µν[α| κ|β] 2 − [δ [γ δ δδ λ](∇ S − 2Sκ S ) − δ [γ δ δδ λ](∇ S − 2Sκ S )] 4 α β [µ| γ δλ|ν] γδ λκ|ν] µ ν [α| γ δλ|β] γδ λκ|β] while 1 D = 0 12 αβµν As a result, the Curvature tensor can be expressed in completely physical and geometrical terms as

κ κ κ Rαβµν =4(h[α Eβ][µ hν]κ − u[αEβ][µ uν]) + 2(αβ u[µHν]κ + µν u[αHβ]κ )− 5 1 − κ(g T − g T ) + κT g g − g g − α[µ ν]β β[µ ν]α 6 α[µ ν]β 2 β[µ ν]α 2 − [δ [γ δ δδ λ](∇ S − 2Sκ S ) − δ [γ δ δδ λ](∇ S − 2Sκ S )] 4 α β [µ| γ δλ|ν] γδ λκ|ν] µ ν [α| γ δλ|β] γδ λκ|β] (3.18)

We can also express the Ricci tensor in physical terms, if one considers that it can be split in a symmetric and an antisymmetric part.

Rαβ = R(αβ) + R[αβ] We know form the reversed ﬁeld equations that the symmetric part can be written as follows κ R = κT − T g (αβ) αβ 2 αβ

32 while, form the Contracted Algebraic Bianchi identities -from the equation (2.48)-, we know that the antisymmetric part will be

ν ν R[αβ] = −∇ν Sαβ + 2∇[αSβ] − 2Sαβ Sν As a result, the Ricci tensor will take the form 1 R = κ T − T g − ∇ S ν + 2∇ S − 2S ν S (3.19) αβ αβ 2 αβ ν αβ [α β] αβ ν Of course, the Ricci tensor has no geometrical terms -no terms related to the “electric” and “magnetic” parts, that is- because it is a trace of the Curvature tensor and all of these terms belong to its trace-free components. Finally, we must notice that the terms related to torsion can be written in respect to the spin density, if one uses the reversed Cartan equations from relation (3.12).

3.2.5 The Effective Field Equations So far, we have presented the field equations of the Einstein-Cartan theory of gravity as they should be: equations that couple geometrical terms from the Riemann-Cartan geometry, with physical terms. However, this approach is not the only, nor the easier to work with. What is more usual is to turn towards the field equations of the General Relativity and try to alter them so that they will accept physical terms related to an Einstein-Cartan universe. These equations are called effective and are very popular among those who study astrophysical or cosmological problems in the Einstein- Cartan theory since they are much closer to the classical theory of gravity with which they feel familiar. A simple way to construct effective field equation for the Einstein-Cartan theory is to take out all torsion-related terms from the left hand of the first field equation (equation (3.8)), translate them into spin-related terms via the reversed Cartan equation (equation (3.12)) and add them to right hand of the first equation as pseudo-matter terms. As a result, one will have the Einstein equation -instead for its Einstein-Cartan counterpart- with an effective energy-momentum tensor that will include both the matter field terms as well as the new pseudo-matter terms; these terms, although they seen to be associated to the spin of the physical system, they are actually representing the changes made by the torsion in a Riemann space-time. Their purpose is to introduce the torsion to the Riemann space-time through a side-way. The form of this Einstein equation will be 1 R˜ − Rg˜ = κT eff = κ(T˜ + U ) αβ 2 αβ αβ αβ αβ ˜ here Tαβ is the typical energy-momentum tensor we know from General Relativity, while Uαβ is a pseudo-energy-momentum tensor concerning the spin-related terms, that are actually torsion- related. Many forms have been proposed for this tensor, some of whom we will discuss in the next section.

One interesting fact about these effective equations is that they can be derived through an Einstein-Hilbert action as the equations we have presented before; in fact, they are derived through an Einstein-Hilbert action similar to ours, not a relativistic one. The main difference is that in this Einstein-Hilbert action the Ricci scalar that is used is given by the alternative form we have discussed 3. If this Ricci scalar is placed in the Einstein-Hilbert action, then the variance in respect to the metric will give us the effective Einstein field equation and not the one we have originally calculated -the variance in respect to the metric must always result to the Cartan equation [11; 12].

3This alternative form is ˜ ˜ α α α κβ R = R − 4∇αS − 4S Sα − C κβ C α

33 3.3 Matter Fields

We have introduced the Energy-Momentum density tensor and the Spin density tensor, two tensors that give us the properties of the matter fields -their energy, mass, momentum, etc.- and allow us to couple these matter fields to geometry. Both of them appear as variances of a Lagrangian density that described the physical system and offers us any information we need to know of it. In this section, we will discuss a little more of those quantities.

3.3.1 The Matter Lagrangian Speaking generally again, we could say that the Matter Lagrangian emerges from the first thermodynamic law, which governs the thermodynamic properties of the system, such as its internal energy, its entropy, its pressure, its density and all other factors that might be of interest in a certain system. But, what we are interested in is a Lagrangian that would describe a system suitable for cosmological solutions of the Einstein-Cartan theory: a spinning non-perfect fluid that would dominate the whole universe. Such a fluid can be described by a number of Lagrangian densities and many of them have been proposed in the bibliography. The difference between all of these is found on the formulation of the spin in the Lagrangian. But, we could go even further and notice that the spin -as a quantum quantity- can be approached both microscopically and macroscopically, something which allows us to understand the presence of so many Lagrangians. What we will try to do here is to give some examples of Lagrangian densities that have been proposed and discuss the properties of each one.

The Weyssenhoff Fluid Lagrangian The Weyssenhoff fluid is a perfect fluid that is constituted by spinning particles. It is the natural extension of the perfect fluid and can be described by the following thermodynamic law 1 1 d = T dS − P d + (α ˙ A α − α˙ A α )dsαβ ρ 4 α Aβ β Aα where is the internal energy -which is given as a function of the usual density, the specific entropy and the spin-, T is the temperature, S is the entropy, P is the pressure, ρ is the usual density of matter and energy, sαβ is a function of the spin -in the form of an antisymmetric tensor of second A order-, and α a is defining an orthonormal set of tetrads, which allows us to study the spin and A β A the angular velocity of the fluid -withα ˙ α = u ∇βα α being its derivative in respect to the proper time or proper distance. This thermodynamic law expresses the conservation of energy in the fluid. The tetrads we have just defined obey the condition

A α ααAβ = ηαβ where ηαβ is the Minkowski metric, and have two sets of indices: the holonomic ones -the α and β- that stand for the space-time dimensions and take the values 0, 1, 2, 3, and the unholonomic ones -the A- that stand for the spin conditions and take the values 1, 2, 3, 4. If the tetrads are constructed so that α4α = uα, then they are attached to the co-moving frame and we can use only two values for their unholonomic indices; in that case, the spin function can take the form

1 2 1 2 sαβ = k(χ)(α αα β − α βα α) where k(χ) is a scalar function of some scalar field χ. Using these, we can construct a matter Lagrangian for the spinning perfect fluid. First, we take the kinetic energy of this fluid 1 E = (α ˙ A α − α˙ A α )sαβ = −ρk(χ)α ˙ 2 α1α (3.20) KIN 4 α Aβ β Aα α and from this we calculate the matter Lagrangian √ √ α √ α √ α LM = − −gρ(1 + ) + λ2∂α ( −gρu ) + λ3 −gX uα + λ4 −gs uα+ √ 1α 2 AB α (3.21) + −g[−ρk(χ)α α˙ α + λ (αAαα B − ηAB )]

34 where Xα is the particle number, sα is the entropy vector, λ2 λ3, λ4 are lagrangian multipliers that correspond to constrains that maintain the conservation laws of the fluid, and λAB are constrains that secure the orthonormality of the tetrads. This Lagrangian that describes a Weyssenhoff fluid is commonly used in several works within the frame of General Relativity, since it does not demand necessarily the presence of space-time torsion. Among others, D. Palle (in 1998) [14], Th. Chrobok, H. Herrmann and G. Rückner (in 2002) [15] and S.D. Brechet, M.P. Hobson and A.N. Lasenby (in 2007 and 2008) [16; 17] have used it to describe a relativistic spinning fluid. Yet, its use is common among the analyses within the Einstein-Cartan theory. One could name, among others, A.J. Fenelly, J.C. Bradas and L.L. Smalley (in 1988) [18] -also found in earlier work [19]-, L.L. Smalley and J.P. Krisch (in 1994) [20; 21] and B. Vakily and S. Jalalzadeh (in 2013) [22]. N.J. Poplawski’s and others work also presents similarities [23; 24; 10]. The analysis we have presented here was largely taken from A.J. Fenelly, J.C. Bradas and L.L. Smalley (in 1988) and K. Pasmatsiou (in 2014) [12]; however, it is probably M. Gasperini whom we should thank for this contribution [25]. However, there is another definition of this matter Lagrangian, by A.F. da F. Teixeira (in 1988) [26], slightly different from the above, that results however in the description of the very same fluid. This Lagrangian density has the following form √ α ˙ AB α LM = −g[−F (ρ, S) + EKIN − V + λρ∇α(ρu ) + λsS + λ (αAαα B − ηAB )] (3.22)

2 1α where F (ρ, S) = ρ(1 + ) in the internal energy density functional, EKIN = kρα˙ αα is the kinetic 2 2 1 2 1[α |2|β] energy density functional, V = 2nk ρ α [αα β]α α is the potential energy density and λρ, λs and λAB are lagrangian multipliers. The similarities to the previous analysis are obvious. What is also important to notice is that this is a macroscopic approach of the spin.

The Microscopically Derived Lagrangian As we pointed out before, the spin is a quantum quantity, since its presence is a quantum microscopic phenomenon. As a result, it seems interesting that -so far- the Lagrangians we saw were constructed from a macroscopic approach of the spin. Of course, these are not the only examples of Lagrangians; several have argued that such a Lagrangian for a ﬂuid made of spinning particles can be derived only by quantum mechanical approaches. As a result, one should start with the Dirac equation in a curved space-time

α A iγ e αdAΨ − mΨ = 0 where Ψ is the wave-function of the spinning particles, m is their mass, γα are the Dirac γ matrices i -derived from the Pauli matrices- and d = ∂ − ωαβ iγ γ is the derivative in respect to the A A 4 A [α β] αβ spin connection ω A. The Dirac equation is used since it is the relativistic expression for particles with spin 1/2; and our universe is dominated by such particles -e.g. electrons, protons and neutrons. From here, the free Dirac action is Z h√ i i S = −g (Ψ¯ γαeA d Ψ − d Ψ¯ γαeA Ψ) − Ψ¯ Mˆ Ψ d4x Ψ 2 α A A α where Mˆ is the mass expressed as a matrix. This expresses the behaviour of a free spinning particle. Deﬁning the vector 1 Γµ = − ωαβ γ γ 4 µ [α β] that is related to the aﬃne connection through the spin connection, we can write the Lagrangian of the free spinning particle as √ i L = −g (Ψ¯ γµ∂ Ψ − ∂ Ψ¯ γµΨ + Ψ[¯ γµ, Γ ]Ψ) − Ψ¯ Mˆ Ψ (3.23) free 2 µ µ µ

35 What we must do next is to describe the interaction between the spinning particles. Using the spin density matrix sλµν -to which we will refer later on-, we can write down the interaction terms analytically via the Dirac matrices. The Lagrangian that stands for these interactions is ¯ [λ µ ν] λµνσ 0 ¯ λ (2) ¯ [µ ν] Lint = −sλµν (iξΨγ γ γ Ψ + η ξ Ψγσ Ψ) = −s µν (ξ ∇λΨγ γ Ψ) (3.24) where ξ, ξ0 and ξ(2) are the interaction constants. As a result, the microscopic Lagrangian density of a ﬂuid composed of spinning particles -fermions in our case- is the sum of those two.

Lspin = Lfree + Lint According to S. Lucat (in 2014), whose analysis we have followed so far, this microscopic Lagrangian corresponds to the following macroscopic one 1 κ L = ∇ sµ − sµs − κsλµν s (3.25) spin 2 µ 4 µ λµν where sµ is the spin vector [27]. The full matter Lagrangian consists of this and of the classical Lagrangian for a perfect ﬂuid. LM = L˜M + Lspin

The Spinning Fluid Lagrangian Trying to bridge the differences between the macroscopic and the microscopic approaches of the spin, a new set of Lagrangians has shown up. Its importance lies on the fact that it treats the spin as a purely quantum quantity, but it adds it naturally to the classical matter fields terms, such as the classical energy density and pressure -it does not translate it to classical terms as the previous one. The most notable of these seems to be the following one, used by G. de Berredo-Peixoto and E.A. de Freitas (in 2009) and by A.H. Ziaie, P.V. Moniz, A. Ranjbar and H.R. Sepangi (in 2014) [28; 29]. What we assume first is the the Lagrangian density can split in two completely different parts: one for the minimal coupling of the spin, that can be written in terms of a quantum spinning fluid, and one for the induced effects of a spinning fluid, which can be written in terms of a classical perfect fluid. LM = LAC + LSF where LAC is the first one and LSF is the second. The second will take the form of a typical Lagrangian for a perfect fluid, with one small difference: the energy density and the pressure are replaced with the following effective energy density and pressure 1 1 ρ = ρ − κs sµν P = P − κs sµν (3.26) eff 2 µν eff 2 µν that express both the classical energy and pressure of the fluid along with the energy and pressure of the spin 4 As for the first term, that stands for the minimal coupling of the spin, we must run to the quantum theory -the one we have discussed before- and come up with the following term, which will describe all that the classical perfect fluid Lagrangian cannot describe. ¯ 5 µ ˆ LAC =< Ψγ γ Ψ > Sµ (3.27) Usually, the term jα =< Ψ¯ γ5γµΨ > is macroscopically reffered to as the axial spin current.

4As we mentioned before -see page 22- we can write an effective Lagrangian in order to include the cosmological constant in the field equations. If we do that, then the effective energy density and the effective pressure will include another term related to the cosmological constant 1 Λ 1 Λ ρ = ρ − κs sµν + P = P − κs sµν + eff 2 µν κ eff 2 µν κ

36 Other Approaches There have been other attempts to describe such a fluid. Just for the record, we will refer to another one:a semi-classical approach of spin by O.V. Babourova and B.N. Frolov (in 1998) [30]. In their work, they described a dilaton-spin fluid that is Weyssenhoff-like -since it obeys the Frenkel condition, we will see later; if, however, we assume the dilatonic charge of the fluid to be zero, we can end up to a Weyssenhoff-like spinning fluid. The Lagrangian they have chosen has the following form 5 L = −(n, S, s ) + ns uµuν + n(φ + τ + λ )uµ + n(χ + ζ )sµν l (3.28) M µν 8 µν µ µ µ µ µ ν where = (n, S, sµν ) is the internal energy density, sµν is the spin density matrix, n is the particle number density, lν is the material co-frame vector and φµ, τµ, λµ, χµ and ζµ are lagrangian multipliers. This description, according to Babourova and Frolov, arises naturally from the additional “kinetic” energy density of the spin 1 E = ns wµν KIN 2 µν where wµν is the intrinsic “angular velocity” of the spinning particles. If we assume that this “angular velocity” can be defined as 1 w = (α ˙ A α − α˙ A α ) µν 4 µ Aν ν Aµ

A where a µ the tetrads defined earlier, one can easily point out similarities between this description and the previous ones -especially of the Weyssenhoff fluid.

3.3.2 The Energy-Momentum density tensor The Stress-Energy-Momentum density tensor -more simply, the Energy-Momentum tensor- is a tensor defined by the matter Lagrangian that stands for all physical processes of a physical system. In a few words, it describes the nature of the physical system and allows us to study -as specified by the name of the tensor itself- the stress, the energy and the momentum of this system. This combination of all these parameters of the physical system in one tensor makes the connection of physics to geometry a lot of simpler. As one might have noticed from the relativistic field equations -the equation(3.1)-, this tensor can be immediately connected with the curvature -the Ricci tensor and the Ricci scalar, to be more concrete. The Energy-Momentum tensor is defined as follows by the variance of the matter Lagrangian in respect to the metric tensor. 2 δL T = −√ M (3.29) αβ −g δgαβ Now, if we consider a matter Lagrangian of a relativistic fluid, the Energy-Momentum tensor will take the form ˜ Tαβ = ρuαuβ + P hαβ + 2u(αqβ) + παβ (3.30) where ρ is the energy-momentum density, P is the scalar (isotropic) pressure applied by it, qα is the energy flux vector, παβ is the viscosity tensor -a traceless and symmetric tensor that represents the anisotropic pressure- and hαβ = uαuβ + gαβ is the projection tensor. This is the most usual form of the Energy-Momentum tensor in relativistic cosmology; however, it does not describe a fluid with spin. As a result, it cannot be used in the Einstein-Cartan theory. As we have seen before, many Lagrangians exist, that describe a perfect spinning fluid -suitable for our cosmological work. As a result, many energy-momentum tensors will exist. Here, we will examine some of them.

37 The Weyssenhoff Fluid Energy-Momentum Tensor According to S.D. Brechet, M.P. Hobson and A.N. Lasenby, the energy-momentum tensor of a fluid can be extended in order to include the spin of the particles of the fluid the fluid consists. Their κλ proposition was to add the term −2h ∇κsλ(αβ) and consequently give the energy-momentum tensor the following form κλ Tαβ = ρeff uαuβ + Peff hαβ − 2h ∇κ(sλ(αuβ)) (3.31) where sαβ is the canonical spin density tensor of a Weyssenhoff fluid -that we have seen before, but we will discuss further right after- and ρeff and Peff are the effective energy density and pressure -with the presence of spin- we have defined before [16; 17]. D. Palle gives us the very same form of the energy-momentum tensor [14]. Yet, Th. Chrobock, H. Herrmann and G. Rückner give us a slightly sifferent form of the same tensor; they are interested in expressing all of the four initial parts of the relativistic energy-momentum tensors in terms of spin, so that the effective character of this tensor can be shown. Their approach lead to the following relations

κλ 2 κλ ρeff = ρ + 2ωκλs Peff = P + ωκλs 3 (3.32) 2 q = hκ ∇ s λ + Aλs π = 2(σ κ + ω κ)s + h ω sκλ ν ν λ κ λν µν (µ (µ ν)κ 3 µν κλ 1 where σ = h µh ν ∇ u − h ∇κu is the shear tensor and ω = h µh ν ∇ u the vorticity αβ α β (ν µ) 3 αβ κ αβ α β [ν µ] tensor of the fluid -two of the kinematic parameters we will discuss in the next chapter. These effective parts of the energy-momentum tensor will lead, when applied, to the form of the energy- momentum tensor given by Palle, Chrobock et al and Brechet et al. [15]. Yet it is of quite some importance to know how the spin is attached to each of the four parts that describe the typical matter fields. Of course, this energy-momentum tensor is good for a relativistic approach of a spin fluid - exactly what Brechet et al. were interested at. It is not used so much in an Einstein-Cartan approach.

Others, like A.J. Fenelly, J.C. Bradas, L.L. Smalley, J.P. Krisch, etc. that work within the Einstein-Cartan framework have given a generalised form of the energy-momentum tensor, more suitable for this theory [18; 19; 20; 21]. Their approach, coming out of the form they have given to the Lagrangian density, is that the energy-momentum tensor must a have a relativistic part, that will describe a classical perfect fluid, and a spin part, that will describe all the effects the spinning particles will have on the energy and momentum of the fluid. The most suitable form of this tensor is ˜ κ κ κλ κλ Tαβ = Tαβ + ρsκ(αβ)u + ∇κ(ρs (αβ)) − ρwκ(αs β)uλ + ρsκ(αβ)w uλ (3.33) 1 where w = (α ˙ A α − α˙ A α ) is the intrinsic “angular velocity” of the spinning particles. κλ 4 κ Aλ λ Aκ A.F. da F. Teixeira (in 1988) has concluded to a similar energy-momentum tensor, while B. Vakily and S. Jalalzadeh (in 2013) have proposed a slightly different form

˜ µ µ µν µν µ ν µν Tαβ = Tαβ + sµ(αβ)u˙ + ∇µs (αβ) + S (α sβ)µν − sµν(αS β) − w (αsβ)µν u + sµ(αβ)w uν that could be considered an improvement of the previous form [22]. O.V. Babourova and B.N. Frolov, on the other side, have proposed the following form of the energy-momentum tensor.

˜ µ ν Tαβ = Tαβ + nu u ∇ν sµ(αβ) which resembles the one that D. Palle, Th. Chrobock et al. and S.D. Brechet et al. have chosen [30].

K. Pasmatsiou (in 2014), whose analysis we have used quite much, seems to agree with A.J. Fenelly, J.C. Bradas, L.L. Smalley and J.P. Krisch as for the form of the energy-momentum tensor [12].

38 The Microscopically Derived Energy-Momentum Tensor Concerning the ones who have derived the Lagrangian densities microscopically, the energy-momentum tensor they have calculated must also contain microscopical terms. Their analysis implies that the energy-momentum tensor will split in two parts: one will be absolutely classical and will contain all terms related to a relativistic perfect fluid, while the other will contain all terms related to the spin and will be written in terms of the microscopic quantum analysis of it. According to S. Lucat, the spin part of the energy-momentum tensor in respect to microscopic terms will be i κ T Ψ = (Ψ¯ γ d Ψ − d Ψ¯ γ Ψ) + ξ(2)g (Ψ¯ γ[κγλγρ]Ψ)2 (3.34) µν 2 µ ν µ ν 12 µν κλρ −1 It is interesting that the average of this tensor in respect to β = (kBT ) -where T is the temperature and kB is the Planck constant- is 0. < Tµν >β= 0 Also, this tensor can be correlated to a macroscopic one that is similar to the Weyssenhoff fluid energy-momentum tensors we have presented before [27].

The Canonical Energy-Momentum Tensor One problem arising from the previous -the field equations we have calculated before and the energy- momentum tensors we have presented- is that the conservation law for energy and momentum in the Einstein-Cartan theory will not have the same form as in General Relativity. In fact, while in General Relativity we have ˜ β ˜ ∇ Tαβ = 0 , we will see that in the Einstein-Cartan theory the left hand of this equation will not be zero. This was considered a major problem by many who tried to replace the energy-momentum tensor with a canonical one that would have a zero divergence, as the General Relativity would demand. S. Capozziello, V.F. Cardone, E. Piedipalumbo, M.Sereno and A. Troisi (in 2002) offered such an energy-momentum tensor [31]. This form was proposed by Belinfante and Rosenfeld -thus having their name- and is also the one supported by A. Trautman (in 2006) [6]. The Belinfante-Rosenfeld energy-momentum tensor is 1 t = T + ∇ (s µ + sµ + s µ ) (3.35) αβ αβ 2 µ αβ βα α β Finally, S. Lucat (in 2014) proposes a certain form of modified field equations that have isolated the “Riemannian” terms on the left hand [27]; exactly like Poplawski’s work, this can give us an effective energy-momentum tenso, however its form is slightly different:

h ˜ κ ˜ 1 ˜ κ 1 κ κλρi tµν = Tµν + ∇κs (µν) − ∇(µsν) + gµν ∇κs − κsκs − κsκλρs + 2 4 (3.36) 1 1 +κ2 sκλ s − sκs + s s (µ ν)κλ 2 κ(µν) 2 (µ ν) N.J. Poplawski seems to agree with this effective energy-momentum tensor [11; 23; 24]; M. Tsambarlis (in 1981), S. Capozziello et al. (in 2001) and K. Atazadeh (in 2014) have also assumed this form for the energy-momentum tensor in order to use it in the effective field equations [32; 4; 33].

Choosing an Energy-Momentum Tensor Having discussed a great number of energy-momentum tensors existing in the literature, it is high time we chose a form we can use. In fact, choosing a form is harder than it looks; it depends greatly upon the nature of the system one wishes to study. Since, our work so far -and for the next chapter- is quite abstract, it is not

39 necessary to make this decision now. What we will do is to short out two of the presented energy- momentum tensors and use them in order to give some examples. For this case, we will choose the energy-momentum tensor of A.J. Fenelly, J.C. Bradas, L.L. Smalley, J.P. Krisch, etc. (equation (3.33)). We will also use an energy-momentum tensor we can create by using the elements of Th. Chrobock, H. Herrmann and G. R¨uckner; this tensor will have the form

κλ 2 κλ κ λ λ Tαβ =(ρ + 2ωκλs )uαuβ + P + ωκλs hαβ − 2u(α(qβ) + h β)∇λsκ + A sβ)λ)+ 3 (3.37) 2 + π + 2(σ κ + ω κ)s + ω sκλ αβ (α (α β)κ 3 κλ However, in the ﬁfth chapter we will use a very simple energy-momentum tensor: that of the perfect ﬂuid. Tαβ = ρuαuβ + P hαβ This happens because the cosmological model on which we will emphasize in the Friedmann- Lemaˆıtre-Robertson-Walker and this energy-momentum tensor is the only one compatible to it.

3.3.3 The Spin density tensor In the previous section, we discussed the energy-momentum density tensor, that is defined as the variance of the matter Lagrangian in respect to the metric tensor. Yet, we know that the matter Lagrangian has also a variance in respect to the contorsion tensor, since the variance of the Einstein- Hilbert action breaks in those two variances. This second variance is -as expected- relative to the spin of the matter fields described by the matter Lagrangian. So we will call the tensor arising from this the spin density tensor -in reflection of the energy-momentum density tensor. 2 δL s = √ M (3.38) αβγ −g δKαβγ

This is the tensor used before in order to describe the effects of the spin on the total energy and momentum of the fluid. It is interesting that if we consider certain particles in our fluid -that is with a specific spin, as we did before- we can calculate the spin density tensor through Quantum Mechanics. For example, if we use leptons (mostly electrons and positrons) and quarks (mostly protons, anti-protons, neutrons and anti-neutrons) that all are fermions and have spin 1/2, we can use the Lagrangian density that is derived from the Dirac equation via the covariant derivative of a spinor in respect to the affine connection. Thus, we can take the spin density tensor as i c sαβγ = ~ Ψ¯ γ[αγβγγ]Ψ (3.39) 2 It is important to say here that this spin density tensor is fully antisymmetric, while the original spin density tensor shares only the symmetries of the torsion tensor [23].

The Spin Matrix and the Frenkel Condition As we have seen, the spin density tensor is of third order, thus possessing too many components. In order to simplify this and come up with less components, we have introduced the Weyssenhoff fluid, a fluid composed of spinning particles -the very one we spoke of before. This choice of a fluid allows us to break the spin density tensor in a second order tensor we call the spin matrix, or the canonical spin density tensor; the way though which this is done is called Frenkel condition. The canonical spin density tensor is an antisymmetric tensor of second order that is defined as

sαβγ = uγ sαβ (3.40) where uγ is the velocity vector.

40 The Frenkel condition, which holds for this tensor, arises naturally from a rigorous variation of the matter Lagrangian It requires the intrinsic spin of a matter ﬁeld to be spacelike in the rest frame of the system -usually the ﬂuid- we study; this is expressed with the following relation

β sαβu = 0 (3.41) also known as the point-particle approximation. This condition also serves -as we will explain in the next chapter- as the conservation of the spin length of each particle in the fluid [11; 12; 16; 18; 20; 23]. One interesting note is that the Frenkel condition -and the subsequent decomposition of the spin density tensor- does not allow any contractions of the original spin density tensor; the canonical spin density tensor is antisymmetric and, as a result, has no contractions at all. The fact that the spin density tensor has no contraction allows the Cartan equation to take the simpler form that we have seen. Furthermore, it means that the torsion tensor is also traceless in this case -something that one can easily check from the Cartan equations. In other words, one can say that it the conservation of the spin length of the particles that calls for the Frenkel conditions, while another might say that a preferred geometrical condition -a traceles torsion tensor- resulted to them. One must also notice that since the canonical spin density tensor is antisymmetric, it possesses only six components. As a result, we can maintain all necessary information even if we write it as a vector: the spin pseudo-vector 1 sˇ = sβγ (3.42) α 2 αβγ This pseudo-vector can be affiliated with the Pauli-Lubanski pseudo-vector for the spin in Quantum mechanics, that is defined as 1 sˇ = J βγ pµ α 2 αβγµ where J βγ is the three-dimensional angular momentum tensor and pµ is the momentum vector. 5

Geometric Classification of Torsion and Different Types of Spin Since the Einstein-Cartan-Sciama-Kibble theory of gravity was fully constructed, the main hypoth- esis as for what physical variable the torsion would stand for was spin. Yet, it is not yet clear if the usual quantum spin is this physical variable, or if the spin density tensor stands for something else. Many believe that it stands for the angular momentum in general and not only the spin; some believe that in the spin density tensor we have nothing more than the typical matter fields expressed in a different way; others would say that it hides something completely different -maybe some exotic forms of matter, or the infamous “quintessence”. So far little steps have been made into answering the question “What the spin density tensor really stands for?” While introducing the torsion tensor in the first chapter, we have referred to a decomposition of it by S. Capozziello, G. Lambiase and C. Stornaiolo (in 2001), in which the torsion tensor was devided in three parts -the V-torsion, the A-torsion and the T-torsion- and each of these parts was associated with a different property of the torsion tensor. S. Capozziello et al. moved even further in classifying the several cases that we meet the torsion tensor in the literature and the way it is used, according to which part of these three is dominant [4]. According to their analysis, the V-torsion is used as the spin source in a Schwarzschild space- time and as a source of anisotropies in a Friedmann-Lemaˆıtre-Robertson-Walker space-time. Its contribution to the canonical energy-momentum tensor would -in these cases- be of the form 8 4 8 4 V t = S S − g S Sν = κ2s s − κ2g s sν (3.43) αβ 3 α β 3 αβ ν 3 α β 3 αβ ν 5The momentum vector and the angular momentum tensor will be defined and discussed in the section for the conservation laws. As for the three-dimensional angular momentum tensor, it is derived from the angular momentum H γ γ tensor in this way: Jαβ = Jαβγ dj , where dj is the unit vector of a three-dimensional hyper-surface in the Riemann-Cartan space-time.

41 In the same manner, the A-torsion is used in description of classical Dirac particles, as well as fermions; the most usual studies of the Weyssenhoff fluid include this -quite reasonably, since the torsion pseudo-vector used here exists in a Weyssenhoff fluid, while the torsion vector does not. The contribution of the A-torsion to the canonical energy-momentum tensor is

A ˇ ˇ ˇ ˇν tαβ = 2Sα Sβ + gαβ Sν S (3.44)

We must notice here that these two parts of the torsion survive in a generalised cosmological model; as a result they are the only two reliable for cosmological work. The third part, the T-torsion does not appear in such models, yet it is used in simple supergravity theories as a Rarita-Schwinger spinor. Due to its complexity -and the many diﬀerent forms it might take- we will not discuss it here.

3.4 Conservation Laws

What is very important for Physics is the emergence of conservation laws in nature. We must know that a certain size we measure -like energy, momentum, mass, charge, spin, etc.- is conserved throughout the space-time. These conservation laws arise naturally from the Lagrangian densities we use and should be respected by any physical theory. They are, in a manner of speaking, pillars of any theoretical structure we have constructed or we are about to construct. Here, we would like to show the form the conservation laws for energy, momentum and spin take in the Einstein-Cartan theory.

3.4.1 The Conservation of Energy and Momentum It is self-proved that all energy and momentum of a closed physical system must be conserved; this comes as an immediate result of the symmetry we assume in time and space. In more simple words, the total amount of energy -as well as the total amount of momentum- in a physical system, that does not interact with another, remains the same throughout all the four-dimensional space-time in which this system is enclosed, despite all possible changes that can take place in it -in respect of time or space. In General Relativity, this conservation is portrayed by the so-called Energy-Momentum Con- servation Laws. These laws are easily parametrised in one relation by using the Energy-Momentun density tensor, that stands for all forms of energy, momentum and stress of the system we have chosen to study. This relation is ˜ ˜αβ ∇βT = 0 (3.45) Breaking this simple relation in terms of time-coordinates and space-coordinates, we can obtain the conservation laws for energy and momentum respectively, separated. It is easy to prove this conservation laws using only geometrical terms. Beginning with the second Bianchi identity ˜ ˜ ∇[αRβγ]µν = 0 we can generate the following relation 1 ∇˜ R˜α = ∇˜ R˜ β β 2 α and, using the deﬁnition of the Einstein tensor, we can obviously see that

˜ ˜αβ ∇βG = 0 (3.46)

Using now the Einstein equations without the presence of a cosmological constant (equation (3.1)), we obtain the equation (3.46) - which is the expression of the conservation laws [2; 8]. However, if we assume a Riemann-Cartan space-time, the previous conservation laws cannot take the simple form they have in the Riemannian case. This is easy to understand by looking the second

42 Weitzenböck identity in a Riemann-Cartan manifold (equation (2.63)). Of course, we can use the same path as before in order to determine the relation that expresses the conservation laws of energy and momentum in such a space-time. We simply begin from the twice contracted differentiating Weitzenböck identity, which can take the following form 1 ∇ν R − ∇ R = 2(S λµR − Sλµν R ) (αν) 2 α α λµ αλµν We observe that the modified Einstein tensor has appeared on the left hand of this relation. As a result, we can write 1 ∇ν G = ∇ν R − Rg = 2(S λµR − Sλµν R ) (3.47) αν (αν) 2 αν α λµ αλµν Finally, we use the modified Einstein equations -without the presence of a cosmological constant, as posed before- from equation (3.8) and obtain the relation expressing the conservation of energy and momentum in a Riemann-Cartan space-time 2 ∇βT = (S λµR − Sλµν R ) αβ κ α λµ αλµν If we use the equations (3.18) and (3.19), for the curvature and the Ricci tensor expressed in geometry and matter terms, and the equation (3.12) for the equivalence of torsion to spin, we can write this as 1 ∇βT = −s κλR − sκλν R + Rs − sλR (3.48) αβ α κλ ακλν 2 α (αλ) This is the mathematical expression for the Energy-Momentum Conservation Laws in an Einstein- Cartan universe. It is obvious that, unlike its General Relativity counterpart, this relation has a non-zero right hand; on the contrary, its right part is dominated by the spin density coupled to the curvature tensor, implying that in Einstein-Cartan theory the energy and the momentum are conserved, but this conservation of theirs relies strongly on the structure of the space-time [34; 35; 36; 37].

It is obvious -even more obvious than before- that the presence of torsion plays an important role 2 in the conservation of energy and of momentum. In fact, the terms uα(S λµR − Sλµν R ) κ α λµ αλµν 2 -when projecting along the velocity- and h κ(S λµR − Sλµν R ) -when projecting vertically κ α α λµ κλµν to the velocity- are the main diﬀerence we observe in these equations with the relativistic ones. The other, quite minor, diﬀerences are the presence of the spin via the spin-related terms of the energy-momentum tensor in both equations.

We should notice here, that the eﬀective and the canonical energy-momentum tensors follow the relativistic conservation law β ∇ tαβ = 0 with the covariant derivative of the Riemann-Cartan space-time [6].

Three specific forms of the Energy and Momentum Converation In order to understand better what we have explained here, we will have a look at three examples of energy-momentum tensors, for which we will calculate the exact conservation laws for energy and for momentum. These tensors are given by equations (3.31), (3.37) and (3.33) respectively; the first is the energy-momentum tensor of a Weyssenhoff perfect fluid, the second is a generalised approach of the same tensor, and the third is a different description of the Weyssenhoff fluid. First of all, in order to derive the energy and the momentum conservation laws separately, we must project the equation (3.48) in respect to time and space; the first is achieved via an inner

43 product with the velocity vector, ua, and the second via an inner product with the projection tensor, hab = gab + uaub. If we proceed to the first of these projections, we have a scalar equation 1 uα∇βT = −uαs κλR − sκλν R uα + Ruαs − uαsλR αβ α κλ ακλν 2 α (αλ) and if we proceed to the second, we have a vector equation 1 h µ∇βT = −h µs κλR − h µsκλν R + Rh µs − h µsλR α µβ α µ κλ α µκλν 2 α µ α (µλ) The first is the conservation law for energy, also known as the continuity equation; the second is the conservation law for momentum, that serves as a generalised Newton’s second law. Using the energy-momentum tensor of S.D. Brechet, M.P. Hobson and A.N. Lasenby from equation (3.31), along with the Frenkel condition from equation (3.41) since we have a perfect Weyssenhoff fluid, we have the following expressions for the continuity equation

α (κ λ) κλ α β ρ˙ + Θ(ρ + P ) − 2u u A ∇κsλα − 2h u ∇ ∇κ(sλ(αuβ)) = 0 (3.49) and the following for the conservation of momentum µ ν λ (ρ + P )Aα + DαP − 2hα sλ(µAν)∇ u = 0 (3.50) µ α µ α where Θ = ∇ uµ the expansion/contraction parameter of the space-time and A = u ∇µu the acceleration. On the other hand, using the form of this tensor that describes a non-perfect fluid, as given by Th. Chrobock, H. Herrmann and G. Rückner in equation (3.32) and (3.37), and the Frenkel condition expressed in the equation (3.41), the continuity equation will be 8 ρ˙ + Θ(ρ + P )+D qα − 2A qα − σ παβ + (ω sκλ)· + Θω sκλ+ α α αβ κλ 3 κλ (3.51) κλ κ λ κ λ αβ κ κ + Dκ(∇λs ) + Θu ∇ sκλ + 2A ∇ sκλ + 2σ (σ(α + ω(α )sβ)κ = 0 and the conservation of momentum will become 4 (ρ + P )A +D P +q ˙ + Θq + (σ + ω )qβ + Dβπ + π Aβ+ α α <α> 3 α αβ αβ αβ αβ 8 2 + ω sκλA + D (ω sκλ) + (hκ ∇ s λ + Aλs )·+ (3.52) 3 κλ α 3 α κλ α λ κ λα 4 + Θ(hκ ∇ s λ + Aλs ) + 2(σ + ω )(hκβ∇ s λ + A sλβ) = 0 3 α λ κ λα αβ αβ λ κ λ Finally, using the energy-momentum tensor for a spinning fluid given by A.J. Fenelly, J.C. Bradas, L.L. Smalley, J.P. Krisch, etc. in equation (3.33), we can take the third form for the continuity equation α α αβ κ κλ κλ α β ρ˙ + Θ(ρ + P )+Dαq − 2Aαq − σαβ π + (sκ(αβ)u − wκ(αs β)uλ − sκ(αβ)w uλ)u ∇ ρ+ κ λ α β κ α β + ρ(u − wκλu )u ∇ sκ(αβ) + ρs (αβ)u ∇ uκ+ α β κ κλ κλ α β + u ∇ ∇κ(ρs (αβ)) − ρ(wκ(αs β) + ρsκ(αβ)w )u ∇ uλ− α λ β κλ κλ β β − ρu u (wκ(α∇ s β) + s (α|∇ wκ|β) + sκ(αβ)∇ wκλ) = 1 = −uαs κλR − sκλν R uα + Ruαs − uαsλR α κλ ακλν 2 α (αλ) (3.53) and the third form of the equation of momentum conservation 4 (ρ + P )A + D P +q ˙ + Θq + (σ + ω )qβ + Dβπ + π Aβ+ α α <α> 3 α αβ αβ αβ αβ µ β κ µ β κ µ β κλ µ β κλ + hα ∇ (ρsκ(βµ)u ) + hα ∇ ∇ (ρsκ(µβ)) + hα ∇ (ρwκ(β s µ)uλ) + hα ∇ (ρsκ(βµ)w uλ) = 1 = −h µs κλR − h µsκλν R + Rh µs − h µsλR α µ κλ α µκλν 2 α µ α (µλ) (3.54)

44 In order to make the comparison with the relativistic expressions of the conservation laws, it is essential to present them here: the equation for the conservation of energy -known as the continuity equation- is 0 ˜ ˜ α ˜α αβ ρ + Θ(ρ + P ) + D qα − 2A qα − σ˜αβ π = 0 while the equation for the conservation of momentum -that resembles the relativistic Navier-Stokes equations- is

0 4 (ρ + P )A˜ + D˜ P + q + Θ˜ q + (˜σ +ω ˜ )qβ + D˜ βπ + π A˜β = 0 α α <α> 3 α αβ αβ αβ αβ ˜ where Θ the contraction/expansion parameter,σ ˜αβ the shear tensor,ω ˜αβ the vorticity tensor and ˜ Aα the acceleration vector in a Riemannian space-time -so that they will be discrete from those of an Einstein-Cartan space-time. It is clear that the conservation laws for energy and momentum are much more complicate in the Einstein-Cartan theory than in General Relativity. Not only extra terms appear on the right hand of the equations, that are related to the torsion, but also the energy-momentum tensor usually takes a very large and complex form so that it can also include the effects of spin. All these make the conservation laws of the Einstein-Cartan theory very difficult to work with -unlike their relatively easy counterparts in General Relativity. We must notice something more as well: when using a Weyssenhoff fluid, the extra torsion- related terms on the right hand of both equations vanish due to the Frenkel condition. This not only explains why the Weyssenhoff fluid is so frequently used in studies of Einstein-Cartan cosmology, but it also allows us to make a remark on the nature of the Weyssenhoff fluid. As we know, it is a fluid that is microscopically constructed by spinning particles -mots usually fermions; however, macroscopically it should behave as a “normal” fluid. As a result, several of the kinematic equations or the conservation laws that concern it are relatively easy -in comparison to other fluids used in the Einstein-Cartan theory- and look much more like the relativistic equations.

3.4.2 The Conservation of Spin What we explored so far was the conservation law for the energy and momentum, a law similar to its counterpart in General Relativity. This is only natural, since the energy-momentum tensor is the only physics-related variable used in both theories. However, in the Einstein-Cartan theory, since we have a non-vanishing torsion tensor, that has itself Weitzenb¨ock identities too, we must notice that an extra conservation law appears: the conservation law for spin. This is also natural, since the spin density tensor becomes the second physics-related variable of the Einstein-Cartan theory, next to the energy-momentum tensor. In the same way as before, we can use the Cartan equations and the Weitzenb¨ock identity for the torsion tensor and calculate the equation that expresses this conservation law. From equation (2.61), by contracting indices α and µ, we can take the relation

ν ∇ Sβνα = 0 and contracting it once more, ν ∇ Sν = 0 If we take the divergence of the spin density tensor and the Cartan equations, we will have 2 ∇ν s = − (∇ν S − ∇ S + g ∇ν S ) αβν κ βνα α β αβ ν and substituting our previous results into it, we reach the ﬁnal form of the conservation law for the spin [34; 35; 36; 37]. 2 ∇ν s = − ∇ S (3.55) αβν κ α β

45 A specific form of the Spin Conversation Right before we discussed the conservation law for energy and momentum; after presenting its general form, we also presented a number of examples by using specific energy-momentum tensors. All of them were derived for a spinning fluid, most notably the Weyssenhoff fluid. This system is one of the most usual for studying the spin and its connection to geometry. As a result, we consider presenting the specific form the conservation law for the spin takes in it. First of all, we begin by decomposing the spin density tensor as in equation (3.40)

sαβν = sαβuν

We know that in order for such a system to exist, the Frenkel condition must be valid

β sαβu = 0 or, in other words, the torsion tensor to be completely traceless. This would turn the conservation law from equation (3.55) into ν ∇ (sαβuν ) = 0 and via simple calculations s˙αβ + Θsαβ = 0 (3.56) ds wheres ˙ = αβ . This is the general form of the conservation law for the spin of a Weyssenhoﬀ αβ ds ﬂuid. However, we can write it in a simpler way, if we use the magnitude of the canonical spin density tensor 1 s2 = s sαβ 2 αβ Applying this to the equation (3.56), we can get a scalar equation for the conservation of spin

s˙ + Θs = 0 (3.57) which resembles the continuity equation in a homogeneous and isotropic universe.

3.4.3 The Conservation of Angular Momentum As we have just mentioned, spin is too a quantity that is conserved in physical systems. Its conservation is associated with the conservation of angular momentum, which in General Relativity is expressed as ˜ γ ˜ ∇ Mαβγ = 0 (3.58) ˜ ˜ ˜ where Mαβγ = (xα − x˘α)Tβγ − (xβ − x˘β)Tαγ is the angular momentum tensor -and xα andx ˘α are two coordinate vectors of the space-time. This tensor is obviously antisymmetric in “α” and “β” indices. It is possible to deﬁne this tensor in the Einstein-Cartan theory as well, proposing we have the energy-momentum density tensor.

Mαβγ = (xα − x˘α)Tβγ − (xβ − x˘β)Tαγ + sαβγ (3.59) where xα andx ˘α two arbitrary vectors. We must notice that the spin density tensor is added as well. This happens for a very speciﬁc reason: In General Relativity, the spin -a microscopical quantity- is not used as a property of the matter ﬁelds; in Einstein-Cartan theory, the spin plays a very important role, since it is coupled to space-time torsion. On can say that this adding of the spin density tensor in the angular momentum tensor -in the Einstein-Cartan theory- is similar to the coupling of spin to the angular momentum in Quantum Mechanics.

46 Then, using the relation for the conservation of energy and momentum (equation (3.47)), we can conclude to a relation for the conservation of the angular momentum as well. This relationship must have the form

γ γ ∇ Mαβγ =∇ [(xα − x˘α)Tβγ − (xβ − x˘β)Tαγ + sαβγ ] = 2 = [(x − x˘ )(S λµR − Sλµν R ) − (x − x˘ )S λµR − Sλµν R )]+ (3.60) κ α α β λµ βλµν β β α λµ αλµν γ γ γ + [Tαγ ∇ (xα − x˘α) − Tβγ ∇ (xβ − x˘β)] + ∇ sαβγ

γ where the term ∇ sαβγ can be replaced by using the Cartan equation. We can now make use of the specific form of the energy-momentum tensor we have -from equation (3.33) or from equation (3.37), for example- and find the relation of the conservation of spin. In the case of the Weyssenhoff fluid -the energy-momentum tensor is given by the equation (3.31)-, the Conservation Law takes the form

γ γ λ κ ∇ sαβγ = ∇ (sαβuγ ) = 2(Γ [α|κs|β]λ + p[αuβ] + q[αuβ]) (3.61)

β µ where pα = (ρeff +Peff )uα +u uµ∇ sαβ is the momentum vector. In other cases, the conservation law has a much more complex form. From here, we may conclude that

dsαβ = 2(Γ[α sβ]γ uλ + p[αuβ] + q[αuβ]) (3.62) ds γλ where s is the proper distance -or the proper time. This relation expresses the conservation of spin [α β] [α β] in terms of the canonical spin density tensor sαβ [23]. Notice that the terms 2p u + 2q u work as the antisymmetric part of the energy-momentum tensor, implying a deeper relationship between the spin density and the energy, momentum and stress density.

3.5 The Equations of Motion

The importance of the geodesic equations in relativistic physics is that they give the equations of motion of test particles that are not affected by any external forces (e.g. an electromagnetic field); in General Relativity, this is easy to think as well as to prove, since the space-time is Riemannian and no other fields appears that gravity, which is the curvature of the space-time itself. In the Riemann-Cartan space-time, things are quite more complicated: we must choose between the two equations we have just derived -the equations (2.28) and (2.30). But the choice is not easy and clear, since the torsional term appears. If we choose, as in the Riemann space-times, the curve with zero “acceleration” -since we want the test particles to have zero acceleration, as a result of their zero affection by external forces-, we end up with this extra term, which will be considered as a “force” applied by the space-time onto the particle in order to express the effects of the torsion to its motion. On the other hand, if we choose a geodesic curve -since we want the test particle to travel along the shortest path-, then we are forced to accept that our force-free motion does not have a zero acceleration. And there is another problem lurking: choosing one of them -regardless of how careful we have been when doing so- does not mean we can actually describe the motion of a spinning particle. The torsion of the space-time, that is included in the auto-parallel equation, would effect all the particles of the space-time uniformly, without taking notice of their spin; in the same time, the geodesic equation that includes no torsion, would have no effect on the spinning particles, despite the fact that a torsional space-time would be expected to change a lot their motion. According to V. de Sabbata, there is only one way to solve both of these problems: to adopt ourselves a force term that would couple torsion to the spin of the particle, if the latter is spinning, and would become zero, if the particle is spin-free. So, the equations of motion should take the form

d2xα dxβ dxγ + Γ˜α + F α = 0 (3.63) ds2 βγ ds ds

47 where F a is a force term related to the spin of the particle and including all interaction of this spin with the geometry of the space-time. It is usually taking the form 1 dt Z √ F α = (ˇsα −g)dx3 m ds α wheres ˇ is the spin pseudo-vector of the particle, g = det|gαβ| is the determinant of the metric tensor and m is the mass of the particle. This equation seems to solve the problem of choice, but allows a force term to re-appear in the theory, while the General Relativity has managed to vanish it. Of course, this is a problem arising from the choice of the speciﬁc space-time [9]. Another approach could be that of the Mathisson-Papapetrou-Dixon equations for the motion of a spinning particle in a curved space-time, which have been calculated for a Riemann space-time. Their original form of these equations is d dsαβ 1 muα + u = − Rα sµν uβ ds β ds 2 βµν dsαβ dsβµ dsαµ + uαu − uβu = 0 ds µ ds µ ds It has been proved that these equations can be generalised for a Riemann-Cartan space-time and work properly as the equations of motion of a test particle in the Einstein-Cartan theory; this work we will present right away [6; 35; 36].

3.5.1 The Mathisson-Papapetrou-Dixon Equations First of all, the equations of motion that M. Mathisson, A. Papapetrou and W.G. Dixon proposed are derived from the conservation law for energy and momentum. What we do is to form the following integrals Z Z Z T˜µν d3x T˜µν (xα − x˘α)d3x T˜µν (xα − x˘α)(xβ − x˘β)d3x ... which are non-zero in the general case; for a single-point particle at least some of these integrals must be non-zero. Assuming a single-pole particle, which has only mass, we will have only the ﬁrst of them Z M αβ T˜µν d3x = − u0 From here we end up to its equation of motion -which is none else than the geodesic one- and an equation for the conservation of its only property: the mass duα dm d M 00 + Γ˜α uµuν = 0 = = 0 ds µν ds ds u0 However, if we assume a pole-dipole particle, which is a spinning particle, we use the angular momentum tensor Z M λµν = −u0 (xλ − x˘λ)T˜µν d3x and the spin tensor Z Z sµν = (xµ − x˘µ)T˜ν0d3x − (xµ − x˘µ)T˜µ0d3x where M 0µν = 0 and sµν = −sµν by deﬁnition. Using the conservation law -from equation (3.44)- we can calculate the equations dsαβ u[α dsβ]0 u[α + 2 + 2 Γ˜[α − Γ˜0 M β]µν = 0 ds u0 ds µν u0 µν d M α0 + Γ˜α M µν − ∂ (Γ˜α )M βµν = 0 ds u0 µν β µν uα M αβγ = −sα[βuγ] + s0(βuγ) u0

48 The independent unknown variables we use in this system are the three spatial components of the velocity vector, the three spatial components of the spin tensor and the component M 00. We introduce the scalar 1 u m = (M 00 + Γ˜0 sµ0uν ) + α uν ∇ sα0 (u0)2 µν u0 ν that is equivalent to the mass of the particle -plus some terms relevant to the spin; we also deﬁne the momentum as dsαβ pα = muα + u β ds that possesses one more term -related to the spin-, if compared to the classical deﬁnition of momentum. By introducing these “mass” and “momentum” for the particle, we can end up to the following set of equations

dp d dsαβ 1 = muα + u = − Rα sµν uβ ds ds β ds 2 βµν dsαβ dsαβ dsβ]µ − 2p[αuβ] = + 2u[αu = 0 ds ds µ ds that are the Mathisson-Papapetrou-Dixon equations of motion for a spinning particle [37]. It is obvious that is the spin tensor goes to zero, we return to the geodesic equation that serves as equation of motion for non-spinning particles [2; 8]. Another way of resulting to these equations is via the use of the following Lagrangian density 1 1 L = mg uµuν − Γ˜α s µuν (3.64) 2 µν 2 µν α where m is the mass and sµν the spin of the pole-dipole particle, deﬁned as before. By following the standard formulation, we create the Euler-Lagrange equations that turn out to be the ﬁrst of the Mathisson-Papapetrou-Dixon equations. The second is derived in a similar way as before [39].

The question now is how do we generalise this method in order to obtain the same equations in a Riemann-Cartan space-time. The ﬁrst idea is to use the same Lagrangian density and construct the Euler-Lagrange equations; the result is indeed the generalised form of the Mathisson-Papapetrou- Dixon equations. We write the action of our Lagrangian as

α µ L = eµ pαu

α where pα = muα is the standard momentum of the particle and eµ is a tetrad ﬁeld, like the one we have used before in order to deﬁne our metric tensor. In order to include the spin of the particle, αβ we assume the presence of a gauge potential A µ to which the spin of the particle is coupled. As a result, the Lagrangian is rewritten as 1 L = e αp uµ − Aαβ s uµ (3.65) µ α 2 µ αβ As a result, if we follow the usual formulation we can write down the Euler-Lagrange equations as

1 ds 1 ds ∂ (e α)p uµ − ∂ Aαβ s uµ = e α αβ − Aαβ αβ [ν µ] α 2 [ν µ] αβ µ ds 2 µ ds The gauge potential is found to be equal to the affine connection and the Euler-Lagrange equations turn out to be -as expected- the first of the generalised Mathisson-Papapetrou-Dixon equations d dsαµ h dsµν 1 i muα + u = S α muµ + u − Rα sµν uβ (3.66) ds µ ds µβ ν ds 2 βµν dsαµ where now the momentum is defined as pα = muα + u [39; 40; 41; 42]. µ ds

49 In order to ﬁnd the second equation, we can either turn to the Ruthian analysis of the same Lagrnagian, or to require the Lorentz gauge we have used to be invariant. As a result, we must have

α µν [µ ν] u ∇αs = 2p u (3.67) dsαµ and if pα = muα + u , then we conclude to the second of the Mathisson-Papapetrou-Dixon µ ds equations [37]. dsαβ dsαβ dsβ]µ − 2p[αuβ] = + 2u[αu = 0 (3.68) ds ds µ ds One thing we can notice is that only the first equation changes form in the Einstein-Cartan theory; the second remains as it is. Probably this is an outcome of the different role of each equation. While the first is the actual equation of motion, providing us with the time evolution of the particle momentum, the second works mostly as a condition that controls the evolution of spin -that is probably the reason it resembles the conservation law for spin density. In the light of these, we can notice that the added term in the first equation actually reflects the coupling of torsion to the particle spin, via its momentum; this proves once more that the torsion does not effect to its particle in the same way, but in respect to its spin.

3.5.2 The Dixon-Souriau Equations Let us now imagine of an even more generalised picture: a massive, charged and spinning particle. This particle will move under the eﬀect of gravity -that is gravity-, torsion -that will be coupled to its spin- and the electromagnetic ﬁeld -that is governed by the Maxwell equations. We know that the Faraday tensor is given as

Fµν = 2∇[µAν] where Aµ is the electromagnetic potential. Furthermore, we know that the Lagrangian for a charged particle in a Riemann space-time is written as 1 L = mg uµuν + qA uµ 2 µν µ where q is the charge of the particle. As a result, it is easy to expand our Lagrangian for a charged spinning particle as 1 L = e αp uµ − Aαβ s uµ + qA uµ (3.69) µ α 2 µ αβ µ Following the lagrangian formulation, as before, we will end up to the following expansion for the first or our equations d dsαµ h dsµν 1 i 1 muα + u = S α muµ + u − Rα sµν uβ + qu F αβ + ksµν DαF (3.70) ds µ ds µβ ν ds 2 βµν β 2 µν and by the demand for the gauge potential to be invariant, we obtain the expansion for the second of our equations dsαβ dsβ]µ + 2u[αu + 2ksµ[αF β] = 0 (3.71) ds µ ds µ qg where k = , with g the gyromagnetic ration of the particle [43]. 2m We see that spin couples with the Faraday tensor, as we should expect. It is widely known that spinning particles are affected by the electromagnetic fields in a variety of ways -like the precession caused on a particle spin under the effect of a magnetic field. This explains why the electromagnetic field is included in the second equations too, since this one is focused completely on the evolution of spin.

50 3.5.3 Supplementary Conditions Taking a step back and looking again at the generalised Mathisson-Papapterou-Dixon equations, we cannot but think again that only the one of them serves as an equation of motion; the second is mainly a condition. Its main purpose is to govern the evolution of spin and supply it to the ﬁrst equation. However, it is not the only conditional equation. In the literature a lot of supplementary conditions have been used, all of whom are derived from the assumption the the mass of the particle and the length of its spin must be preserved. Furthermore, all of them make sure that the Mathisson-Papapetrou-Dixon equations are valid and that the following are true

α α α 2 u uα = −1 pαu = −m pαp = −m for each test particle -these can also been considered as normalisation conditions for the test particles. The ﬁrst that we meet is the Corinaldesi-Papapetrou condition, which demands that

αβ s ξβ = 0 (3.72) where ξβ is a Killing vector of the space-time; this condition may lead to the following integral of motion 1 pµξ − sµν ∇ ξ = 0 µ 2 ν µ that makes sure the world-line the particle centre of mass is a representing path. Also, this condition can be written in a more general form as

αβ s τβ = 0 where τβ is a tangent vector of congruence, that -in stationary space-times- is directed along the time-like Killing vector ξβ. The second is the Mathisson-Pirani condition, that demands

αβ s uβ = 0 (3.73) which denotes a transport according to the Fermi-Walker mechanism; this condition makes sure that the intrinsic centre of mass is determined in the rest frame of the body. The third is the Dixon-Tulezykyjew condition that demands

αβ s pβ = 0 (3.74) and provides us with the information that the intrinsic centre of mass moves relative to the rest frame of the body. These three conditions are used separately, with the last two to be eﬀective in the case of a spinning body, or of a group of spinning particles. In simple occasions, the one can replace the other; yet, there is a wide discussion as whether this is valid, since in more complex cases -eg. the ultra-relativistic limit or the non-stationary space-times- only one survives [44].

51 Chapter 4

The Kinematics and the Dynamics of the Einstein-Cartan Theory

4.1 The Kinematic Parameters

In order to discuss the kinematics of an Einstein-Cartan universe, we must first have defined the kinematic parameters, the geometrical/physical quantities that we will use in order to describe the Einstein-Cartan universe from this perspective. Then, we will discuss their evolution throughout the space-time. These are defined by the main kinematic quantity, the covariant derivative of the velocity, which can decompose as 1 ∇ u = Θh + σ + ω − A u (4.1) β α 3 αβ αβ αβ α β α µ ν where Θ = ∇ uα is the trace, σαβ = D<β uα> = hα hβ ∇<ν uµ> is the symmetric part minus µ ν µ the trace, ωαβ = D[β uα] = hα hβ ∇[ν uµ] is the antisymmetric part, Aα =u ˙ α = uµ∇ uα is the derivative of the velocity in respect to the proper time or proper distance (the acceleration), and hαβ = gαβ +uαuβ is the projection tensor. The above quantities are spatial and can easily be proved to fulfil the following relations

β β β α hαβu = 0 σαβ u = ωαβ u = 0 Aαu = 0 (4.2)

Using the Ricci identities (equation (2.28)), we can derive equations for the evolution of the following three variables in respect to the proper distance or the proper time: Θ, which can show us the contraction or the expansion of the space-time, σab, which can give us information about the shear of the space-time, and ωab, which indicates the vorticity of the space-time [45]. Of course, the decomposition we have suggested here is not the only one suggested in the bibliography. Mason and Tsambarlis (in 1981) have decomposed the covariant derivative of the velocity vector in the following way 1 ∇ u = ∇˜ u + 2h S µuν = Θ˜ h +σ ˜ +ω ˜ − A˜ u + 2h S µuν β α β α µα βν 3 αβ αβ αβ α β µα βν

˜ ˜ α ˜ µ ν ˜ where Θ = ∇ uα is the trace,σ ˜αβ = D<β uα> = hα hβ ∇<ν uµ> is the symmetric part minus the ˜ µ ν ˜ ˜ 0 ˜ µ trace,ω ˜αβ = D[β uα] = hα hβ ∇[ν uµ] is the antisymmetric part and Aα = u α = uµ∇ uα is the acceleration - all of them, in relativistic terms. This decomposition allows the following analysis in terms similar to the General Relativity, thus making the comparison of the two theories easier [46]. But this is not our case. We can, nevertheless, refer for a moment to the relations between the relativistic kinematic variables and the Einstein-Cartan ones. It is easy, by just using the deﬁnition of the covariant

52 derivative and the decomposition of the aﬃne connection to the Levi-Civita connection and the contorsion tensor -from equation (2.8)-, to conclude to the following relationships

˜ α Θ = Θ + 2S uα 1 σ =σ ˜ − 2 h κh λSµ + h Sµ u αβ αβ (α β) (κλ) 3 αβ µ κ λ µ ωαβ =ω ˜αβ − 2h[α hβ] S κλ uµ ˜ Aα = Aα

We notice that the expansion/contraction parameter, the shear tensor and the vorticity tensor change through the addition of a torsion-related term; this will also be depicted in their propagation and constraint equations. Only, the acceleration remains the same in both universes, since it is time-like and related to external ﬁelds.

Let us now move on with the evolution of the kinematic parameters. First of all we use the Ricci identities from the equation (2.46), along with the following identity that holds for the tangent vector uα, denoting the possible decompositions of its covariant derivatives,

µ µ µ u ∇µ∇βuα = ∇β(u ∇µuα) − ∇µuα∇βu (4.3)

µ where ∇β(u ∇µuα) = ∇βAα, in order to take the following relation

µ µ µ ν µ µ µ u ∇µ∇βuα = ∇βAα − ∇µuα∇βu − Rαµβν u u − 2Sνα u ∇ν uβ + ∇αuµA uβ

Then, we substitute the covariant derivative of the velocity vector with its decomposition from the equation (4.1), and we have

µ · µ ν ν µ µ u ∇µ∇βuα = (∇βuα) =∇βAα + Rα(µ|β|ν)u u − 2Sµα u ∇ν uβ + A uβ∇αuµ− 1 2 2 − Θ2h − Θσ − Θω − (4.4) 9 αβ 3 αβ 3 αβ µ µ µ µ − σαµσ β − ωαµω β − σαµω β − ωαµσ β

This relation can stand as the evolution of the quantity ∇βuα in respect to the proper distance or the proper time. As a result, we can use to it to draw any information about the propagation of the three kinematic variables [47].

4.1.1 The Raychaudhuri Equation Substituting the “α” and “β” indices in the equation (4.3), we can easily end up to the propagation equation for the Θ scalar. 1 uα∇ Θ = Θ˙ = −R uαuβ − Θ2 − 2(σ2 − ω2) + ∇αA + AαA − 2S uα∇ν uβ (4.5) α (αβ) 3 α α αβν 1 1 where σ2 = σ σαβ and ω2 = ω ωαβ. This equation -the ﬁrst of the propagation equations we 2 αβ 2 αβ need- is concerning the contraction and the expansion of the space-time and is known as the Landau- Raychaudhuri equation; its importance in the cosmological problems is great and undoubted. Of course, when we usually run across the Raychaudhuri equation in a cosmological problem, it does not have the compact geometrical form we have presented here; this is a strictly mathematical formulation, possessing the slightest possible tie with physics. As a result, even the term that α β α ν β connects physics and geometry -the term −R(αβ)u u − 2Sαβν u ∇ u is introduced as a mathematical expression and nothing more. Of course we have proved that this expression stands for the gravitational energy of a physical system and we have given its equivalent form in physics in the equation (3.40); furthermore, we know that the torsion tensor can be replaces any time with by the

53 spin density tensor from equation (3.15) If we use this, we can write the Raychaudhuri equation in the following, more “physics-like” way

˙ α β 1 1 2 2 2 α α Θ = −κ Tαβ u u − T − Θ − 2(σ − ω ) + ∇ Aα + A Aα+ 2 3 (4.6) 1 1 +κuαs ν ∇ uβ + κAαs − κuαs Θ αβ ν 2 α 2 α

We must notice that, in a Riemannian space-time, the Raychaudhuri equations have the form:

0 1 Θ˜ = −R˜ uαuβ − Θ˜ 2 − 2(˜σ2 − ω˜2) + ∇˜ αA˜ + A˜αA˜ αβ 3 α α or 0 1 1 Θ˜ = − κ(ρ + 3P ) − Θ˜ 2 − 2(˜σ2 − ω˜2) + ∇˜ αA˜ + A˜αA˜ 2 3 α α 1 Thus, the main differences of the equations in the two space-times is the addition of a “torsion” α ν β term - the term −2Sαβν u ∇ u - and the replacement of the Ricci tensor with the symmetric part of its Riemann-Cartan counterpart, which leads to the addition of “spin” terms -according to the energy-momentum tensor that we will use [45]. We must also notice something else. In relativistic cosmology, the Raychaudhuri equation is an indirect proof for the existence of a “Big Bang Singularity”. If we omit all terms relevant to the presence external fields -the terms that include the acceleration- and assume that the vorticity of the universe is zero, we can obviously see that Θ approaches −∞ and, consequently, all geodesic curves must shear off in a certain point in the past: the “singularity”. This is not so easy to prove in the Einstein-Cartan theory. Firstly, we cannot -as we will see later- demand a zero vorticity; secondly, we cannot omit the terms related to the spin; finally, we cannot be sure whether we can omit the terms related to the presence of external fields, since we are not sure whether we should be using the geodesic or the autoparallel curves - besides, both of them are causing trouble when trying to avoid the presence of a force applied on the test particle. As a result, the “Big Bang Singularity” -which appears as a one-way street in General Relativity- could bot be what we think it is, in an Einstein-Cartan universe; it might have happened in a very different way in a very different time; or, it might never have existed.

Three specific cases of the Raychaudhuri equation To make our point more obvious, we can try to give an even more specific form for the Raychaudhuri equation, one that we can obtain by using a specific energy-momentum tensor and replace it in the equation we have derived. Since we have not specified the energy-momentum tensor we use in this work, it is good to show this replacement for three of the dominant forms of the energy-momentum tensor; these forms are given from equation (3.31), when we consider a Weyssenhoff perfect fluid, and (3.37) and (3.33), when we consider a generalised Weyssenhoff fluid. If we use the energy-momentum tensor of equation (3.31), we can see that the first of the algebraic field equations is written as 1 1 R uαuβ = κ T uαuβ − T = κ(ρ + 3P ) + κhκλuµ∇ s (αβ) αβ 2 2 eff eff κ λµ 1 1 where ρ = ρ − κsµν s and P = P − κsµν s ; we also use the Frenkel condition, since we eff 2 µν eff 2 µν work with a perfect Weyssenhoff fluid. As a result the Raychaudhuri equation will be written as 1 1 Θ˙ = − Θ2 − 2(σ2 − ω2) + ∇αA + AαA − κ(ρ + 3P ) + 2κ2s2 + κhκλuµ∇ s (4.7) 3 α α 2 κ λµ 1Once more, if we wish to add the cosmological constant in the Raychaudhuri equation we can use the effective energy density and pressure and write 1 1 2 Θ˜ 0 = − κ(ρ + 3P ) − Θ˜ 2 − 2(sigma˜ − omega˜ 2) + ∇˜ αA˜ + A˜αA˜ + Λ 2 3 α α

54 The second case is a generalised Weyssenhoff fluid, as described by the energy-momentum tensor of equation (3.37). In the same way, the first of the algebraic field equations becomes 1 1 R uαuβ = κ T uαuβ − T = κ(ρ + 3P ) + 2κω sκλ (4.8) (αβ) αβ 2 2 κλ Again, we will also use the Frenkel condition from equation (3.41). As a result, the Raychaudhuri equation will take the form 1 1 Θ˙ = − Θ2 − 2(σ2 − ω2) + ∇αA + AαA − κ(ρ + 3P ) + 2κω sκλ (4.9) 3 α α 2 κλ Finally, the third case is another expression for the generalised Weyssenhoff fluid, as it is given from the equation (3.33). The first algebraic field equation is

α β α β 1 R(αβ)u u = κ Tαβ u u − T = 2 (4.10) 1 = κ(ρ + 3P ) + κsκ uαuβ∇ ρ + κρuαuβ∇ sκ + κρs wκλu uαuβ 2 (αβ) κ κ (αβ) κ(αβ) λ This time we will not use the Frenkel condition, since we give a generalised expression of the ﬂuid. Eventually, the Raychaudhuri equation becomes 1 Θ˙ = − Θ2 − 2(σ2 − ω2) + ∇αA + AαA − 3 α α 1 − κ(ρ + 3P ) + κsκ uαuβ∇ ρ + κρuαuβ∇ sκ + κρs wκλu uαuβ+ (4.11) 2 (αβ) κ κ (αβ) κ(αβ) λ 1 1 + κuαs ν ∇ uβ + κAαs − κuαs Θ αβ ν 2 α 2 α Of course, if we were to apply the Frenkel condition in this case too, only one spin-related term α β κ would survive: the term κρu u ∇κs (αβ). This application would make the equation (4.10) look like the equation (4.8) that was derived through a much simpler energy-momentum tensor.

4.1.2 The Shear Propagation Equation As before, we can take the symmetric part of the equation (4.3) and subtract the trace, thus we will ﬁnd the equation 1 uµ∇ σ = ∇ A + R uµuν − R uµuν h − 2Sµ uν ∇ u + µ αβ <β α> (α|µ|β)ν 3 µν αβ ν<α ν β> 1 2 + (∇ u Aµu + ∇ u Aµu − uν ∇ u Aµh ) − Θσ − σ σµ − ω ωµ 2 α| µ β β µ α ν µ αβ 3 αβ <α|µ |β> <α|µ |β> µ µ ν µ where ∇αuµA uβ +∇βuµA uα −u ∇ν uµA hαβ = 2A<αAβ>. As a result, we have the propagation equation of the shear tensor.

2 µ µ σ˙ αβ = − Θσαβ − σµ<ασ β> − ωµ<αω β> + ∇<αAβ> + A<αAβ>+ 3 (4.12) 1 +2R uµuν − 2S ν uµ∇ u − R uµuν h (α|µ|β)ν µ<α ν β> 3 (µν) αβ This equation is not used as much as the Raychaudhuri equation; it possesses, however, a lot of interest since the shear tensor brings information about the tidal forces that act on the cosmic ﬂuid and all the distortions caused on the space-time. µ ν What seems interesting in the shear propagation equation is the term R(α|µ|β)ν u u ; this term relates the energy and momentum of the physical system with the curvature of the space-time, but not in the straight way we have presented in the Section 3.1. This indicates that this term stands for much more than just this connections. Using the decomposition of the curvature tensor into

55 geometric and matter terms we have proposed (in equation (3.18)), we can simplify this particular term in the following way 1 1 R uµuν = E − (g R uµuν − R uµu − R uν u − R ) − Rh (4.13) (α|µ|β)ν (αβ) 2 αβ (µν) (βµ) α (αν) β (αβ) 3 αβ This allows us to connect this term immediately with the physical characteristics we are interested in; we also have the chance to connect it with the curvature at great distances, as it is expressed with the “electric” part of the Weyl tensor. Using the equation (4.12), along with the equations (3.15) and (3.17), we can replace the terms 1 2R uµuν an − R uµuν h , and using the equation (3.12), we can replace the torsion (α|µ|β)ν 3 (µν) αβ tensor, so that the shear propagation equation will take the form 2 σ˙ = − Θσ − σ σµ − ω ωµ + ∇ A + A A − E − αβ 3 αβ µ<α β> µ<α β> <α β> <α β> (αβ) 1 1 1 − κT uµuν g − T uµuν h − κT + κT u uµ+ (4.14) 2 µν αβ 3 µν αβ 2 αβ (α|µ |β) 1 1 +κuµs ν ∇ u + κuµs σ − κs A µ<α| ν |β> 2 µ αβ 2 <α β> We observe that almost every part of this equation is space-like. This means that multiplying κ it with the projection tensor hα would not make a signiﬁcant change; on the other hand, it could κ λ simplify some of the terms. Thus, we proceed into the multiplicationσ ˙ <αβ> = hα hβ σ˙ κλ, taking the following expression 2 σ˙ = − Θσ − σ σµ − ω ωµ + D A + A A − E − <αβ> 3 αβ µ<α β> µ<α β> <α β> <α β> <αβ> 5 1 − κT uµuν h − κh µh ν T + (4.15) 6 µν αβ 2 α β µν 1 1 +κuµs ν ∇ u + κuµs σ − κs A µ<α| ν |β> 2 µ αβ 2 <α β> This form of the shear tensor propagation equation -although slightly diﬀerent from the previous- is more comfortable and more usual in the literature.

We know that, in a Riemann space-time, the shear propagation equation will have the following form:

0 2 σ˜ = − Θ˜˜ σ − σ˜ σ˜µ − ω˜ ω˜µ + D˜ A˜ + A˜ A˜ + <αβ> 3 αβ µ<α β> µ<α β> <α β> <α β> 1 +2h κh λR˜ uµuν − R˜ uµuν h α β κµλν 3 µν αβ or

0 2 1 σ˜ = − Θ˜˜ σ − σ˜ σ˜µ − ω˜ ω˜µ + D˜ A˜ + A˜ A˜ − E˜ + κπ <αβ> 3 αβ µ<α β> µ<α β> <α β> <α β> αβ 2 αβ [45]. It is obvious that the main diﬀerence between those two -as with the Raychaudhuri equation- is the presence of torsion in the space-time. The terms that express the connection of geometry to κ λ ˜ µ ν matter, energy and momentum in the Riemannian space-time -notably the terms 2hα hβ Rκµλν u u 1 and − R˜ uµuν h - are replaced, in the Riemann-Cartan space-time, with their counterparts -the 3 µν αβ 1 terms 2h κh λR uµuν and − R uµuν h , that involve torsional features- and a new term α β κµλν 3 (µν) αβ ν µ arises concerning the torsion -being the term 2Sµ<α| u ∇ν u|β>. This second appearance of torsion in a propagation equation proves, beyond any shadow of a doubt, that the torsion of space-time plays a major role in the evolution of the kinematic parameters.

56 Three specific cases of the Shear Propagation Equation The propagation equation for the shear tensor, much alike the Raychaudhuri equation, includes several matter-related terms. These terms can be specified, if a certain energy-momentum tensor is applied. As we did before for the Raychaudhuri equation, we will use three of the given forms for the energy-momentum tensor -the equations (3.31), (3.37) and (3.33)- and replace the matter-related terms, so that the shear propagation equation will take an even more familiar form. Firstly, we will use the equation (3.31) that describes a perfect Weyssenhoff fluid; along with it, we will use the Frenkel condition from equation (3.41). The third of the algebraic field equations will be 1 1 h µh ν R = κh µh ν T − h T ) = κ(ρ − P )h − 2κh µs Dλu α β (µν) α β µν 2 αβ 2 eff eff αβ (α| λµ |β) 1 1 where ρ = ρ − κs sµν and P = P − κs sµν . Using this and the first of the equations eff 2 µν eff 2 µν -that we have calculated before-, we conclude to the form of the shear propagation equation

2 µ µ σ˙ <αβ> = − Θσαβ − σµ<ασ β> − ωµ<αω β> + D<αAβ> + A<αAβ> − E<αβ> + 3 (4.16) 5 + hκλuµ∇ (s )h − κh µs Dλu 3 κ λµ αβ (α| λµ |β) Something we should notice here is that there are no typical matter terms present; this is simple to explain, since the energy-momentum tensor we have used is one for a perfect fluid. Going on, we will use the energy-momentum tensor given by the equation (3.37) and describing a non-perfect Weyssenhoff fluid. The third of the algebraic field equations becomes 1 1 h µh ν R = κh µh ν T − h T ) = κ(ρ−P )h +κπ +2κω sκλh +2κ(σ κ+ω κ)s α β (µν) α β µν 2 αβ 2 αβ αβ κλ αβ (α (α β)κ From this as well as the first of the algebraic field equations and the Frenkel condition, we can find the form of the shear propagation equation for this energy-momentum tensor

2 µ µ σ˙ <αβ> = − Θσαβ − σµ<ασ β> − ωµ<αω β> + D<αAβ> + A<αAβ> − E<αβ> + 3 (4.17) 1 1 + κπ + (σ κ + ω κ)s + h ω sκλ 2 αβ (α (α β)κ 3 αβ κλ Finally, we will use the energy-momentum tensor for a generalised Weyssenhoff fluid, as given by the equation (3.33). The third of the algebraic field equations is written as 1 h µh ν R = κh µh ν T − h T ) = α β (µν) α β µν 2 αβ 1 = κ(ρ − P )h + κπ + κρh µh ν s uκ + κρD sκ + κh µh ν s ∇ ρ− 2 αβ αβ α β κ(µν) κ (αβ) α β κ(µν) κ µ ν κλ µ ν κλ − ρhα hβ wκ(µs ν)uλ + ρhα hβ sκ(µν w uλ In the same manner as before, the shear propagation equation will take the form 2 σ˙ = − Θσ − σ σµ − ω ωµ + D A + A A − E − <αβ> 3 αβ µ<α β> µ<α β> <α β> <α β> <αβ> 1 κ µ ν κλ µ ν µ ν + κπ + κ∇κ(s )u u h + κρs w u (u u h + κh h )+ 2 αβ µν αβ κ(µν) λ αβ α β (4.18) µ ν κ κ κλ µ ν +κρsκ(µν)hα hβ + D (ρs (αβ)) − κρwκ(µs ν)uλhα hβ + 1 1 +κuµs ν ∇ u + κuµs σ − κs A µ<α| ν |β> 2 µ αβ 2 <α β> Again, we can plainly see that if we apply the Frenkel condition only two of the spin-related terms κ µ ν κ κ would survive: the terms κ∇κ(s µν )u u hαβ and D (ρs (αβ)). As a result, this equation would look very much like the equation (4.15), despite having been derived from another energy-momentum tensor; the only difference would be the presence of the viscosity tensor in the equation (4.17), that could disappear if we demanded a perfect Weyssenhoff fluid.

57 4.1.3 The Vorticity Propagation Equation

Another of the kinematic parameters that have been discussed before is the vorticity tensor, ωαβ = µ ν hα hβ ∇[ν uµ], that stands for the rotational characteristics of the space-time. Again, this tensor and its propagation equation are not as usual in cosmology as the Raychaudhuri equation, however their importance is not little; this antisymmetric tensor allows us to describe the rotation of the cosmic ﬂuid and the role this rotation plays in its structure and evolution. To determine the evolution of the vorticity tensor in respect to the proper distance or proper time -the propagation equation- we will follow exactly the same idea we followed in calculating the previous equations , beginning with the equation (4.3). If we take the antisymmetric part of that equation, we will have the following relation

µ µ ν ν µ u ∇µωαβ =∇[βAα] + R[α|µ|β]ν u u − 2Sαβ u ∇ν uµ+ 1 2 1 + (∇ u Aµu − ∇ u Aµu ) − Θω − (σ ωµ − ω σµ ) 2 α µ β β µ α 3 αβ 2 αµ β αµ β

µ µ µ µ µ where ∇αuµA uβ − ∇βuµA uα = 0 and σαµω β − ωαµσ β = 2ω[α|µσ |β]. Furthermore, from the (equation (3.18)), we can show that

µ ν R[α|µ|β]ν u u = E[αβ] (4.19) And, ﬁnally, the torsion tensor can be replaced with the spin density tensor, via the equation (3.12). As a result, the vorticity propagation equation takes the form

2 µ 1 µ ν ω˙ αβ = − Θωαβ − ω[α|µσ |β] + hα hβ ∇[ν Aµ] + E[αβ] + 3 2 (4.20) 1 1 +κuµs ν ∇ u + κuµs ω − κs A µ[α| ν |β] 2 µ αβ 2 [α β] As we have said before, the vorticity tensor is antisymmetric, having thus only six components. Consequently, we can write it in the form of a vector, the vorticity pseudo-vector 1 ω = ωβγ (4.21) α 2 αβγ With this, we can simplify the vorticity propagation equation into a vector equation. It will have the following form

2 β 1 β γ 1 βγ ω˙ α = − Θωα − σαβ ω − αβγ ∇ A + αβγ E + 3 2 2 (4.22) 1 1 1 + κ uµs βν ∇ uγ + κuµs ω − κ sβAγ 2 αβγ µ ν 2 µ α 4 αβγ Once more, we notice that every term of the equation is space-like; so, the multiplication with κ the projection tensor hα will not make a diﬀerence, but it could clarify several things. Considering that, we proceed to multiply the equation (4.21) with the projection tensor, as we did before with the shear propagation equation -the equation (4.13). The result is

2 β 1 β γ 1 βγ ω˙ <α> = − Θωα − σαβ ω − αβγ D A + αβγ E + 3 2 2 (4.23) 1 1 1 + κ uµs βν D uγ + κuµs ω − κ sβAγ 2 αβγ µ ν 2 µ α 4 αβγ Now, comparisons with the literature will be much more comfortable.

Again, we know that in the Riemann space-time, the vorticity propagation equation is

0 2 1 ω˜ = − Θ˜˜ ω − σ˜ ω˜β − D˜ βA˜γ <α> 3 α αβ 2 αβγ

58 [45]. We can tell from the first look, that the main difference of the two equations is the three extra terms included by the Einstein-Cartan theory in order to express the effect of the torsion -the 1 1 1 terms κ uµs βν D uγ + κuµs ω − κ sβAγ . However, we have another very important 2 αβγ µ ν 2 µ α 4 αβγ 1 addition: the term Eβγ , which is taken from the antisymmetric components of the “electric” 2 αβγ part of the Weyl tensor; this terms does not appear in the relativistic equation, since the “electric” and “magnetic” parts are symmetric. The existence of antisymmetric components for those two fields rely -as one should expect- in the existence of torsion in the space-time, that makes once more itself noticeable. In fact, it is quite interesting that even if we consider a Weyssenhoff fluid -which 1 1 1 means that the spin-related terms κ uµs βν D uγ + κuµs ω − κ sβAγ would become 2 αβγ µ ν 2 µ α 4 αβγ 1 zero, the torsion-originated term Eβγ would remain. 2 αβγ It is interesting that the vorticity equation in both theories -the General Relativity and the Einstein-Cartan- does not include any terms relative to the matter, energy and momentum of the 1 space-time -although, we should notice that in Einstein-Cartan theory the term Eβγ could be 2 αβγ related to matter fields, yet we will presume it does not for the moment. As a result, the presence of spin in an Einstein-Cartan universe is introduced only through the torsion tensor. This brings a rather peculiar conclusion, to which we had referred a little earlier: it is not possible to have zero vorticity in a space-time that includes torsion. In relativistic cosmology, it is easy to consider a space-time with zero initial vorticity; in that case, and having excluded all terms related to external fields -the terms including the acceleration-, we haveω ˙ <α> = 0, which means that no vorticity will appear. In the Einstein-Cartan theory though, even if we consider a zero initial vorticity and exclude 1 (somehow) the terms related to external fields, we will end toω ˙ = Eβγ − Sβγν uµD u , <α> 2 αβγ αβγ ν µ since torsion must always be non-zero in this space-time; even if we consider a Weyssenhoff fluid as generator for torsion, which would omit the spin-related terms due to the Frenkel condition, the term 1 Eβγ would remain, stating that vorticity can be generated by the gravitational field alone. 2 αβγ 4.1.4 The Constraints The three equations we have seen so far are the main kinematic equations, because they show the propagation of the three main kinematic parameters -the contraction/expansion scalar, the shear tensor and the vorticity vector. However, they cannot be alone; they need constraints. The equations for the constraints are the space-like projection of the Ricci identities, in the same manner as the propagation equations are the time-like projection. Following the same thinking, we must acquire three equations of constraints: a scalar, a vector and a tensor one. First of all, we must take the Ricci identities (equation (2.46)) and project them in the three- dimensional space, thus multiplying them with the projection tensor.

κ λ µ κ λ µ ν κ λ µ ν hα hβ hγ (∇κ∇λuµ − ∇λ∇κuµ) = hα hβ hγ Rκλµν u − 2hα hβ hγ S κλ ∇ν uµ (4.24) Then, by taking the trace, the symmetric and the antisymmetric parts of this relation, we can conclude to the equations of the constraints. The ﬁrst is the constraint for the shear tensor; it is derived if we consider the trace of the equation (4.13) over the indices “κ” and “λ”. Its ﬁnal form will be 2 Dβσ = D Θ + Dβωγ − 2 Aβωγ − 2 Hβγ − h βR uν − 2h βS ν ∇ uγ (4.25) αβ 3 α αβγ αβγ αβγ α βν α βγ ν and, by using the equations (3.12) and (3.16) for the replacement of the torsion and Ricci tensors respectively, it can become

β 2 β γ β γ βγ β µ D σαβ = DαΘ + αβγ D ω − 2αβγ A ω − 2αβγ H − hα Tβµ u − 3 (4.26) 1 1 −2κh β s ν ∇ uγ + sγ ∇ u − s Θ α βγ ν 2 β γ 2 β

59 The second is the constraint for the “magnetic” part of the Weyl tensor -or, to be honest, for its symmetric components. This is taken from the symmetric part of the equation (4.13), minus the trace of it, and its form is

µ ν κ ν µ H(αβ) = µν<αD σ β> + D<αωβ> + 2A<αωβ> − 6hαβµνκS D u (4.27) After using the reversed Cartan equation (equation (3.12)), it can become 1 1 H = Dµσν + D ω + 2A ω − κ sµνγ D u + sµDν u − sν σµ (αβ) µν<α β> <α β> <α β> µν<α| γ |β> 2 |β> 2 |β> (4.28) And, ﬁnally, the third is the divergence of the vorticity vector. Obviously, this is derived from the antisymmetric part of the equation (4.13) and it will

β β β D ωαβ = A ωαβ − 12S Dβuα

In order to simplify it, -as we simpliﬁed the vorticity propagation equation- and transform in into a scalar one, we will use the deﬁnition of the vorticity vector (equation (4.11); the new form of this constraint will be α α γ β α D ωα = A ωα − 6αβγ S D u (4.29) or, if we use the equation (3.12),

α α γ β α D ωα = A ωα − 6καβγ s D u (4.30)

In the General Relativity, these three equations have the form 2 D˜ βσ˜ = D˜ Θ˜ + D˜ βω˜γ − 2 A˜βω˜γ − q αβ 3 α αβγ αβγ α ˜ ˜ µ ν ˜ ˜ Hαβ = µν<αD σ˜ β> + D<αω˜β> + 2A<αω˜β> ˜ α ˜α D ω˜α = A ω˜α

[45]. It is obvious that the main difference of the equations in the Einstein-Cartan theory is the addition of a term related to the torsion tensor and, via the Cartan equation, to the spin density tensor. Yet, two more differences appear. Firstly, any spin terms involved in the energy-momentum tensor will appear in the first constraint, while in the relativistic equations we do not have similar terms. Secondly, the antisymmetric components of the “magnetic” part of the Weyl tensor are involved to the second constraint, while in the relativistic equation no such term exists -since the “electric” and “magnetic” part in a Riemann space-time are symmetric.

Three specific forms for the First Constraint In the same manner with the Raychaudhuri and the shear propagation equations, we will replace the energy-momentum tensor in the first constraint by using three specific forms of it -from equations (3.31), (3.37) and (3.33). This will give the constraint a more specific and more familiar form and will allow us to study even better. The first case we will study is the perfect Weyssenhoff fluid; this one is described by the energy- momentum tensor of the equation (3.31). According to it, the second of the algebraic field equations is β µ β µ κ hα R(βµ)u = κhα Tβµ u = κD sκα Using this, along with the Frenkel condition from equation (3.41), the first constraint will become 2 Dβσ = D Θ + Dβωγ − 2 Aβωγ − 2 Hβγ + κDκs − 2κh βs Aγ (4.31) αβ 3 α αβγ αβγ αβγ κα α βγ

60 The second case is that of the non-perfect Weyssenhoff fluid, as expressed via the energy- momentum tensor of the equation (3.37). With this one, the second algebraic field equation becomes

β µ β µ κλ λ hα R(βµ)u = κhα Tβµ u = −κqα − κhακ∇λs − κA sλα From this and the Frenkel condition, we can see that the ﬁrst constraint will take the following form

β 2 β γ β γ βγ D σ = DαΘ + D ω − 2 A ω − 2 H − κq − αβ 3 αβγ αβγ αβγ α (4.32) κλ λ β γ −κhακ∇λs − κA sλα − 2κhα sβγ A Finally, we will study the generalised Weyssenhoff fluid described by the energy-momentum tensor of the equation (3.33). The second algebraic field equation will take the form

β µ β µ hα R(βµ)u = κhα Tβµ u = κλ β µ κ β µ κ β µ κλ = −κqα − κhακ∇λs + κρhα u ∇κs (βµ) + κhα u s βµ∇κρ + κρhα u sκ(βµ)w uλ And using it, the first constraint can be written as 2 Dβσ = D Θ + Dβωγ − 2 Aβωγ − 2 Hβγ − κq + αβ 3 α αβγ αβγ αβγ α β µ κ β µ κ β µ κλ + κρhα u ∇κs (βµ) + κhα u s βµ∇κρ + κρhα u sκ(βµ)w uλ− (4.33) 1 1 − 2κh β s ν ∇ uγ + sγ ∇ u − s Θ α βγ ν 2 β γ 2 β Again, we can notice that if the Frenkel condition is applied, only two terms would survive: the term β µ κ β µ κλ κρhα u ∇κs (βµ) and the term κρhα u sκµβw uλ -only this part of it and no the whole. This would make it look more like the equation (4.30), with one major significance: the presence of the energy flux vector. Yet, this could be omitted, if we assumed a perfect Weyssenhoff fluid.

4.2 Electrodynamics and Magnetohydrodynamics

Having studied the kinematic parameters and their evolution in the Einstein-Cartan theory, we can easily move on to study the hydrodynamics of this universe. In fact, the propagation equation (4.5), (4.13) and (4.22), and the constraint equations (4.25), (4.27) and (4.29), along with the conservation equations -derived from the relation (3.48)- are exactly the equations we need for this study; they describe the motion of self-gravitating system we have assumed to be a non-perfect fluid. However, we can go a step further than that: we can attempt to involve the electromagnetic processes in our analysis. The equations we will conclude will be a generalisation of the equations we have examined so far, only with this important addition. One can easily go back to the hydrody- namical approach if he demands that the electric and magnetic field are zero. However, we consider their involvement quite important, since it will allow us to understand the differences a magnetized fluid presents in the Einstein-Cartan theory. In other words, it will allow us to study the coupling of spin and torsion to the electric and magnetic fields.

4.2.1 Maxwell Equations and Conservation Laws As expected, we will begin our studies from the general description of the electrodynamics in a Riemann-Cartan space-time. As in Riemannian space-time, the electromagnetic phenomena are described by a set of equations concerning the propagation and the constraining of the electric and the magnetic field; a description that originates in the work of J.C. Maxwell. Of course, we know that these two fields are simply two different sides of the same coin, yet until today we tend to study them as discrete quantities as long as we remain within the classical regime. First, we consider the electromagnetic potential Aα and the two fields, Eα and Bα, that are defined from it: α α ˙α α αβγ E = −∇ Φ + A B = ∇βAγ (4.34)

61 where Φ = A0 the time component of the electromagnetic potential -also known as scalar potential. All these are uniﬁed in the antisymmetric ﬁeld tensor

γ Fαβ = 2∇[αAβ] = 2u[αEβ] + αβγ B (4.35) that we have seen in section 2.4; it is known as Faraday tensor. From here, we can see that the electric and the magnetic ﬁeld vectors are given as 1 E = F uβ B = F βγ (4.36) α αβ α 2 αβγ and that they are both space-like -so they are vertical to the velocity vector.

α α Eα u = 0 Bαu = 0

2 From here, we can calculate the Maxwell equations, that is the propagation equations and the constraints for the electric and the magnetic ﬁeld vectors. First, we can express this evolution of their in respect to the Faraday tensor. First of all, we assume the source of the electromagnetic ﬁeld to be the current vector Jα , which can decompose in the electric charge density µ and the spatial current Jα as

Jα = µuα + Jα (4.37)

-from here, we can tell that α Jα u = 0 since this is a space-like vector field by definition. A usual form the spatial current takes is Jα = ςEα , where ς is the conductivity of matter; this relates the electric current of matter to the electric field applied to it. In simpler words, this form is valid only when it is the electric field causing the motion of electric charges in a matter field. As a result, the current vector is written as

Jα = µuα + ςEα (4.38)

The current vector is associated to the endogenous property of the matter fields to interact electro- magnetically; as a result, they can be considered as generators of the electromagnetic fields and not just in interaction with them -something essential for the connection with the General Relativity or the Einstein-Cartan theory of gravity. The evolution of the Faraday tensor -which describes the behaviour of the electromagnetic field as a whole- relies on two things: the source of the field and the geometry of the space time. We have just examined the first of them; as for the second, the most suitable description is the one derived from the Bianchi identities -or the Weitzenböck, since we work on a Riemann-Cartan space-time. As a result, the first of the basic equations for the evolution of the electromagnetic field is known to be β ∇ Fαβ = Jα (4.39) which is identical to the relativistic one; this equations secures the generation of the electromagnetic field from the matter fields. In order to construct the second, we will apply the differentiating Weitzenböck identity on the Faraday tensor, so that we will have

µ ∇[αFβγ] = S[αβ Fγ]µ (4.40) which presents a difference with its relativistic counterpart: while in General Relativity, the right hand of this relation is zero, in the Einstein-Cartan theory the torsion appears there. This is apparently natural, since the torsion alters the behaviour of any vector field, yet its presence here implies a lot of things for the evolution of the magnetic fields -as we will see in a while. Usually, the second equation stands for the symmetries of the space-time concerning the existence of the

2This is the ﬁrst similarity we see with the “electric” and the “magnetic” ﬁelds of the Weyl tensor.

62 electromagnetic potential; here, things are slightly different, since this equations does not secure by itself that the electromagnetic vector can exist in such a space-time. If we attempt to project these two equations along time and space, we can obtain the propagation and the constraint equations for the electric and the magnetic field vectors separately; thus, the Maxwell equations will take a more familiar form. We begin by projecting the equation (4.37) along the velocity vector and vertically to its; if we do so, we can easily end up to the following two relations for the electric fields vector: the first denotes its time evolution 2 E˙ = − ΘE + (σ + ω )Eβ + AβBγ + DβBγ − J (4.41) <α> 3 α αβ αβ αβγ αβγ α and the second denotes its spatial constraining

α α D Eα + 2ω Bα = µ (4.42) These two equations represent the Maxwell-Ampere law and the Gauss’s law for the electric field respectively; they plainly show that this field is generated by the electric charges and currents and its evolution is related to them. Moving on, we project the relation (4.38) vertically to the velocity vector resulting to the following two equations for the magnetic field vector: its time evolution is given as 2 B˙ = − ΘB + (σ + ω )Bβ + AβEγ + DβEγ − 2 Sβγµ(E + uν Bκ) (4.43) <α> 3 α αβ αβ αβγ αβγ αβγ µ µνκ while its spatial constraint is

α α αβ [γ µ] µ D Bα − 2ω Eα = −2(αβγ S µ u B − Sµ B ) (4.44) These two equations stand for the Faraday-Henri and the Gauss’s laws for the magnetic field respectively; they show the geometric peculiarity of the magnetic field that is not directly generated from the matter fields, but follows a more geometric behaviour; in other words, they prove the non-existence of magnetic monopoles within the classical region. In order to compare our results to those of the relativistic theory, we present the relativistic form of the equations, which is

0 2 E = − Θ˜ E + (˜σ +ω ˜ )Eβ + A˜βBγ + D˜ βBγ − J <α> 3 α αβ αβ αβγ αβγ α ˜ α α D Eα + 2˜ω Bα = µ 0 2 B = − Θ˜ B + (˜σ +ω ˜ )Bβ + A˜βEγ + D˜ βEγ <α> 3 α αβ αβ αβγ αβγ ˜ α α D Bα − 2˜ω Eα = 0 [45]. It is only natural that the torsion tensor did not appear in the first two equations, while it is natural it appeared in the last two -because it has already made an appearance in the equation (4.38); what seems interesting here is the way it is coupled to the electric and the magnetic field vectors. Looking at the relation (4.41), we can see that the torsion interacts with the Lorentz force -the force “felt” by a charged particle when in the presence of an electromagnetic field; the same result can be extracted from the constraint equation too, but quite more difficultly. This result comes is agreement with our knowledge and understanding of physical phenomena, since we know that spin is interacting with the electromagnetic fields through the Lorentz force. Another interesting result we can conclude to from this equations is the generation of magnetic fields. In the standard theory -the General Relativity- the electric field is generated from the electric charges and currents of the matter fields; then the magnetic field appears due to the presence of the electric one -after all, they are bound to each other. This can be easily proved from the equations if we set all initial electric and magnetic fields to be zero, yet we assume the pres- ˙ ence of charges and currents; the electric field will appear first due to the equation E<α> = −J α, ˙ ˜β γ ˜ β γ and the magnetic will follow due to the equation B<α> = αβγ A E + αβγ D E . However, in

63 our case -the Einstein-Cartan theory- the magnetic fields can also be generated via the torsion; βγµ ν κ the added terms −2αβγ S (Eµ + µνκu B ) in the right hand of the propagation equation and αβ [γ µ] µ −2(αβγ S µ u B − Sµ B ) in the right hand of the constraint, make this plausible. Of course, the need for -at least- an initial electric fields still holds, but the magnetic field will present a completely different behaviour; it could be amplified or dumped very quickly, depending on the torsion of the space-time.

The next step in describing the electromagnetic phenomena in a Riemann-Cartan space-time -as in any similar space-time- is to consider the Lagrangian density and the energy-momentum tensor that includes the effects of the electromagnetic fields. We will follow the classical approach that is also followed in General Relativity: the Lagrangian density for the electromagnetic field will be 1 L = − F F µν + J Aµ em 4 µν µ so the energy-momentum tensor will take the form 1 T (em) = −F F µ − F F µν g (4.45) αβ αµ β 4 µν αβ If we replace the Faraday tensor with its decomposition in electric and magnetic field vectors as in equation (4.34), we can take another form for the energy-momentum tensor, which resembles a lot the form of a fluid. 1 1 T (em) = (E2 + B2)u u + (E2 + B2)h + 2P u + Π (4.46) αβ 2 α β 6 αβ (α β) αβ 1 1 where (E2 +B2) is the energy density, (E2 +B2) the mean pressure, P = EβBγ the Poynting 2 6 α αβγ vector -acting as the energy flux- and Παβ the anisotropic stress -acting as the viscosity- of the electromagnetic field. The similarities to a non-perfect fluid are obvious; in fact, these four quantities can function as add-ins in any energy-momentum tensor describing a non-perfect fluid. We should also note that this energy-momentum tensor is traceless

α T α = 0 and is usually called Maxwell stress tensor. Beginning from this energy-momentum tensor and the two Maxwell equation -the equation (4.37) and the (4.38)- we can also calculate the conservation laws for energy and momentum for an electromagnetic ﬁeld. The calculation are somehow trivial and follow the same idea as with the conservation laws for a ﬂuid -see Section 2.4. The result will be 5 ∇βT (em) = −F J µ + S µF F νβ (4.47) αβ αµ 2 αβ µν In the relativistic theory, we have the following conservation law

˜ β (em) µ ∇ Tαβ = −Fαµ J 5 It is obvious that the term S µF F µβ represents the changes caused in the conservation of 2 αβ µν electromagnetic energy and momentum by the form and structure of a torsional space-time.

Two specific forms for the Conservation of Energy, Momentum and Spin We can make a small pause here -as we have done before- so that we can examine the changes caused in the conservation laws for energy, momentum and spin of a non-perfect fluid. We will use the results we have derived from this section, along with the conservation laws we have derived for specific cases in the Section 2.4. What is more important in this note is to understand even better the coupling of spin to the electromagnetic field.

64 We consider a non-perfect Weyssenhoff fluid that is described by the energy-momentum tensor of equation (3.37); this is the most generalised energy-momentum tensor we can study, so it presents us with the most complex and the most complete behaviour of such a fluid. Of course, the Weyssenhoff fluid itself is a simplification of a much wider case: that of a spinning fluid for which we know nothing but the way it couples to matter -such as the one described by the energy-momentum tensor of equation (3.33). Our choice of this description -while we used both of them in our previous examples- resembles the necessity for a greater specification in our needs: the Weyssenhoff fluid and the point-particle description are very effective in a natural conception of a spinning fluid, but also very easy to handle. Consequently, it is a very good choice-for the moment, despite the assumptions it makes. Since we know how its part of the energy-momentum tensor for an electromagnetic field works, we can easily attach it to the hydrodynamic parts of the energy-momentum tensor and add it to the equations (3.51) and (3.52) we have already concluded. What we must take under serious consideration, though, is the addition of some more terms concluded from the conservation of the electromagnetic energy and momentum -the terms related to the right hand of the equation (4.45). It is easy to compute that

α β α u Fαβ J = −Eα J µ β µ ν hα Fµβ J = −µEα + αµν J B and, by using the equation (3.40), 5 uαS µF F νβ = − κs u[µPν] αβ µν 2 µν κ µ νβ β hα Sκβ Fµν F = κsαβP

So, all we have to do is to add those results to our previous -that from equations (3.51) and (3.52). So, the equation for the conservation of energy for a magnetized ﬂuid is 8 ρ˙ + Θ(ρ + P )+D qα − 2A qα − σ παβ + (ω sκλ)· + Θω sκλ+ α α αβ κλ 3 κλ κλ κ λ κ λ αβ κ κ + Dκ(∇λs ) + Θu ∇ sκλ + 2A ∇ sκλ + 2σ (σ(α + ω(α )sβ)κ− (4.48) 5 − E J α − κs u[αPβ] = 0 α 2 αβ In the same manner, the equations for the conservation of momentum for a magnetized ﬂuid is 4 (ρ + P )A + D P +q ˙ + Θq + (σ + ω )qβ + Dβπ + π Aβ+ α α <α> 3 α αβ αβ αβ αβ 8 2 4 + ω σκλA + D (ω σκλ) + (hκ ∇ s λ + Aλs )· + Θ(hκ ∇ s λ + Aλs )+ 3 κλ α 3 α κλ α λ κ λα 3 α λ κ λα κβ λ λβ µ ν µ + 2(σαβ + ωαβ )(h ∇λsκ + Aλs ) − µEα − αµν J B + κsαµP = 0 (4.49)

µ ν It is interesting to notice that the terms −µEα + αµν J B are the the most usual expression of the Lorentz electromagnetic force i a covariant formulation.

Another thing we can do is to extend the equation (3.62) for the conservation of spin in order to include the electromagnetic effects; it is already clear, after all, that spin and electromagnetic fields interact. Looking at the form of this equation, we can see that the momentum and the energy flux of the matter fields are included; these are the terms we can alter in order to obtain the form of the spin conservation law for a magnetized fluid. First of all, we know that the Poynting vector behaves vary similarly to the energy flux, so it will be added under the same conditions; second, we understand

65 the energy density and pressure that are used in the momentum vector must also include the electric and magnetic field magnitudes. Eventually, the momentum vector will take the form 2 8 pα = (ρ + P )uα + (E2 + B2)uα + ω s uα + u u ∇µsαβ 3 3 κλ κλ β µ while the conservation law for the canonical spin density tensor will hold its old form, plus a new term for the Poynting vector. dsαβ = 2(Γ[α sβ]γ uλ + p[αuβ] + q[αuβ] + P[αuβ]) (4.50) ds γλ From here, we can see the role the electromagnetic fields play in the evolution of the spin density. What is rather interesting, though, is the non-coupling of spin with any of the electromagnetic terms. Obviously, the electric and magnetic field -exactly like the energy density and the pressure- contribute to the evolution of spin density, but they do not interact directly with it while in this contribution; this interaction is presented as coupling in other equations, because it explains other phenomena. The only exception appears to come via the Poynting vector in equations (4.47) and (4.48). This is an interesting result as well; the Poynting vector plays the role of the energy flux vector for the electromagnetic fields. So, the spin coupling to it functions as if the spin regulates the flux of the electromagnetic fields. This also seems quite reasonable if one things that the spin density -via the torsion tensor- is coupled to the Lorentz force in equation (4.42); in other words, the spin density interacts with -or regulate- the effects of the electromagnetic field on a charged particle towards a specific direction. We can easily extract the conclusion that a the torsion of the space-time, or the spin of the matter fields, acts on the charged particles towards the direction that is defined by the electromagnetic fields themselves -either through the Lorentz force, or through the Poynting vector 3. Another interesting comment is the fact that the spin coupling to the Poynting vector has a different effect on the continuity equation and on the Navier-Stokes’ one; in fact, it has the exactly opposite result. In the first case, the spin appears to accelerate the evolution of the energy density, while in the second it appears to resist to the effects of the pressure.

We could also consider a diﬀerent type of energy-momentum tensor, one that describes a generalised spinning ﬂuid like the equation (3.33). In this case, it is not as easy to obtain a relation for the conservation of spin, but we can generalise our results for the conservation of energy and momentum; this time our basis are the equations (3.53) and (3.54).

Finally, we should consider the conservation of charge, one of the most important conservation laws in physics. In General Relativity, this conservation is derived from the the ﬁrst of the Maxwell equations -given in terms of the Faraday tensor- and the antisymmetry of the Faraday tensor. It is easy to reach the following equation ˜ α ∇ Jα = 0 which expresses the conservation of the electric current; by projecting this one towards the velocity vector and using the decomposition of the current vector, we can easily get to 0 ˜ ˜ α ˜ α µ + Θµ + D Jα + AαJ = 0 which is the continuity equation for charge -the relation that assures us of the charge conservation. This is one of the few relations that remain the same in the Einstein-Cartan theory, obviously because the equations used to derive it are not aﬀected by the torsion. As a result, the conservation of the current vector is α ∇ Jα = 0 (4.51) and the equation for the conservation of charge that is taken from here, is α α µ˙ + Θµ + D Jα + AαJ = 0 (4.52) 3In comparison, one can see the coupling of spin to the Faraday tensor in the Dixon-Souriau equations of motion for a charged spinning particle -the equations (3.70) and (3.71)

66 4.2.2 Propagation of Electromagnetic Waves One of the most important results of Maxwell’s theory on electromagnetic fields is the existence of electromagnetic waves; that is the wave-like propagation of the electric and the magnetic fields along the vacuum space. The existence of these waves is undoubted in the General Theory of Relativity. What we will attempt to do is to produce similar waves in the Einstein-Cartan theory. The idea is pretty much the same, since it depends on the propagation equations of the electric and magnetic fields we have already presented above -the equations (4.40) and (4.42) respectively. What we need to do first is to include the energy-momentum-stress terms for the electromagnetic fields in our algebraic field equations. We will again use a non-perfect Weyssehoff fluid, as described by the energy-momentum tensor of the equation (3.37), plus the terms related to the energy, momentum and stress of an electromagnetic field. By simply adding these terms to the energy-momentum tensor we have h 1 i h 1 2 i T = ρ + (E2 + B2) + 2ω sκλ u u + P + (E2 + B2) + ω sκλ h − αβ 2 κλ α β 6 3 κλ αβ 2 − 2u (q + P + hκ ∇ s λ + Aλs ) + π + Π + 2(σ κ + ω κ)s + ω sκλ (α β) α β) λ κ β)λ αβ αβ (α (α β)κ 3 κλ Then, we apply the relation (3.15), (3.16) and (3.17) to this energy-momentum tensor and we result to the following algebraic equations 1 1 1 R uαuβ = κ T uαuβ − T = (ρ + 3P ) + (E2 + B2) + 2κω sκλ (4.53) (αβ) αβ 2 2 2 κλ

β µ β µ κλ λ hα R(βµ)u = κhα Tβµ u = −κ(qα + Pα ) − κhακ∇λs − κA sλα (4.54)

1 h µh ν R =κh µh ν T − h T ) = α β (µν) α β µν 2 αβ 1 1 1 = κ ρ − P + E2 + B2 h + κ(π + Π ) + 2κω sκλh + 2κ(σ κ + ω κ)s 2 3 3 αβ αβ αβ κλ αβ (α (α β)κ (4.55)

As it happened with the conservation laws before, it is easy to see that the extra terms have followed their matter counterparts.

What we need to do right after is to show the coupling of curvature -or gravity, as we know- to the electromagnetic ﬁeld. For this reason, we will use the Ricci identities for a tensor ﬁeld -from the section 2.2.2. We begin as

κ κ κ 2∇[α∇β]Fµν = RαβµκF ν + RαβνκFµ − 2Sαβ ∇κFµν If we contract along the indices “β” and “ν”, we will have

ν ν ν κ κ νκ ∇α∇ Fµν − ∇ ∇αFµν = −R αµκF ν − RακFµ − 2Sα ∇κFµν and if we contract along the indices “β” and “µ”, we will have

µ µ ν κ κ µκ ∇α∇ Fµν − ∇ ∇αFµν = −R µακF ν − RµκFα − 2Sα ∇κFµν

ν µ By using the equation (4.38) we can replace the terms ∇α∇ Fµν and ∇α∇ Fµν with the terms ν ∇αJµ and ∇ν Jα respectively; then, by using the equation (4.39) we can rewrite the terms ∇ ∇αFµν µ and ∇ ∇αFµν . After that, we change the index “µ” of the ﬁrst equation to “β” and the index “ν” of the second one to “β” again. If we subtract the second equation form the ﬁrst, we will result to the following interesting expression for the Faraday tensor

µ µ ν µ µν ∇ ∇µFαβ = −2R [αβ]ν F µ + 2R[α|µF |β] − 4Sα ∇ν Fµβ + 2∇[βJα] (4.56)

67 that shows most beautifully the coupling of the electromagnetic fields to the curvature and the torsion of space-time. If we wish to replace the curvature, Ricci and torsion tensors from equations (3.18), (3.19) and (3.12) respectively, we will see the coupling of the electromagnetic fields to long-run gravity -via the “electric” and “magnetic” parts of the Weyl tensor- and to the energy, momentum, stress and spin of the matter fields -that “create” the curvature and torsion. We can see this result with the electric and the magnetic fields separately if we deconstruct the Faraday tensor; although it is much easier to use the Ricci identities for a vector field -from equation (2.40)- separately to each of those vectors. The results would be

β β 2∇[µ∇ν]E = R E − 2S ∇βE α αµν β µν α (4.57) β β 2∇[µ∇ν]Bα = R αµν Bβ − 2Sµν ∇βBα These equations are supplying us with an extra constraint -a second-order constraint- for the electromagnetic ﬁeld

In order to find the propagation equations for the electromagnetic fields, we will use these two equations we have resulted to along with the four Maxwell equations. Firstly, we will take the time derivative -again the second-order time derivative- of the equations (4.40) and (4.42); we will follow the procedure only for the electric field vector, since it is exactly the same in both cases. 2 2 E¨ = − Θ˙ E − ΘE˙ + (σ ˙ +ω ˙ )Eβ + (σ + ω )E˙β+ <α> 3 α 3 a αβ αβ αβ αβ β γ ˙ β γ β ˙γ + ˙αβγ A B + αβγ A B + αβγ A B + β γ β γ · ˙ + ˙αβγ D B + αβγ (D B ) − J α

ν We know that the three-dimensional Levi-Civita symbol is given as αβµ = αβµν u , so its temporal ν β γ · β ˙γ β γ derivative will be ˙αβµ = αβµν A . We also know that αβγ (D B ) = αβγ (D B + ΘD B ). Finally, we use the equations (4.41), (4.42), (4.43), (4.9), (4.17), (4.23) and (4.57) to replace the temporal derivatives of the magnetic field, the expansion/contraction scaler, the shear tensor and the vorticity tensor and the spatial derivatives of the electric and the magnetic field vectors. The result is the wave equation for the electric field vector that goes as 1 1 3 E¨ − DµD E = κ(ρ + 3P )E + κπ Eβ − κ Bβqγ + <α> µ α 3 α 2 αβ 2 αβγ 10 3 + κω sκλE + κ(σ κ + ω κ)s Eβ − κ Bβ(D sγλ + A sγλ)+ 3 κλ α (α (α β)κ 2 αβγ λ λ β γλ µν β ˙ + 2καβγ B s Aλ + κµν<αs Aβ>B − Dαµ − ΘJα − Jα + 5 1 4 + σ − ωγ − Θh E˙β + σ + ωγ − Θh Eβ− αβ αβγ 3 αβ 3 αβ αβγ 3 αβ 4 2 1 − σ µσ Eβ + Eβσγµω + σ2 − ω2 E + ω ω Eβ + AµA E − <α β>µ αβγ µ 3 3 α 3 <α β> µ α 5 2 − Aβγµν D E + 2AβD E + BβDγ Θ + A˙ βBγ + BβD ω − 2 αβγ µ ν <α β> 3 αβγ αβγ <α β> 3 − Bβγµν D ω + 2ωβD B − 2 σβ D<γ Bµ> + AβBγ + 2 αβγ µ ν <α β> αβγ µ αβγ 7 4 + Aβω B + Bβω A − 3AβB ω + 3 AβσγµB − 3 β α 3 β α β α αβγ µ β β β γµν β γµν β − E(αβ) E − H(αβ) B + αβγ (E Eµν − 3B Hµν ) − RαβE (4.58)

µ ν µ ν µ where Rαβ = hα hβ Rµν + Rαµβν u u + ∇µuα∇βu − Θ∇βuα the spatial Ricci tensor, that we will discuss in the following chapter. In an analogous way, we can result to the wave equation for

68 the magnetic ﬁeld vector 1 1 3 B¨ − DµD B = κ(ρ + 3P )B + κπ Bβ + κ Eβqγ + <α> µ α 3 α 2 αβ 2 αβγ 10 3 + κω sκλB + κ(σ κ + ω κ)s Bβ + κ Eβ(D sγλ + A sγλ)− 3 κλ α (α (α β)κ 2 αβγ λ λ 2 − 2κ EβsγλA − κ sµν A Eβ − µω + 2 AβJ γ + DβJ γ + αβγ λ µν<α β> 3 α αβγ αβγ 5 1 4 + σ − ωγ − Θh B˙β + σ + ωγ − Θh Bβ− αβ αβγ 3 αβ 3 αβ αβγ 3 αβ 4 2 1 − σ µσ Bβ + Bβσγµω + σ2 − ω2 B + ω ω Bβ + AµA B − <α β>µ αβγ µ 3 3 α 3 <α β> µ α 5 2 − Aβγµν D B + 2AβD B − EβDγ Θ − A˙ βEγ − EβD ω + 2 αβγ µ ν <α β> 3 αβγ αβγ <α β> 3 + Eβγµν D ω − 2ωβD E + 2 σβ D<γ Eµ> − AβEγ − 2 αβγ µ ν <α β> αβγ µ αβγ 7 4 − Aβω E − Eβω A + 3AβE ω − 3 AβσγµE − 3 β α 3 β α β α αβγ µ β β β γµν β γµν β − E(αβ) B + H(αβ) E + αβγ (B Eµν + 3E Hµν ) − RαβB (4.59)

These two wave equations are presenting the behaviour of the electric and magnetic field over time and their wave nature. The electric and magnetic field are propagated throughout a curved and torsional space-time under the restrictions set by these two equations; which are but a derivation of Maxwell’s classical theory on the behaviour of the electromagnetic fields. What we can see in these equations is that the curvature and torsion of the space-time, as well as its kinematic variables affect the propagation of the electromagnetic waves. Specifying that even further, the local spatial curvature -expressed by the spatial Ricci tensor-, the tidal gravitational forces and the gravitational waves -expressed by the “electric” and “magnetic” parts of the Weyl tensor- are driving the propagation of the electric and magnetic field by coupling themselves to the propagated vector fields; the kinematic variables also couple to the electric and the magnetic field vectors guiding theme along the space-time; finally, the matter field terms -the energy density, the pressure, the energy flux, the viscosity and the spin density- also contribute to this propagation, since the electromagnetic waves travel through the matter as well. These couplings -especially the coupling to gravitational terms- is one of the main arguments in favour of a deeper connection between the gravitational and the electromagnetic interactions -as described by the Einstein’s and the Maxwell’s theories respectively, or the classical extensions of those. An interesting realisation concerns the symmetry of the equations. One can easily notice that the main difference between the two equations is the addition of “sources”; in the first equation, regarding ˙ the propagation of the electric field, the role of the sources is played by the terms −Dαµ−ΘJα −Jα , while in the second one, regarding the propagation of the magnetic field, this role is played by the 2 term − µω + 2 AβJ γ + DβJ γ . Assuming that the electric charges and currents of the 3 α αβγ αβγ matter fields vanish, then the two equations will be completely symmetric; one would easily transcend from the first to the second by substituting Eα with Bα and Bα with −Eα , and from the second to the first by the exactly opposite actions. This not only proves the inner connections between the electric and the magnetic field -that are actually two different faces of the same elementary interaction-, but also stands as an argument againts the existence of magnetic monopoles -at least within the limits of the classical theory 4. Finally, the two wave equations we have calculated present a similar form to the wave equations presented in the relativistic theory [50; 51]. There are two main differences between the two cases,

4The inner connection between the electric and the magnetic ﬁeld can be also shown by the Maxwell equations -the equations (4.40), (4.41), (4.42) and (4.43)- that also have a symmetric form. The inexistence of the magnetic monopoles is also proved by these equations much better than its was proved here, since only the two of them -the Maxwell-Ampere and the Gauss’s laws for the electric ﬁeld- contain “source” terms.

69 the first concerning the antisymmetric parts of the tidal gravitational fields -the “electric” and the “magnetic” parts of the Weyl tensor- and the second concerning the presence of spin terms coupled to the electric and the magnetic fields. The first is not really different, since the “electric” and “magnetic” parts of the Weyl tensor were present in the relativistic case as well. The second also seems natural, since the spin terms are coupled to the electric and the magnetic field vectors in the same way the “normal” mater field terms are; but this mostly a peculiarity of the description we have chosen for the matter fields, since the Weyssenhoff fluid is a simplifying case with spin-related terms added naturally to the energy density, the pressure, the energy flux and the viscosity of the fluid.

4.2.3 Kinematics of a Charged Fluid Let us assume a ﬂuid that behaves as we have described before, but contents also electric charge of density µ and is completely unconductible -so we consider the ﬂuid to have null conductivity. The current vector is simply given as Jα = µuα and the conservation of charge equation is

µ˙ + Θµ = 0

It is easy to rewrite the Maxwell equations in this case, since only the first of them changes as follows 2 E˙ = − ΘE + (σ + ωγ )Eβ + AβBγ + DβBγ (4.60) <α> 3 α αβ αβγ αβγ αβγ We wish to examine the kinematic behaviour of charged fluid; the only way to do so is by the conservation laws and the kinematic equations. In order to keep our results general, we will use a generalised Weyssenhoff fluid described by the energy-momentum tensor of equation (3.37) -as we did before in order to calculate the conservation laws.

As long as the conservation laws are concerned, the equations (4.47) and (4.48) we have resulted before remain the same with two minor exceptions; since the spatial electric current is zero, the α µ ν terms Eα J and αµν J B vanish. As for the kinematic equations, the only thing we have to do is add the extra terms related to the energy and momentum of the electromagnetic ﬁeld. For the three propagation equations we will simply have 1 1 Θ˙ = − Θ2 − 2(σ2 − ω2) + ∇αA + AαA − κ(ρ + 3P ) − κ(E2 + B2) + 2κω sκλ (4.61) 3 α α 2 κλ

2 µ µ σ˙ <αβ> = − Θσαβ − σµ<ασ β> − ωµ<αω β> + D<αAβ> + A<αAβ> − E<αβ> + 3 (4.62) 1 1 + κ(π + Π ) + (σ κ + ω κ)s + h ω sκλ 2 αβ αβ (α (α β)κ 3 αβ κλ

2 β 1 β γ 1 βγ ω˙ <α> = − Θωα − σαβ ω − αβγ D A + αβγ E + 3 2 2 (4.63) 1 1 1 + κ uµs βν D uγ + κuµs ω − κ sβAγ 2 αβγ µ ν 2 µ α 4 αβγ In the same manner, the constraints will be 2 Dβσ = D Θ+ Dβωγ −2 Aβωγ −κ(q +P )−κh ∇ sκλ−κAλs −2κh βs Aγ (4.64) αβ 3 α αβγ αβγ α α ακ λ λα α βγ

70 1 1 H = Dµσν + D ω + 2A ω − κ sµνγ D u + sµDν u − sν σµ (αβ) µν<α β> <α β> <α β> µν<α| γ |β> 2 |β> 2 |β> (4.65)

α α γ β α D ωα = A ωα − 6καβγ s D u (4.66) We notice that only three of them have actually changed and this was -as stated- only by the addition of the terms relevant to the energy and momentum of the electromagnetic field. Further- more, one can state that for slow and non-rotational motions of the fluid, the electric charges are not moving, producing thus no current. In this case, the magnetic field vector should vanish; this means we should also omit the Poynting vector and rewrite the above equations only with the presence of the electric field.

4.2.4 Kinematics of a Magnetized Fluid Following, we will give another more important example of the electromagnetic fields applied to matter fields; we will assume a fully conductive Weyssenhoff fluid that has no displacement current. From the Ohm’s law for the conduction current

β γ Jα = ς(Eα + αβγ u B ) and the assumption that the conductivity ς is inﬁnite, we result to

β γ Eα + αβγ u B = 0 (4.67) which relates the electric and the magnetic field vectors. From here, it is obvious that the description of this fluid will be quite simpler, since only one of the two fields is necessary. We usually choose the magnetic field; thus, this description is known as Magnetohydrodynamics. Consequently, from the four Maxwell equations we will need only two: the propagation equation for the electric field -also known as the Faraday-Henri law- and for the magnetic field -also known as Maxwell-Ampere law. After substituting the electric field form equation (4.59), the first takes the form 2 B˙ = − ΘB + (σ + ωγ )Bβ − γκλ[Aβu B ) + Dβ(u B )] (4.68) <α> 3 α αβ αβγ αβγ κ λ κ λ ; after neglecting the displacement current, the second takes the form

β γ β γ αβγ D B − Jα + αβγ A B (4.69)

. We notice that the spin-related terms do not survive in this case.

As before, in order to study the kinematic behaviour of this magnetised Weyssenhoff fluid, we need to examine the conservation and the kinematic equations of it. We will assume again the energy-momentum tensor of equation (3.37) and we will move to our calculations as before. First of all, if we wish to write the conservation equations, we simply have to omit the terms 5 concerning the electric field; notably the terms E J α and κs u[αPβ] from the continuity equation α 2 αβ µ and the terms µEα and κsαµP from the Navier-Stokes’ equation. An additional equation can be easily extracted from the conservation of charge; it will be

α α D Jα + AαJ = 0 (4.70)

As for the kinematic equations, we will write them again following the results we gave before. Firstly we will write the propagation equations 1 1 Θ˙ = − Θ2 − 2(σ2 − ω2) + ∇αA + AαA − κ(ρ + 3P ) − κB2 + 2κω sκλ (4.71) 3 α α 2 κλ

71 2 µ µ σ˙ <αβ> = − Θσαβ − σµ<ασ β> − ωµ<αω β> + D<αAβ> + A<αAβ> − E<αβ> + 3 (4.72) 1 1 + κ(π + Π ) + (σ κ + ω κ)s + h ω sκλ 2 αβ αβ (α (α β)κ 3 αβ κλ

2 β 1 β γ 1 βγ ω˙ <α> = − Θωα − σαβ ω − αβγ D A + αβγ E + 3 2 2 (4.73) 1 1 1 + κ uµs βν D uγ + κuµs ω − κ sβAγ 2 αβγ µ ν 2 µ α 4 αβγ and then we will show the constraints 2 Dβσ = D Θ + Dβωγ − 2 Aβωγ − κq − κh ∇ sκλ − κAλs − 2κh βs Aγ (4.74) αβ 3 α αβγ αβγ α ακ λ λα α βγ

1 1 H = Dµσν + D ω + 2A ω − κ sµνγ D u + sµDν u − sν σµ (αβ) µν<α β> <α β> <α β> µν<α| γ |β> 2 |β> 2 |β> (4.75)

α α γ β α D ωα = A ωα − 6καβγ s D u (4.76) We notice, in both sets of equations, that the effects of the electromagnetic field are minimized in comparison to the results of the previous example; to specify this statement, we see that the terms related to the electric field and the Poynting vector have vanished. This happens only due to the relation between the electric and the magnetic field we have found -the equation (4.59).

It is interesting to show that a great part of the relativistic analysis remains in this one as well. For this we can use the Raychaudhuri equation -after all it is the only one that includes the magnetic field. From here it is easy to prove that the stream lines of the cosmic fluid are related to the magnetic field. In fact, as with the relativistic case, the magnetic lines seem here to freeze within the fluid, following the stream lines (see [49; 52; 53] for the relativistic case); the only thing that can break this is the vorticity that is necessarily present and that carries the spin with it.

4.3 The “Electric” and “Magnetic” Parts of the Weyl Tensor

In a previous section of this chapter, we surveyed the kinematic equations that are derived by the time-like and the space-like components of the Ricci identities (equation (2.46)); these were respectively the propagation equations and the constraints of the main kinematic parameters we have presented in the beginning of the section. We can, however, consider another case of kinematic equations, the ones that are derived by the time-like and the space-like components of the Bianchi identities (equation (2.65)). These equations will present the evolution throughout time and space of some geometric variables. In order to choose the geometric variables whose evolution we want to study, we must consider what kind of such variables we have in hand. The easiest set is the “electric” and the “magnetic” part of Weyl tensor, that are defined by the equations (2.58) and (2.59). The two symmetric tensors are expressing the form of curvature in great distances - in other words, the gravitational field in great distances. The derivation of the propagation equations and the constraints of the “electric” and the “magnetic” parts -in other words, the equations of time and space evolution respectively- is easy proposing we use the special form of the generalised Bianchi -also known as Weitzenböck- identities for the Weyl tensor (equation (2.66)). This relation can also be written in respect to the physical parameters instead of the geometrical ones, if we use the equation (3.19) in order to replace the Ricci

72 tensor and the Ricci scalar with the energy-momentum tensor and the torsion tensor. In this way, the Weitzenb¨ock identities for the Weyl tensor will look like 1 ∇ν W =κ ∇ T + g ∇ T − ∇ ∇ν S + 2∇ ∇ S − ∇ (S Sν )+ αβµν [α β]µ 3 µ[α β] [α β]µν [α β µ] [α β]µν κ ν κ + 4Sν[α Rβ]κµ + 2Sαβ Rκµ

We can also use the equation (3.12) in order to replace the torsion tensor with the spin density tensor and take the following relation 1 4 1 ∇ν W =κ ∇ T + g ∇ T − κ ∇ ∇ν s − ∇ (s sν ) + ∇ ∇ s − ∇ ∇ s − αβµν [α β]µ 3 µ[α β] [α β]µν [α β]µν 6 [α β µ] 3 µ [α β] 1 1 1 1 − 2κ s κ − s δ κ + δ κs R ν − κ s κ − s δ κ + s δ κ R ν[α 2 ν [α 2 ν [α β]κµ αβ 2 α β 2 β α κµ (4.77)

This form is a lot more helpful, if we want to derive the equations in respect to physical parameters immediately [16; 45; 53]. To be honest, the transformation from geometry terms could be continued if we used equation (3.18) and (3.19) to replace the curvature and the Ricci tensor. But the sole gain of this step would be the addition of complexity to our relations; it would not change anything in the following process. So, in order to keep a low degree of diﬃculty in our study, we will leave them be. The only 1 1 thing we should have in mind when we see the terms −2κ s κ − s δ κ + δ κs R ν and ν[α 2 ν [α 2 ν [α β]κµ 1 1 −κ s κ − s δ κ + s δ κ R in the propagation and constraint equations for the “electric” and αβ 2 α β 2 β α κµ “magnetic” parts, is that they consist of coupling terms: spin density couple to itself, spin density coupled to energy and momentum, and spin density coupled to the “electric” and “magnetic” ﬁelds.

A small comment we could make here is the obvious fact that the equations we will derive right after, are justifying the name of the two tensors. Their form and structure will remind us quite much of the Maxwell equations for the electromagnetic field. In that case, we have two time-evolution equations and two space-evolution equations for the electric and the magnetic field vectors; in this case, we will have four similar equations for the evolution of the gravitational field.

4.3.1 The Propagation Equations In order to acquire the propagation equations for the “electric” and the “magnetic” parts of the Weyl tensor, we need to project the equation (4.33) along the velocity vector and decompose it properly. To do so, we follow an easier path: the projections we will take will have the form 1 h hγ u ∇κW αβ + ∇[αRβ] + δ [α∇β]R− α<µ ν> β γκ γ 6 γ λακ β λβκ α αβκ − 2(S R κγλ + S R κγλ + S Rκγ ) = 0

1 hγ hρ hσ ∇κW αβ + ∇[αRβ] + δ [α∇β]R− ρσ<µ ν> α β γκ γ 6 γ λακ β λβκ α αβκ − 2(S R κγλ + S R κγλ + S Rκγ ) = 0

The ﬁrst of these will serve as the propagation for the “electric” part of the Weyl tensor, while the second will transform itself to the propagation equation for the “magnetic” part [15]. The calculations are long but somehow trivial [16; 53].

73 If we make the proper replacements as before, the first of these will become h 1 h hγ u ∇κW αβ + κ ∇[αT β] + δ [α∇β]T + α<µ ν> β γκ γ 3 γ 4 1 + κ ∇[α|∇λs|β] + ∇[α(sβ] sλ) − g ∇[α∇βsλ] + ∇ ∇[αsβ] + γλ γλ 6 γλ 3 γ 1 1 1 1 i + 2κ sλ[α|κ + sλg[α|κ + gλκs[α Rβ] + κ sαβκ − sαgβκ + sβgακ R = 0 2 2 κγλ 2 2 κγ and by further computations, we will finally have our first propagation equation:

˙ µ ν µ µ ν ν E<αβ> = − ΘEαβ + µν<αD H β> + 3σ<α Eβ>µ + µν<α 2A Hβ> − ωµEβ> + 5 − κh hγ u ∇[µT ν] + κh T˙ − µ<α β> ν γ 6 αβ 4 1 − κh hγ u ∇[µ|∇λs|ν] + ∇[µ(sν] sλ) − g ∇[µ∇ν sλ] + ∇ ∇[µsν] − (4.78) µ<α β> ν γλ γλ 6 γλ 3 γ 1 1 − 2κh hγ u sλ[µ|κ + sλg[µ|κ + gλκs[µ Rν] − µ<α β> ν 2 2 κγλ 1 1 − κh hγ u sµνκ − sµgνκ + sν gµκ R µ<α β> ν 2 2 κγ

In the same manner, the second equation can become h 1 hγ hρ hσ ∇κW αβ + κ ∇[αT β] + δ [α∇β]T + ρσ<µ ν> α β γκ γ 3 γ 4 1 + κ ∇[α|∇λs|β] + ∇[α(sβ] sλ) − g ∇[α∇βsλ] + ∇ ∇[αsβ] + γλ γλ 6 γλ 3 γ 1 1 1 1 i + 2κ sλ[α|κ + sλg[α|κ + gλκs[α Rβ] + κ sαβκ − sαgβκ + sβgακ R = 0 2 2 κγλ 2 2 κγ and with further computations, it will transform into the second propagation equation:

˙ µ ν µ µ ν µ ν H<αβ> = − ΘHαβ − µν<αD E β> + 3σ<α Hβ>µ − µν<α 2A Eβ> + ω Hβ> + 1 − κ D[ρT σ] − κ h[ρ Dσ]T − ρσ<α β> 3 ρσ<α β> 4 1 − κ hγ hρ hσ ∇[µ|∇λs|ν] + ∇[µ(sν] sλ) − g ∇[µ∇ν sλ] + ∇ ∇[µsν] − ρσ<α β> µ ν γλ γλ 6 γλ 3 γ 1 1 − 2κ hγ hρ hσ sλ[µ|κ + sλg[µ|κ + gλκs[µ Rν] − ρσ<α β> µ ν 2 2 κγλ 1 1 − κ hγ hρ hσ sµνκ − sµgνκ + sν gµκ R ρσ<α β> µ ν 2 2 κγ (4.79)

We know that the relativistic form of the propagation equation for the “electric” part is

˜0 ˜ ˜ ˜ µ ˜ ν µ ˜ ˜µ ˜ ν ˜ ν E <αβ> = − ΘEαβ + µν<αD H β> + 3˜σ<α Eβ>µ + µν<α 2A Hβ> − ω˜µEβ> + 5 − κh hγ u ∇˜ [µT ν] + κh T 0 µ<α β> ν γ 6 αβ and for the “magnetic” part

˜ 0 ˜ ˜ ˜ µ ˜ν µ ˜ ˜µ ˜ ν µ ˜ ν H <αβ> = − ΘHαβ − µν<αD E β> + 3˜σ<α Hβ>µ − µν<α 2A Eβ> +ω ˜ Hβ> + 1 − κ D˜ [ρT σ] − κ h[ρ D˜ σ]T ρσ<α β> 3 ρσ<α β>

74 It is easy to observe the main differences of the two sets of equations. Again -as in the previous section- we have to replace all Riemannian terms with their Riemann-Cartan counterparts. Fur- thermore, the torsion of the space-time comes with a direct effect via its presence in the terms µ κ ν κ µν κ λ κ λ κ +2u (2Sν<α Rβ>κµ + S<αβ> Rκµ) and −<α| (Sλµ Rνκ|β> + Sλν Rµκ|β> + Sµν Rκ|β>) respectively; at the same time, the spin of the matter fields is also present in the equations through its appearance in the energy-momentum tensor -see Section 3.2. This means simply that spin and torsion have an effect on the evolution of the curvature of space-time -and the gravitational field, as a result- in great distances. To put in simpler words, the large scale structure of an Einstein-Cartan universe, as well as the tidal forces and the gravitational waves in it, are all effected by the existence of torsion and spin; something that is quite expectable, judging on the previous results.

A specific form of the Propagation Equations for the “Electric” and the “Magnetic” field tensors As in the previous sections, we can here present here the form of the two propagation equations for a specific physical system described by the energy-momentum tensor of the equation (3.37). The choice of this energy-momentum tensor alone instead of the three tensors we used so far for our examples is not by chance; the non-perfect Weyssenhoff fluid described by it is a specification considering the generalised case -where the spin density tensor is not so easily specified-, while it is a generalisation of most cases studied in the literature -e.g the perfect Weyssenhoff fluid described by the equation (3.31). As a result, the form of the equations we will present are easy to extract, as well as easy to understand and to compare with their counterparts of General Relativity. We begin, by noting that, since the Frenkel condition is valid -this is a Weyssenhoff fluid, after all-, the torsion-related terms that were added in the two equations are zero

µ κ ν κ 2u (2Sν<α Rβ>κµ + S<αβ> Rκµ) = 0 µν κ λ κ λ κ −<α| (Sλµ Rνκ|β> + Sλν Rµκ|β> + Sµν Rκ|β>) = 0

This is easy to prove and makes things a lot easier. Then, if we assume the equation (3.37) for the energy-momentum tensor, we have the following propagation equation for the “electric” part of the Weyl tensor. 1 4 1 1 E˙ = − ΘE + DµHν − κ(ρ + P )σ − κσ ω sκλ − D q − hκ D (∇λs )− <αβ> αβ µν<α 2 αβ 3 αβ κλ 2 <α β> 2 <α β> κλ 11 1 − Θhκ u + σκ u + ωκ u + uκσ ∇λs − (s D Aµ − AµD s )− 2 3 <α β> <α β> <α β> αβ κλ 2 µ<α β> <α β>µ 1 2 − κπ˙ + κ(σ ˙ κ +ω ˙ κ)s + κ(σ κ + ω κ)s ˙ + κ(u A ω sκλ + h ω˙ sκλ + h ω s˙κλ)− 2 αβ (α (α β)κ (α (α β)κ 3 (α β) κλ αβ κλ αβ κλ 1 1 1 − κΘπ + κΘ(σ κ + ω κ)s − κΘh ω sκλ+ 6 αβ 3 (α (α β)κ 18 αβ κλ h 1 1 1 i + 2AµHν − ωµ E ν + κπ ν + κ(σ + ω )sνκ + κs (σνκ + ωνκ) + µν<α β> β> 2 β> 2 β>κ β>κ 2 β>κ 1 1 1 + 3σ µ E − κπ + κ(σ κ + ω κ)s + κs (σ κ + ω κ) + <α β>µ 6 β>µ 6 β> β> µκ 6 β>κ µ µ 1 + 2 h ν ωµ + κσ ω sκλ 3 µν<α β> αβ κλ (4.80)

In the same manner, we have the following equation for the propagation of the “magnetic” part of

75 the Weyl tensor 3 3 H˙ = − ΘH − DµHν + 3σ µH − ω q − ω hκ ∇λs + <αβ> αβ µν<α β> <α β>µ 2 <α β> 2 <α β> κλ 1 h + κ Dµπν − (Dµσνκ + Dµωνκ)s − (Dµσ + Dµω )sνκ− 2 µν<α β> β>κ β>κ β>κ νκ νκ µ µ νκi − (σ + ω )D sβ>κ − (σβ>κ + ωβ>κ)D s − 1 h1 1 i − κω sκλ Θh µ + σ µ + ω µ uν + Θhµν + σµν + ωµν u + 6 κλ µν<α 3 β> β> β> 3 β> 1 + κ h ν ω Dµsκλ + h ν s Dµωκλ − 2 µν<α β> κλ β> κλ 1 1 − 2AµEν − κσµ qν − κσµ hκν ∇λs + ωµH ν µν<α β> 2 β> 2 β> κλ β> (4.81) It is obvious in both cases that the propagation of the gravity field in large scales is associated with the form and evolution of the spin density, especially the way it couples to the main kinematic features of space-time. This last notice comes from the simple observation that the canonical spin density tensor is multiplied to the kinematic parameters Θ, σαβ and ωαβ ; it also becomes rather important since the spin density does not behave similarly to the other physical quantities -e.g the energy density ρ or the pressure P - - as one would expect, but it is affiliated to the geometry of the space-time. Of course one can extract a small similarity in the behaviour of the quantity mu µ ∇ sµν and of the energy flux qµ and of the quantity ∇ sαβ and of the derivative of the viscosity µ ∇ παβ ; this allow us to think that the canonical spin density tensor is actually functioning as the antisymmetric part of our energy-momentum tensor.

Only in order to give a mean of comparison, we can present the form of the propagation equations with the energy-momentum tensor of a non-perfect ﬂuid -from equation (3.30)- applied to it: the one for the “electric” part will be

0 1 1 1 0 1 E˜ = −Θ˜ E˜ + D˜ µH˜ ν − κ(ρ + P )˜σ − D˜ q − κA˜ q − κπ − κΘ˜ π + <αβ> αβ µν<α β 2 αβ 2 <α β> <α β> 2 αβ 6 αβ h 1 i 1 + 2A˜µH˜ ν − ω˜ E˜ ν + κπ ν + 3˜σ µ E˜ − κπ µν<α β> µ β> 2 β> <α β>µ 6 β>µ and the one for the “magnetic” part

0 3 1 H˜ = −Θ˜ H˜ − D˜ µE˜ν + 3˜σ µH˜ − ω˜ q + κ D˜ µπν − <αβ> αβ µν<α β <α β>µ 2 <α β> 2 µν<α β 1 − 2A˜µE˜ ν − κσ˜µ q +ω ˜µH˜ ν µν<α β> 2 β> µ β> [45]. It is easy to spot the main diﬀerences between these two equations and the (4.49) and (4.50) ones; writing them down and commenting on them is rather diﬃcult and without real meaning. The only thing we can point here -again- is the coupling of the spin density to matter and its behaviour as the antisymmetric part of the energy-momentum tensor.

4.3.2 The Constraints As with the kinematic parameters, the two propagation equations are not enough; we also need two constraint equations, one for each of the two tensors. To ﬁnd them, we will follow the same method. First of all, we will project the equation (4.33) vertically to the velocity vector and decompose the Weyl tensor to its “electric” and “magnetic” part. This can be done easily via the equations 1 h hγ ∇κW αβ + ∇[αRβ] + δ [α∇β]R− αµ β γκ γ 6 γ λακ β λβκ α αβκ − 2(S R κγλ + S R κγλ + S Rκγ ) = 0

76 1 uγ hρ hσ ∇κW αβ + ∇[αRβ] + δ [α∇β]R− ρσµ α β γκ γ 6 γ λακ β λβκ α αβκ − 2(S R κγλ + S R κγλ + S Rκγ ) = 0

Again, the first of these will provide us with the constraint for the “electric” part of the Weyl curvature, while the second will become the constraint for the “magnetic” part [16; 53]. Beginning with the first of them, we can rewrite it as h 1 h hγ ∇κW αβ + κ ∇[αT β] + δ [α∇β]T + αµ β γκ γ 3 γ 4 1 + κ ∇[α|∇λs|β] + ∇[α(sβ] sλ) − g ∇[α∇βsλ] + ∇ ∇[αsβ] + γλ γλ 6 γλ 3 γ 1 1 1 1 i + 2κ sλ[α|κ + sλg[α|κ + gλκs[α Rβ] + κ sαβκ − sαgβκ + sβgακ R = 0 2 2 κγλ 2 2 κγ by using the equation (4.20); further computation can lead us to the following form of our first constraint equation -the one for the “electric part”: 8 DβE = − 3H ωβ + σβ Hγδ − κDβT + κD T − αβ αβ αβγ δ αβ 3 α γ [µ| λ |β] [µ β] λ 4 [µ β λ] 1 [µ β] − κhαµh β ∇ ∇ s γλ + ∇ (s γλs ) − gγλ∇ ∇ s + ∇γ ∇ s − 6 3 (4.82) 1 1 − 2κh hγ sλ[µ|κ + sλg[µ|κ + gλκs[µ Rβ] − αµ β 2 2 κγλ 1 1 − κh hγ sµβκ − sµgβκ + sβgµκ R αµ β 2 2 κγ In the same manner, the second of these equations will become 1 uγ hρ hσ Big[∇κW αβ + κ ∇[αT β] + δ [α∇β]T + ρσµ α β γκ γ 3 γ 4 1 + κ ∇[α|∇λs|β] + ∇[α(sβ] sλ) − g ∇[α∇βsλ] + ∇ ∇[αsβ] + γλ γλ 6 γλ 3 γ 1 1 1 1 i + 2κ sλ[α|κ + sλg[α|κ + gλκs[α Rβ] + κ sαβκ − sαgβκ + sβgακ R = 0 2 2 κγλ 2 2 κγ and, if we complete our calculations, we will bring it to the following form: 1 DβH =3E ωβ − σβ Eγδ − κ uγ hρ hσ ∇[µT β] − κ hρ uµDσT − αβ αβ αβγ δ ρσµ α β γ 3 ρσµ α 4 1 − κ uγ hρ hσ ∇[µ|∇λs|β] + ∇[µ(sβ] sλ) − g ∇[µ∇βsλ] + ∇ ∇[µsβ] − ρσµ α β γλ γλ 6 γλ 3 γ 1 1 − 2κ uγ hρ hσ sλ[µ|κ + sλg[µ|κ + gλκs[µ Rβ] − ρσµ α β 2 2 κγλ 1 1 − κ uγ hρ hσ sµβκ − sµgβκ + sβgµκ R ρσµ α β 2 2 κγ (4.83)

As before, we can compare these equations with the relativistic ones, which are 8 D˜ βE˜ = −3H˜ ω˜β + σ˜β H˜ γδ − κD˜ βT + κD˜ T αβ αβ αβγ δ αβ 3 α and 1 D˜ βH˜ = 3E˜ ω˜β − σ˜β E˜γδ − κ uγ hρ hσ ∇˜ [µT β] − κ hρ uµDσT αβ αβ αβγ δ ρσµ α β γ 3 ρσµ α

77 The image is almost the same as before: all Riemann-Cartan terms have been replaced with their Riemannian counterparts, while all the spin-relate terms have been deleted. Notably, the β µ ν κ λ κ λ κ γ µν λ κ β κ β terms 2hα u u (Sλβ Rµκν +Sλµ Rβκν +Sβµ Rκν ) and −3hα γ u (Sβµ Rνκλ +Sβν Rµκλ + κ Sµν Rκλ) respectively, and the spin of the matter ﬁelds included in the energy-momentum tensor were the terms denoting the eﬀects caused by the torsion of the Riemann-Cartan space-time; all of them are no longer present in an Einstein universe.

A specific form of the Constraints for the “Electric” and the “Magnetic” field tensor To make things a little more clear as long as the matter fields are concerned, we can present the form of the constraints after replacing the matter fields terms with the energy-momentum tensor of a non-perfect Weyssenhoff fluid -from equation (3.37). As we have said before, this choice is the simplest, the most natural and the easiest one to understand and compare with the relativistic equations. As we explained before, since the Frenkel condition is valid -because this is a Weyssenhoff fluid-, the additional terms related to torsion are zero.

β µ ν κ λ κ λ κ 2hα u u (Sλβ Rµκν + Sλµ Rβκν + Sβµ Rκν ) = 0 γ µν λ κ β κ β κ −3hα γ u (Sβµ Rνκλ + Sβν Rµκλ + Sµν Rκλ) = 0 This is the first step towards simplification and better comparison to the equations of General Relativity. Following the steps we followed before, we can easily -via long, but trivial calculations- the constraint for the “electric” part of the Weyl tensor 1 2 1 1 DβE = − 3H ωβ = κD ρ − κD (ω sκλ) + κσ qβ + κσ κ∇λs − αβ αβ 2 α 3 α κλ 2 αβ 2 α κλ 1 β β κ β κ κ κ β − κD παβ + κ(D σ(α + D ω(α )sβ)κ + κ(σ(α + ω(α )D sβ)κ+ 2 (4.84) 1 + κu (Θω sκλ +ω ˙ sκλ + ω dotsκλ)+ 3 α κλ κλ κλ 3 3 + σβ Hµν − κωβqµ − κωβhµκ∇λs αβµ ν 2 2 κλ and the one for the “magnetic” part of the Weyl tensor is 8 DβH =3E ωβ + κ(ρ + P )ω + κω sκλω − αβ αβ α 3 κλ α 1 h β µ (κ1 µ)β µ)β µ)β λ κµ β λ − καβµ D q + u Θh + σ + ω ∇ sκλ + h D (∇ sκλ)− 2 3 (4.85) h 1 i − σβ Eµν + κπµν − κ(σ(mu + ω(µ )sν)κ − αβµ ν 2 κ κ 1 1 − κπ ωβ + κ(σ κ + ω κ)s ωβ + κω ω sκλ 2 αβ (α (α β)κ 3 α κλ Again, it is easy to spot the terms added by as effects of spin and torsion -even in this simplified energy-momentum tensor. We can also observe the coupling of spin density to pure kinematic quantities, as well as its behaviour representing the antisymmetric components of the energy-momentum tensor.

Thinking again of the comparison with the constraints derived from General Relativity, we present them here 1 1 1 3 D˜ βE˜ = −3H˜ ω˜β + κD˜ ρ − κD˜ βπ + κσ˜ qβ + σ˜β H˜ µν − κω˜βqµ αβ αβ 3 α 2 αβ 2 αβ αβµ ν 2 1 1 1 D˜ βH˜ = 3E˜ ω˜β + κ(ρ + P )˜ω − κ D˜ βqµ − σ˜β E˜µν + κπµν − κπ ω˜β αβ αβ α 2 αβµ αβµ ν 2 2 αβ [45].

78 4.4 Comparison to Other Analyses

The kinematic equations to which we have concluded are not the only ones proposed for an Einstein- Cartan universe -or for the approach of spin in curved space-times in general. Depending on the analysis followed, many equations have been found and it is essential to present a few of them here, as a form of comparison between our method and theirs. We will keep the comparisons to the three main kinematic equations -the Raychaudhuri, the shear propagation and the vorticity propagation ones- since they have been derived and studied in the majority of the relative, while they are of the most interesting and important results of our analysis, as well as a quite typical example of the whole analysis.

4.4.1 The Relativistic Approach of Spinning Fluid Kinematics The first interesting case is the analysis by D. Palle (in 1998) and by S.D. Brechet, M.P. Hobson and A.N. Lasenby (in 2007), who studied a Weyssenhoff fluid -that is a continuous macroscopic medium, characterised on microscopic scales by the spin of the matter fields. Its effective energy-momentum tensor is σλ σλ ˜ Tµν = (ρs + Ps)uµuν − Psgµν − 2(g + u )∇σ[u(musν)λ] where ρs is the spin density, Rs is the spin pressure and sµν is the canonical spin density tensor. As a consequence, it can be described in terms of the General Relativity, rather than in the Einstein- Cartan theory of gravity. Brechet, Hobson and Lasenby, however, proved that spin appears appears even in this treatment; this has also been proved by D. Palle in 1998, but they have presented a much more compact approach. According to D. Palle, the Raychaudhuri equation is 1 Θ˙ = − Θ2 + ∇˜ Aλ + 2(ω2 − σ2 − A2) − κ(ρ + 3P ) (4.86) 3 λ , the vorticity propagation equation is 2 1 ω˙ = − Θω + D˜ ρAσ + (σ λ + κs λ)ω (4.87) <µ> 3 µ 2 ρσµ µ µ λ and the shear propagation equation is

2 ˜ λ λ σ˙ <µν> = − Θσµν + D<µAν> − A<µAν> − σ<µ σν>λ + ω<µων> + κσ<µ sν>λ− 3 (4.88) 1 − h (Θ − 2σ2 + 2ω2 + 2κ2s2) 3 µν [13; 15; 55]. According to Brechet et al., the Raychaudhuri equation is 1 Θ˙ = − Θ2 + ∇˜ Aλ + 2(ω2 − σ2 − A2) − κ(ρ + 3P + 8ωλsˇ ) (4.89) 3 λ s s λ

λ where ωλ is the vorticity vector,s ˇλ is the spin density pseudo-vector and κ(ρs + 3Ps + 8ω Sλ ) = ˜ µ ν Rµν u u is the coupling of matter, energy, momentum and spin to geometry, as results from the energy-momentum tensor above. Following, the vorticity propagation equation is 2 1 ω˙ = − Θω + D˜ ρAσ + σ λω (4.90) <µ> 3 µ 2 ρσµ µ λ and the shear propagation equation is 2 σ˙ = − Θσ + D˜ A − A A − σ λσ + ω ω − E + κ(σλ s − ω sˇ ) <µν> 3 µν <µ ν> <µ ν> <µ ν>λ <µ ν> µν <µ ν>λ <µ ν> (4.91) [15; 16].

79 We must notice that the spin appears only in two of these equations: the shear propagation and the vorticity propagation ones for D. Palle, and the Raychaudhuri and the shear propagation ones for Brechet et al. At first, it might sound strange that Brechet et al. do not include the effects of the spin in the vorticity propagation, since it is one the major effects of the spin in our analysis. However, the fact that this analysis was conducted in a Riemann space-time -and probably with the natural relativistic demand for a “Big Bang Singularity”-, while ours was coupling the spin to geometrical torsion -which is zero in a Riemann space-time- might explain this.

The third comparison we will do is with the propagation equations of the kinematic parameters that were given by Th. Chrobock, H. Hermann and G. Rückner (in 2002) for a non-perfect Weyssenhoff fluid that is described by the following energy-momentum tensor

Tµν = ρeff uµuν + Peff hµν − 2u(µqν) + πµν 2 where ρ = ρ + 2ω sκλ, P = P + ω sκλ, q = −(hκ ∇ s λ + Aλs ) and π = −2(σ κ − eff κλ eff 3 κλ ν ν λ κ κλ µν (µ 2 ω κ)s − h ω sκλ; this tensor can be transformed to the previous one used by D. Palle and (µ ν)κ 3 µν κλ Brechet et al., but we will present Chrobock et al. results completely separated by the previous two analyses, since themselves have showed them this way. First of all, the Raychaudhuri equation is 1 1 Θ˙ = − Θ2 − κ(ρ + 3P ) + 2ω sκλ − 2(σ2 − ω2) + D˜ κA (4.92) 3 2 κλ κ , the shear propagation equation is 2 σ˙ = − Θσ + D˜ A + A A + E + <αβ> 3 αβ <α β> <α β> αβ 1 + ω µ(ω − κs ) − σ µ(σ + κs ) − h [2(ω2 − σ2) + κs ωκλ + D˜ κA ] (β α)µ α)µ (β α)µ α)µ 3 αβ κλ κ (4.93) and the vorticity propagation equation is 2 5 1 ω˙ = − Θ ω − κsˇ + 2(ω − κsˇ )σ µ + D˜ κAλ + 2κs µω (4.94) <α> 3 α 2 α µ µ α 2 ακλ α µ [14]. We must notice that the vorticity propagation equation is not given by Chrobock et al. in this form, but in its full form -as a tensor equation; we have transformed it to this by using the equation (4.20) so that comparisons with our or others’ results will be easier. What we may notice for the work of Chrobock et al. is that they come to very similar results compared to the work of D. Palle and Brechet et al. Yet, they present some interesting diﬀerences, the most important of whom is that they include the spin eﬀects in all three equations -while both D. Palle and Brechet et al. included only in two of them.

4.4.2 The Einstein-Cartan Approach of Spinning Fluid Kinematics The second comparison we will make is with the results posed by K. Pasmatsiou (in 2014). K. Pesmatsiou follows an analysis very similar to the one of N.J. Poplawski (in 2010, 2011, 2012 and 2013); she formulates the field equations in such a way that the first set does not include any torsion, being same to the field equations of General Relativity. However, the Lagrangian density and the energy-momentum tensor used by her are the same we have posed in our work. This contradiction is easily explained if someone considers that K. Pasmatsiou uses the alternative expression of the curvature tensor and wishes to express all the equations in a Riemann-Cartan space-time, as they would be viewed by a observer who understands himself as being in a Riemann space-time.

80 According to her method, the Raychaudhuri equation has the form 1 1 1 8 Θ˙ = − Θ2 +D˜ Aκ +2(Ω2 −σ2 −A2)− κ(ρ+3P )+κΩκλs +κ2ρ2sκλs − κωκλs − (sκu )2 3 κ 2 κλ κλ 2 κλ 3 κ (4.95) where ρ is the matter density and P is the pressure of the cosmic fluid, ωκλ is the vorticity tensor of the Riemann-Cartan space-time, Ωκλ is the vorticity tensor of the Riemann space-time, sκλ 1 is the canonical spin density tensor, s is the spin pseudo-vector and κ(ρ + 3P ) − κΩκλs + κ 2 κλ 1 8 κ2ρ2sκλs + κωκλS + (sκu )2 = R˜ uκuλ is the coupling of matter, energy, momentum and κλ 2 κλ 3 κ κλ spin to geometry, as it is derived from the energy-momentum tensor that is used. Following, the shear propagation equation is 2 1 1 1 1 ω˙ = − Θω − D˜ µAν +σ ωµ − κhµ (ρ κλs )− κρωµσ + κ2ρ2 κλs s µ (4.96) <α> 3 α 2 αµν αµ 4 α µ κλ 4 αµ 8 µ κλ α and the vorticity propagation equation is 2 1 1 1 σ˙ = − Θσ −σ σµ −ω ω +A A − κ2ρs s µ− κωµ s + κs µs <αβ> 3 αβ µ<α β> <α β> <α β> 2 <αµ β> 2 <α βµ> 2 <α βµ> (4.97) [11]. What is interesting in this analysis is that -like ours- uses the Einstein-Cartan theory of gravity, with a completely different approach: the first of the field equations is Riemannian, with no torsion terms. Furthermore, it uses the same Lagrangian density and energy-momentum tensor with us. All these allow us a straight comparison -alike the relativistic analysis. What we see, of course, is that the spin of the matter fields plays a very important role in the three kinematic equations - this is, after all, the great difference with the relativistic formulation of a spin fluid we have seen in Brechet et al. However, all these terms concerning the spin are not derived through the torsion of the space-time, but through a modified energy-momentum tensor which includes the spin density. As a result, there is no clear presence of geometric torsion in the equations, as it happens in our analysis. This, as we have already explained, happens because of the different approaches. K. Pasmatsiou’s approach is following an observer that “lives” in a Riemann-Cartan space-time, but “knows” that he “lives” in a Riemann space-time; as a result, he cannot understand the presence of torsion and tries to explain its effects through the relativistic theory. On the other hand, we follow an observer that “lives” in a Riemann-Cartan space-time and “knows” it; so, he understands not only the spin of the matter fields, but also the torsion of the space-time and explains what he observes from this perspective [11]. What might be interesting in this analysis is that K. Pasmatisou, having followed this set-up, calculated the direct difference between the four kinematic parameters -the expansion/contraction parameter, Θ, the shear tensor, σαβ , the vorticity vector, ωα, and the acceleration, Aα- of an Einstein-Cartan universe and of a relativistic one. The relations that show this difference are the following differential equations

˙ ˙ α Θ = θ + 2s uα ˙ 1 β µν 1 β 1 2 2 µν β ω˙ <α> = Ω<α> − κh α(ρβ sµν ) − κρΩ σαβ + κ ρ β sµν sα 4 4 8 (4.98) 1 σ˙ = Σ˙ − κρs µs <αβ> <αβ> 2 <α β>µ

Aα = αα where θ,Ωα,Σαβ and αα are used here to denote the kinematic parameters in General Relativity -in order to keep K. Pasmatisou’s original notation. It is quite interesting that the acceleration does not change from the one universe to the other -something that we have not implied either. It is also interesting to see that, writing the time derivatives of those three parameters -and consequently the

81 three previous equations- in this way, we can make our comparisons a lot easier. Here, we can see the terms added in order to “transpose” from the relativistic universe to the Einstein-Cartan one; those terms are very close to the terms we have proposed as additional in the equations (4.5), (4.8) and (4.12) -although not the same, since we have not referred to the Einstein frame at all. This also comes to agreement with the decomposition of the covariant derivative of the velocity vector that Mason and Tsambarlis have proposed (in 1981) [47].

A similar Raychaudhuri equation has been presented by A.J. Fenelly, J.P. Krisch, J.R. Ray and L.L. Smalley (in 1991), using too a perfect Weyssenhoff fluid; they have been using the same La- grangian density and -consequently- the same energy-momentum tensor as K. Pasmatsiou. Their approach -also within the Einstein-Cartan framework- resulted to the following Raychaudhuri equation 1 1 1 1 1 8 1 Θ˙ = − Θ2−σ σαβ + ω + κs ωαβ + κsαβ − w + κs sαβ + (Sαu )3− κ(ρ+3P ) 3 αβ αβ 2 αβ 2 2 αβ 2 αβ 3 α 2 (4.99) A A where wαβ =e ˙[α eβ]A the spin angular momentum -with eα being the tetrads that define the spin [18].

Another proposition for the propagation equations has been made by Th. Chrobock, H. Hermann adn G. R¨uckner (in 2002); they have moved from the relativistic form of the equations we presented before to their form in the Einstein-Cartan theory. They used the following energy-momentum tensor µ ν α β Tµν = ρu u + P hµν + 2u(µ|u ∇β(sα|ν)u ) which looks very much alike the energy-momentum tensor Brechet et al. have used in their relativistic analysis, but without the spin aﬀecting the energy density and the pressure -since this is in the Einstein-Cartan theory, according to Chrobock et al. So the Raychaudhuri equation is 1 1 Θ˙ = − Θ2 − κ(ρ + 3P ) + 2ω sκλ + 2(ω2 − σ2) + ∇κA + 2uκS µ∇ uλ (4.100) 3 2 κλ κ λκ µ , the shear propagation equation is 2 σ˙ = − Θσ + D A + A A + E + <αβ> 3 αβ <β α> <α β> αβ 1 + ω µ(ω − κs ) − σ µ(σ − κs ) − h [2(ω2 − σ2) + κs ωµν + ∇µA ] (β α)ν α)µ (β α)µ α)µ 3 αβ µν µ (4.101) and the vorticity propagation equation is 2 1 1 1 ω˙ = − Θω + 2ω σµ + DµAν − Eµν + κ πµν + uκSµ ∇λuν (4.102) <α> 3 α µ α 2 αµν 2 αµν 4 αµν αµν κλ

αβ α β κ αβ β κα where π[µν] = 2E[µν] +2uαuβ(2∇[µSν] +∇ Sµν −2Sµν Sκ −2S [µ|κ S|ν] is the antisymmetric part of the anisotropic pressure tensor that exists only in the Einstein-Cartan theory. We must notice again that the vorticity propagation equation is given by Chrobock et all in its full form -the tensor form; yet, we have transformed it to its equivalent vector equation via the equation (4.20), so that comparisons can be made easily -exactly as before [14]. What we notice in this form of the propagation equations is that they are constructed explicitly for the Einstein-Cartan universe -much like ours. As a result, the present great similarities.

82 Chapter 5

Constructing a Cosmological Model in the Einstein-Cartan Theory

5.1 The Spatial Curvature

Since we have split the space-time in space and time and studied the evolution of all geometrical and physical variables in respect to the one or to the other separately, one could suggest the existence of a three-dimensional spatial curvature. This curvature reﬂects the four-dimensional curvature of the space-time as it is observed from a three-dimensional hyper-surface that acts as rest frame. We can express this spatial curvature through a curvature tensor that is deﬁned as follows

γ δ κ λ Rαβµν = hα hβ hµ hν Rγδκλ − DµuαDν uβ + Dν uαDµuβ (5.1) where the ﬁrst term expresses the intrinsic curvature of the space-time, the one depicted by the curvature tensor, and the second term stands for the extrinsic curvature, that is caused by the motions of the space-time [45]. Using the decomposition of the curvature tensor in Ricci and Weyl tensor ﬁelds from the equation (2.52) and the space-like components of the decomposition of the covariant derivative of the velocity vector (equation (4.1)) -by taking away the acceleration-, we have 1 R =h γ h δh κh λW − (h h λh δR − h h κh δR − h h λh γ R + h h κh γ R )+ αβµν α β µ ν γδκλ 2 αµ ν β (λδ) αν µ β (κδ) βµ ν α (λγ) βν µ α (κγ) 1 1 1 + R(h h − h h ) + h γ h δh κh λ(σ D − σ D ) + h γ h δh κh λ D− 3 αµ βν αν βµ 4 α β µ ν γδ[κ |σ|λ] λκ[γ| σ|δ] 12 α β µ ν γδκλ 1 1 − Θ2(h h − h h ) − Θ[h (σ + ω ) + h (σ + ω ) − h (σ + ω ) + h (σ + ω )]− 9 αµ βν αν βµ 3 αµ βµ βν βν αµ αµ αν βµ βµ βµ αν αν

− (σαµ + ωαµ)(σβν + ωβν ) + (σαν + ωαν )(σβν + ωβµ) 1 1 where the antisymmetric terms h γ h δh κh λ(σ D −σ D ) and h γ h δh κh λ D 4 α β µ ν γδ[κ| σ|λ] κλ[γ| σ|δ] 12 α β µ ν γδκλ are zero. From here, we can use the deﬁnition of the “electric” part of the Weyl tensor (equation (2.56)) and the equation (3.17) of the algebraic ﬁeld equations, in order to write the upper relation in a more compact form -and with more references to physics. Thus, we have the three-dimensional

83 curvature tensor 5 1 R = γ κE + κT (h h − h h ) − Θ2(h h − h h )+ αβµν αβ µν γκ 6 αµ βν αν βµ 9 αµ βν αν βµ 1 − κ(h h δh λT − h h δh κT − h h γ h λT + h h γ h κT )− 2 αµ β ν δλ αν β µ δκ βµ α ν γλ βν α µ γκ (5.2) 1 − Θ[h (σ + ω ) + h (σ + ω ) − h (σ + ω ) + h (σ + ω )]− 3 αµ βν βν βν αµ αµ αν βµ βµ βµ αν αν

− (σαµ + ωαµ)(σβν + ωβν ) + (σαν + ωαν )(σβµ + ωβµ)

As with the original curvature tensor, we can consider the contractions of this tensor. The ﬁrst of them -a second-order tensor- takes the form:

µ µν 5 1 κ λ κλ 2 2 Rαβ = R αµβ = h Rαµβν = Eαβ + κT hαβ − κTκλ (hα hβ − hαβh ) − Θ hαβ− 3 2 9 (5.3) 1 2 − Θ(σ + ω ) + (σ2 − ω2)h + σ σµ − ω ωµ + 2σ ωµ 3 αβ αβ 3 αβ µ<α β> µ<α β> µ[α β] while the second -the trace of the first- is: 2 R = κ(5T − 2hκλT ) − Θ2 + 2(σ2 − ω2) (5.4) κλ 3 If we combine those two relations, by substituting the Ricci scalar in the first with the use of the second, we can take a very useful relation between the these two contractions, which is known as Gauss-Codacci equation. 1 1 1 R = Rh +E − κh κh λT − Θ(σ +ω )+σ σµ −ω ωµ +2σ ωµ (5.5) αβ 3 αβ αβ 2 α β κλ 3 αβ αβ µ<α β> µ<α β> µ[α β] An interesting result here is the great similarity of these equations to the relativistic ones. Aside form the fact that the curvature tensors include torsion-related terms -that describe the effects of torsion to the space-time- the energy-momentum tensor includes spin-related terms -that describe the effects of spin to energy and momentum-, there is no other difference in these equations; no extra terms have been added, as usually happened. So, in order to get from the relativistic spatial curvature to the one of the Einstein-Cartan theory, the only thing we have to do is replace the terms concerning the Riemann space-time and the typical matter fields with those concerning the Riemann-Cartan space-time and the spinning matter fields.

The Spatial Ricci Identity Since we have defined a spatial curvature tensor for the three-dimensional space -excluding completely the time dimension- we are forced to present also the most important identity following the curvature tensor: the identity that defines it. The Ricci identity is given -for the four-dimensional space-time- in equation (2.29) for a scalar field and in equations (2.32) and (2.33) -or (2.40)- concerning a vector fields; it is this second form of them that helps us in defining curvature. The question arising naturally is which is its form in the three-dimensional space. One can easily think that the breaking of space-time affects this identity by adding terms relevant to the geometry of the space-time; terms like the kinematic parameters. Taking in consideration the antisymmetric of the identity too, we easily conclude that the vorticity tensor will appear. Indeed, if we have a look at the Riemannian Ricci identity for the three-dimensional space, we can see this very picture.

˜ ˜ ˜ν 2D[αDβ]uµ = R µαβuν − 2˜ωαβ u˙ µ ˜ ˜ µ ˜µ ν µ 2D[αDβ]u = −R ναβu − 2˜ωαβ u˙

84 Even the identity for a scalar ﬁeld possesses this extra term ˜ ˜ ˙ D[αDβ]Φ = −ω˜αβ Φ In the Riemann-Cartan geometry, these equations must also contain the torsion tensor -as denoted by the four-dimensional equations in the section 2.2. It is easy to conclude to their form, if we consider that γ δ µ γ δ µ D[αDβ]uµ = hα ∇[γ|(hβ ∇|δ]uµ) D[αDβ]u = hα ∇[γ|(hβ ∇|δ]u )

If we go on with the calculations, that are quite easy since ∇αhµν = ∇alpha(uµuν ), we can easily take the form of the Ricci identity for a vector fields in the three-dimensional Riemann-Cartan space as ν κ λ ν 2D[αDβ]uµ = R µαβuν − 2ωαβ u˙ µ − 2hα hβ Sκλ Dν uµ (5.6) µ µ ν µ κ λ ν µ 2D[αDβ]u = R ναβu − 2ωαβ u˙ − 2hα hβ Sκλ Dν u Also, if we take the trivial case for a scalar field, we will have ˙ κ λ ν D[αDβ]Φ = −ωαβ Φ − hα hβ Sκλ Dν Φ (5.7) The importance of these equations -especially the ones considering an arbitrary vector fields- is the presence of the time derivative of this field on the right hand of them. If we choose this arbitrary field to be the velocity vector -something we do most of the time-, then we will have the acceleration derived naturally from the spatial Ricci identities. This easily leads to the conclusion that several phenomena that are included in the general theory -covering all four dimensions- and need not the addition of an external field, can be handled only through such an addition in the three-dimensional space 1.

5.2 The Friedmann-Lemaˆıtre-Robertson-Walker Model

What cosmological work needs is an axiomatic principle in order to build its theory and a model in order to derive its necessary equations, that will allow us to study the evolution of the universe. Without those two, no cosmological theory can be constructed and no arguments can be said about “how the universe works in large scale”. The principle that all cosmological models begin with is known as Cosmological Principle and is summarizing the simple idea that the distribution of matter -and energy, and momentum- in the universe in homogeneous and isotropic. Thus, no irregularities should be produced in a large scale approach of the universe as a whole. Of course, small irregularities have been observed even in large scale structures -from the Cosmic Microwave Background Radiation- and this principle seems a little worn and torn. If, however, we rephrase in a way that will exclude the small irregularities and act more as a guideline, we can still use it with remarkable results. This reformulation is that “no observer has a privileged position or course in the universe, so the three-dimensional space-time must -in large scale approach- be homogeneous and isotropic”. As for the cosmological model, what we need is to solve the field equations for a homogeneous and isotropic universe and find the metric that describes its space-time; from this metric we can derive any equation we need in order to study a certain parameter of the universe. The Friedmann-Lemaˆıtre-Robertson-Walker model is one of the simplest cosmological solutions of the relativistic field equations, that gives the most realistic results compared to the universe we live in. It was proposed independently by Alexander Friedmann in 1922 and 1924 and Georges Lemaˆıtrein 1927; the metric was derived by Howard P. Robinson and Arthur G. Walker in 1935. Its key element is the fact that the space-like components of the metric are homogeneously evolving in respect to time, due to their multiplication with the scale factor. As a result, all quantities we will study will be scalar functions of time alone; all vector and tensor fields that cannot be reduced into a scalar field are automatically set to zero. This is also a result of the homogeneity and the isotropy of the universe we have demanded in order to get this solution. 1This conclusion holds true for both the Riemann and the Riemann-Cartan space-times and the two theories respectively.

85 5.2.1 The Metric, the Torsion and the Matter Fields The simplest way to describe a universe with a homogeneous and isotropic distribution of matter, energy and momentum, is via a simple energy-momentum tensor, like the one that describes an ideal ﬂuid. Tαβ = ρuαuβ + P hαβ

This tensor is diagonal and has the form Tαβ = diag[−ρ, P, P, P ]. If we try to solve the field equations of the General Relativity (equation (3.1)) for this tensor, we will end up with the following linear element ds2 = −dt2 + a(t)2[(1 − kr2)−1dr2 + r2dθ2 + r2 sin2 θdφ2] (5.8) h a(t)2 i from where we have the following metric tensor g = diag − 1, , a(t)2r2, a(t)2r2 sin2 θ . αβ 1 − kr2 Here, we must note that a = a(t) is called the scale factor and is a function of time only, that governs the scale of the three-dimensional space -as it is multiplied to the space-like components of the metric tensor; on the other hand, k is the spatial curvature parameter that can take only three values, −1 if the universe is open (its geometry is hyperbolic), 0 if the universe is flat (its geometry is euclidean), and +1 if the universe is closed (its geometry is spheric) 2. This metric is called Friedmann-Robertson-Walker metric, since it describes the Friedmann-Lemaˆıtre-Robertson-Walker model [45; 56]. Beginning from this metric tensor and using equation (2.6), we can easily prove that the Christof- fel symbols will have the form a˙ a˙ Γ˜i = Γ˜i = δi Γ˜0 = − g (5.9) 0j j0 a j ij a ij where the latin indices -i and j- stand for the three-dimensional space. These Christoffel symbols agree -as it was expected- with the ones obtained by the relativistic analysis of a Friedmann model. The reason is simple: we have used an ideal fluid to describe the universe, without adding any spin effects. This choice comes almost naturally, since the Friedmann models do not allow any vector or tensor physical quantity to “live” in this universe -resulting that the spin density tensor cannot interfere to the energy and momentum of the universe [56]. But, what about the spin of the physical system and the torsion of the space-time? It is proved -by A.V. Minkevich and A.S. Garkun (1998)- that the Friedmann models can be constructed in an Einstein-Cartan universe only as long as the torsion has one degree of freedom and fulfils the condition S0ij = gij φ(t) where φ = φ(t) is a function of time similar to the scale factor, that can be considered the generator of torsion; our belief is that it could be somehow connected to the spin of the matter-fields [57]. This condition can be written in a covariant form as

α α Sµν = 2u[µhν] φ(t) (5.10) from where we can also obtain the torsion vector of this universe

Sα = 3uαφ(t) (5.11)

Using the Cartan equations (equation (3.11)), we ﬁnd out the spin density tensor must also have one degree of freedom and a similar form 8 s α = u δ αφ(t) (5.12) µν κ [µ ν] 2So far, the observations are converging in the belief that the geometry of the universe is euclidean -or very close to it-, so the spatial curvature takes the values k = 0.

86 12 while the spin density vector is s = s α = u φ(t). Since the spin density vector -as well as the µ µα κ µ torsion vector- is parallel to the velocity vector, it is also interesting to show its normalisation: 144 12 s sµ = − φ(t)2 s uµ = − φ(t) (5.13) µ κ2 µ κ These information could give us the full picture for the aﬃne connection of a Friedmann model.

Additionally, these three relations we have ended up to -the equation (5.8) for the metric tensor, the equation (5.9) for the Christoffel symbols, and the equation (5.10) for the torsion tensor- can be derived via another mechanism, which is the “mathematically correct” way of deriving the Friedmann models. In order to acquire a homogeneous and isotropic space-time, we demand that the metric tensor will be constant in respect to the Lie differentiation along all the Killing vector fields xia of this space-time. Lξgαβ = 0 (5.14) We also demand that the affine connection of the space-time will be constant in respect to the Lie differentiation along the same Killing vector fields.

µ LξΓ αβ = 0 (5.15) In General Relativity, the affine connection is the Levi-Civita connection, identified with the Christof- fel symbols; as a result is can be related to the metric tensor via the equation (2.6). This leads to the use of only dynamical variable: the metric tensor. However, the metric tensor is not the only dynamical variable of the Einstein-Cartan theory, since the affine connection is not identified with the Christoffel symbols. As a result, a second dynamical variable arises: the contorsion tensor, which is the additional part of the connection -the torsion tensor can also be used instead of the contorsion. Consequently, the second condition turn into

LξCαβγ = 0 (5.16) If both (5.14) and (5.16) hold for an Einstein-Cartan universe, the latter is homogeneous and isotropic. It has been proved by M. Tsambarlis (in 1977) that a perfect Weyssenhoff fluid cannot serve as the matter field source of such a universe [58]; however, the ordinary perfect fluid we have used, along with the φ scalar field can do this work. So the Christoffel symbols form equation (5.9) and the torsion tensor given in equation (5.10) fulfil these conditions.

5.2.2 The Evolution of the Universe The expansion or the contraction of the universe -its evolution on scalar terms, in other words- is described by the quantity Θ, as we already have figured out; the evolution of Θ is given by the Raychaudhuri equation and it is this property of Θ that makes the Raychaudhuri equation so important for cosmology. With this as a beginning, we have figured out that the scale factor of the Friedmann models should be related to Θ. Following the definition of the latter, and considering the presence of the scale factor in the metric tensor and the linear element, we manage to find the following relation between the two quantities. a˙ Θ = 3 a There is, however, a small problem: neither Θ, nor the scale factor, are observable and measurable quantities. As a result, a cosmological theory built in respect to them will be useless in anything concerning the observational data, since it will not be comparable to them. The only way to solve this problem is to relate them with a third quantity that is observable and expresses the same property of the universe. This third quantity is obviously the Hubble constant, that is defined from the relation v = H(t)d

87 where v is the velocity and d is the distance of a certain spot of the universe in respect to an observer. The relation between the Hubble constant and the other two sizes is Θ a˙ H = = (5.17) 3 a It is important to notice that all of these quantities are functions only of time -an important detail for a Friedmann model. Finally, we can also define a deceleration parameter by using the Hubble constant, or the scale factor. aa¨ H˙ q = − = − 1 + (5.18) a˙ 2 H2 This parameter can easily measure the deceleration -or the acceleration- of the universe evolution, since it includes the second derivative of the scale factor. As we just mentioned, the evolution of the expansion/contraction parameter, that can eventually show us the evolution of the universe, is expressed by the Raychaudhuri equation. This equation, as we have written it in equation (4.5) is given in unobservable terms -since it includes the expansion/contraction parameter, but it does not include the Hubble parameter. However, it is easy to transpose it into a relation of observable terms, using the sizes we have just introduced. The resulting form of the Raychaudhuri equation is 1 qH2 − 6φ(t)H = κ(ρ + 3P ) + 2(σ2 − ω2) − 2S uα∇ν uβ − ∇αA − AαA 6 αβν α α The same equation could be derived by the equation (5.4) for the scalar spatial curvature [51; 56]. To make this equation even more compatible to the Friedmann models, we could omit the terms that involve the shear, the vorticity and the acceleration; this would hold the principle we have set, that no tensor or vector field could disturb the homogeneity and the isotropy of the universe. However, as we have already said when presenting the original form of the Raychaudhuri equation, it is not easy to decide whether the acceleration is zero or not in an observer’s course -since we are not sure which of the two curves, the geodesic or the autoparallel, should the observer follow-, nor is it acceptable to avoid any vorticity terms -since vorticity is produced in the Einstein-Cartan universe whether we want it or not. This is actually the interesting with the form of this equation in the Einstein-Cartan theory. If we take this very equation from the relativistic Friedmann model, we will have the following 1 qH2 = κ(ρ + 3P ) + 6Hφ (5.19) 6 since there is no spin in “normal” relativistic matter and all terms related to acceleration -that is to the effects of external fields- and to vorticity should be omitted. Even the shear could be omitted, since the Friedmann models hold true only for a universe that is completely isotropic and homogeneous -at least, as far as the three-dimensional space is concerned. The presence of all other terms is what suggests a universe where the Big Bang might have happened, or might have happened in a completely different way and under completely different circumstances -eg. the moment it happened might be different, etc.

5.2.3 The Friedmann Equations The Friedmann-Robertson-Walker model is one of the simplest cosmological solutions of the relativistic field equations, that gives the most realistic results compared to the universe we live in. Its key element is the fact that the space-like components of the metric are homogeneously evolving in respect to time, due to their multiplication with the scale factor. As a result, all quantities we will study will be scalar functions of time alone; this is also a result of the homogeneity and the isotropy of the universe we have demanded in order to get this solution. By assuming that the Christoffel symbols are as in equation (5.9) and the torsion tensor as in equation (5.10), we can calculate the curvature tensor and its contractions, by their definitions

88 -equations (2.42), (2.46) and (2.49). If we do so, we will see that the Einstein tensor is 1 G = R − Rg = αβ (αβ) 2 αβ h a¨ a˙ 2 k a˙ 3 i ha¨ a˙ 2 k a˙ i = η 2 − − + 3φ˙ − 3 φ − φ2 − δ0 δ0 − 2 − 2 + 6φ˙ + 3φ2 + 6 φ αβ a a a2 a 2 α β a a a2 a (5.20)

From here, it is easy to conclude to the Friedmann equations for the evolution of the universe by separating the time and the space components; each one will give us one of the Friedmann equations, which must have the following form a¨ k 1 a˙ + = − κ(ρ + 3P ) + 2 φ + φ2 (5.21) a a2 6 a a¨ a˙ 2 k 1 a˙ 7 − − = − κ(ρ + P ) + 6 φ + 3φ˙ + φ2 (5.22) a a a2 2 a 2 If we combine the two of them, we can end up with the following one a˙ 2 1 a˙ 5 = κρ − 5 φ − 3φ˙ − φ2 (5.23) a 3 a 2 We can also write those equations in terms of the Hubble constant, if we use the relation (5.17); the form of the equations now will be 1 a˙ k H2 = − κ(ρ + 3P ) + 2 φ + φ2 − (5.24) 6 a a2 1 a˙ 7 k H˙ = − κ(ρ + P ) + 6 φ + 3φ˙ + φ2 + (5.25) 2 a 2 a2 and the combination of them 1 a˙ 5 H˙ + H2 = κρ − 5 φ − 3φ˙ − φ2 (5.26) 3 a 2 We have to notice that the second of the Friedmann equations is none else but the Raychaudhuri equation -given by relation (5.19), in the case of the Friedmann models. It is interesting to show here the conservation laws for the energy density, the momentum and the spin density will take for an ideal fluid. It is easy to find them, if we apply the energy-momentum density tensor and the spin density tensor we have calculated for the Friedmann model into the equations (3.48) -for the energy and momentum conservation- and (3.55) -for the spin conservation- and project along the velocity vector or vertically to it, when needed. The first of the conservation laws, know as continuity equation, will be 4 κρ˙ = −κΘ(ρ + P ) − 4φ(ρ + 3P ) − φΘ2 (5.27) 3 4 where the terms −4φ(ρ + 3P ) − φΘ2 exist only as long as the space-time has non-zero torsion. It 3 is interesting that although the energy-momentum tensor did not imply any coupling of the φ scalar field to matter -as a Weyssenhoff fluid energy-momentum tensor implies for a coupling of spin to matter-, this coupling appears in the equation for the energy conservation. In the same way, the relation for the conservation of momentum will take the form 16 D P = − h κR uλφ = 0 (5.28) α κ α (κλ)

κ λ since -from equation (3.16), applied to the perfect ﬂuid- hα R(κλ)u = 0. Finally, we can get the relation for the conservation of spin, which simulates quite much the continuity equation

˙ α φ + Θφ = ∇ Sα

89 α which we simplify by using the fact that ∇ Sα = 0, which has already been proved -see the section 3.4.2 for the conservation of spin. φ˙ + Θφ = 0 (5.29) Having found the Raychaudhuri equation for the Friedmann-Robertson-Walker model, it is time to ﬁnd the last but most important equations of this model: the Friedmann equations. Those two equations, derived originally -for the relativistic theory- by Alexander Friedmann in 1922, and separately by Georges Lemaˆıtrein 1927. Their great importance is that they prove beyond any shadow of a doubt -and in spite of what Einstein proposed at ﬁrst- that the universe evolves with time, and more concretely that its radius increases over time.

In order to make the comparison, the two equations that bare the name of Friedmann, in General Relativity have the following form:

2 1 a˜0 = κρa˜2 − k (5.30) 3 2 1 a˜0 + 2˜aa˜00 = − κ(ρ + P )˜a2 + k (5.31) 2 or, in respect to the Hubble constant, 1 k h2 = κρ − 3 a2 0 1 k h = − κ(ρ + P ) + 2 a2 a˜0 wherea ˜ the relativistic scale factor and h = the relativistic Hubble parameter. These two a˜ equations can also be derived by the Raychaudhuri equation and the three-dimensional spatial curvature scalar, if we apply to them the Friedmann model symmetries. Also, the conservation equations for energy and momentum are

0 ρ + θ(ρ + P ) = 0 (5.32) ˜ DαP = 0 (5.33)

[45; 56]. We can easily see the diﬀerence between the equations we have calculated and these ones: the spin-related (or torsion-related) φ ﬁeld is missing from all of them, making them much simpler. Of course, this is only natural, since the Friedmann model in General Relativity has nothing to do with spin, while the very space-time of General Relativity has nothing to do with torsion.

5.3 Solutions to the Friedmann Model

We already defined our cosmological model -a Friedmann-Robertson-Walker model- and presented its basic equations -the Friedmann equations and the conservation laws; now, we need to solve them. In order to do this, we will need one more equation, so that the system will be closed. This equation is none else than the equation of state that depends on the form of matter we want. In relativistic cosmology, we usually discuss about five types of matter: the dust fluid, the radiation fluid, the stiff matter, the inflation and the vacuum. Each one of them possesses a different equation of state that combines the pressure of the cosmic fluid to its energy density. Furthermore, those types of matter are chosen not only because of their simplicity, but also because they give solutions fitting to the stages of evolution of the universe. For example, the era we are supposed to live in now -according to the standard cosmological model- is called “Dust Era”, since the cosmic fluid has quite many similarities with a dust fluid and the space-time evolves appropriately; before this, we were supposed to live in the “Radiation Era”, because the radiation dominated the cosmic fluid altering the behaviour of the space-time evolution.

90 Here, we will discuss all these ﬁve cases for a universe that is described not by the theory of General Relativity, but for one that is described by the Einstein-Cartan theory. The three equations we will need for this task are (i) the continuity equation

a˙ ρ˙ + 3 (ρ + P ) + 4φ(ρ + 3P ) = 0 a , (ii) the spin conservation equation a˙ φ˙ + 3 φ = 0 a and (iii) one of the Friedmann equations, which we choose to be one of the following

a¨ 1 = − κ(ρ + 3P ) a 6 a˙ 2 1 = − κ(ρ + P ) − 3φ˙ a 2 We insist on using both of them, because combined they can give us the deceleration parameter a˙ -from equation (5.18). Notice that we have used the identity Θ = 3H = 3 ; we have also assumed a φ that << 1, meaning that the φ field is quite weak in comparison to the expansion of the universe, Θ so that we can exclude second order terms -like φΘ2 or φ2 3. These three equations, along with the equation of state, provide us with the systems of equations we must solve in order to discuss the evolution of the space-time in each case. Looking the three equations we realise that only the two of them are related to the matter fields -and, as a result, take different forms for each case. The spin conservation equation remains the same for any fluid we choose, possessing thus a unique solution for all physical systems. If we proceed in solving it, we will conclude that the φ scalar field is given by the very simple relation

a 3 φ = φ 0 (5.34) 0 a which reminds us the relation between the energy density and the scale factor in the case of a dust ﬂuid in General Relativity. Notice that φ0 and a0 are not the initial conditions, but the value of the φ ﬁeld and the scale factor in a chosen moment -usually the present day or the moment of change between the cosmological eras.

Something we must note before we continue is that the solutions we will present can be used as separate eras for the total evolution of the universe; we usually imagine the universe as being a synthesis of at least three eras, from the five we have studied above. After the “Big Bang”, the universe is dominated by a slowly sliding scalar field, which causes the inflation and the exponential expansion of the universe; this is called the “Inflation era”. Right after, the universe is dominated by relativistic particles and photons -what we may call radiation; this causes the evolution of the universe according to the second scenario we have seen and, as a result, this is called the “Radiation era”. During this era, the light nuclei are formed. But, as the universe expands, its temperature decreases causing the radiation to fall back and the dust to come forth; the epoch that follows where a dust fluid dominates the universe is called the “Dust era”. At this era, the fully ionised atoms and molecules are paired with the free electrons that have been decelerated. The main addition to this classical model is the more recent era, during which the universe has an accelerating expansion. This expansion is usually justified by the assumption of a a new type

3This assumption is not generally correct; we use it because it simplifies the calculations we need to do. However, we can find a valid justification for it: if we assume that the spin-related φ field is relatively small, the results we will derive from the solutions of these equations will be comparable to the results extracted from the relativistic solutions of the Friedmann models. Notice that the Raychaudhuri equations has become almost the same with its relativistic counterpart. Furthermore, if the φ field is indeed related to the spin of the matter fields -something we have no reason to doubt- it is only natural that it will be relatively small in large scale approaches of the universe.

91 of matter that evolves dynamically, known as “dark energy”, or by the presence of a scalar field, known as the cosmological constant. The problem that we face is how do we attach all these eras to one another, since each of them is described by a different equation of state and is provided with a different evolution of its scale factor. Yet we know that the universe is one and -according to our beliefs- must evolve smoothly and pass smoothly from one era to to another. Consequently, we have to assume certain conditions for the transition among the different cosmic eras. The most usual of them are the junction conditions. We assume that the space-time is divided in as many sub-space-times as the eras the universe evolves through; these sub-space-times are separated from one another by four-dimensional super- surfaces that have zero width and are called transition super-surfaces. The junction conditions secure that the transition through these super-surfaces will be smooth and all physical and geometrical quantities will be continuous. The first of them concerns the energy density and results to the following equality ρ+ − ρ− = 0 which means that the energy density of the first space-time (ρ+) must be equal to the one of the second space-time (ρ−). The second concerns the scale factor and results to a similar equality,

a+ − a− = 0

This equality holds not only for the scale factor, but also for the Hubble parameter -or the expansion/contraction parameter. Of course, we can choose an equality of the metric tensor of the ﬁrst space-time (g+) to the one of the other (g−) as a second condition; in this case, however, the equality will not hold for the Hubble parameter.

5.3.1 The Dust Solution The dust is characteristic for the complete absence of pressure; a dust ﬂuid has no pressure by itself. As a result, the equation of state for such a ﬂuid is

P = 0 (5.35) which is probably the simplest equation of state one can think of. Beginning from this, the continuity equation will take the form a˙ ρ˙ + 3 + 4φ ρ = 0 (5.36) a and its solution is proved to be a 3 R ρ = ρ 0 e4 φdt (5.37) 0 a In other words, the energy density of a dust fluid depends on both the scale factor and the φ field. We have to notice that the dependence from the scale factor preserves the form it has in the relativistic theory; if we set φ = 0, we can obtain the relation between the energy density and the scale factor we have in the relativistic case of a dust fluid. Meanwhile, the Friedmann equation will be

a˙ 2 1 = − κρ − 3φ˙ (5.38) a 2 and the Raychaudhuri equation a¨ 1 = − κρ (5.39) a 6 If we combine the two of them and use the equation (5.18), we can have the deceleration parameter for a dust ﬂuid 1 qdust = −2 (5.40) φ˙ 1 + κρ

92 This expression for the deceleration can give us a lot of useful information concerning the couple of ˙ the spin-related φ field to the energy density of the matter fields. If φ < κρ, then the qdust < 0, ˙ which means that the universe is acceleratingly expanded; if φ > κρ, then qdust > 0, which means that the universe expansion is decelerated; and if φ˙ = κρ, the the deceleration parameter goes to infinity, which means that either the scale factor is infinite or its derivative is zero -something that can hold true for a sort of bounce. In a Riemann universe dominated by dust, the deceleration parameter will have the valueq ˜dust = 1 − ; this is a constant and denotes the ongoing acceleration of the universe expansion. 3 5.3.2 The Radiation Solution Of course, the radiation has some pressure depending on its energy density; the equation of state that describes better than others this effect is 1 P = ρ (5.41) 3 and, consequently, the continuity equation will have the form a˙ ρ˙ + 4 + 8φ ρ = 0 (5.42) a which can be solved in a 4 R ρ = ρ 0 e8 φdt (5.43) 0 a Again, we notice that the energy density depends on the scale factor in a way similar to that in the relativistic case of a radiation fluid; if we set φ = 0, then we will take exactly that law of evolution. Furthermore, we notice that the coupling of the φ field to the energy density of radiation is stronger than that of its coupling to the energy density of dust. The Friedmann equation will become a˙ 2 2 = − κρ − 3φ˙ (5.44) a 3 and the Raychaudhuri equation will take the form a¨ 1 = − κρ (5.45) a 3 From here, we calculate the deceleration parameter by using the relation (5.18). 1 qrad = −2 (5.46) φ˙ 1 + κρ Like before, the decelaration parameter can tell us quite much for the coupling of the phi field to the matter fields. The three case we can distinguish are more or less the same as with the dust ˙ fluid: if φ < κρ, then the qrad < 0, which means that the universe is acceleratingly expanded; if ˙ ˙ φ > κρ, then qrad > 0, which means that the universe expansion is decelerated; and if φ = κρ, the the deceleration parameter goes to infinity, which means that either the scale factor is infinite or its derivative is zero -something that can hold true for a sort of bounce. Again, the deceleration parameter for a radiation-dominated relativistic universe would beq ˜rad = 2 − ; this means that the acceleration of the universe expansion is greater in the radiation era rather 3 than in the dust era.

It is interesting that the deceleration parameter for the dust and the radiation ﬂuid presents so much similarities. In fact, the only thing that changes is the absolute value of the parameter.

93 5.3.3 The Stiff Matter Solution The case of stiff matter is quite different from the previous two; a stiff matter fluid is described by an equation of state such as P = ρ (5.47) where pressure is exactly equal to the energy density, and is best understood as a cold gas of baryons. This kind of fluid was used by Zel’dovich in order to describe a primary phase of the universe. In this case, the continuity equation will take its simplest form a˙ ρ˙ + 6 + φ ρ = 0 (5.48) a that has a solution of the form a 6 R ρ = ρ 0 e16 φt (5.49) 0 a Again, we notice that, if the φ field is set to zero this equation resembles the evolution law for a stiff matter fluid in relativistic hydrodynamics; also, the coupling of the φ field is stronger here than any of the previous cases. The Friedmann equation takes the form a˙ 2 = −κρ − 3φ˙ (5.50) a and the Raychaudhuri equations takes the form a¨ 4 = − κρ (5.51) a 6 From here, if we use the relation (5.18), we can find the deceleration parameter for a stiff matter fluid. 4 1 qstiff = − (5.52) 6 φ˙ 1 + κρ Once more, we see that we can distinguish three different case for the coupling of the φ field to the energy density of the matter fields. And once more, these three cases are the same. First of all, if ˙ φ < κρ, then the qstiff < 0, which means that the universe is acceleratingly expanded; second, if ˙ ˙ φ > κρ, then qsitff > 0, which means that the universe expansion is decelerated; and last, if φ = κρ, the the deceleration parameter goes to infinity, which means that either the scale factor is infinite or its derivative is zero -something that can hold true for a sort of bounce. In the relativistic case, a universe dominated by stiff matter would have a deceleration parameter 2 q˜ = − -the same number as with the radiation. This means that the universe expands stiff 3 acceleratingly but not more acceleratingly as if it was radiation-dominated.

5.3.4 The Inflation Solution The “Inflation era” in the standard cosmological model is the short era after the initial singularity during which the universe dimensions grew extremely great. The type of matter that dominates the universe at this case arises from a slowly sliding scalar field and is described by the following equation of state P = −ρ (5.53) where the energy density is almost constant. As a result, the continuity equations takes the form ρ˙ + 8φρ = 0 (5.54) where we can see that the scale factor plays no role at all, unlike the spin-related φ field; the solution to this continuity equation is φ 8 a 24 ρ = ρ 0 = ρ 0 (5.55) 0 φ 0 a

94 if we use the equation (5.29) for the evolution of the φ field. As a result of this relation between the spin density and the scale factor, we see here the first difference between the analysis within the General Relativity framework and the Einstein-Cartan one. In relativistic cosmology, the continuity equation has the form ρ˙ = 0 which leads to a constant energy density -and a constant pressure, due to the equation of state. In the Einstein-Cartan cosmology, the inflationary matter does not have a constant energy density, but an energy depending on the spin density -or the φ field, to be more precise; as a result, since this field is evolving in time in respect to the scale factor, the energy density of the inflationary matter is slowly evolving in time -as we can plainly see through the −24 exponent over the scale factor. In the same manner, the Friedmann equation becomes a˙ 2 = −3φ˙ (5.56) a and we see that the matter terms have vanished -opposite to the continuity equation; the Raychaud- huri equation, on the other hand, becomes a¨ 2 = κρ (5.57) a 6 As a result, the deceleration parameter from equation (5.18) is κρ qinfl = (5.58) φ˙ This once, the deceleration parameter is much simpler; it is also different to the three previous cases, κρ since we can distinguish only two cases: if < 0, then qinfl < 0, so the universe is acceleratingly φ˙ κρ expanded; if, however, > 0, then qinfl > 0, so the universe is expanding deceleratingly. φ˙ In the relativistic case, the deceleration parameter for inflation is approaching minus infinity, a˜˙ 2 since its denominator - - is zero. This eventually means that the universe expands with almost a˜ infinite acceleration; after all, that is the meaning of inflation.

5.3.5 The Curvature Solution The spatial curvature, if it is the only “force” acting on the universe, has a negative pressure much like the inflation; only, that this pressure is quite larger than the pressure of inflationary matter. The equation of state for a fluid representing the curvature is 1 P = − ρ (5.59) 3 and, consequently, the continuity equation will take the form a˙ ρ˙ + 2 ρ = 0 (5.60) a in which the terms containing the phi field have vanished, as if they play no role at all. This evolution law is similar to many evolution laws in the relativistic analysis -e.g. the evolution of the energy density of dust or radiation. As a result, we will be able to see many similarities. The solution to this is a 2 ρ = ρ 0 (5.61) 0 a The Friedmann equation will be written as a˙ 2 1 = − κρ − 3φ˙ (5.62) a 3

95 and the Raychaudhuri equation can be written as a˙ = 0 (5.63) a This last result means that the deceleration parameter is zero

qcurv = 0 (5.64) as if the universe is stationary. Obviously, a solution with spatial curvature alone means that the universe does not evolve acceleratingly. There are implications that such a universe is dominated by cosmic strings. It is interesting that in both cases -the relativistic and the Einstein-Cartan one- the scale factor is proportional to time t a = a0 t0 This means that the domination of the universe by cosmic strings is equivalent to a stable expansion. Such an expansion could be modelled by an Anti-deSitter universe.

5.3.6 The Vacuum Solution When we refer to a vacuum universe, we demand that this universe has zero energy density -and a zero pressure. In this case, the Friedmann equation will be

a˙ 2 = −3φ˙ (5.65) a and the Raychaudhuri equation will be a˙ = 0 (5.66) a From here, we can follow two different paths: the first relies on a full nullification of every field, including the φ field; the second proceeds with the elimination of matter fields only, while leaving the φ field untouched. Both of them have their interesting features and we will examine them carefully. In the first case, we will have a constant scale factor, since

a¨ =a ˙ = 0

From this we can conclude to the necessity for negative spatial curvature. This solution can be derived from General Relativity as well and is known as the Milne universe; in this universe, the energy density and pressure are zero, the cosmological constant is zero and the metric is given as

ds2 = dt2 − t2(dχ2 + sinh2χdθ2 + sinh2χsin2θdφ2) where χ = sinh−1r. In the second case, the scale factor is not constant but is depending on time via the φ field and is given as 9 R φdt a = a0e which means that the universe will evolve exponentially. This solution brings to mind the deSit- ter universe, a solution of the Einstein field equations that assumes only a non-zero cosmological rΛ constant. In our case the exponent 9 R φdt works as the exponent t of the deSitter universe; in 3 other words, the φ field -or the spin density- is playing the part of the cosmological constant.

96 5.3.7 The Static Solution Finally, the static solution resembles a universe that has no evolution; it was proposed by A. Einstein in 1917 as his preferred cosmology. The spatial curvature is zero and the energy density is constant 2Λ ρ = E (5.67) κ where ΛE is the cosmological constant of the Einstein solution -the pressure is constant as well. The Hubble parameter of this universe is identically zero and, consequently, the scale factor is constant. The Einstein solution is the only universe ﬁtting in this case, but it is only of historical importance, since it comes up with a number of inconsistencies: (i) it does not explain the red-shift, (ii) it does not explain the Cosmic Microwave Background, (iii) it needs a mechanism in order to recreate matter, in spite of the energy and momentum conservation laws, and (iv) it does not fully agree with the Friedmann models. As a result, it has not practical use.

5.4 Comparison to Other Analyses

Having examined the Friedmann model in the Einstein-Cartan theory and presented its main solutions, we can discuss the existing results in the literature concerning the spinning fluid cosmologies, emphasizing to the simplest model -the Friedmann. As with the kinematic equations, we will discuss the other analyses and compare them to ours in two stages: first we will present the attempts to derive a spinning fluid Friedmann cosmology from the General Relativity, then we will follow the results given in the Einstein-Cartan framework.

5.4.1 The Relativistic Approach of Spinning Fluid Cosmologies To start with, the only research within the relativistic frame -of those available to us- to go as far as to construct a cosmological model with a spinning fluid was that of S.D. Brechet, M.P. Hobson and A.N. Lasenby (in 2008). They use a perfect Weyssenhoff fluid that is easy to handle and they apply the cosmological principle quite easily as well, since they work within General Relativity. They Friedmann equations they derive are a˙ 2 1 1 3k = κ(ρ − s2) + σ2 − + Λ (5.68) a 3 3 a2 a¨ 1 4 4 1 = − κ(ρ + 3P ) + κ2s2 − σ2 − Λ (5.69) a 6 6 6 2 1 1 where s2 = s sµν the spin density, σ2 = σ σµν the shear and Λ the cosmological constant. 2 µν 2 µν It is interesting that S.D. Brechet et al. choose to maintain the shear in the Friedmann equations, despite the fact the any such tensor vanishes in the Robertson-Walker space-time; this choice of theirs must be based on the addition of spin. To be honest, the addition of spin density in these equations is weird by itself, since the canonical spin density tensor they use should also vanish in a Friedmann model. Yet, they preserve it through the following argument: While the macroscopic spin averaging leads to a vanishing expectation value for the canonical spin density tensor < sµν >= 0 , it does not lead to a locally vanishing expectation value for the spin density squared scalar D1 E < s2 >= s sµν = 0 2 µν Whether there is indeed a physical meaning behind this, or whether the Friedmann model is indeed compatible to it, we will not discuss here; yet, we can still declare that we do not fully agree with this treatment. However, we have to note that the numerical results they provide from these equations are very significant [16].

97 5.4.2 The Einstein-Cartan Approach of Spinning Fluid Cosmologies As we have said, N.J. Poplawski (in 2010 and in 2011) proposed that the Friedmann-Robertson- Walker metric is valid for the Einstein-Cartan theory as well. So, in order to derive the Friedmann equations in this theory, he substituted the metric and the relativistic energy-momentum tensor in the field equations; the effective field equations he used have the form 1 R˜ − Rg˜ = κ(T + U ) αβ 2 αβ αβ αβ and are leading to the following expressions, that are the Friedmann equations for the Einstein- Cartan theory, according to Poplawski: 1 1 a˙ 2 = κ ρ − κs2 a2 − k (5.70) 3 4 1 1 a˙ 2 + 2aa¨ = − κ ρ + P − κs2 a2 + k (5.71) 2 4 1 where s2 = s sαβ [22; 23]. It is interesting that this formulation of the Friedmann model in the 2 αβ Einstein-Cartan theory resembles the formulation of S.D. Brechet et al. in General Relativity; even the Friedmann equations are similar. A similar approach -quite simplest- belongs to K. Atazadeh (in 2014); the Friedmann equations he concludes -in a more or less similar manner- are very close to the relativistic equations. a˙ 2 3 3 + = κρ (5.72) a a2 tot 1 where ρ = ρ − κs2. It is found that the spin density can be written in respect to the number n tot 4 of spinning particles 1 1 s2 = < s sµν >= 2 < n2 > 2 µν 8~ and that the energy density connected to it evolves as 1 1 ρ = κs2 = κ B−2/(1+w)ρ2/(1+w)a−6 s 4 32 ~ w 0

−3(1+w) while the typical energy density evolves as ρ = ρ0a ; so, the spin density becomes almost zero very quickly, leaving the typical matter to dominate [32] 4. This view is shared by Capoziello et al. (in 2002). According to them, the Einstein tensor for a spinning ﬂuid in the Einstein-Cartan theory is 1 h a¨ a˙ 2i ha¨ a˙ 2i G = R − Rg = η 2 − − δ0 δ0 − 2 (5.73) αβ αβ 2 αβ αβ a a α β a a a˙ 2 h a¨ a˙ 2i where G = 3 its time-like component and G = 2 − η its space-like components 00 a ij a a ij -in the form of a three-dimensional tensor. From here, we can get the Friedmann equations, that are similar to the one Poplawski has calculated. a¨ 1 = − κ(ρ + 3P ) a 2 eff eff a˙ 2 k = κρ − a eff a2 where ρeff and Peff are deﬁned as in equation (3.26). The continuity equation is given as

ρ˙eff + 3H(ρeff + Peff ) =ρ ˙ + 3H(ρ + P ) − κ(s ˙ + 3Hs)s = 0 (5.74)

4We have to notice that the Friedmann equations given by K. Atazadeh are valid for an open uiverse, or -in other words- for constant curvature k = 1

98 √ r1 where s = s2 = s sµν [29]. 2 µν G. de Berredo-Peixoto and E.A. de Freitas (in 2009) come to a similar result as well, using a quantum-mechanical approach for the spin and introducing the spin current. However, they seem to agree with the previous analyses that the addition of spin density in energy density and pressure -via the eﬀective energy density and pressure- is the solution in order to ﬁnd the Friedmann equations [27]. A. Trautman (in 2006) had concluded to an interesting result. He claims that no more than one of the components of the canonical spin density tensor can survive in a Friedmann model; this one is s23 = s So the Friedmann equation is given as

<˙ 2 1 s2 − κ<2ρ + κ = 0 (5.75) < 2 < where < is the scale factor. He also gives the following simpler conservation laws for the mass and the spin separately 4 4 M = πρ<3 = const. ς = πs <3 = const. 3 3 23 [6]. His approach is shared with that of C.G. Boehmer (in 2006), who gives some more complex conservation laws Z t Z t M(t) = 4πρτ 2dτ. ς(t) = 16π2s2τ 2dτ 0 0 and adds the comsological constant in his analysis [59]. Another cosmological model has been proposed by A.J. Fenelly, J.C. Bradas and L.L. Smalley (in 1988); their approach is based on the Weyssenhoﬀ ﬂuid as well and has the following generalised Friedmann equation

a¨ 1 1 1 B = − κs sµν − κ(ρ + 3P ) + κρe2ae [αβ] sαβ (5.76) a 3 µν 6 2 Notice that this is not exactly a Friedmann equation; A.J. Fenelly et al. do not use the Weyssenhoff fluid in a Robertson-Walker space-time, respecting the fact the symmetries of this space-time do not allow a Weyssenhoff fluid. However, if one tries to simplify their equation into a Friedmann one, then the last term will necessarily vanish and we will obtain the Fridemann equation given by N.J. Poplawski or K. Atazadeh -with zero constant curvature [17]. One last case we can examine is this of A.H. Ziaie, P.V. Moniz, A. Ranjbar ans H.R. Sepangi (in 2014). Their approach is slightly different, despite the fact they use Weyssenhoff fluid as well. Through Nother’s theorem, they decompose the spinning fluid Lagrangian in two parts, the first of which represents the minimal coupling of spin to matter and the second the induced effects of a spinning fluid. As a result, the first is associated with an axial spin current jµ and the second with the canonical spin density tensor sµν -both of these are considered to be functions of time. Consequently, the continuity equation is given as 1 3 ρ˙ + 3H(ρ + P ) = κ2 12jj˙ + ss˙ + Hs2 (5.77) SF SF SF 4 4

2 µ where ρSF and PSF are the energy density and the pressure of the spinning ﬂuid, j = j jµ and 1 s2 = sµν s -as usual. The Friedmann equations are found to be 2 µν 1 1 3H2 − Λ = κ4 − 3j2 − s2 + κ2ρ = κρ (5.78) 16 2 SF tot 1 1 −2H˙ − 3H2 + Λ = κ4 − 3j2 − s2 + κ2P = κP (5.79) 16 2 SF tot

99 What is interesting in these equations are the addition of both the axial spin current anf the spin density; one would expect that the axial spin current -being parallel to the velocity vector- could be there, while the spin density -being vertical to it- could not [28].

A problem appears with all these formulations of the Friedmann equations; all of them use the effective field equations and the perfect Weyssenhoff fluid as a matter source, so they respect the Frenkel condition. However, as we have discussed before, the cosmological principle does not allow any tensor fields to survive unless they have some timelike symmetry and they can be projected along the velocity vector -which means as long as they evolve in time and they do not break the symmetrical time evolution of the three-dimensional space; so, the canonical spin density tensor -or the spin pseudo-vector- of the Weyssenhoff fluid does not survive in a Friedmann universe. In other words, the non-vanishing components of the torsion tensor are not relevant to the ones associated to the Weyssenhoff fluid -that can be shown by the geometric classification of the torsion S. Capozziello, G. Lambiase and C. Stornaiolo [4]. So, all these analyses do not seem valid under this light. The only way that the Weyssenhoff fluid can be preserved in an Einstein-Cartan cosmology would be to avoid the Friedmann models; A.J. Fenelly et al. (1988) and others (see A.J. Trautman, 1973 or A.K. Raychaudhuri, 1979) have used models similar to the Friedmann ones, but not the same, so they could include the point-particle description of spin into the matter fields. Some of the Bianchi models can also be used in this direction [60].

But these opinions are no the only ones. K. Pasmatsiou (in 2014) used the Gauss-Codacci equations she had calculated and the two conditions we have referred to (equations (5.12) and (5.14)), and resulted to the following Friedmann equations for an Einstein-Cartan universe.

a˙ 2 1 3 8 4 k = κρ − κ2ρ2s sµν + (ˇs nµ)2 + sˇ sˇµ − (5.80) a 3 8 µν 9 µ 9 µ a2 a¨ 1 κ2 8 k = − κ(ρ + 3P ) + s sµν − (ˇs nµ)2 + (5.81) a 6 6 µν 9 µ a2 µ wheres ˇµ is the spin pseudo-vector and n a vector that represents the velocity in the three- dimensional space. It is interesting to notice that K. Pasmatsiou uses the typical energy density and pressure -that of the matter fields; she introduces the spin through the presence of torsion in the space-time and that is why her equations are that different from the previous. Of course, that can be transformed into those ones naturally [11]. However, K. Pasmatsiou remains within the framework of the effective field equations and the Weyssenhoff fluid, while still using the Friedmann models. So, her results are not compatible to the cosmological principle, as well.

Some research has headed towards a different direction by abandoning the Weyssenhoff fluid. A.V. Minkevich and A.S. Garkun has done a similar work -out of which we have chosen the torsion constraint we used in order to construct the Friedmann model in this work. They have tried to generate torsion from a scalar field that can survive in the Robertson-Walker space-time and can be induced into the Friedmann equations quite easily; this scaler field can function as a spin density. A.V. Minkevich (in 2009 and in 2011) gave the following form of the Friedmann equations

2 a˙ k 1 ρ 1 1 2 + = κ + (1 − 2κb2) (5.82) a a 3 b2 24 b1b2 a¨ 1 ρ + 3P 1 1 2 = − κ + (1 − 2κb2) (5.83) a 6 b2 24 b1b2 where b1 and b2 are two indeﬁnite dimensionless parameters, related to this scalar ﬁeld we have discussed in the following way

2 1 − 2κb2 κ(ρ − 3P ) (S2 ) = − 12b1b2 12b2

100 1 1 where S2 the only component of the torsion vector that survives. So, the additional term (1− 24 b1b2 2 2κb2) is directly related to the space-time torsion [61; 62]. A.V. Minkevich, A.S. Garkun and V.I. Kudin (2013) are in complete agreement to this result [63]. S. Capozziello has also done work in this region -away from the Weyssenhoﬀ ﬂuid. S. Capozziello, R. Cianci, C. Stornaiolo and S. Vignolo (in 2008) presented a work in the region of f(R) theory of gravity with torsion that included the Friedmann equations in this theory. It is easy to convert them to Einstein-Cartan theory if we choose f(R) = R, which is the trivial case of this theory. The equations they give are

a˙ 2 k γ ρ 1 1φ˙ 2 a˙ φ˙ + = − f(T ) − − (5.84) a a2 4 φ 12 4 φ a φ a¨ γ ρ 1 1φ˙ 2 1 φ¨ 1a˙ φ˙ = − − f(T ) + − − (5.85) a 4 φ 12 2 φ 2 φ 2 a φ where γ the adiabatic index of the equation of state -taking values as 0 ≤ γ ≤ 1-, f(T ) a function α of the energy-momentum scalar T = T α -related to the function f(R) of the Ricci scalar- and φ the auxiliary scalar field related to torsion -or to spin. If we decide to make the conversion to Einstein-Cartan theory, then f(T ) = T = ρ − 3P And, finally, the torsion vector is given in respect to the φ scalar as 3 ∂ φ S = α (5.86) α 2 φ Looking at the Friedmann equations, we can make an important notice: the terms related to torsion are very similar to the terms related to curvature, if one decides to replace the φ scalar to the scale factor. That could mean that this φ field works for torsion as the scale factor work for curvature -they denote the time evolution of those two quantities [64]. Another interesting case is M. Tsambarlis’ work. In 1977, he came to the conclusion that no Friedmann model can be constructed within the Riemann-Cartan framework if the Weyssenhoff fluid is used as a matter source; he suggested that the “spinning particle” description of torsion is not compatible to symmetric solutions, like the Robertson-Walker metric [58]. So, in 1981 he presented the idea of modified Friedmann equations that can be compatible to this theory. These equations are constructed by the very same conditions, if one considers an effective energy-momentum tensor as the Belinfante-Rosenfeld one; the metric must have the form

ds2 = dt2 − <(t)2(dx2 + dy2 + dz2) that is quite similar to the Robertson-Walker metric. Then, using the effective energy-momentum tensor, we conclude that the Ricci tensor must have the form 1 R˜ = κ(ρ + P )u u + κ(ρ − P )g (5.87) αβ eff eff α β 2 eff eff αβ And finally, the modified Friedmann equations are revealed to have the following form 1 3<¨ = − κ(ρ + 3P )< (5.88) 2 eff eff 1 <<¨ + <˙ 2 + 2k = κ(ρ − P )<2 (5.89) 2 eff eff where k is the constant curvature of the cosmic hypersurfaces. Tsambarlis also gives the following conservation law d (ρ <3) = −3<2ρ (5.90) d< eff eff However, an interesting case is that the effective energy density and the effective pressure Tsam- barlis is using are completely different from the previous ones -the ones used within the Weyssenhoff

101 fluid description of torsion. According to him, the torsion tensor has the following non-vanishing components 1 2 3 S01 = S02 = S03 = FS[123] = f (5.91) where F and f two scalar arbitrary function of time; this can also be written as 2 V S γ = δ γ u F AS γ = γδu f (5.92) αβ 2 [β α] αβ αβ δ which comes in full agreement with what S. Capozzielo et al. have claimed in 2001 [4]. It also looks very much like the choice A.V. Minkevich and A.S. Garkun have made -the one we followed [50]. So, the effective energy density and the effective pressure have the form 4 4 ρ = ρ + κF 2 + 3κf 2 P = P + κF 2 − κf 2 (5.93) eff 3 eff 3

102 Chapter 6

Conclusions

6.1 The Main Results

What we have presented so far is not really a Cosmology of an Einstein-Cartan universe, as the title implies; it is more of an introduction to it. It tries to formulate this theory of cosmology in a way that it will come opposite to the relativistic theory. The main concepts we analysed so far are a) the geometry of the space-time this theory is constructed, b) the field equations of this theory and the connection of the space-time geometry with physics, and c) the kinematics of this universe; we have also made a step into constructing a Friedmann-Lenaˆıtre-Robertson-Walker model -without getting however deep into its solutions. The analysis is obviously similar to the the relativistic one, since the method used is the same; that is the reason for the constant comparison between them. As it was made clear from the very beginning, the geometry of this space-time -the Riemann- Cartan space-time- does not accept the demand of the General Relativity for a symmetric connection and a zero torsion tensor. As a result, the whole structure of the space-time changes, as torsion makes its appearance. What we see as very interesting is that torsion effects the parallel transport of vectors in this space-time, resulting to unclosed curves, there where -in a Riemann space-time- we should see closed curves. It also effects the free-falling curves, since the geodesic curves of a Riemann space-time -that had zero “acceleration” and were the shortest paths between two points of the space-time- split in two completely different curves: the geodesic ones, that are the shortest paths, and the autoparallel ones, that have zero “acceleration”. The problem with both of them is that you cannot exclude the effect of torsion in a test particle motion, that appears as a force term, although external force fields -through the non-zero “acceleration”- do not interfere. Moving further, we can see that the main geometric quantities -the Curvature tensor, the Ricci tensor, the Weyl tensor- not only are affected by the presence of torsion, but they are also presenting a rather odd decomposition in three parts. The first is the “Riemann” one, that comes only in respect to the Christoffel symbols and represents the curvature that arises naturally from the space-time; the second is the “Cartan” one, that is the same in form with the first one, but in respect only to the contorsion tensor, and represents the “curvature” that is added by the non-zero torsion; the third is the “Coupling” one, that combines the Christoffel symbols with the contorsion tensor and represents the “mixed curvature” added by the co-existence of curvature and torsion. The result of this is that, when physics is connected to geometry -through the field equations-, we cannot formulate the energy and momentum of the matter fields separately from the spin, since torsion cannot be completely separated from the curvature. Thus, we come up to two sets of field equations: the first is similar to the Einstein equations of General relativity, with the main difference that torsion is included in the geometrical terms and spin is included in the energy-momentum ones; the second, which is known as Cartan equation, is coupling the spin of the matter fields to the torsion of space-time alone and is quite supporting to the first one. Having formed these equations, we can know study some of the direct physical effects of torsion. This study begins with the fact that the energy-momentum density tensor must include the spin density tensor, since the Lagrangian includes the spin as well. Trying to find the conservation laws

103 for the energy, the momentum and the angular momentum (the spin) of the matter fields, we realise that torsion is playing a major part, both directly and indirectly. The same image continues as we are introduced into the kinematics of this theory. We observe that both directly -through the constant presence of the torsion tensor in our equations- and indirectly -through the silent, but also constant, presence of spin in the energy-momentum tensor- the torsion of the space-time, that is coupled with the spin of the matter fields, is effecting the behaviour of this universe. Two of the most important results one could think of are a) the necessary appearance of vorticity in such a universe, even if there are no external fields or initial vorticity in the cosmic fluid -or the space-time-, and b) the suggestion, from the form of the Raychaudhuri equation, that there may not be a “Big Bang Singularity”, since we cannot exclude at the same time the effects of torsion, the appearance of vorticity and the external fields. Finally, we come before the task of applying all these tools into a “realistic” cosmological model, with that being the simplest case of a heterogeneous and isotropic universe described by the Robertson-Walker metric and containing only a perfect fluid. We see that if a such a universe is to exist within the Einstein-Cartan theory of gravity, where torsion is present as well, the torsion must also be restricted as the curvature was. This restriction ends up to the generation of torsion from a scalar field, that can function as a scalar spin density in the matter fields. Using the first of the field equations, we easily conclude to the two Friedmann equations and observe the differences with their relativistic counterparts; these differences are discussed even further in a small study of the deceleration parameter for several equations of state, that is for several different cosmic fluids.

6.2 Cosmological Implications

That were the main results that occurred from this work; not as much as one would think of -or hope for- when starting it, but not so unimportant. What comes to mind is that “if this theory can queestion Big Bang itself from the very beginning, what could it do later?” Well, many answers have been posed, since many tried to find out what is the horribly brand new about this theory, when compared to its older and much more famous brother: the General Relativity. Throughout my work, I have collected a number of researches that explored the limits of this theory. Since the results are quite interesting and since the cosmological models constructed by several researchers have been presented in the end of the thesis, I thought it would be only right to refer to several of the pathways the results of those models had opened. The first, to which we ourselves were introduced, was the suggestion that Big Bang might never have happened; or it might have happened in a completely different manner -and at a completely different time- than the one we think we know. Following that, one must see that the evolution of the universe could shiver before this theory; epochs of the universe life that have been considered safe and sound so far, should be put under the microscope again, under very different circumstances -the inflation would probably be the first... Having tasted ourselves the differences the deceleration parameter shows during the cosmic eras in relativistic and Einstein-Cartan geometry, we can dare express our view that there far more differences to be explored if the Friedman equations are to be solved properly. And then, the structure formation too and the dark energy scenarios should also be reconsidered, since the presence of torsion would probably affect their work in quite an unpleasant way. Not a few of researches seem optimistic about the chance of linking torsion to the so questionable cosmological constant. Suddenly, a lot of paths are open for new research entirely on the other bank of the river. So long that this bank is safe and sound as well. For Poplawski’s numbers tell us that we don’t have many chances to detect torsion in our universe -at least not in the coming years. The spin density parameter is calculated to be at about 10−70. And in that perspective, -and taking into consideration that this thesis was written in the echo of the gravitational waves discovery- I do not think that the oldest brother is insecure at any way from this prematurely grown-old youngster...

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