Field Equations of General Relativity and Einstein-Cartan Theory

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MASARYKOVA UNIVERZITA PŘÍRODOVĚDECKÁ FAKULTA ÚSTAV MATEMATIKY A STATISTIKY

Diplomová práce

BRNO 2018 RADEK SUCHÁNEK MASARYKOVA UNIVERZITA PŘÍRODOVĚDECKÁ FAKULTA ÚSTAV MATEMATIKY A STATISTIKY

Field equations of General relativity and Einstein-Cartan theory

Diplomová práce Radek Suchánek

Vedoucí práce: prof. RNDr. Jan Slovák, DrSc. Brno 2018 Bibliograﬁcký záznam

Autor: Radek Suchánek Přírodovědecká fakulta, Masarykova univerzita Ústav matematiky a statistiky

Název práce: Field equations of General relativity and Einstein-Cartan theory

Studijní program: Matematika

Studijní obor: Geometrie

Vedoucí práce: prof. RNDr. Jan Slovák, DrSc.

Akademický rok: 2017/2018

Počet stran: 10 + 42

Klíčová slova: variace; funkcionál akce; princip stacionární akce; ča- soprostor; rovnice geodetik; metrický tenzor; konexe; Levi-Civitova konexe; Christoffelovy symboly; Rieman- nova křivost; Ricciho křivost; skalární křivost; obecná teorie relativity; stress-energy tenzor; Einsteinovy rovnice pole; linearizovaná gravitace; ADM 3+1 formalismus; diferenciální forma; forma konexe; forma křivosti; forma torze; Einstein-Cartanova teorie; metrický-aﬁnní prostor; Riemann-Cartanův prostor; spin tenzor; Einstein- Cartanovy rovnice pole Bibliographic Entry

Author: Radek Suchánek Faculty of Science, Masaryk University Department of mathematics and statistics

Title of Thesis: Field equations of General relativity and Einstein- Cartan theory

Degree Programme: Mathematics

Field of Study: Geometry

Supervisor: prof. RNDr. Jan Slovák, DrSc.

Academic Year: 2017/2018

Number of Pages: 10 + 42

Keywords: variation; action functional; principle of stationary action; spacetime; geodetic equations; metric tensor; connection; Levi-Civita connection, Christoffel symbols; Riemannian curvature; Ricci curvature; scalar curvature; general theory of relativity; stress-energy tensor; Einstein field equations; linearized gravity; ADM 3+1 formalism; differential form; connection form; curvature form; torsion form; Einstein-Cartan theory; metric-affine space, Riemann-Cartan space; spin tensor; Einstein-Cartan field equations Abstrakt

V této diplomové práci se budeme věnovat zejména matematickým aspektům Einsteinovy obecné teorie relativity (GR) a Einstein-Cartanovy teorie (EC), které se zabývají popisem gravitace. Každou z těchto teorií je možné studovat pro dimenzi n 6= 4, my však svou pozornost zaměříme pouze na čtyřdimenzionální časoprostor. Gravitační pole je popisováno pomocí nedegenerovaného metrického tenzorového pole signatury (3,1), které je kovariantně konstantní. Pomocí principu stacionární akce je možné odvodit rovnice popisující dynamiku grativačního pole. V případě GR předpokládáme, že je časoprostor pseudo-Riemannovská hladká varieta vybavená Levi-Civitovou konexí, tj. konexí s nulovou torzí. EC předpoklad nulové torze nepoužívá, čímž se konexe stává další proměnnou ve funkcionálu akce, z nějž se rovnice pole odvozují. Je-li však torze nulová, pak se EC redukuje na GR. Z tohoto důvodu se dá Einstein-Cartanova teorie považovat za zobecnění Einsteinovy teorie relativity.

Abstract

In this thesis we will mainly study the mathematical aspects of gravitational theories, namely the Einstein general theory of relativity (GR) and the Einstein- Cartan theory (EC). It is possible to study both GR and EC in dimensions other then four nevertheless, we will be focused on the case of four-dimensional spacetime. Gravitational field is described with a non-degenerate metric tensor field with signature (3,1) which is covariantly constant. Using the principle of stationary action, it is possible to derive the equations describing the dynamics of the gravitational field. In GR, the spacetime is assumed to be a pseudo-Riemannian smooth manifold equipped with the Levi-Civita connection, i.e. the torsion of connection vanishes. In EC, the requirement for the torsion tensor to vanish is dropped and the connection occurs as another independent variable in the action functional from which the field equations are derived. If the torsion vanishes then EC reduces to GR. This is the reason to consider Einstein-Cartan theory to be a generalization of Einstein’s theory of relativity.

Poděkování

Na tomto místě bych chtěl poděkovat svým rodičům, jmenovitě taťkovi a mamce, za obrovskou podporu, které se mi z jejich strany dostalo. Bez jejich podpory by pro mne studium matematiky nebylo možné. Můj upřímný vděk patří také mému školiteli Janu Slovákovi za odborné vedení, věcné připomínky a všechny konzultace, které byly bez vyjímky prostě super. Ještě dlouho se budu podivovat nad tím, jak velké štěstí mám, že jsem mohl být jeho studentem. Dále bych chtěl velice poděkovat Milanu Petríkovi, kterému jsem opravdu zavázán za všechno, co pro mne udělal. Mé díky patří i bratrovi Ottíkovi, který mne vždy dokázal povzbudit. Zítra s ním půjdu do hospody a užijeme si spoustu srandy. Bude určitě rád, že jsem odevzdal. Možná mě i pozve na pivo a nakládaný hermelín. Svůj vděk bych chtěl vyjádřit všem blízkým přátelům, kteří stáli při mně a podporovali mě v době psaní diplomové práce. Martinovi Kubečkovi, On- drovi Hulíkovi, Páje Francírkovi, Ondrikovi Komárků, Leoškovi Kajzarů, Lence Michalkové, Marušce Bakušce, Otce Botce a v neposlední řadě Jiříkovi Jandovi, mému prvnímu kamarádovi na vysoké škole. Tento seznam jmen není uspořádán podle toho, jak mám koho rád. Mám je všechny rád hodně. Snad jsem na nikoho nezapomněl. Na koho jsem zapomněl, může mi to oplatit v jeho/její závěrečné práci.

Prohlášení

Prohlašuji, že jsem svoji diplomovou práci vypracoval samostatně s využitím informačních zdrojů, které jsou v práci citovány.

Brno 4. ledna 2018 ...... Radek Suchánek Contents

Introduction...... ix

Notation summary...... x

Chapter 1. Principle of stationary action...... 1 1.1 Functionals and variations...... 1 1.1.1 Proper time and action for geodesics...... 3 1.2 Derivation of geodesic equations...... 4 1.3 Geodesic invariance...... 7

Chapter 2. General theory of relativity...... 9 2.1 Derivation of the Einstein ﬁeld equations...... 9 2.1.1 Variation of the determinant...... 11 2.1.2 Covariant derivative and variation of Γ ...... 13 2.1.3 Variation of scalar curvature...... 14 2.1.4 Field equations in vacuum...... 16 2.1.5 Field equations in the presence of matter source...... 16 2.2 Conservation of energy-momentum...... 18 2.3 Linearized gravity...... 20 2.3.1 Christoffel symbols...... 21 2.3.2 Curvatures...... 22 2.3.3 Linearized ﬁeld equations...... 23 2.4 ADM 3 + 1 formalism...... 23 2.4.1 Foliation of the spacetime...... 24 2.4.2 Splitting of the metric...... 25 2.4.3 Spatial covariant derivative...... 27 2.4.4 Curvatures of foliation and ADM action...... 29

Chapter 3. Einsten-Cartan theory of gravity...... 31 3.0.1 Variations of frames and k-forms...... 33 3.1 Connection, curvature, torsion...... 34 3.1.1 Metric-afﬁne space...... 36 3.1.2 Riemann-Cartan space...... 38 3.1.3 Hodge duals...... 38

– vii – 3.2 Field equations...... 39 3.2.1 Vacuum ﬁeld equations...... 39 3.2.2 Field equations in non-vacuum...... 40

Seznam použité literatury...... 42 Úvod

In 1916 A. Einstein introduced his general theory of relativity (GR) which became one of the most successful physical theories. General relativity is a theory of gravity. The gravitational field is described with a non-degenerate metric tensor field with signature (3,1). The spacetime is defined as a pseudo-Riemannian manifold equipped with the metric compatible and torsion free Levi-Civita connection. In 1922 E. Cartan came with new gravitational theory that suggested to ease the assumption of vanishing torsion while keeping the requirement of metric compatible connection. As a consequence, the connection might not be symmetric which influences the form of the field equations. In empty space, the field equations remain unchanged and EC reduces to GR. Using the principle of stationary action, the laws that govern most of the observable physical phenomenons can be derived. So called action, corresponding to a specific functional at hand, is varied with respect to the quantity we want to investigate and a variational problem is formulated. By solving the problem, equations describing the dynamics of the varied quantity can be obtained. Following this procedure, we firstly derive the geodesics equations in chapter one. Moreover, investigation of invariance of the corresponding action functional will naturally lead us to the notion of isometry. This serves as a motivational exercise before we move to the second chapter in which the Einstein field equations are derived. We then move to linearized gravity that may be used to facilitate computations in finding approximate solutions to the original field equations. Further in the chapter we present the Arnowitt-Deser-Misner (ADM) 3 + 1 formulation of gravity. This formalism is focused on splitting the Lorentzian metric into the timelike and spacelike parts. Brief discussion about the possibility of such a splitting is involved. ADM version of the action functional is given. The third chapter is focused on the second of the above mentioned theories of gravity, the Einstein-Cartan theory (EC). Using the language of tensor-valued differential forms, the field equations are presented in the context of metric-affine and Riemann-Cartan spaces.

– ix – Notation summary

All the objects we are dealing with are assumed to be smooth.

R real numbers , ∂ i i, ∂i, ∂xi partial derivative in the direction of -th coordinate vector field ;i, ∇i the covariant derivative in the direction of i-th coordinate vector field δ variation sign i δ j Kronecker delta tr trace i Γ jk connection coefficients, Christoffel symbols d exterior derivative ∇ covariant derivative D exterior covariant derivative i R jkl Riemann curvature i R jil,Rici j Ricci curvature i j Ri j,S scalar curvature exp the exponential map i ω j connection form i Ω j curvature form Θi torsion form i Q jk torsion tensor i κ j contorsion tensor i j Ti j,T stress-energy tensor, tensor valued 4-form l sk tensor of intrinsic angular momentum, spin tensor Ωk(M) space of k-forms on M

– x – Chapter 1

Principle of stationary action

Using the principle of stationary action, also called Hamilton’s principle, we can derive the laws that govern most of the observable physical phenomenons. For us, the main application of this principle would be to derive the Einstein ﬁeld equations. Before we do this, a motivational and a bit easier example would be appropriate. Nice and interesting candidate is the derivation of the equations describing the path that would take a particle in a spacetime that is not assumed to be flat, i.e. the geodesics equations of the Lorentzian manifold. Moreover, investigation of invariance of the corresponding action functional will naturally lead us to the notion of isometry. So, our aim is to solve a certain variational problem and in order to do this we ﬁrstly introduce the basic concepts, notations and the necessary results of the variational calculus [12], [13].

1.1 Functionals and variations

Let γ : [0,1] → R4 be an arbitrary smooth function. We may consider a new function 4 γε : [0,1] → R given by

γε = γ + εϕ , (1.1) where ε is a real parameter and ϕ : [0,1] → R4 is another smooth function satisfying the boundary condition ϕ(0) = ϕ(1) = 0. We define the variation of a function γ, denoted δγ, as d δγ = γ | (1.2) dε ε ε=0 which is simply the function ϕ. To generalize the notion of variation, e.g. for tensor fields, we simply follow the idea that the varied object is given by adding some small quantity of the same type as the original object to the original object. The variation then would be given as in (1.2). Something slightly different is happening when we are dealing with objects that depends on the varied quantities. This is the case of functionals as we shall see in the upcoming definition.

– 1 – Chapter 1. Principle of stationary action 2

Let S: C∞(R4) → R be a mapping from the space of smooth functions1 into the reals, S = S(γ). Such a mapping is called a functional. If we consider a family of new functions2 as given in (1.1) then we may understand S to be a functional of two variables S = S(γ,ε). Differentiating with respect to ε at 0 gives rise to the notion of variation of a functional, denoted δS, given by3 d δS = S(γ + εϕ)| , (1.3) dε ε=0 where ϕ is defined as in (1.1). If we take the polynomial expansion of S(γ + εϕ) in ε then the definition (1.3) tells us that δS is the coefficient of linear term in the expansion. It is among the basic results of the variational calculus that

1. δS is unique, if it exists;

2. δ is a linear operator on the space of functionals and behaves as derivative, thus satisﬁes the product rule and the chain rule;

3. if S is a functional given by integration then δ commutes with the integration4.

The third item in the above list is crucial for us and we are going to use it frequently in our computations. The stationary point of a functional S is given as a solution to the equation5

δS = 0 . (1.4)

The functional under consideration may be dependent on more than one function and even on more general quantities such as tensor ﬁelds. This is the case of Einstein-Hilbert action which is implicitly dependent on the metric ﬁeld. In the situation where the functional depends on more than one quantity we shall specify with respect to which quantity the variation is considered. The last thing we need is the fundamental lemma of variational calculus.

n Lemma 1.1.1. Let f ∈ C∞(Ω) where Ω ⊂ R . If the following integral Z f ϕdnx (1.5) Ω vanishes for arbitrary ϕ ∈ C∞(Ω) with compact support then f = 0.

1For our needs it is sufficient to consider the smooth functions defined on R4 even though the domain of S may be a more general space of functions. 2By letting ε to be a parameter of this family. 3This definition corresponds to the Gateauxˆ derivative of a functional. 4What me mean can be vaguely expressed as: δS = δ R f d µ = R δ f d µ 5Note that (1.4) is a generalization of the equation describing the stationary points of functions of real variables in classical analysis. Chapter 1. Principle of stationary action 3

1.1.1 Proper time and action for geodesics Aim of this section is to find the equations of the path that a particle would take if moving from a given point A to a given point B through a timelike region6 of the spacetime in the absence of any kind of matter fields. These are precisely the geodesic equations inside the timelike region of the manifold under consideration. It is possible to derive these equations from a specific functional called the geodesic action of a test particle. The Hamilton’s principle tells us that stationary points of such a functional correspond to the path we are looking for7. Willing to obtain the equations which determine the geodesics uniquely we need to parametrize γ with respect to the proper time τ 8 so we need to define this quantity and we will do it implicitly. Let gi j be a metric field on M with signature (−,+,+,+) and coordinates such that c = 1 9. The proper time differential is given by q i j dτ = gi j dx dx (1.6) and the proper time interval, an object independent of the specific additive con- stant10, measured along timelike γ can then be defined as Z ∆τ = dτ . (1.7) γ

Note that (1.6) and (1.7) are invariant with respect to the Lorentz transformations 11 and thus are the correct quantities representing the time independently of our choice of description of the spacetime. We proceed with the deﬁnition of the action functional. Let γ : [0,1] → M be a smooth curve inside the light cone between two spacetime events A and B, parametrized with the proper time τ. The action for geodesics12 is deﬁned as the integral of the proper time differential along such a path γ which plays a role of the functional variable s Z i j g dγ dγ S = −gi j dτ . (1.8) γ dτ dτ

6i.e. interior of the light cone 7This path, in general, is not a straight line since the vacuum ﬁeld equations allows for a non-trivial solutions and thus non-vanishing curvature of spacetime 8Proper time is a time that an observer would measure with his own time measuring instrument and is the only time that has a physical meaning. Compare this with the timelike variable t, i.e. the time coordinate of the spacetime, which does not generally have a physical interpretation of the time that an observer would measure with respect to his frame of reference. The quantity that does have this interpretation is precisely the proper time τ. 9Here the constant c refers to the speed of light 10This constant correspond to the initial setting of the time measuring clocks. 11This can be shown easily by a direct computation and a similar computation is done in the following section concerning the geodesic invariance. 12measuring the length of a given geodesic Chapter 1. Principle of stationary action 4

The minus sign in front of gi j is included due to the timelike character of the integrated paths to obtain a positive value under the square root. The upper index on the left-hand side of (1.8) denotes the dependence of S on the metric and this notation will be useful for us in the next section considering isometries. The geodesics are precisely the stationary points of this functional, i.e. solutions to the equation

δSg = 0 , (1.9) where δSg is a variation with respect to γ. From a mathematical viewpoint we cannot be sure that the stationary points are extremes of a given functional and thus we should compute the second variation to ensure ourselves that (1.9) leads to geodesics being the minima of S. Nevertheless, using in advance the knowledge that the solution of (1.9) is physically meaningful and hence what we are looking for13, we will skip the second variation part of the computation.

1.2 Derivation of geodesic equations

Using the deﬁnition (1.8) the equation (1.9) reads as s Z i j g d dγε dγε δS = ( −gi j dτ)|ε=0 (1.10) dε γ dτ dτ s i j Z d dγε dγε = ( −gi j )|ε=0 dτ (1.11) γ dε dτ dτ i j Z d dγε dγε −1 dε (gi j dτ dτ )|ε=0 = q dτ . (1.12) 2 γ dγ i dγ j −gi j dτ dτ

i j If we use the δ notation to denote δ(gi jγ˙ γ˙ ) as the variation with respect to γ and dγ denote the derivative dτ simply by γ˙ so that we can write

i j i j d dγε dγε δ(gi jγ˙ γ˙ ) = (gi j )| (1.13) dε dτ dτ ε=0 then (1.12) becomes

i j d dγε dγε i j −1 Z (gi j )|ε=0 −1 Z δ(gi jγ˙ γ˙ ) dε dτ dτ dτ = dτ . (1.14) q p i j 2 γ dγ i dγ j 2 γ −gi jγ˙ γ˙ −gi j dτ dτ

13as is the case also with other physically motivated variational problems corresponding to different action functionals such as the Einsten-Hilbert action (which we will meet in the next chapter) Chapter 1. Principle of stationary action 5

If we let our coordinates be such that the tangent ﬁeld along γ is parametrized with the proper time and normalized, meaning g(γ˙,γ˙) = −1 holds along γ14, then the variational problem (1.9) is equivalent to Z g i j δE = δ(gi jγ˙ γ˙ )dτ = 0 . (1.15) γ

This leads us to denote by Eg = Eg(γ) the following functional Z g i j E = gi jγ˙ γ˙ dτ . (1.16) γ

In other words, the solution to (1.9) and the corresponding investigation of the invariance of geodesics will be done by examining this simpler functional, that is we are looking for the solution of

δEg = 0 . (1.17)

Let us note that reducing the variational problem (1.9) to (1.17) is not only a computational simplification but actually enables us to find the geodesics15. Let us proceed with the variation of Eg with respect to γ. Using the product rule and linearity satisfied by δ, is Z g i j δE = δ(gi jγ˙ γ˙ )dτ (1.18) γ Z i j i j i j = (δgi jγ˙ γ˙ + gi jδγ˙ γ˙ + gi jγ˙ δγ˙ )dτ . (1.19) γ

Let us examine the varied quantities in the above integral. The inﬁnitesimally varied γ is expressed in some chosen coordinates as

i i i γε = γ + εδγ , (1.20) where δγi amounts, in accordance with (1.2), to δγi = ϕi for some smooth ϕ d satisfying the same conditions as in (1.1). Then we can describe γ˙ε = dτ γε as d γ˙i = (γi + εϕi) (1.21) ε dτ = γ˙i + εϕ˙ i , (1.22)

i d i where we denoted ϕ˙ = dτ ϕ . Thus the variation δγ˙ is

i d i δγ˙ = (γ˙ )|ε=0 dε ε (1.23) = ϕ˙ i .

14The minus sign comes from the assumption about our curve to be timelike. 15since S does not determine them uniquely due to dependency on a chosen parametrization Chapter 1. Principle of stationary action 6

We observe that δγ induces the variation δγ˙ = ϕ˙ of the tangent ﬁeld along γ. Because gi j = gi j(γ(τ)): Tγ(τ)M × Tγ(τ)M → R depends on γ, the variation of gi j will be non-trivial. Using the expansion of gi j(γ(τ) + εϕ(τ)) we directly compute the coefﬁcient of the linear term, i.e. the variation δgi j d δgi j = [gi j(γ(τ) + εϕ(τ))] dε ε=0 (1.24) k ∂ = ϕ gi j . ∂xk We use (1.23) and (1.24) to rewrite (1.19) Z ∂ g k ˙i ˙ j ˙ i ˙ j ˙i ˙ j δE = (ϕ k gi jγ γ + gi jϕ γ + gi jγ ϕ )dτ (1.25) γ ∂x Z ∂ k ˙i ˙ j ˙ i ˙ j = (ϕ k gi jγ γ + 2gi jϕ γ )dτ (1.26) γ ∂x Z ∂ d k ˙i ˙ j i ˙ j i ˙ j 1 = (ϕ k gi jγ γ − 2ϕ (gi jγ ))dτ + 2[ϕ gi jγ ]0 (1.27) γ ∂x dτ Z ∂ d k ˙i ˙ j i ˙ j = (ϕ k gi jγ γ − 2ϕ (gi jγ ))dτ (1.28) γ ∂x dτ Z ∂ ∂ k ˙i ˙ j i ˙k ˙ j ¨ j = (ϕ k gi jγ γ − 2ϕ ( k gi jγ γ + gi jγ ))dτ . (1.29) γ ∂x ∂x In the above computation we integrated by-parts when getting from (1.26) to (1.27). Note that the last term in (1.27) vanished due to the assumption that ϕ is trivial in the endpoints of γ. Reindexing the last expression and splitting the middle term gives Z ∂ ∂ ∂ g i ˙k ˙ j ˙k ˙ j ˙ j ˙k ¨ j δE [γ] = ϕ ( i gk jγ γ − k gi jγ γ − j gikγ γ − 2gi jγ )dτ (1.30) γ ∂x ∂x ∂x Z 1 ∂ ∂ ∂ i ¨ j ˙ j ˙k = −2 ϕ (gi jγ + ( k gi j + j gik − i gk j)γ γ )dτ . (1.31) γ 2 ∂x ∂x ∂x Since we require δEg[γ] = 0 it is now enough to use the lemma 1.1.1 and the fact that ϕ is an arbitrary variation of γ. We conclude that

j 1 ∂ ∂ ∂ j k gi jγ¨ + ( gi j + g − g )γ˙ γ˙ = 0 . (1.32) 2 ∂xk ∂x j ik ∂xi k j We observe that the bracketed expression in (1.32) are precisely the Christoffel symbols of the first kind Γik j = Γi jk and recall that the Christoffel symbols of the l second kind Γ jk are given by raising the first index with the inverse metric. In terms of the metric coefficients l 1 li ∂ ∂ ∂ Γ = g ( gi j + g − g ) . (1.33) jk 2 ∂xk ∂x j ik ∂xi k j Multiplying (1.32) with the inverse metric and reindexing conveniently once again k enables us to write the last result in a compact form using Γ i j as k k i j γ¨ + Γ i jγ˙ γ˙ = 0 (1.34) which are the geodesic equations. Chapter 1. Principle of stationary action 7

1.3 Geodesic invariance

We are interested in the question of how the geodesics are affected by the action of Diff(M), the group of diffeomorphisms16 of M, and what is the subgroup that preserves the geodesics. The answer can be found through investigation of the invariance of the functional Eg with respect to transformations given by elements of Diff(M). Consider an arbitrary diffeomorphism Φ: M → M which can be described in coordinates (xi) by n functions yk = Φk(xi) where n = dimM. We know that the ˙ = ˙i ∂ transformation rule for coefﬁcients of the contravariant vector ﬁeld γ γ ∂xi is ∂xi γ˙˜i = γ˙k (1.35) ∂yk

i ∂x (0,2) where ∂yk is the inverse Jacobi matrix. The rule for transformation of metric i ∂x g ﬁeld coefﬁcients involves the inverse of ∂y j , one for each of the indices of i j

∂yk ∂yl g˜i j = g (1.36) ∂xi ∂x j kl Putting the rules (1.35) and (1.36) together we see how acting by Φ on M results in the following transformation of the integrated function in Eg 17

k l i j i j ∂y ∂y ∂x ∂x k l g˜i jγ˙˜ γ˙˜ = g γ˙ γ˙ (1.37) ∂xi ∂x j ∂yk ∂yl kl k l k l = δi δ jgklγ˙ γ˙ (1.38) i j = gi jγ˙ γ˙ , (1.39) which means that the Eg is diffeomorphism invariant. Further, if we rewrite this functional in a coordinate-free fashion as Z Eg = g(γ˙,γ˙)dτ (1.40) γ then the invariance of Eg can be neatly restated as Z Z ∗ g(γ˙(t),γ˙(t))dt = Φ g(Φ∗γ˙(t),Φ∗γ˙(t))dt (1.41) γ γ ∗ where Φ∗γ˙ is the push-forward of the vector ﬁeld γ˙ and Φ g is the pull-back of the metric g with respect to Φ. We can rewrite this equivalently in an even more compact form18

∗ Eg(γ) = EΦ g(Φ(γ)) . (1.42)

16We allow also the diffeomorphisms that are deﬁned only localy on M since all our considerations are local. 17 i Here and later in the text as well, we abuse our notation a bit with δ j being the Kronecker delta (which equals to 1 if i = j and 0 otherwise). Also recall that Eg is given in (1.16). 18In (1.42) we see the justiﬁcation of so far unutilized notation with g as the upper index of E. Chapter 1. Principle of stationary action 8

From (1.42) we may conclude the following statement. If the functional Eg has a smooth curve γ as a stationary point, in other words if γ is a geodesic for g, ∗ then Φ(γ) is a stationary point of EΦ g and thus a geodesic for Φ∗g. Note that this does not imply in general that Φ(γ) is a geodesic for g. Nevertheless, we can immediately read from (1.42) what is the condition on the diffeomorphism to preserves the geodesics, namely

Φ∗g = g (1.43) which is the condition on Φ to be an isometry. The set consisting of all isometries of the spacetime is the Poincaré group which has as a subgroup the Lorentz group O(3,1) , preserving the metric. The Poincaré group can be expressed as R3 1 o O(3,1), the semi-direct product of the group of translation of Minkowski space with the Lorentz group. To sum up, we have shown that considering a certain functional, we can derive the equations predicting the path that a particle would take in a vacuum. Those are precisely the equations of geodesics of the spacetime under consid- eration19. Further investigation of the invariance of the action Eg with respect to diffeomorphisms led us naturally to the notion of isometry as precisely those diffeomorphisms that preserves the scales20 and hence the geodesics.

19Let us comment briefly on the fact that we searched for the stationary points of the functional, yet we know that the geodesics are minimizers between all possible paths from A to B. It is possible to use similar approach as in classical analysis to show that those stationary points are actually minimums. 20including the proper time Chapter 2

General theory of relativity

2.1 Derivation of the Einstein ﬁeld equations

In this section we will derive the Einstein field equations (EFE) of general theory of relativity (GR). We will firstly derive the equations only for the vacuum case. Then we will introduce the stress-energy tensor corresponding to the presence of matter fields which will enable us to describe the field equations in the general case. Let us begin by considering the following functional L known as the Einstein-Hilbert action 1 Z p L = S −detgd4x , (2.1) 2κ where detg is the determinant of a pseudo-Riemannian metric tensor gi j with (−,+,+,+) signature1, S is the scalar curvature and κ is a constant2. We assume the integral domain of L to be compact3. Before we move further, let us ponder a bit about the specific form of the functional (2.1) we will be dealing with in the subsequent chapters. Firstly we reflect that if we desire the laws of gravity to hold good irrespectively of the chosen frame of reference we need our functional to be invariant with respect to a general coordinate transformation. For if this is satisfied, the resulting equations and physical theory based on them will be invariant as well4. Secondly, we realize that the metric field is the mathematical object chosen to describe gravity and the dynamical properties of spacetime. Willing to derive the physical laws using the principle of stationary action, we assume the variation to be done with respect to the metric field. Therefore, the volume form we wish to integrate to obtain the

1 the minus sign appearing once in the signature of gµν is the reason for considering√ the square root of minus the determinant in the integral (2.1) since we wish the value of −detg to be always positive 2 = 8πG G c given by κ c4 where is the gravitational constant and is the speed of light measured in vacuum 3This assumption will be used in the computation together with the Stoke’s theorem to ensure two things: vanishing of certain divergence terms and the possibility to apply the fundamental lemma of variational calculus in the form given in the previous chapter. 4The feature of a physical theory to be invariant with respect to a group of general coordinate transformations is called a generally covariant theory.

– 9 – Chapter 2. General theory of relativity 10 action must be constructed from√ the metric. Every metric yields a natural volume form, which in our case is −detgd4 x, invariant with respect to a general coordinate transformation. This seems to be a good√ choice to start with. Though the invariance is a very much desired property, −detgd4 x is not enough to define a reasonable action functional because it does not contain any partial derivative of the metric which seems to be a key element in the physical equations describing some dynamical properties. Moreover, due to the Einstein’s principle of equivalence [6], at any given point the coordinates can be chosen such that the metric is the Minkowski one, gi j = ηi j, and the first partial derivatives vanishes ∂kgi j = 0. So √in order to get some non-trivial equations we are led to consider, in addition to −detg, some other invariant scalar quantity which contains the second order derivatives of the√ metric. The most simple choice here is the scalar curvature. The expression S −detgd4x is then the volume form satisfying our requirements since it is given by a second order differential operator and is invariant under the coordinate transformations. Moreover, should we investigate further the possi- bilities√ of defining a correct volume form we would arrive to the conclusion that S −detgd4x is essentially the only reasonable choice up to addition of and multi- 1 plication by a real constant. Finally, the multiplicative constant 2κ is determined in such a way so that the laws of gravity derived from the action functional reduces to the Newtonian case in non-relativistic scenarios. Now we shall proceed with the derivation of the field equations. Inspiration was taken from [5]. —————————————————————————- Using the principle of least action with respect to (2.1), we are led to the following equation δ L = 0 , (2.2) where δ L is understood relative to the inverse metric tensor gi j and the variation is assumed to be zero on the boundary. We might be curious and ask why do we choose to vary with respect to the inverse metric and not the metric gi j. It is only a matter of comfort to choose the inverse metric in our computation and we will see in the section below that this choice is convenient. Let us proceed with computing δ L using the notation and facts introduced in the previous chapter 1 Z p δ L = δ S −detg d4x (2.3) 2κ 1 Z p p = δ S −detg + Sδ −detg d4x . (2.4) 2κ Thus we have the following expression to work with p p δ S −detg + Sδ −detg (2.5)

From the two summands above, let us start with rewriting p Sδ −detg (2.6) since it is more simple to deal with. Chapter 2. General theory of relativity 11

2.1.1 Variation of the determinant In the ensuing computation we need to use the following formula for the variation of the determinant of an invertible matrix

δ detA = detAtrA−1δA , (2.7) where A is a regular matrix and trA denotes the trace of A. By δ detA we mean, accordingly to our use of the variation symbol δ, d δ detA = (detA )| , (2.8) dε ε ε=0 where Aε = A+εδA. To show that (2.7) holds we use the following argumentation. We realize that detAε is a polynomial in ε and, since we wish to compute δ detA, we are interested in terms of the ﬁrst degree in ε only. Step 1: consider the simpler case A = E, where E is the identity matrix and Eε = E + εδE looks as follows   1 + εδE1,1 εδE1,2 ··· εδE1,n  εδE2,1 1 + εδE2,2 ··· εδE2,n  E =   . ε  . . .. .  (2.9)  . . . .  εδEn,1 εδEn,2 ··· 1 + εδEn,n

When the diagonal elements of Eε are multiplied we get

(1 + εδE1,1)(1 + εδE2,2)...(1 + εδEn,n) . (2.10)

The first degree terms of the above expression are of the form εδEi,i and thus collecting them corresponds to the trace of δE, i.e. d [(1 + εδE )(1 + εδE )...(1 + εδEn,n)]| = trδE . (2.11) dε 1,1 2,2 ε=0 Moreover, it is not possible to find the first degree terms in any other way then from the product of the diagonal elements of Eε because every other term in detEε arises as a product of at least two factors of the form (0 + εδEi, j), thus being of degree higher then 1. Hence we have proved the formula (2.7) for the case A = E. Step 2: for the general non-singular A we compute directly d δ detA = [det(A + εδA)]| (2.12) dε ε=0 d = [det(A(E + εA−1δA))]| (2.13) dε ε=0 d = detA [det(E + εA−1δA)]| . (2.14) dε ε=0 Since δA represents some general matrix, A−1δA can also be thought of as being a variation of E so that we can apply what has been already proven in step 1 to get Chapter 2. General theory of relativity 12 the desired result d δ detA = detA [(E + εA−1δA)]| (2.15) dε ε=0 = detAtr(A−1δA) . (2.16)

Using formula (2.7) on the metric tensor gives i j δ detg = detg g δgi j , (2.17) √ and we can compute the variation of δ −detg as p 1 δ −detg = − √ δ detg (2.18) 2 −detg 1 i j = − √ detg g δgi j (2.19) 2 −detg 1p = −detggi jδg . (2.20) 2 i j To proceed with variation of the determinant we will make use of the following i j relation between δgi j and δg

i j i j gi jδg = −g δgi j (2.21) which we can derive by varying the equation for rising up indices of the metric tensor

i j ik l j g = g gklg . (2.22)

Applying δ, using the Leibniz rule and the symmetry of g and δg we get

i j ik l j δg = δ g gklg (2.23) l j ik ik l j ik l j = gklg δg + g δ (gkl)g + g gklδg (2.24) j ik ik l j i l j = δk δg + g δ (gkl)g + δl δg (2.25) i j ik l j i j = δg + g δ (gkl)g + δg . (2.26)

Thus we have

i j ik l j δg = −g δ (gkl)g . (2.27)

Multiplying with gi j from the left gives the desired relation

i j k l j gi jδg = −δ j δ (gkl)g (2.28) kl = −g δgkl . (2.29)

Substituting (2.21) into (2.20) we conclude p 1p δ −detg = − −detgg δgi j . (2.30) 2 i j Chapter 2. General theory of relativity 13

Finally, after multplication with the scalar curvature, we obtain the following equivalent form of (2.6) p 1 p Sδ −detg = − Sg −detgδgi j . (2.31) 2 i j We will proceed with rewriting p δ S −detg , (2.32) the ﬁrst summand of (2.5). To do this, it will be useful to ﬁrstly investigate some k aspects of the Christoffel symbols Γi j and then compute the covariant derivative k of δΓi j.

2.1.2 Covariant derivative and variation of Γ In this part we derive important fact about the variation of Christoffel symbols. Particularly, Christoffel symbols are connection coefficients in a specific choice of coordinates and they do not transform tensorially which can be seen from the upcoming computation. Let us emphasize also that knowing the covariant derivative of the varied Christoffel symbols we can avoid expanding everything in terms of the varied metric and swiftly express the important terms using Riemann, Ricci and scalar curvature. Recall that the transformation rule between coordinate vector fields is given by ∂ ∂xn ∂ = , (2.33) ∂yi ∂yi ∂xn ∂ ∂yk ∂ = . (2.34) ∂x j ∂x j ∂yk We proceed with computing the transformation rule for connection coefficients using the defining equation5

∂ k ∂ ∇ ∂ j = xΓ i j k , (2.35) ∂xi ∂x ∂x and rules given by (2.33) and (2.34). We will do this explicitly to see two things: non-tensoriality of the Christoffel symbols and where the obstruction for these objects to be tensorial arises. Let us compute n ∂ ∂ xΓ i j n = ∇ ∂ j (2.36) ∂x ∂xi ∂x ∂yk ∂ = ∇ ∂ j k (2.37) ∂xi ∂x ∂y ∂ 2yk ∂ ∂yk ∂yl ∂ = i j k + j i ∇ ∂ k (2.38) ∂x ∂x ∂y ∂x ∂x ∂yl ∂y ∂ 2yk ∂ ∂yk ∂yl ∂ = + Γ m . (2.39) ∂xi∂x j ∂yk ∂x j ∂xi y lk ∂ym

5 k Here the left lower index in xΓ i j is a book-keeping of the coordinates with respect to which the symbols are considered. Chapter 2. General theory of relativity 14

∂ xp Expressing ∂y j , using the transformation rules and applying d gives

∂ 2yk ∂xr ∂yk ∂yl ∂xs Γ nδ p = δ p + Γ m δ p , (2.40) x i j n ∂xi∂x j ∂yk r ∂x j ∂xi y lk ∂ym s therefore, the transformation rule for Christoffel symbols can be written as

2 k p k l p p ∂ y ∂x ∂y ∂y ∂x Γ = + Γ m . (2.41) x i j ∂xi∂x j ∂yk ∂x j ∂xi y lk ∂ym

Notice that the first part of formula (2.41) is the aforementioned obstruction for Γ to transform as (1,2)−tensor. Further observe that this obstruction is dependent purely on the coordinate change and not on the connection coefficients. Thus, the rule for difference of two connections will have the obstruction term vanished and the transformation will be tensorial. Understanding variation of Christoffel symbols as a difference of connections enables us to use general rules for covariant derivative of (1,2)−tensor field δΓ 6

k k k m m k m k ∇l δΓi j = δΓi j + ΓlmδΓi j − Γil δΓm j − Γ jlδΓim . (2.42) ,l

2.1.3 Variation of scalar curvature To compute the varied scalar curvature we will vary the Riemann curvature and contract the result. We may describe (1,3)-Riemann curvature tensor as

l l m l Ri jk = Γi[k, j] + Γi[kΓ j]m . (2.43)

By the product rule of δ, variation of (2.43) yields

l l l m l m l m l m l δ Ri jk = δΓik − δΓi j + δΓikΓ jm + ΓikδΓ jm − δΓi jΓkm − Γi jδΓkm . (2.44) , j ,k

Note that this equation contains covariant derivatives of the varied Christoffel symbols and the ﬁrst two terms are hint to compute the following difference of ∇(δΓ) l l l l m m l m l ∇ j δΓik − ∇k δΓi j = δΓik + Γ jmδΓik − Γi jδΓmk − Γk jδΓim , j (2.45) l l m m l m l − δΓi j + ΓkmδΓi j − ΓikδΓm j − Γ jkδΓim ,k l l l m l m = δΓik − δΓi j + Γ jmδΓik − ΓkmδΓi j , j ,k (2.46) m l m l − Γi jδΓmk + ΓikδΓm j .

6Let us recall that the index symbol ,l denotes the partial derivative in the direction of the l-th ﬁeld and not the covariant derivative (denoted ;l). Chapter 2. General theory of relativity 15

Between the steps (2.45) and (2.46) we have used the fact that the Levi-Civita connection is torsion-free, i.e. that the Christoffel symbols are symmetric in the k k lower indices Γi j = Γ ji. Comparing (2.46) with (2.44) makes us notice that the varied Riemann curvature can be neatly described as

l l l δ Ri jk = ∇ j δΓik − ∇k δΓi j , (2.47) which we can use to compute variation of the Ricci curvature by contracting (2.47) with respect to l and j. This is called the Palatini identity

l l δ Rici j = ∇l δΓi j − ∇ j δΓil . (2.48)

Let us continue with computation of variation of the scalar curvature by contracting the Ricci curvature with the inverse metric and then varying the outcome i j δ S = δ g Rici j (2.49) i j i j = Rici j δg + g δ Rici j (2.50) i j i j h l l i = Rici j δg + g ∇l δΓi j − ∇ j δΓil (2.51) where we have used (2.48) in substitution for δ Rici j. Relabeling the indices conveniently and using the fact that the connection is Levi-Civita, thus satisfying ∇g = 0, we can rewrite the previous result as

i j il j i j l δ S = Rici j δg + ∇ j g δΓil − g δΓil . (2.52)

The second term on the right-hand side of (2.52) will have no contribution in the varied action√ integral (2.4) due to the Stoke’s theorem for the following reasons. Because −detg is a scalar invariant it is covariantly constant and thus commutes il j i j l with the covariant derivative. Hence, if we multiply the term ∇ j g δΓ − g δΓ √ il il with −detg we can write

p il j i j l hp il j i j l i −detg∇ j g δΓil − g δΓil = ∇ j −detg g δΓil − g δΓil . (2.53)

Now the bracketed expression on the right side of (2.53) can be thought of as a vector ﬁeld, say ξ,

j p il j i j l ξ = −detg(g δΓil − g δΓil) . (2.54) The divergence of a vector ﬁeld can be expressed via the covariant derivative as j divξ = ∇ jξ , (2.55) which is a well-deﬁned invariant operation which enables us to use the Stoke’s theorem if (2.53) is taken as an integrand in (2.4). Indeed, we have the following formula which holds generally for volume forms

divξωg = d(ιξ ωg) , (2.56) Chapter 2. General theory of relativity 16

√ 4 where ωg = −detgd x is the metric volume and ιξ ωg is the interior product of ωg with ξ 7. Therefore we observe that the second summand in (2.52), occurring in the integral (2.4) as multiplied by ωg, corresponds to the exterior derivative of some 3-form. Applying the Stoke’s theorem leads to integration over the boundary of the integral domain. This integral will be zero since the variation vanishes on the boundary as we assumed in the beginning. Consequently, we can express (2.32) with the ∇ j part being neglected for the reasons given above as

p p i j δ S −detg = Rici j −detgδg + divergence terms . (2.57)

2.1.4 Field equations in vacuum Finally, if we substitute (2.57) and (2.31) into (2.4), omitting the divergence terms8, we can rewrite the equation δ L = 0 as follows

Z 1 p (Ric − Sg ) −detgδgi j d4 x = 0 . (2.58) i j 2 i j √ Because g is non-degenerate −detg is always non-zero and since (2.58) holds for an arbitrary variation δgi j, the fundamental lemma of variational calculus implies that the bracketed expression equals to zero. In other words, we conclude that the Einstein field equations in vacuum, that is, the equations describing the dynamics of the metric tensor field in the absence of any matter field are of the form 1 Ric − Sg = 0 . (2.59) i j 2 i j

2.1.5 Field equations in the presence of matter source Our next aim is to formulate the Einstein field equations that describes spacetime with the presence of matter fields. The notion of matter is understood as non- graviton particles, i.e. only non-gravitational fields are considered9. The matter is mathematically described by a collection of tensor fields and is represented in the action functional by considering another term inside the integral (2.1). Let us denote this new term by Lmatter. We assume that Lmatter depends on the matter fields and their derivatives, on the metric gi j and also on the first and second 10 derivatives of the metric, gi j,k and gi j,kl, respectively . We leave the concrete form of Lmatter unspecified for our aim is not to investigate some particular case. Action functional (2.1) then becomes

Z 1 p L = ( S+L ) −detgd4x , (2.60) 2κ matter

7 i.e. the contraction of ωg with respect to a vector field ξ 8As we have explained, they would be integrated down to zero should we consider them in δ L 9Electromagnetic field is an example of a non-gravitational field. 2 10 g = ∂gi j g = ∂ gi j Recall the notation: i j,k ∂xk and i j,kl ∂xk∂xl . Chapter 2. General theory of relativity 17

To obtain the ﬁeld equations we proceed in the same manner as for the vacuum case, i.e. by solving the variational problem

δ L = 0 , (2.61) where δ is understood with respect to gi j. Using the properties satisﬁed by δ, we compute Z 1 p 0 = δ( S+L ) −detgd4x (2.62) 2κ matter Z 1 p p = [δ( S −detg) + δ(L −detg)]d4x . (2.63) 2κ matter Since we have already shown that the integral of the ﬁrst summand in (2.63) can be reduced in accordance with (2.58), we may continue as follows Z 1 p p 0 = [δ( S −detg) + δ(L −detg)]d4x (2.64) 2κ matter Z 1 1 p i j p 4 = [ (Rici j − Sgi j) −detgδg + δ(L −detg)]d x . (2.65) 2κ 2 matter So now we are interested in the term p δ(Lmatter −detg) . (2.66) √ The variation δ −detg is given by (2.30) and thus p p p δ(Lmatter −detg) = δLmatter −detg + Lmatterδ −detg (2.67) p 1 p i j = δL −detg − gi jL −detgδg . (2.68) matter 2 matter

The last step is to rewrite δLmatter. The variation with respect to the inverse metric amounts to δL δL = matter δgi j (2.69) matter δgi j

δLmatter 11 where δgi j is a symbolical expression of the generalized Euler-Lagrange equation for Lmatter. If we substitute (2.69) in (2.68) then (2.66) becomes

p δLmatter p i j 1 p i j δ(L −detg) = −detgδg − gi jL −detgδg . (2.70) matter δgi j 2 matter We have computed the variations of all terms in (2.60) and so putting (2.65) together with (2.70), we see that the equation δL = 0 is equivalent to Z 1 1 δLmatter 1 p i j 4 [ (Rici j − Sgi j) + − gi jL ] −detgδg d x = 0 (2.71) 2κ 2 δgi j 2 matter

11We wanted to avoid this notation as long as possible due to its confusing appearance. Never- theless, the notation is widely used in the physics community and it is convenient for us to use it in this part of text as well. Chapter 2. General theory of relativity 18

Using the fundamental lemma of variational calculus we conclude

1 1 δLmatter 1 (Rici j − Sgi j) + − gi jL = 0 , (2.72) 2κ 2 δgi j 2 matter which we rewrite so that the terms corresponding to the vacuum equations are on the left side and matter terms together with the multiplicative constant 2κ are on the right

1 δLmatter Rici j − Sgi j = κ(−2 + gi jL ) . (2.73) 2 δgi j matter The result (2.73) are Einstein ﬁeld equations in non-vacuum. The bracketed expression on the right side of the equation is taken as a deﬁnition for the Hilbert stress-energy tensor Ti j, that is

def δLmatter Ti j = −2 + gi jL . (2.74) δgi j matter

The requirement that Ti j is a tensorial quantity implies certain limits on the possible form of Lmatter. In other words, we are not allowed to choose the matter term arbitrarily since we want the equation (2.75) to be invariant with respect to all coordinate changes. We may also go the other way around and ﬁrstly choose Ti j and then ask what are the possible Lmatter satisfying (2.74). Having introduced Ti j, the Einstein-ﬁeld equations (2.73) takes the following, more familiar, form 1 Ric − Sg = κT (2.75) i j 2 i j i j 8πG and we recall that κ = c4 .

2.2 Conservation of energy-momentum

In this section we want to show that the stress-energy tensor satisﬁes

i j ∇ jT = 0 . (2.76)

This result corresponds to a local conservation of energy and momentum and is one of the desirable properties of general theory of relativity as far as physical consistency is concerned. Interestingly, the physically important conservation of Ti j can be computed merely by applying differential geometric operations and results which is the path we will follow in the derivation below. For the sake of conciseness of the proceeding computation, we will use the comma notation ;i to denote ∇i, the covariant derivative in the direction of i-th coordinate vector ﬁeld. Let us begin the computation with recalling the second Bianchi identity

Ri jkl;m +Ri jlm;k +Ri jmk;l = 0 . (2.77) Chapter 2. General theory of relativity 19

Since the ﬁeld equations are expressed using the Ric and S, we may try to obtain some conditions on these curvature tensors arising from the identity. Using the antisymmetry in the last two indices of the Riemann curvature, we can rewrite (2.77) as

Ri jkl;m +Ri jlm;k −Ri jkm;l = 0 . (2.78)

Because we assume the metric to be covariantly constant, we may contract (2.78) with gik to get

ik k k k 0 = g (Ri jkl;m +Ri jlm;k −Ri jkm;l) = R jkl;m +R jlm;k −R jkm;l (2.79) k = Ric jl;m +R jlm;k −Ric jm;l . (2.80)

Further contracting the ﬁrst two indices with gl j to involve the Scalar curvature gives

l j k l kl l 0 = g (Ric jl;m +R jlm;k −Ric jm;l) = Ric l;m +R lm;k −Ric m;l (2.81) kl l = S;m +R lm;k −Ric m;l . (2.82)

Let us do a bit of index gymnastics to conveniently rewrite the last two terms in kl (2.82). We ﬁrstly handle the middle term R lm;k. Observe that

i j ji R kl = −R kl (2.83) holds true as a simple consequence of antisymmetry of the Ri jkl in the ﬁrst two indices since

i j ai b j R kl = g g Rabkl (2.84) ai b j = −g g Rbakl (2.85) ji = −R kl . (2.86) We can use (2.83) in its contracted version

lk kl R km = −R km (2.87) li k = −g R ikm (2.88) li = −g Ricim (2.89) l = −Ric m . (2.90)

Thus we have

kl k R lm;k = −Ric m;k (2.91) which allows us to simplify (2.82) to

k S;m −2Ric m;k = 0 . (2.92) Chapter 2. General theory of relativity 20

We can rewrite (2.92) using the mixed metric tensor in the following way 1 gk S −2Rick = 2( Sgk − Rick ) . (2.93) m ;k m;k 2 m m ;k Multiplying (2.93) with gml yields 1 1 2gml( Sgk − Rick ) = ( Sgkl − Rickl) . (2.94) 2 m m ;k 2 ;k Therefore (2.78) is equivalent to 1 (Rickl − Sgkl) = 0 . (2.95) 2 ;k We recognize that the bracketed expression in (2.95) is the contravariant form of the left side of the ﬁeld equations introduced in this text in their covariant form as 1 Ric − Sg = T . (2.96) i j 2 i j i j Multiplying the equations with gkigl j to raise both of the indices gives 1 1 Rickl − Sgkl = gkigl j(Ric − Sg ) . (2.97) 2 i j 2 i j Finally, applying the covariant derivative on (2.97) and using (2.95) implies the desired result

1 i j 0 = (Rici j − Sgi j) = T (2.98) 2 ;i ; j

2.3 Linearized gravity

We will begin our computation with the assumption of having an exact solution of the vacuum field equations, the flat space Minkowski metric ηi j, to which we add a perturbation term, say γi j, multiplied by a very small constant ε. This is called a perturbation of the exact solution. We want to understand such a choice as a description of the spacetime in which a gravitation field propagates, yet without occurrence of any matter. In general, a perturbation of the metric can be written as 2 gi j = ηi j + εγi j + O(ε ) , (2.99) where 0 < ε 1 and O(ε2) are terms of order higher then one in ε. We want to consider only the terms up to the first order at most and thus we will write the perturbation of ηi j simply as 0 gi j = ηi j + εγi j . (2.100) In a similar manner, whenever some terms occur in our computations with ε in power higher than one we will simply omit those and write equality, keeping in mind that the equality holds modulo ε2. Neglecting the non-linear terms of Chapter 2. General theory of relativity 21 the space-time metric, which is called the linearized theory of gravity, enables us to simplify the form of Einstein field equations. This is used to find approximate solutions of otherwise complicated partial differential equations. So our aim is to describe the linear perturbation (linearization) of Christoffel symbols, Riemann, Ricci and scalar curvature, which will be denoted with the primed notation. Sub- sequently we will obtain the linearized Einstein field equations.

2.3.1 Christoffel symbols In pseudo-Riemannian geometry we can describe the Christoffel symbols solely via the metric tensor12. The Christoffel symbols of the ﬁrst kind

l Γi jk = gilΓ jk (2.101) are given by the metric as 1 Γ = g + g − g . (2.102) i jk 2 i j,k ik, j jk,i 0 Plugging in the expression for gi j introduced in (2.100), we get 1 Γ0 = η + εγ + (η + εγ ) − η + εγ . (2.103) i jk 2 i j i j ,k ik ik , j jk jk ,i

Since we want to consider coordinates in which ηi j remain constant we may conclude ε Γ0 = γ + γ − γ . (2.104) i jk 2 i j,k ik, j jk,i In further computations we will use the Christoffel symbols of the second kind, i i j 0 Γ jk. Raising of indexes of γi j is done by η . On the other hand, Γi jk are connection 0 coefﬁcients associated with the perturbed metric gi j and thus we need to raise the indexes with the inverse of the perturbed metric, g0i j. Hence it will be useful to ﬁnd the expression for g0i j in terms of ηi j and γi j. We can derive it with the help of the following simple trick

k j j k j k j (η + εγ)ik (η − εγ) = δi − εηikγ + εη γik j j j = δi + ε(γi − γi ) j = δi .

So with respect to our interest being bounded to constant and ﬁrst order terms we have

g0i j = ηi j − εγi j . (2.105)

12 Here gi j is a general metric tensor, not the perturbation of ηi j, meaning (2.102) holds in general Chapter 2. General theory of relativity 22

Let us note that from (2.105) wee see that using the inverse of the metric to raise indexes is consistent with our O(ε2) modular arithmetic13. Now we can proceed with computing the linearized Christoffel symbols of the second kind keeping in mind that our choice of frames is such that the symbols associated with the Minkowski ηi j vanishes. Substituting (2.104) and (2.105) in

0i 0il 0 Γ jk = g Γl jk (2.106) we get ε Γ0i = (ηil − εγil)(γ + γ − γ ) (2.107) jk 2 l j,k lk, j jk,l ε = ηil(γ + γ − γ ) (2.108) 2 l j,k lk, j jk,l ε = (γi + γi − γ i) . (2.109) 2 j,k k, j jk,

2.3.2 Curvatures Starting with perturbed Riemann curvature we will further describe Ricci and scalar curvature. In the chosen coordinates, the Riemann (1,3) curvature tensor can be deﬁned as i i i i m i m R jkl = Γ jl,k − Γ jk,l + Γ kmΓ jl − Γ lmΓ jk , (2.110)

0i so the perturbation R jkl can be computed as

0i 0i 0i 0i 0m 0i 0m R jkl = Γ jl,k − Γ jk,l + Γ kmΓ jl − Γ lmΓ jk (2.111) 0i 0i = Γ jl,k − Γ jk,l (2.112) where we got the second row by remembering that two consequent primed Christoffel symbols produce ε2. Using (2.109) we can further compute ε ε R0i = (γi + γi − γ i) − (γi + γi − γ i) (2.113) jkl 2 j,l l, j jl, ,k 2 j,k k, j jk, ,l ε = (γi − γi + γi − γi + γ i − γ i ) . (2.114) 2 j,lk j,kl l, jk k, jl jk, l jl, k The ﬁrst two terms in the last equality are symmetric in their second partial derivatives and hence can be canceled out which yields ε R0i = (γi − γi + γ i − γ i ) . (2.115) jkl 2 l, jk k, jl jk, l jl, k The Ricci curvature is deﬁned as the following contraction of the Riemann tensor k Rici j = R ik j , (2.116)

13 i j 1 We ought to be careful when omitting some terms for should g involve ε then we could easily made a mistake since raising the indexes would lower the powers of ε. Nevertheless, this would not be the case since in (2.105) we expressed gi j without the reciprocals of ε. Chapter 2. General theory of relativity 23 and so we can compute the perturbation of the Ricci curvature using (2.115) ε Ric0 = R0l = (γk − γk + γ k − γ k ) . (2.117) i j ik j 2 j,ik k,i j ik, j i j, k The last step towards the linearized field equations is expression for perturbed 0 0 scalar curvature S which we get by contracting Rici j 0 0i j 0 S = g Rici j (2.118) ε = ηi j(γk − γk + γ k − γ k ) (2.119) 2 j,ik k,i j ik, j i j, k ε = (γik − γk i + γi k − γi k ) . (2.120) 2 ,ik k, i k, i i, k Putting together what we have prepared so far, i.e. the results (2.117) and (2.120), 0 we can compute the perturbation Gi j of the Einstein tensor Gi j which is defined as the left-hand side of the Einstein field equations 1 G0 = Ric0 − S0 g0 (2.121) i j i j 2 i j 1 = Ric0 − S0(η + εγ ) (2.122) i j 2 i j i j 1 = Ric0 − S0 η (2.123) i j 2 i j ε = (γk − γk + γ k − γ k ) 2 j,ik k,i j ik, j i j, k ε (2.124) − (γk − γk + γ k − γ k ) 4 j,ik k, ji jk, i i j, k ε = (γk − γk − γ k + 2γ k − γ k ) (2.125) 4 j,ik k,i j i j, k ik, j jk, i

2.3.3 Linearized ﬁeld equations Finally, the linearized ﬁeld equations in the sense of Einstein general theory of relativity can be expressed in the following form k k k k k ε(γ j,ik − γ k,i j − γi j, k + 2γik, j − γ jk, i) = 32πTi j , (2.126) where Ti j is the Hilbert stress-energy tensor introduced in the previous section.

2.4 ADM 3 + 1 formalism

In the sequel we wish to describe the Arnowitt-Deser-Misner (ADM) 3+1 formulation of gravity. This formalism is focused on splitting the relativistic spacetime and the corresponding Lorentzian metric into the timelike and spacelike subspaces. Since the existence of such a splitting is not automatically assured we ﬁrstly we want to express the condition on the existence of spacetime foliation in a concise way using differential forms. The notation will be such that the Greek indices α,β, µ,ν,... will denote the spacetime components and Latin indices i, j,k,... will refer to the spatial components. Majority of this section is based on [4] Chapter 2. General theory of relativity 24

2.4.1 Foliation of the spacetime Let us assume coordinates t,y1,y2,y3 on a four dimensional Lorentzian manifold M14 = ∂ 15 = ∂ ,i = , , 16 such that ∂t : ∂t is timelike and ∂i : ∂yi 1 2 3 are spacelike local vector i ﬁelds in TM. Let α = α t,y be a 1-form on M such that α(∂t) = 1 and α(∂i) = 0 for i = 1,2,3, i.e. α is the ﬁrst element of the coframe given by coordinates the t,yi. Consider the following equation

α (X) = 0 (2.127) which defines in each point p of the manifold a three dimensional subspace ∆t = {X ∈ TpM|α(X) = 0} of the tangent space at p if we fix the time variable t. Considering the system of all such subspaces yields a three dimensional distribution ∆ ⊂ TM which can be used to define a foliation of the manifold. Therefore, we are interested in conditions on α to ensure integrability of ∆ and, using integral submanifolds of the distribution, existence of locally defined spacetime foliation. Those conditions can be deduced from the Frobenius theorem. Version of the theorem for differential forms states that given a distribution ∆ on a manifold M and k the system I = {α ∈ Ω (M)|α (X1,...,Xk) = 0 ∀X ∈ ∆,∀k = 1,...,4} of all differential forms vanishing on ∆, the condition on I to be closed under exterior differentiation

dI ⊆ I (2.128) implies integrability of the distribution. In our situation, I is generated by α only, in which case (2.128) is equivalent to the differential of α being collinear with α itself. This can be expressed by the following equation

dα ∧ α = 0 . (2.129)

On the other hand, having an integrable distribution ∆ as the kernel of α yields dα = 0 on ∆. Indeed, 1) by assumption, both ﬁelds are such that α (X) = α (Y) = 0 and 2) integrability of ∆ ensures that the Lie bracket of X,Y ∈ ∆ satisﬁes [X,Y] ∈ ∆ and thus α ([X,Y]) = 0. Recalling the formula for exterior differentiation of one- forms

dα (X,Y) = Xα (Y) +Yα (X) − α ([X,Y]) (2.130) implies that the left hand side of (2.130) vanishes on ∆, hence, (2.129) holds true. We conclude that the existence of a foliation of M given by the kernel of α is equivalent to (2.129) being satisﬁed. As parametrizing curves for the foliation leaves ∆ we can take the integral curves of time-like ∂t. This curves enable us to connect two nearby subspaces which differs by time shift and have ﬁxed space variables. On the other hand, we would like to connect subspaces given by two different points of the manifold with identical time

14 So that M is equipped with a pseudo-Riemannian metric gµν with (−,+,+,+) signature 15 That is, ∂t satisﬁes g(∂t ,∂t ) < 0 16 Meaning ∂i satisﬁes g(∂i,∂i) > 0 for all i. Chapter 2. General theory of relativity 25 coordinate. Existence of the aforementioned integral curves is locally guaranteed by Picard-Lindelöf theorem17. The foliation of M enables us to compute splitting of the metric with respect to the three dimensional spacelike subspace and one dimensional timelike subspace. This is our next aim.

2.4.2 Splitting of the metric µ ν Our goal now is to express the metric g = gµν dx dx with respect to the spacetime foliation. Let us assume that ∆ is integrable and that the corresponding integral submanifolds Σt define a foliation of M so that we can introduce local coordinates µ µ i i x = x t,y on M where y are coordinates on Σt for each fixed t and t is understood as a parameter for the family of foliation leaves Σt. Here ∂t,∂i are basis of timelike and spacelike subspace, respectively, and ∂α refers to the coordinate basis of the whole spacetime. With the xµ coordinates comes up a vector field τ tangent to the µ ∂x i µ time curves and having components τ µ = . Denote ei = ∂y and eµ = ∂x . Let ∂t µ ∂xµ i ∂yi µ n be components of a unit vector field n normal to Σt for all t. This vector field i µ ν satisfies the condition eµ n = 0 and thus for nµ = gµν n the following holds µ nµ ei = 0 , (2.131) for all i, since gµν is non-degenerate. We can decompose τ with respect to the basis ∂t,∂i as

µ µ i µ τ = Nn + N ei , (2.132) where N arises as the normalizing function for n and is called lapse. Functions Ni,i > 1, are components of τ in ∆ and are called spatial shift. Consequently, coordinate 1-forms dxµ can be expressed in the following way

µ µ µ i dx = τ dt + ei dy (2.133) µ i µ µ i = Nn + N ei dt + ei dy (2.134) µ µ i i = Nn dt + ei dy + N dt . (2.135) µ ν Inserting (2.135) in g = gµν dx dx we get

µ ν µ µ i i ν ν j j gµν dx dx = gµν Nn dt + ei dy + N dt Nn dt + e j dy + N dt (2.136) µ ν 2 µ ν j j = gµν n n (N dt) + gµν Nn e j dt dy + N dt ν µ i i µ ν i i j j + gµν Nn ei dt dy + N dt + gµν ei e j dy + N dt dy + N dt (2.137) µ ν 2 ν i i = gµν n n (N dt) + nν ei N dt dy + N dt µ j j µ ν i i j j (2.138) + nµ e j N dt dy + N dt + gµν ei e j dy + N dt dy + N dt .

17The theorem provides not only the existence but also uniqueness of this ﬁrst order initial value problem. Chapter 2. General theory of relativity 26

µ ν Observe that gµν n n = −1 since n is normal to the hypersurface given by timelike vector ﬁeld and thus is itself timelike. Moreover, using orthogonality condition (2.131) makes two middle terms in (2.138) vanish so we obtain

µ ν 2 µ ν i i j j gµν dx dx = −(N dt) + gµν ei e j dy + N dt dy + N dt . (2.139) In the last step we introduce coefﬁcients which capture spacelike properties of g

µ ν hi j = gµν ei e j (2.140) and, therefore, induces metric ﬁeld on Σt. Finally, the metric decomposition can be written in the form

2 2 i i j j g = −N dt + hi j dy + N dt dy + N dt . (2.141)

Using (2.141) we shall compute the time-time, time-space and space-space components of g by evaluating in ∂t,∂i basis. Let us begin with time-time component

g00 = g(∂t,∂t) 2 2 k k l l = −N dt + hkl dy + N dt dy + N dt (∂t,∂t)

2 k l 2 = −N + hklN N dt (∂t,∂t) 2 k l = −N + hklN N 2 l = −N + NlN .

i We have used the fact that dt,dy is dual basis to ∂t,∂i and also that hi j is projection 18 of gµν onto Σt and serves as an induced metric . We proceed with time-space components

g0i = g(∂t,∂i) 2 2 k k l l = −N dt + hkl dy + N dt dy + N dt (∂t,∂i) k l = hklN dt dy (∂t,∂i) l = Nlδi = Ni .

Finally we compute purely spatial components of the metric gi j = g ∂i,∂ j 2 2 k k l l = −N dt + hkl dy + N dt dy + N dt ∂i,∂ j k l = hkl dy dy ∂i,∂ j k l = hklδi δ j = hi j .

18 So that hi j can be used to lower indices of spatial objects Chapter 2. General theory of relativity 27

The last result is rather expected since the induced metric hi j should, by the ADM construction, capture the space-space properties of the original spacetime metric gµν . Let us summarize the above results in the matrix notation

 2 i  −N + NiN N1 ···Nn  N1  g =  .  . (2.142) . hi j Nn

2.4.3 Spatial covariant derivative Our next aim is to relate the ADM 3+1 decomposition of the metric field to define covariant differentiation on Σt and, consequently, use this to define curvatures of Σt. To define the spatial covariant derivative we will use the projection operator µ hν . To define it, we firstly introduce the 4-dimensional extension of hi j as

hµν = gµν + nµ nν . (2.143)

Raising up the ﬁrst index gives the following tensor

µ µα hν = g hαν (2.144) µα = g (gαν + nα nν ) (2.145) µ µ = δν + n nν . (2.146)

This operator acts on a contravariant vector ﬁeld X = (X µ ) as

µ ν ν ν µ hν X = X + n nµ X (2.147) (2.148) and thus behaves as the identity on X which is tangent to Σt since, in this case, the µ µ µ condition X nµ = 0 is satisfied. On the other hand, if X is time-like, i.e. X nµ = X µ ν µ ν holds, then hν X = 0. So hν X is a spatial vector field. Similarly for covariant Xµ , µ the expression hν Xµ is a spatial covector field. Therefore we recognize that the operator

µ µ µ hν = δν + n nν (2.149) acts as a projection on the spatial hypersurface. It naturally follows that the covariant derivative associated with the spacetime metric gµν can be projected µ via hν on the spatial hypersurface. In this way we can deﬁne the spatial covariant derivative D of a covariant vector ﬁeld tangent to Σt as

α β Dµ Xν = hµ hν ∇α Xβ . (2.150)

In the next step we will define the covariant derivative of functions and contravariant vector fields. We would be able, then, to extend the definition on arbitrary tensors. As a consequence, the definition of curvature of spacetime with respect Chapter 2. General theory of relativity 28 to the ADM formalism naturally follows. So, we prescribe the way D acts on an arbitrary function f

α Dµ f = hµ ∇α f (2.151) and we require of D to satisfy the product rule. It is possible now to determine how D acts on spatial contravariant vector ﬁelds. In order to do that consider µ µ spatial Y and compute the covariant derivative of scalar function XµY . Firstly, using (2.151), we get

ν α ν Dµ (XνY ) = hµ ∇α (XνY ) ν α α ν (2.152) = Y hµ ∇α Xν + Xν hµ ∇αY .

Secondly, we can apply product rule and (2.150) to get

ν ν ν Dµ (XνY ) = Y Dµ Xν + Xν DµY ν β α ν = Y hν hµ ∇α Xν + Xν DµY (2.153) β α ν = Y hµ ∇α Xν + Xν DµY .

ν β β We have used the fact that Y hν = Y for spatial Y. Comparing the last two results gives

ν α α ν β α ν Y hµ ∇α Xν + Xν hµ ∇αY = Y hµ ∇αXν + Xν DµY , (2.154) which is equivalent to

α ν ν Xν hµ ∇αY = Xν DµY . (2.155)

Here we need to be careful because the last equation is true only for Xν tangent 19 to Σt . Nevertheless, since h is surjective, we can express any spatial Xν as a projection of a suitable spacetime Yβ , meaning

β Xν = hν Yβ . (2.156)

Thus, (2.155) can be rewritten as

β α ν β ν Yβ hν hµ ∇αY = Yβ hν DµY . (2.157)

Becuase (2.157) is satisﬁed for arbitrary Yβ satisfying (2.156), we can omit Yµ on both sides of (2.157). Also recall that image of D is tangential to Σt which allows β us to omit projection hν on the right-hand side of (2.157). Therefore, we arrive at the formula for three-dimensional covariant derivative on Σt, applicable on any spacelike contravariant vector ﬁeld

β β α ν DµY = hν hµ ∇αY . (2.158)

19 ν In the sense that Xν n = 0 Chapter 2. General theory of relativity 29

By similar considerations as used in the previous paragraphs, we can extend D 20 ν1...νn on any spatial tensor Tµ1...µm of arbitrary rank by projecting all indeces

ν1...νn β ν1 νn γ1 γm δ1...δn Dα T ... = h h ...h h ...h ∇ T ... . (2.159) µ1 µm α δ1 δn µ1 µm β γ1 γm Important property that must be satisﬁed by D to be consistent with the rest of the notions developed in this section is the compatibility condition with respect to the spatial metric Dhµν = 0. We can directly verify that this metricity condition µ ν holds. Firstly we observe that hν n = 0 since µ ν µ µ ν µ µ hν n = (δν + n nν )n = n − n = 0 , (2.160) which is an expected result because h should map anything orthogonal to Σt on the zero vector. Recalling the deﬁnition (2.143) of hµν and the fact that gµν is covariantly constant with respect to ∇, we compute

ψ α β Dφ hµν = hφ hµ hν ∇ψ hαβ (2.161) = hψ hα hβ (g + n n ) φ µ ν ∇ψ αβ α β (2.162) = hψ hα hβ (n n ) φ µ ν ∇ψ α β (2.163) = hψ hα hβ n n + hψ hα hβ n n φ µ ν β ∇ψ α φ µ ν α ∇ψ β (2.164) = 0 . (2.165) In the last step we have used the Leibniz rule satisﬁed by ∇ and the result (2.160) making both of the summands vanish.

2.4.4 Curvatures of foliation and ADM action Now we will use the spatial covariant derivative D to deﬁne curvatures of the (3) δ foliation, namely extrinsic curvature of Σt and tensor Rαβγ . The extrinsic curvature tensor Kµν , which provides measurement of local non-triviality of the foliation, is deﬁned as the spatial covariant derivative of the normal

Kµν = −Dµ nν . (2.166)

21 The extrinsic curvature Kµν is symmetric . The 3-dimensional curvature tensor (3) δ Rαβγ can be deﬁned as anticommutator of the spatial covariant derivatives D

(3) δ Rαβγ Xδ = (Dγ Dβ − Dβ Dγ )Xα . (2.167) The curvature tensors (2.166) and (2.167) can be related to the Riemann curvature tensor Rαβγδ through the Gauss-Codazzi equation

µ ν φ ψ (3) hα hβ hγ hδ Rµνφψ = Rαβγδ + Kαγ Kβδ − Kαδ Kβγ . (2.168)

20 µ which can be deﬁned as T being invariant with respect to the projection hν in all indices 21To see this is true requires a bit of playfulness and some further investigation of the covariant derivative of the normal. Chapter 2. General theory of relativity 30

(3) From (2.168) we can read that Rαβγδ consists of the projection of Rµνφψ and two extra terms. Those extra terms arise as a consequence of the extrinsic properties of embedding the three-dimensional foliation leaves in M. (3) µν (3) Contracting Rαβγδ with h yields the spatial scalar curvature S. After further investigation of the above deﬁned tensors we can arrive at the following expression (3) for the scalar curvature S in terms of S and Kαβ

(3) αβ α β S = S+Kαβ K − Kα Kβ + (divergence terms) , (2.169) where the non-divergence terms deﬁnes the Arnowitt-Deser-Misner Lagrangian LADM

(3) αβ α β LADM = S+Kαβ K − Kα Kβ . (2.170)

Now we are ready to formulate the ADM action functional which will be the last step in our brief exploration of this 3 + 1 formalism 1 Z √ L = L N dethd3 ydt + (surface term) , (2.171) ADM 16π ADM where N is the lapse function22 and deth is the determinant of non-degenerate spatial metric hµν . The integral is considered over a compact domain. As we have seen in the previous section, the field equations being the solution to δLADM = 0, where δ is understood with respect gµν , will not depend on the surface term if the variation is assumed to vanish on the boundary of the integral domain. Varying the action (2.171) with respect to gi j amounts to independent variation i with respect to the lapse N, spatial shift N and the spatial metric hi j. Each of the independent variations will provide a set of equations describing the dynamics of the corresponding quantity, leading to the Einstein’s field equations in the 3 + 1 form. Let us note also without further explanation that, after ignoring the surface term, the action LADM can be investigated to determine the canonical momenta conjugate to hi j. The canonical momenta can be used to find the Hamiltonian form of the action functional (2.171) which leads to the Einstein’s field equations in the 3 + 1 form as well.

22introduced in the beginning of this section as a normalizing function Chapter 3

Einsten-Cartan theory of gravity

In this chapter, we wish to introduce the Einstein-Cartan1 theory which can be considered as a slight generalization of the general theory of relativity as formulated by A. Einstein. This chapter is primarily based and stands on the resultson given in [2], [3]. The approach without involving the differential forms can be found in [8]. The generalization is given by not assuming the connection to have vanishing torsion. Easing the assumption that the torsion should be zero forces the right-hand side of the ﬁeld equations to contain some anti-symmetric terms since in this case the Ricci tensor is not necessarily symmetric. Let us recall some basic assumptions. Let M be a four-dimensional spacetime manifold, meaning M is smooth and equipped with a non-degenerate symmetric 2 bilinear form gi j with signature (−,+,+,+) . Let all objects of interest be inﬁnitely differentiable as well. One of the underlying idea in EC is to consider general 3 i i i frames (ei) and the dual coframes (θ ), i.e. 1-forms satisfying θ e j = δ j for all i, j. Since a coframe is a basis of the cotangent space, (θ i) will be called a frame as well ( ∂ ) unless some ambiguity should arise. Unlike for coordinate frames ∂xi , elements of (ei) might not commute, meaning [ei,e j] 6= 0. This means that (ei) may not be given by local coordinates. In other words, if (xi) are local coordinates then their exterior derivative give rise to frame ( f i) = (dxi) implying that d f i = 0 which is i not true for a general frame (θ ). Bracketed expressions such as (ei) corresponds to ordered set of elements. If the brackets are omitted then we mean, for example, a single vector being the element of the ordered set. We keep in mind that the Ein- stein summation convention is used throughout the whole text. The most compact notation carefully utilized here is given by omitting the brackets and indexes, e.g. we denote by e the whole frame e = (ei).

1shortly referred to as EC 2Most of the upcoming notions and theory can be generalized to arbitrary dimension n without major difﬁculties as is done in [2] but since we decided to work in four dimensions throughout the whole text we will stick to this case. 3In four dimensions, these general frames are called tetrads or vierbeins and in higher dimensions those are called vielbeins.

– 31 – Chapter 3. Einsten-Cartan theory of gravity 32

Frames e and θ can be described with respect to the coordinate frames as

j ∂ ei = E , (3.1) i ∂x j i i j θ = D j dx (3.2) for some suitable E,D. Using e and θ we can describe arbitrary tensor ﬁeld on M. The metric tensor can be written as i j g = gi jθ θ . (3.3)

Here θ iθ j is an abbreviation of the symmetric tensor product. From now on, small latin indices i, j,k ∈ {1,...,4} will correspond to the four-dimensional spacetime manifold M. Capital latin indices I,J,K will correspond to other spaces. Following Trautman’s papers [2], [3] based on Cartan’s approach, we want to involve the language of tensor valued differential forms in our description. In order to do this, we need to know how a general linear transformation4 of the spacetime frames acts on higher order tensors. Therefore we need a linear representation of Gl(4,R) which amounts to a homomorphism

σ : Gl(4,R) → Gl(N,R) . (3.4) The tangent map to σ at the identity is an algebra homomorphism

σ∗ : End(4,R) → End(N,R) (3.5) where gl(4,R) = End(4,R) and gl(N,R) = End(N,R) are the corresponding Lie al- gebras. Let A ∈ End(4,R). The composition

expt(−) σ End(4,R) / Gl(4,R) / Gl(N,R) , (3.6) induces a Lie algebra representation. To see this, we recall that End(4,R), being an algebra, has the underlying vector space structure and thus we can identify ∼ TeEnd(4,R) = End(4,R). So, the representation is given by taking the tangent map to σ ◦ exp at the identity

d dt σ(expt(−))|t=0 End(4,R) / End(N,R) . (3.7) Since for the matrix Lie groups we have 1 1 exptA = I +tA + (tA)2 + (tA)3 + ... (3.8) 2 3! we can write d d 1 1 σ(exptA)| = σ ( (I +tA + (tA)2 + (tA)3 + ...)| ) (3.9) dt t=0 ∗ dt 2 3! t=0 = σ∗A , (3.10)

4an element of Gl(4,R) Chapter 3. Einsten-Cartan theory of gravity 33 where I is the identity matrix. The induced algebra homomorphism (3.7) can thus j J j be represented by a matrix which we denote σi = (σIi ) and write accordingly

d j σ(exptA)| = σ Ai . (3.11) dt t=0 i j N To see how this index notation works, say we pick a basis (eI ) in R then we can j 5 express the I-th coefficient of σi with respect to this basis as a 1-form given by j J j σi (eI) = σIi eJ . (3.12) i Let (ei) and (θ ) be some local fields of frames of the tangent and cotangent bundles of M. The frames may change from point to point and hence we consider i a = (a j): M → Gl(4,R) (3.13) corresponding to a field of frame transformations. Then a transforms e and θ as

0 j ei = e jai , (3.14) 0i −1 i j θ = (a ) jθ . (3.15)

6 N If α is a tensor-valued differential form , α ∈ Ω(M) ⊗R R then we say that α is of type σ if a change of frames transforms α as α0 = σ(a−1)α . (3.16)

Since θ i is a 1-form and transforms according to (3.15) it is said to be, with respect to the above deﬁnition, of type σ = id.

3.0.1 Variations of frames and k-forms We abuse our notation a bit and use the symbol A again to denote the field of infinitesimal generators of frame transformations on M, that is i A = (A j): M → End(4,R) . (3.17) Using the exponential map we define

a(t) := exptA: M → Gl(4,R) (3.18) which enables us to deﬁne the variation of frames and of arbitrary tensor-valued form α induced by an inﬁnitesimal change of frames as d δθ = (a(t)−1θ)| , (3.19) dt t=0 d δα = σ(a(t)−1α)| (3.20) dt t=0 5This is the same procedure as expressing the i-th coordinate of an arbitrary 1-form by evaluating the form on the i-th vector of the basis. N 6Observe that it is enough to consider only R to be the codomain of differential forms under N consideration since any tensor space is isomorphic to R for a suitable N. Chapter 3. Einsten-Cartan theory of gravity 34 where d d 1 (a(t)−1θ)| = ((I −tA + (tA)2 − ...)θ)| (3.21) dt t=0 dt 2 t=0 = −Aθ (3.22) and so (3.19) amounts to

i i j δθ = −A jθ (3.23) Similarly, d d 1 σ(a(t)−1α)| = σ ((I −tA + (tA)2 − ...)α)| (3.24) dt t=0 ∗ dt 2 t=0 = −σ∗Aα (3.25) and thus (3.20) equals to

J J j i I δα = −σIi A jα . (3.26)

3.1 Connection, curvature, torsion

The covariant derivative of e j in the direction of ei, denoted ∇ei e j or simply ∇ie j, is given by

k ∇ie j = Γ jiek . (3.27)

7 i A linear connection on M is given by a set of vector-valued 1-forms ω j. These can 8 i be described with respect to the connection coefﬁcients Γk j and the ﬁeld frame θ as follows

i i k ω j = Γk jθ . (3.28) Changing θ in accordance with (3.14) and (3.15) results in the following transformation of the connection forms

i 0k i k i akω j = ωka j + da j . (3.29) Since the connection form has a great importance in our considerations, we will derive and investigate a bit the transformation formula (3.29). This will be done in k N the next section devoted to metric-afﬁne spaces. Let α ∈ Ω (M) ⊗ R be a k-form of type σ, α = (αI). The covariant exterior derivative of a α with respect to the j connection ωi is deﬁned as

I I Ii j J Dα = dα + σJ jωi ∧ α . (3.30)

7 i ∗ Note that the 1-forms ω j corresponds to ∇e j ∈ T M ⊗ TM. 8These connection coefﬁcients represent the same objects introduced in the context of Levi- Civita connection in the previous chapter, whereby called Christoffel symbols. Chapter 3. Einsten-Cartan theory of gravity 35

N The (k +1)-form Dα is again of type σ. Note that for f ∈ C∞(M)⊗R , i.e. a 0-form, we have I i I D f = θ ∇i f . (3.31)

Following the deﬁnition (3.20), the variation of ω is d δω = σ(a(t)−1ω)| , (3.32) dt t=0 which amounts to i i δω j = DA j , (3.33)

i Curvature 2-form Ω = (Ω j) on M is deﬁned as

i i i k Ω j = dω j + ωk ∧ ω j . (3.34)

Observe that Ω is of type ad, where ad: Gl(4,R) → End(4,R) is given by adg(h) = ghg−1. The torsion 2-form Θ = (Θi) on M is given as the exterior derivative of the frame ﬁelds i i i j Θ = dθ + ω j ∧ θ (3.35) which is of type id. With respect to this formulation of geometric objects using differential forms, the ﬁrst and second Bianchi identities are expressed as i DΩ j = 0 , (3.36) i i j DΘ = Ω j ∧ θ . (3.37)

i i The curvature tensor R jkl and the torsion tensor Q jk on M are given as a solution to the equations 1 Ωi = Ri θ k ∧ θ l , (3.38) j 2 jkl 1 Θi = Qi θ j ∧ θ k , (3.39) 2 jk respectively. Let us mention that if the covariant derivative is the Levi-Civita one then it is not a difficult task using the definitions given above to explicitly compute i i that R jkl amounts to the Riemann curvature tensor and Q jk vanishes. Since, if we recall that the torsion tensor can be expressed with respect to the coordinate frame as the commutator of the connection coefficients i i i Q jk = Γ jk − Γk j , (3.40)

i i we can directly observe that Q jk does not vanish if and only if Γ jk is not symmetric in its lower indexes9. 9which it is for the Levi-Civita connection Chapter 3. Einsten-Cartan theory of gravity 36

3.1.1 Metric-affine space Our next aim is to define a Riemann-Cartan space. In order to do so we firstly introduce the notion of a metric-affine space and the corresponding metric-affine geometry, of which the Riemann-Cartan geometry is a special case. A metric-affine space is a smooth manifold M equipped with a metric field g and a linear connection ω. Note that the connection is not assumed to be metric nor to have vanishing torsion, thus g and ω are, in general, independent of each other. Yet, we may associate to g the unique Levi-Civita connection which comes with any metric field. We will denote this uniquely determined connection by ω¯ to emphasize that ω is a different object. The Levi-Civita ω¯ is determined by the following two conditions

D¯ gi j = 0 , (3.41) i i j dθ + ω¯ j ∧ θ = 0 , (3.42) where D¯ denotes the exterior covariant derivative with respect to ω¯ . Equations (3.41) corresponds to the metricity condition, i.e. requirement that the metric be covariantly constant. Comparing (3.42) with the definition (3.35), we see that the latter of the above equations expresses vanishing of the torsion 2-form. The curvature 2-form of ω¯ is in accordance with (3.34) given by ¯ i i i k Ω j = dω¯ j + ω¯k ∧ ω j , (3.43) Having two different connection forms naturally leads us to consider the difference i i i κ j = ω j − ω¯ j , (3.44) which is a 1-form of type ad. This 1-form is called the contorsion tensor and has two interesting properties. The first thing to observe from (3.44) is that the contorsion tensor is precisely the quantity that measures how much a general connection ω differs from the Levi-Civita ω¯ . Secondly, as the name indicates, κ is a tensorial quantity. We have already seen in the previous chapter devoted to derivation of the Einstein field equations that the Christoffel symbols, even though considered in a more restrictive case of a pseudo-Riemannian geometry, do not transform as tensors. To see that this is the case for the connection 1-forms as well, we observe that the covariant derivative of an element of a frame field e can be written as j ∇ei = e j ⊗ ωi (3.45) and that for an arbitrary smooth function f and a vector field v we may use the Leibniz rule satisfied by D ∇( f v) = f ∇v + v ⊗ d f . (3.46) Thus considering a frame transformation (3.14) we have in accordance with (3.45)

0 0 0 j ∇ei = e j ⊗ ω i (3.47) k 0 j = eka j ⊗ ω i . (3.48) Chapter 3. Einsten-Cartan theory of gravity 37

By the Leibniz rule, the covariant derivative of transformed frame is

0 j ∇ei = ∇(e jai ) (3.49) j j = ∇(e j)ai + e j ⊗ e j dai (3.50) k j k = ek ⊗ ω j ai + ek ⊗ dai (3.51)

Comparing (3.48) and (3.51) we have

k 0 j k j k eka j ⊗ ω i = ek ⊗ ω j ai + ek ⊗ dai . (3.52)

Considering only the components in equation (3.52) and multiplying by a−1 from the left we see that (3.52) is equivalent to

0 j −1 k k j −1 k k ω i = (a ) jω j ai + (a ) j dai . (3.53)

Let us rewrite (3.53) in a more concise and comprehensive manner as

ω0 = a−1ωa + a−1 da . (3.54)

From this equation we may observe two things. Firstly, the second summand in (3.54) is the obstruction term responsible for the connection form to not transform as a tensor. Secondly, because two different connections ω1 and ω2 would have the same obstruction term in their transformation rule, their difference would make this obstruction terms vanish. That is, the transformation rule for the difference of two connections is

0 0 −1 −1 −1 −1 ω1 − ω2 = a ω1a + a da − a ω2a + a da (3.55) −1 = a (ω1 − ω2)a (3.56) and hence ω1 − ω2 is a tensorial quantity. This is the second reason why the i contorsion tensor k j as introduced in (3.44), is considered. Moreover, from (3.56) i we see that k j is indeed a 1-form of type ad in the sense of (3.16). We can use this 1-form to describe the torsion and curvature 2-forms of ω

i i j Θ = κ j ∧ θ , (3.57) i ¯ i ¯ i i k Ω j = Ω j + Dκ j + κk ∧ κ j . (3.58)

i Similarly, lowering the upper index of κ j with g

k κi j = gikκ j , (3.59) we may express the exterior covariant derivative of g as

Dgi j = −(κi j + κ ji) . (3.60) Chapter 3. Einsten-Cartan theory of gravity 38

3.1.2 Riemann-Cartan space A metric-afﬁne space with a connection compatible with the metric, i.e.

Dgi j = 0 , (3.61) is called a Riemann-Cartan space. Equation (3.61) has two immediate consequences. Firstly due to (3.60) we get antisymmetry of κ

κi j = −κ ji (3.62) and similarly for the curvature 2-form we have

Ωi j = −Ω ji . (3.63)

The connection is completely determined by the metric and the torsion tensor. This means that if we have a metric deﬁned on our manifold then for each choice of the torsion tensor there is a unique D satisfying (3.61). Using purely covariant form of the torsion tensor

m Qi jk = gimQ jk (3.64) we can describe κ as 1 κ = (Q + Q + Q )θ k . (3.65) i j 2 ik j jik ki j Due to (3.44) the connection ω is thus given by 1 ω = ω¯ − (Q + Q + Q )θ k . (3.66) i j i j 2 ik j jik ki j Equation (3.66) can be interpreted similarly as (3.44), namely that a general metric compatible connection ω differs from the Levi-Civita ω¯ by a quantity which is determined by the torsion tensor Q.

3.1.3 Hodge duals In the proceeding paragraphs we will use the Hodge duals of the coframe ﬁelds to describe the action functional and corresponding ﬁeld equations in a concise way. Thus we introduce the following differential forms

η = ?1 , ηi = ?θ i , ηi j = ?(θ i ∧ θ j) , (3.67) ηi jk = ?(θ i ∧ θ j ∧ θ k) , ηi jkl = ?(θ i ∧ θ j ∧ θ k ∧ θ l) , Chapter 3. Einsten-Cartan theory of gravity 39 where ? represents the Hodge dual operator. The metric gi j is used to lower the kl indexes of the above deﬁned forms, e.g. ηi j = gikg jlη . Moreover, the following relations hold m m m m m θ ηi jkl = δl ηi jk − δk ηli j + δ j ηkli − δi η jkl , l l l l θ ∧ ηi jk = δkηi j + δ jηki + δi η jk , k k k (3.68) θ ∧ ηi j = δ j ηi − δi η j , j j θ ∧ ηi = δi η .

3.2 Field equations

Let M be a metric-affine four-dimensional spacetime manifold. Firstly let us describe the case for empty space and then naturally extend it to the case where matter fields occur in the spacetime. We have already seen in the previous chapters that the essential information about the field equations is contained in the Lagrangian density, i.e. the integrand of a specific action functional. We consider the following 4-form

1 j 8πL = η ∧ Ωi (3.69) 2 i j which is a differential form of the Lagrangian density used in the previous chapter, meaning, (3.69) contains essentially the same information about the ﬁeld equations i j as η S. However, the difference here is that the scalar curvature S = g Rici j arises from possibly non-symmetric Ricci tensor Rici j as we do not assume the connection to be Levi-Civita.

3.2.1 Vacuum ﬁeld equations We follow the results given in [3] that by variation of (3.69) with respect to the metric g and connection ω independently10, using the principle of stationary action, we obtain the vacuum equations in the form

1 j (gi jη k − gikη − g jkη i) ∧ Ωl = 0 (3.70) 2 l l l k and

l Dηk = 0 . (3.71)

Equation (3.70) corresponds to variation with respect to the metric gi j and (3.71) is l obtained from the variation with respect to the connection ωk . Together they are the Einstein-Cartan ﬁeld equations in vacuum.

10meaning that one is ﬁxed when the other is varied Chapter 3. Einsten-Cartan theory of gravity 40

Using the relations (3.68), equation (3.70) may be rewritten in terms of the generally asymmetric Ricci and scalar curvature as

1 j k j k j (δ Rickl −Ric −Ric )η = 0 , (3.72) 2 i kl ki ki which can be recognized as a generalization, with respect to non-vanishing torsion tensor, of the vacuum ﬁeld equations11 of GR.

3.2.2 Field equations in non-vacuum We proceed with the case of non-vacuum spacetime. For this, let us consider a general Lagrangian 4-form

k Lmatter(gi j,θ ,αI,DαI) (3.73)

k 12 depending on the metric (gi j), frame (θ ), a tensor-valued p-form (αI) and its derivative (DαI). The quantity corresponding in the field equations to the presence of matter fields is denoted T i j and is introduced in the same manner as in the previous chapter. Specifically, T i j is a tensor-valued 4-form given by varying Lmatter with respect to the metric gi j. What can be considered as a novelty in the Einstein-Cartan theory, compared to l the general theory of relativity, is the tensor of intrinsic angular momentum sk given i by the variation of Lmatter with respect to the connection form ω j. Let us note that 13 the variation of Lmatter with respect to a general matter field αI results in the equations of motion of the matter field, denoted simply LI. Putting all pieces together and varying the total action functional Z (L + Lmatter) (3.74) independently with respect to the metric and connection and by the virtue of the principle of stationary action and Stoke’s theorem we arrive at the Einstein-Cartan field equations describing the geometry of metric-affine spacetime in the presence of a matter field

1 j (gi jη k − gikη − g jkη i) ∧ Ωl = −8πT i j (3.75) 2 l l l k and

l l Dηk = −8πsk . (3.76) Our final task is to formulate the field equations for a Riemann-Cartan space. Therefore, we assume the spacetime to be equipped with the metric field g, connection form ω and that the connection is compatible with the metric, i.e. (3.61)

11as presented in the previous chapter 12This p-form corresponds to the presence of matter in spacetime. 13unspeciﬁed due to arbitrariness of α Chapter 3. Einsten-Cartan theory of gravity 41 is satisﬁed. From these assumptions it follows the antisymmetry of the curvature 2-form Ωi j and the vanishing of the covariant derivative of ηi jkl

−Ωi j = Ω ji , (3.77) Dηi jkl = 0 .

If we introduce the 3-form τ given by the stress-energy tensor T via the relation

i τ j = ηiTj , (3.78) we can formulate the ﬁeld equations in the following neat form 1 η ∧ Ω jk = −8τ , (3.79) 2 i jk i k ηi jkΘ = 8πsi j . (3.80) Seznam použité literatury

[1] Einstein, Albert (1916), The Foundation of the General Theory of Relativity, An- nalen der Physik. 354 (7): 769. [2] Trautman, Andrzej. Einstein-Cartan theory. arXiv preprint gr-qc/0606062 (2006). [3] Trautman, Andrzej. On the structure of the Einstein-Cartan equations. In Symp. Math., vol. 12, pp. 139-162. 1973. [4] Padmanabhan, Thanu. Gravitation: foundations and frontiers. Cambridge Uni- versity Press, 2010. [5] Carroll, Sean M. Spacetime and geometry. An introduction to general relativity. Vol. 1. 2004. [6] Will, C.M. Living Rev. Relativ. (2006) Dostupné z https://doi.org/10.12942/lrr-2006-3 [7] Misner, Charles W., Kip S. Thorne, and John Archibald Wheeler. Gravitation. Princeton University Press, 2017. [8] Hehl, Friedrich W., et al. General relativity with spin and torsion: Foundations and prospects. Reviews of Modern Physics 48.3 (1976): 393. [9] H. B. Lawson, The Qualitative Theory of Foliations, (1977) American Mathemat- ical Society CBMS Series volume 27, AMS, Providence RI. [10] Steven Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, (1972) John Wiley and Sons, New York ISBN 0- 471-92567-5. [11] Landau, Lev Davidovich, and Evgenii Mikhailovich Lifshitz. The classical theory of ﬁelds. (1971). [12] Parr, R. G., Yang, W. Density-Functional Theory of Atoms and Molecules.. New York: Oxford University Press (1989). pp. 246–254. ISBN 978-0195042795. [13] Gelfand, I. M.; Fomin, S. V. Calculus of variations, translated and edited by Richard A. Silverman (Revised English ed.). Mineola, N.Y.: Dover Publica- tions (2000) [1963], pp. 184-190. ISBN 978-0486414485

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