Outline Introduction: A Useful Method Tetrad Formalism 1 The Spin 2 Field Conclusion Bibliography Spinors in Curved Space Erik Olsen December 5, 2008 Erik Olsen Spinors in Curved Space Outline Introduction: A Useful Method Tetrad Formalism 1 The Spin 2 Field Conclusion Bibliography Introduction: A Useful Method Tetrad Formalism Tetrads Covariant Derivatives 1 The Spin 2 Field Conclusion Bibliography Erik Olsen Spinors in Curved Space I The solution: The Principle of General Covariance I 1. Ignoring gravity, find the equations of motion I 2. Make the following substitutions: I Lorentz Tensors become Tensor-like objects I Derivatives become Covariant Derivatives I Minkowski tensors (η matrices) become the metric tensor for curved spacetime gµν Outline Introduction: A Useful Method Tetrad Formalism 1 The Spin 2 Field Conclusion Bibliography Spinors in Curved Space I The problem: How to put gravity into a Lagrangian density? Erik Olsen Spinors in Curved Space I 1. Ignoring gravity, find the equations of motion I 2. Make the following substitutions: I Lorentz Tensors become Tensor-like objects I Derivatives become Covariant Derivatives I Minkowski tensors (η matrices) become the metric tensor for curved spacetime gµν Outline Introduction: A Useful Method Tetrad Formalism 1 The Spin 2 Field Conclusion Bibliography Spinors in Curved Space I The problem: How to put gravity into a Lagrangian density? I The solution: The Principle of General Covariance Erik Olsen Spinors in Curved Space I 2. Make the following substitutions: I Lorentz Tensors become Tensor-like objects I Derivatives become Covariant Derivatives I Minkowski tensors (η matrices) become the metric tensor for curved spacetime gµν Outline Introduction: A Useful Method Tetrad Formalism 1 The Spin 2 Field Conclusion Bibliography Spinors in Curved Space I The problem: How to put gravity into a Lagrangian density? I The solution: The Principle of General Covariance I 1. Ignoring gravity, find the equations of motion Erik Olsen Spinors in Curved Space I Lorentz Tensors become Tensor-like objects I Derivatives become Covariant Derivatives I Minkowski tensors (η matrices) become the metric tensor for curved spacetime gµν Outline Introduction: A Useful Method Tetrad Formalism 1 The Spin 2 Field Conclusion Bibliography Spinors in Curved Space I The problem: How to put gravity into a Lagrangian density? I The solution: The Principle of General Covariance I 1. Ignoring gravity, find the equations of motion I 2. Make the following substitutions: Erik Olsen Spinors in Curved Space I Derivatives become Covariant Derivatives I Minkowski tensors (η matrices) become the metric tensor for curved spacetime gµν Outline Introduction: A Useful Method Tetrad Formalism 1 The Spin 2 Field Conclusion Bibliography Spinors in Curved Space I The problem: How to put gravity into a Lagrangian density? I The solution: The Principle of General Covariance I 1. Ignoring gravity, find the equations of motion I 2. Make the following substitutions: I Lorentz Tensors become Tensor-like objects Erik Olsen Spinors in Curved Space I Minkowski tensors (η matrices) become the metric tensor for curved spacetime gµν Outline Introduction: A Useful Method Tetrad Formalism 1 The Spin 2 Field Conclusion Bibliography Spinors in Curved Space I The problem: How to put gravity into a Lagrangian density? I The solution: The Principle of General Covariance I 1. Ignoring gravity, find the equations of motion I 2. Make the following substitutions: I Lorentz Tensors become Tensor-like objects I Derivatives become Covariant Derivatives Erik Olsen Spinors in Curved Space Outline Introduction: A Useful Method Tetrad Formalism 1 The Spin 2 Field Conclusion Bibliography Spinors in Curved Space I The problem: How to put gravity into a Lagrangian density? I The solution: The Principle of General Covariance I 1. Ignoring gravity, find the equations of motion I 2. Make the following substitutions: I Lorentz Tensors become Tensor-like objects I Derivatives become Covariant Derivatives I Minkowski tensors (η matrices) become the metric tensor for curved spacetime gµν Erik Olsen Spinors in Curved Space I No representations of GL(4) act like spinors under an infinitesimal Lorentz transformation Outline Introduction: A Useful Method Tetrad Formalism 1 The Spin 2 Field Conclusion Bibliography Spinors in Curved Space I A major issue: this method does not work for spinors Erik Olsen Spinors in Curved Space Outline Introduction: A Useful Method Tetrad Formalism 1 The Spin 2 Field Conclusion Bibliography Spinors in Curved Space I A major issue: this method does not work for spinors I No representations of GL(4) act like spinors under an infinitesimal Lorentz transformation Erik Olsen Spinors in Curved Space α I Let zX be normal local coordinates to each point in space-time X I @zα(x) @zβ (x) g (x) = X X η (2) µν @xµ @xν αβ Outline Introduction: A Useful Method Tetrad Formalism Tetrads 1 The Spin 2 Field Covariant Derivatives Conclusion Bibliography Tetrad Formalism I @ξα @ξβ g = η (1) µν @xµ @xν αβ where ξ represents a coordinate system under the influence of gravity. Erik Olsen Spinors in Curved Space I @zα(x) @zβ (x) g (x) = X X η (2) µν @xµ @xν αβ Outline Introduction: A Useful Method Tetrad Formalism Tetrads 1 The Spin 2 Field Covariant Derivatives Conclusion Bibliography Tetrad Formalism I @ξα @ξβ g = η (1) µν @xµ @xν αβ where ξ represents a coordinate system under the influence of gravity. α I Let zX be normal local coordinates to each point in space-time X Erik Olsen Spinors in Curved Space Outline Introduction: A Useful Method Tetrad Formalism Tetrads 1 The Spin 2 Field Covariant Derivatives Conclusion Bibliography Tetrad Formalism I @ξα @ξβ g = η (1) µν @xµ @xν αβ where ξ represents a coordinate system under the influence of gravity. α I Let zX be normal local coordinates to each point in space-time X I @zα(x) @zβ (x) g (x) = X X η (2) µν @xµ @xν αβ Erik Olsen Spinors in Curved Space α I Vµ (X ) is a tetrad. α µ 0µ I Fix zX , change x to x I α 0α Vµ ! Vµ (5) Outline Introduction: A Useful Method Tetrad Formalism Tetrads 1 The Spin 2 Field Covariant Derivatives Conclusion Bibliography Tetrad Formalism I Let: α @zX (x) α µ = Vµ (X ) (3) @x x=X β @zX (x) β ν = Vν (X ) (4) @x x=X Erik Olsen Spinors in Curved Space α µ 0µ I Fix zX , change x to x I α 0α Vµ ! Vµ (5) Outline Introduction: A Useful Method Tetrad Formalism Tetrads 1 The Spin 2 Field Covariant Derivatives Conclusion Bibliography Tetrad Formalism I Let: α @zX (x) α µ = Vµ (X ) (3) @x x=X β @zX (x) β ν = Vν (X ) (4) @x x=X α I Vµ (X ) is a tetrad. Erik Olsen Spinors in Curved Space I α 0α Vµ ! Vµ (5) Outline Introduction: A Useful Method Tetrad Formalism Tetrads 1 The Spin 2 Field Covariant Derivatives Conclusion Bibliography Tetrad Formalism I Let: α @zX (x) α µ = Vµ (X ) (3) @x x=X β @zX (x) β ν = Vν (X ) (4) @x x=X α I Vµ (X ) is a tetrad. α µ 0µ I Fix zX , change x to x Erik Olsen Spinors in Curved Space Outline Introduction: A Useful Method Tetrad Formalism Tetrads 1 The Spin 2 Field Covariant Derivatives Conclusion Bibliography Tetrad Formalism I Let: α @zX (x) α µ = Vµ (X ) (3) @x x=X β @zX (x) β ν = Vν (X ) (4) @x x=X α I Vµ (X ) is a tetrad. α µ 0µ I Fix zX , change x to x I α 0α Vµ ! Vµ (5) Erik Olsen Spinors in Curved Space I @xν V 0α = V α (9) µ @x0µ ν Outline Introduction: A Useful Method Tetrad Formalism Tetrads 1 The Spin 2 Field Covariant Derivatives Conclusion Bibliography Tetrad Formalism I @zα V 0α = X −! (6) µ @x0µ @zα @xν V 0α = X −! (7) µ @x0µ @xν @xν @zα V 0α = X (8) µ @x0µ @xν Erik Olsen Spinors in Curved Space Outline Introduction: A Useful Method Tetrad Formalism Tetrads 1 The Spin 2 Field Covariant Derivatives Conclusion Bibliography Tetrad Formalism I @zα V 0α = X −! (6) µ @x0µ @zα @xν V 0α = X −! (7) µ @x0µ @xν @xν @zα V 0α = X (8) µ @x0µ @xν I @xν V 0α = V α (9) µ @x0µ ν Erik Olsen Spinors in Curved Space I α 0α α β zX −! zX = Λβ (X )zX (10) I @ V α(X ) = Λαzβ −! (11) µ @xµ β x @zβ V α(X ) = Λα X (12) µ β @xµ I α α β Vµ (X ) ! Λβ Vµ (X ) (13) Outline Introduction: A Useful Method Tetrad Formalism Tetrads 1 The Spin 2 Field Covariant Derivatives Conclusion Bibliography Tetrad Formalism α I Taking the Lorentz transform of zX Erik Olsen Spinors in Curved Space I @ V α(X ) = Λαzβ −! (11) µ @xµ β x @zβ V α(X ) = Λα X (12) µ β @xµ I α α β Vµ (X ) ! Λβ Vµ (X ) (13) Outline Introduction: A Useful Method Tetrad Formalism Tetrads 1 The Spin 2 Field Covariant Derivatives Conclusion Bibliography Tetrad Formalism α I Taking the Lorentz transform of zX I α 0α α β zX −! zX = Λβ (X )zX (10) Erik Olsen Spinors in Curved Space Outline Introduction: A Useful Method Tetrad Formalism Tetrads 1 The Spin 2 Field Covariant Derivatives Conclusion Bibliography Tetrad Formalism α I Taking the Lorentz transform of zX I α 0α α β zX −! zX = Λβ (X )zX (10) I @ V α(X ) = Λαzβ −! (11) µ @xµ β x @zβ V α(X ) = Λα X (12) µ β @xµ I α α β Vµ (X ) ! Λβ Vµ (X ) (13) Erik Olsen Spinors in Curved Space I µ Vα Bµ = Bα (14) I (1) Under a local Lorentz transformation it will behave as a vector.