Nonlocal Teleparallel Gravity Sebasti an Bahamonde PhD student at - PowerPoint PPT Presentation

Introduction to Teleparallel equivalent of general relativity Nonlocal Teleparallel gravity Conclusions Nonlocal Teleparallel Gravity Sebasti´ an Bahamonde PhD student at Department of Mathematics, University College London. Gravity and Cosmology - Yukawa Institute for Theoretical Physics, Kyoto University 01 March 2018 Based on S. Bahamonde, S. Capozziello, M. Faizal and R. C. Nunes, Eur. Phys. J. C 77 (2017) no.9, 628 1 / 19

Introduction to Teleparallel equivalent of general relativity Nonlocal Teleparallel gravity Conclusions Outline Introduction to Teleparallel equivalent of general relativity 1 Basic concepts in teleparallel gravity Teleparallel gravity vs General Relativity Nonlocal Teleparallel gravity 2 Nonlocal gravity Nonlocal Teleparallel gravity Conclusions 3 2 / 19

Introduction to Teleparallel equivalent of general relativity Basic concepts in teleparallel gravity Nonlocal Teleparallel gravity Teleparallel gravity vs General Relativity Conclusions Tetrad fields Assuming that the manifold is differentiable: Define tetrads (or vierbein) { e a } (or { e a } ) which are the linear basis on the spacetime manifold. At each point of the spacetime, tetrads gives us basis for vectors on the tangent space. Notation: Greek letters → space-time indices; Latin letters → tangent space indices; E aµ is the inverse of the tetrad. Tetrads satisfy the orthogonality condition: E mµ e nµ = δ n m and E mν e mµ = δ ν µ and metric can be reconstructed via g µν = η ab e aµ e bν 3 / 19

Introduction to Teleparallel equivalent of general relativity Basic concepts in teleparallel gravity Nonlocal Teleparallel gravity Teleparallel gravity vs General Relativity Conclusions Tetrad fields Assuming that the manifold is differentiable: Define tetrads (or vierbein) { e a } (or { e a } ) which are the linear basis on the spacetime manifold. At each point of the spacetime, tetrads gives us basis for vectors on the tangent space. Notation: Greek letters → space-time indices; Latin letters → tangent space indices; E aµ is the inverse of the tetrad. Tetrads satisfy the orthogonality condition: E mµ e nµ = δ n m and E mν e mµ = δ ν µ and metric can be reconstructed via g µν = η ab e aµ e bν 3 / 19

Introduction to Teleparallel equivalent of general relativity Basic concepts in teleparallel gravity Nonlocal Teleparallel gravity Teleparallel gravity vs General Relativity Conclusions Tetrad fields Assuming that the manifold is differentiable: Define tetrads (or vierbein) { e a } (or { e a } ) which are the linear basis on the spacetime manifold. At each point of the spacetime, tetrads gives us basis for vectors on the tangent space. Notation: Greek letters → space-time indices; Latin letters → tangent space indices; E aµ is the inverse of the tetrad. Tetrads satisfy the orthogonality condition: E mµ e nµ = δ n m and E mν e mµ = δ ν µ and metric can be reconstructed via g µν = η ab e aµ e bν 3 / 19

Introduction to Teleparallel equivalent of general relativity Basic concepts in teleparallel gravity Nonlocal Teleparallel gravity Teleparallel gravity vs General Relativity Conclusions Tetrad fields Assuming that the manifold is differentiable: Define tetrads (or vierbein) { e a } (or { e a } ) which are the linear basis on the spacetime manifold. At each point of the spacetime, tetrads gives us basis for vectors on the tangent space. Notation: Greek letters → space-time indices; Latin letters → tangent space indices; E aµ is the inverse of the tetrad. Tetrads satisfy the orthogonality condition: E mµ e nµ = δ n m and E mν e mµ = δ ν µ and metric can be reconstructed via g µν = η ab e aµ e bν 3 / 19

Introduction to Teleparallel equivalent of general relativity Basic concepts in teleparallel gravity Nonlocal Teleparallel gravity Teleparallel gravity vs General Relativity Conclusions Connection in Teleparallel gravity Teleparallel gravity (TEGR) is an alternative formulation of gravity which uses tetrads as the dynamical variables. Let us introduce the so-called “Weitzenb¨ ock connection”: Weitzenb¨ ock connection ˜ Γ ρµν = E aρ D µ e aν = E aρ ( ∂ µ e aν + w abµ e bν ) . By using this connection, one can express the torsion tensor as follows Torsion tensor T ρµν = ˜ Γ ρνµ − ˜ Γ ρµν . 4 / 19

Introduction to Teleparallel equivalent of general relativity Basic concepts in teleparallel gravity Nonlocal Teleparallel gravity Teleparallel gravity vs General Relativity Conclusions Connection in Teleparallel gravity Teleparallel gravity (TEGR) is an alternative formulation of gravity which uses tetrads as the dynamical variables. Let us introduce the so-called “Weitzenb¨ ock connection”: Weitzenb¨ ock connection ˜ Γ ρµν = E aρ D µ e aν = E aρ ( ∂ µ e aν + w abµ e bν ) . By using this connection, one can express the torsion tensor as follows Torsion tensor T ρµν = ˜ Γ ρνµ − ˜ Γ ρµν . 4 / 19

Introduction to Teleparallel equivalent of general relativity Basic concepts in teleparallel gravity Nonlocal Teleparallel gravity Teleparallel gravity vs General Relativity Conclusions Connection in Teleparallel gravity Teleparallel gravity (TEGR) is an alternative formulation of gravity which uses tetrads as the dynamical variables. Let us introduce the so-called “Weitzenb¨ ock connection”: Weitzenb¨ ock connection ˜ Γ ρµν = E aρ D µ e aν = E aρ ( ∂ µ e aν + w abµ e bν ) . By using this connection, one can express the torsion tensor as follows Torsion tensor T ρµν = ˜ Γ ρνµ − ˜ Γ ρµν . 4 / 19

Introduction to Teleparallel equivalent of general relativity Basic concepts in teleparallel gravity Nonlocal Teleparallel gravity Teleparallel gravity vs General Relativity Conclusions Connection in Teleparallel gravity Teleparallel gravity (TEGR) is an alternative formulation of gravity which uses tetrads as the dynamical variables. Let us introduce the so-called “Weitzenb¨ ock connection”: Weitzenb¨ ock connection ˜ Γ ρµν = E aρ D µ e aν = E aρ ( ∂ µ e aν + w abµ e bν ) . By using this connection, one can express the torsion tensor as follows Torsion tensor T ρµν = ˜ Γ ρνµ − ˜ Γ ρµν . 4 / 19

Introduction to Teleparallel equivalent of general relativity Basic concepts in teleparallel gravity Nonlocal Teleparallel gravity Teleparallel gravity vs General Relativity Conclusions Connection in Teleparallel gravity Teleparallel gravity (TEGR) is an alternative formulation of gravity which uses tetrads as the dynamical variables. Let us introduce the so-called “Weitzenb¨ ock connection”: Weitzenb¨ ock connection ˜ Γ ρµν = E aρ D µ e aν = E aρ ( ∂ µ e aν + w abµ e bν ) . By using this connection, one can express the torsion tensor as follows Torsion tensor T ρµν = ˜ Γ ρνµ − ˜ Γ ρµν . 4 / 19

Introduction to Teleparallel equivalent of general relativity Basic concepts in teleparallel gravity Nonlocal Teleparallel gravity Teleparallel gravity vs General Relativity Conclusions Connection in TEGR ock connection ˜ Γ ρνµ is related to the The Weitzenb¨ Levi-Civita connection Γ ρνµ via Relationship between connections ˜ Γ ρνµ = Γ ρνµ + K ρµν , where K ρµν = 1 2 ( T µρν + T νρµ − T ρµν ) is the contorsion tensor. In this connection, it is easy to verify that the spacetime is globally flat: Curvature in Teleparallel gravity R abµν ( ω abµ ) = ∂ µ ω abν − ∂ ν ω abµ + ω acµ ω cbν − ω acν ω cbµ ≡ 0 . 5 / 19

Introduction to Teleparallel equivalent of general relativity Basic concepts in teleparallel gravity Nonlocal Teleparallel gravity Teleparallel gravity vs General Relativity Conclusions Connection in TEGR ock connection ˜ Γ ρνµ is related to the The Weitzenb¨ Levi-Civita connection Γ ρνµ via Relationship between connections ˜ Γ ρνµ = Γ ρνµ + K ρµν , where K ρµν = 1 2 ( T µρν + T νρµ − T ρµν ) is the contorsion tensor. In this connection, it is easy to verify that the spacetime is globally flat: Curvature in Teleparallel gravity R abµν ( ω abµ ) = ∂ µ ω abν − ∂ ν ω abµ + ω acµ ω cbν − ω acν ω cbµ ≡ 0 . 5 / 19

Introduction to Teleparallel equivalent of general relativity Basic concepts in teleparallel gravity Nonlocal Teleparallel gravity Teleparallel gravity vs General Relativity Conclusions Connection in TEGR ock connection ˜ Γ ρνµ is related to the The Weitzenb¨ Levi-Civita connection Γ ρνµ via Relationship between connections ˜ Γ ρνµ = Γ ρνµ + K ρµν , where K ρµν = 1 2 ( T µρν + T νρµ − T ρµν ) is the contorsion tensor. In this connection, it is easy to verify that the spacetime is globally flat: Curvature in Teleparallel gravity R abµν ( ω abµ ) = ∂ µ ω abν − ∂ ν ω abµ + ω acµ ω cbν − ω acν ω cbµ ≡ 0 . 5 / 19