Bachelor Thesis for the Degree of Physics Fisikako Gradu Amaierako Lana Generalized Geometry for Relativistic Theories of Gravity Author/Egilea: Andoni Royo Abrego Directors/Zuzendariak: J¨urgen Schmidt Jon Urrestilla Oulu, June 2020 / Oulu, 2020 ekaina Abstract The TGR is constructed over a Pseudo-Riemaniann manifolod, where the tor- sion is assumed to vanish. However, there is not any physical argument behind it and when torsion is allowed, the spacetime in which the theory is constructed generalizes, from a V4 to a U4 spacetime, which carries several interesting conse- quences to be analyzed in this work. The most important one is the idea that the spin angular momentum acts, along with mass, as a source of gravitational interaction. In the first part of this work the TGR is formulated from an action principle. Later, the torsionless restriction is removed and the same procedure leads to slightly di↵erent field equations, in which spin is involved. Finally, requir- ing to curved spacetimes to posses symmetry under local Poincar´etransformations, the same U4 spacetime is deduced. Its implications are discussed in the end. Laburpena Erlatibitate orokorraren teoria barietate Sasi-Riemanndarrean eraikitzen da, non definizioz, tortsioa nulua den. Honek, ordea, ez du zertan hala izan, ez baitago tortsioa nulutzat hartzeko arrazoi fisikorik eta tortsioa agertzen denean, V4 espazio-denboratik U4 espazio-denborara igarotzen da, non lan honetan aztertzen diren hainbat ezberdinatasun nabari antzeman daitezkeen. Garrantzitsuena, be- harbada, spin momentu angeluarrak grabitazio iturri bezala jokatzen duela da, masak bezalaxe. Lan honen lehen zatian espazio-denbora Sasi-Riemanndarrean akzio printzio batetatik abiatuz, Einsteinen teoria garatzen da. Ondoren, prozesu berbera jarraitzen da U4 espazio-denboran, spinari lotutako eremu ekuazio berriak lortuz. Azkenik, kurbaduradun espazio-denborari Poincar´etaldearekiko simetria lokala derrigortuz, U4 espazio-denbora topatzen da. Honek eduki ditzakeen ondorio fisikoak eztabaidatzen dira amaieran. Resumen La Teor´ıa de la Relatividad General se construye en una variedad Pseudo- Riemanniana, donde la torsi´on es nula. Sin embargo, no hay ning´un argumento f´ısico que impida la aparici´on de la torsi´on. Esto generaliza el espacio-tiempo V4, convirtiendolo en U4,dondeaparecenciertascaracter´ısticasqueseanalizaneneste trabajo, de las cuales destaca el hecho de que el esp´ın,junto con la masa, act´ua como fuente de gravitaci´on. En la primera parte del trabajo se deduce la teor´ıa de Einstein desde un principio de acci´on y despu´es, se repite el proceso pero esta vez con torsi´on no nula, del cual se obtienen las ecuaciones de campo relacionadas al esp´ın. Finalmente se identifica el espacio-tiempo U4 cuando se requiere que el espacio-tiempo curvo sea invariante bajo el grupo de transformaciones locales de Poincar´e. i Contents 1 Fundamentals of di↵erential geometry and manifolds with torsion and non-metricity 1 1.1 Basics and notation . 1 1.2 Affineconnections............................ 2 1.3 Torsion and Curvature . 4 1.4 Levi-Civita connection . 8 2 General Relativity 9 2.1 The Theory of General Relativity . 10 2.2 Einstein field equations . 11 2.2.1 Einstein-Hilbertaction . 12 2.2.2 Palatini formalism . 14 3 U4 geometry and gravitation in U4 spacetime 16 3.1 Riemann-Cartan spacetime . 17 3.1.1 Geodesics in U4 ......................... 19 3.2 Fieldequations ............................. 21 3.3 Spin and gravitation . 23 4 Poincar´egroup symmetries, conservation laws and sources of grav- ity 25 4.1 Symmetry transformations and Noether currents . 26 4.2 Global Poincar´etransformations in the Minkowski spacetime . 26 4.2.1 Translation symmetry and Canonical energy-momentum ten- sor................................ 27 4.2.2 Lorentz transformations . 28 4.2.3 Spinor fields . 28 4.3 Local Poincar´etransformation . 29 4.3.1 Coupling and the Belinfante-Rosenfeld tensor . 30 5 Conclusions and implications 32 References 34 A Non-coordinate frames 36 ii 1. Fundamentals of di↵erential ge- ometry and manifolds with torsion and non-metricity Most of modern theories of gravity are based on the concept of curvature of the spacetime and for a good understanding of it, I consider of great importance a proper mathematical treatment of many aspects of di↵erential geometry. That is why this first chapter is focused on the definition of some mathematical tools, specially on those that are not usually covered in Geneal Relativiy textbooks, and later bring all the physical intuition. This first chapter is developed for a general n-dimensional manifolds and later sections deal with four dimensional spacetimes of di↵erent types. 1.1 Basics and notation At any point P described by P = xµ in some coordinate system xµ in a n- { } manifold M,itispossibletodefineabasisformedbythetangentvectorseµ µ ⌘ @/@x = @µ, called coordinate basis, which expand the tangent vector space at P , denoted by TP M.Correspondingly,itisdenotedTP⇤ M to the dual vector space formed by all mappings TP M IR, called the cotangent space, and usually chosen the dual basis vectors eµ dx!µ so that the inner product gives ⌘ @x⌫ e ,e⌫ = = δ⌫ . (1.1) h µ i @xµ µ Let the manifold be endowed with a metric tensor field of rank (0, 2), g : TP M TP M IR, which provides the possibility to make measurements on the manifold⇥ and it! is given by g = g (x)dxµ dx⌫ . (1.2) µ⌫ ⌦ Here gµ⌫(x)arecalledthecovariantcomponentsofthemetrictensorfield, which come naturally defined by the product of coordinate basis vectors, gµ⌫(x)= e ,e . Hence, for two arbitrary tangent vectors1 V = V µe , U = U µe T M: h µ ⌫i|x µ µ 2 P µ ⌫ g(V,U)=g(V eµ,U e⌫) µ ⌫ = V U g(eµ,e⌫) µ ⌫ ↵ β = V U g↵(P ) eµ,e e⌫,e (1.3) µ ⌫ h↵ β ih i = V U g↵(P )δµ δ⌫ µ ⌫ = V U gµ⌫(P ) . 1The Einstein sum convention is used during the whole work. 1 1.2 Affine connections Similarly the contravariant components of the metric tensor field can be written like gµ⌫(P )= eµ,e⌫ , h i|P and conclude the important relation between contravariant and covariant com- ponents of the metric tensor: µλ λµ µ g g⌫ = g⌫g = δ⌫ . (1.4) The metric tensor components can be used to move between contravariant and ↵ ↵ λβ λβ ⌫↵ covariant expressions of tensors, for example: T = T λg = T⌫g g . NOTE: in this last expression and here on, I will not explicitly write where the tensor field components are evaluated and it has to be understood that they are fields on the manifold. The metric gµ⌫ is symmetric by definition, and consequently, its eigenvalues real. If the metric is positive-definite it is called a Riemannian metric and the manifold itself is a Riemannian manifold. Similarly, if the metric is nondegen- erate it is called a Pseudo-Riemannian metric and the manifold itself a Pseudo- Riemannian manifold,whichisaslightgeneralizationandwillbedenotedVn.A vector field over a manifold M is defined as a map that smoothly maps a tangent vector to every point in M. In this work I will refer to the set of all vector fields over M by X(M), to the set of all scalar fields over M by F(M)andtothesetof q tensor fields of rank (q, r)overM by Tr(M). 1.2 Affine connections Aconnectionisamathematicalconstructthatspecifieshowtheinformationis transported along curves in a manifold. A general manifold may be endowed with a connection, which is in principle, an independent geometric structure that char- acterizes M. Specifically, if the connection gives information about how tangent spaces are related, it is called an affine connection, which is an essential object to develop di↵erential calculus on smooth manifolds. Let X, Y, Z X(M)and 2 f F(M). Formally, an affine connection maps : X(M) X(M) X(M), denoted2 by Y ,inawaythatthefollowingpropertiesaresatisfied:r ⇥ ! rX (Y + Z)= Y + Z (1.5) rX rX rX Z = Z + Z (1.6) r(X+Y ) rX rY Y = f Y (1.7) r(fX) rX (fY)=X[f]Y + f Y (1.8) rX rX If the connection is applied on two coordinate basis vector fields, the resulting vector field written in the coordinate basis would be 2 1.2 Affine connections e =Γλ e , (1.9) reµ ⌫ µ⌫ λ 3 λ where the n coefficients Γ µ⌫ are called the connection coefficients (and often just connection). For simpler notation, it is usually written just µe⌫ instead of µ rµ eµ e⌫. Now, for two arbitrary vector fields X = X eµ and Y = Y eµ and making user of the properties above: µ µ X Y = (X eµ)(Y eµ) r rµ λ ⌫ λ = X @µY + Y Γ µ⌫ eλ . If X was taken just a coordinate basis vector field, the vector components of the resulting vector field, are (1, 1) tensor field components Y λ = @ Y λ + Y ⌫Γλ . (1.10) rµ µ µ⌫ This is the definition of the usual covariant derivative, and represents the change of the λ component of the vector field Y along the eµ direction. A more intuitive and less mathematical derivation of this can be found in most General Relativity books, like in [9, 17]. For a curve c(t) M,wheret IR , a n d t h e v e c t o r fi e l d X d/dt = (dxµ/dt)(@/@xµ)definedalongthecurve,itissaidthatthevectorfield! 2 Y⌘is parallel transported along it if it does not change its direction, i.e. Y =0, (1.11) rX which can be written like dY µ dx⌫ +Γµ Y λ =0. (1.12) dt ⌫ dt Parallelism in two and three dimesional Euclidean spaces seems a totally natu- ral and intuitive notion. However, in manifolds with curvature there is not a nat- ural way of defining ”parallelism” but the one that the connection itself uniquely provides.