Introduction to Teleparallel equivalent of general relativity Modified teleparallel gravity theories Conclusions Teleparallel gravity and its modifications Sebastian´ Bahamonde [email protected] Department of Mathematics, University College London. Particles and Fields Seminar - Ben Gurion University 04th December 2017 1 / 49 Introduction to Teleparallel equivalent of general relativity Modified teleparallel gravity theories Conclusions Outline 1 Introduction to Teleparallel equivalent of general relativity Basic concepts in teleparallel gravity Gauge structure: translation group Torsion, connection and action Teleparallel gravity vs General Relativity 2 Modified teleparallel gravity theories General properties Modified teleparallel theories of gravity: f(T;B) gravity New classes of modified theories of gravity f(Ti;B) Modified teleparallel theories of gravity: f(T; B; TG;BG) 3 Conclusions 2 / 49 Basic concepts in teleparallel gravity Introduction to Teleparallel equivalent of general relativity Gauge structure: translation group Modified teleparallel gravity theories Torsion, connection and action Conclusions Teleparallel gravity vs General Relativity Outline 1 Introduction to Teleparallel equivalent of general relativity Basic concepts in teleparallel gravity Gauge structure: translation group Torsion, connection and action Teleparallel gravity vs General Relativity 2 Modified teleparallel gravity theories General properties Modified teleparallel theories of gravity: f(T;B) gravity New classes of modified theories of gravity f(Ti;B) Modified teleparallel theories of gravity: f(T; B; TG;BG) 3 Conclusions 3 / 49 Basic concepts in teleparallel gravity Introduction to Teleparallel equivalent of general relativity Gauge structure: translation group Modified teleparallel gravity theories Torsion, connection and action Conclusions Teleparallel gravity vs General Relativity Tetrad fields Assuming that the manifold is differentiable: Define tetrads a (or vierbein) ea (or e ) which are the linear basis on the spacetimef manifold.g f g At each point of the spacetime, tetrads gives us basis for vectors on the tangent space. Notation: Greek letters space-time indices; ! µ Latin letters tangent space indices; Ea is the inverse of the tetrad. ! µ a a µ Then for example ea = Ea dxµ and e = e µdx . µ n n Tetrads satisfy the orthogonality condition: Em e µ = δm ν m ν and Em e µ = δµ 4 / 49 Basic concepts in teleparallel gravity Introduction to Teleparallel equivalent of general relativity Gauge structure: translation group Modified teleparallel gravity theories Torsion, connection and action Conclusions Teleparallel gravity vs General Relativity Tetrad fields Assuming that the manifold is differentiable: Define tetrads a (or vierbein) ea (or e ) which are the linear basis on the spacetimef manifold.g f g At each point of the spacetime, tetrads gives us basis for vectors on the tangent space. Notation: Greek letters space-time indices; ! µ Latin letters tangent space indices; Ea is the inverse of the tetrad. ! µ a a µ Then for example ea = Ea dxµ and e = e µdx . µ n n Tetrads satisfy the orthogonality condition: Em e µ = δm ν m ν and Em e µ = δµ 4 / 49 Basic concepts in teleparallel gravity Introduction to Teleparallel equivalent of general relativity Gauge structure: translation group Modified teleparallel gravity theories Torsion, connection and action Conclusions Teleparallel gravity vs General Relativity Tetrad fields Assuming that the manifold is differentiable: Define tetrads a (or vierbein) ea (or e ) which are the linear basis on the spacetimef manifold.g f g At each point of the spacetime, tetrads gives us basis for vectors on the tangent space. Notation: Greek letters space-time indices; ! µ Latin letters tangent space indices; Ea is the inverse of the tetrad. ! µ a a µ Then for example ea = Ea dxµ and e = e µdx . µ n n Tetrads satisfy the orthogonality condition: Em e µ = δm ν m ν and Em e µ = δµ 4 / 49 Basic concepts in teleparallel gravity Introduction to Teleparallel equivalent of general relativity Gauge structure: translation group Modified teleparallel gravity theories Torsion, connection and action Conclusions Teleparallel gravity vs General Relativity Tetrad fields Assuming that the manifold is differentiable: Define tetrads a (or vierbein) ea (or e ) which are the linear basis on the spacetimef manifold.g f g At each point of the spacetime, tetrads gives us basis for vectors on the tangent space. Notation: Greek letters space-time indices; ! µ Latin letters tangent space indices; Ea is the inverse of the tetrad. ! µ a a µ Then for example ea = Ea dxµ and e = e µdx . µ n n Tetrads satisfy the orthogonality condition: Em e µ = δm ν m ν and Em e µ = δµ 4 / 49 Basic concepts in teleparallel gravity Introduction to Teleparallel equivalent of general relativity Gauge structure: translation group Modified teleparallel gravity theories Torsion, connection and action Conclusions Teleparallel gravity vs General Relativity Tetrad fields Assuming that the manifold is differentiable: Define tetrads a (or vierbein) ea (or e ) which are the linear basis on the spacetimef manifold.g f g At each point of the spacetime, tetrads gives us basis for vectors on the tangent space. Notation: Greek letters space-time indices; ! µ Latin letters tangent space indices; Ea is the inverse of the tetrad. ! µ a a µ Then for example ea = Ea dxµ and e = e µdx . µ n n Tetrads satisfy the orthogonality condition: Em e µ = δm ν m ν and Em e µ = δµ 4 / 49 Basic concepts in teleparallel gravity Introduction to Teleparallel equivalent of general relativity Gauge structure: translation group Modified teleparallel gravity theories Torsion, connection and action Conclusions Teleparallel gravity vs General Relativity Tetrads and Special Relativity c General basis ea satisfies [ea; eb] = f abec , where c µ ν f gc c f ab = Ea Eb (@νe µ @νe ν) is the coefficient of anholonomy (curl of the− basis). a Inertial frame (or holonomic frame) ea0 : f 0 cd = 0 (global property) Special Relativity in Minkowski: Linear frames (trivial frames) gives us a relation between ηab (tangent) and ηµν µ ν a b (spacetime): ηab = ηµνEa Eb and ηµν = ηabe µe ν. In trivial frames: Curvature and torsion are zero. 5 / 49 Basic concepts in teleparallel gravity Introduction to Teleparallel equivalent of general relativity Gauge structure: translation group Modified teleparallel gravity theories Torsion, connection and action Conclusions Teleparallel gravity vs General Relativity Tetrads and Special Relativity c General basis ea satisfies [ea; eb] = f abec , where c µ ν f gc c f ab = Ea Eb (@νe µ @νe ν) is the coefficient of anholonomy (curl of the− basis). a Inertial frame (or holonomic frame) ea0 : f 0 cd = 0 (global property) Special Relativity in Minkowski: Linear frames (trivial frames) gives us a relation between ηab (tangent) and ηµν µ ν a b (spacetime): ηab = ηµνEa Eb and ηµν = ηabe µe ν. In trivial frames: Curvature and torsion are zero. 5 / 49 Basic concepts in teleparallel gravity Introduction to Teleparallel equivalent of general relativity Gauge structure: translation group Modified teleparallel gravity theories Torsion, connection and action Conclusions Teleparallel gravity vs General Relativity Tetrads and Special Relativity c General basis ea satisfies [ea; eb] = f abec , where c µ ν f gc c f ab = Ea Eb (@νe µ @νe ν) is the coefficient of anholonomy (curl of the− basis). a Inertial frame (or holonomic frame) ea0 : f 0 cd = 0 (global property) Special Relativity in Minkowski: Linear frames (trivial frames) gives us a relation between ηab (tangent) and ηµν µ ν a b (spacetime): ηab = ηµνEa Eb and ηµν = ηabe µe ν. In trivial frames: Curvature and torsion are zero. 5 / 49 Basic concepts in teleparallel gravity Introduction to Teleparallel equivalent of general relativity Gauge structure: translation group Modified teleparallel gravity theories Torsion, connection and action Conclusions Teleparallel gravity vs General Relativity Tetrads and Special Relativity c General basis ea satisfies [ea; eb] = f abec , where c µ ν f gc c f ab = Ea Eb (@νe µ @νe ν) is the coefficient of anholonomy (curl of the− basis). a Inertial frame (or holonomic frame) ea0 : f 0 cd = 0 (global property) Special Relativity in Minkowski: Linear frames (trivial frames) gives us a relation between ηab (tangent) and ηµν µ ν a b (spacetime): ηab = ηµνEa Eb and ηµν = ηabe µe ν. In trivial frames: Curvature and torsion are zero. 5 / 49 Basic concepts in teleparallel gravity Introduction to Teleparallel equivalent of general relativity Gauge structure: translation group Modified teleparallel gravity theories Torsion, connection and action Conclusions Teleparallel gravity vs General Relativity Tetrads and Special Relativity a a a a b SR: e0µ = @µx0 and then x Λ bx . The new frame will be ! Lorentz-rotated frame in SR a a b a a b a e = Λ e0 = @ x + w x D x ; (1) µ b µ µ bµ ≡ µ a a e where w bµ = Λ e@µΛb is known as the spin connection which represents inertial effects. du0a Example free particles: inertial frame: dλ = 0. This Eq. is not local Lorentz invariant However, in the anholonomic frame, changing a a b du0a a b µ e µ = Λ be0µ, one gets dλ + w bµu u = 0 which is Lorentz covariant! Spin-connection in SR represent inertial effects and they are very important to the local Lorentz invariance. 6 / 49 Basic concepts in teleparallel gravity Introduction to Teleparallel equivalent of general relativity Gauge structure: translation group Modified teleparallel gravity theories Torsion, connection and action Conclusions Teleparallel gravity vs General Relativity Tetrads and Special Relativity a a a a b SR: e0µ = @µx0 and then x Λ bx . The new frame will be ! Lorentz-rotated frame in SR a a b a a b a e = Λ e0 = @ x + w x D x ; (1) µ b µ µ bµ ≡ µ a a e where w bµ = Λ e@µΛb is known as the spin connection which represents inertial effects.