Spinors in Curved Space

Outline Introduction: A Useful Method Tetrad Formalism 1 The Spin 2 Field Conclusion Bibliography

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.

I (2) Under a general coordinate transformation it will transform as four scalars.

Outline Introduction: A Useful Method Tetrad Formalism Tetrads 1 The Spin 2 Field Covariant Derivatives Conclusion Bibliography Tetrad Formalism

I Let Bµ be a generally covariant vector.

Erik Olsen Spinors in Curved Space I (1) Under a local Lorentz transformation it will behave as a vector.

I (2) Under a general coordinate transformation it will transform as four scalars.

Outline Introduction: A Useful Method Tetrad Formalism Tetrads 1 The Spin 2 Field Covariant Derivatives Conclusion Bibliography Tetrad Formalism

I Let Bµ be a generally covariant vector.

I µ Vα Bµ = Bα (14)

Erik Olsen Spinors in Curved Space I (2) Under a general coordinate transformation it will transform as four scalars.

Outline Introduction: A Useful Method Tetrad Formalism Tetrads 1 The Spin 2 Field Covariant Derivatives Conclusion Bibliography Tetrad Formalism

I Let Bµ be a generally covariant vector.

I µ Vα Bµ = Bα (14)

I (1) Under a local Lorentz transformation it will behave as a vector.

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 Bµ be a generally covariant vector.

I µ Vα Bµ = Bα (14)

I (1) Under a local Lorentz transformation it will behave as a vector.

I (2) Under a general coordinate transformation it will transform as four scalars.

Erik Olsen Spinors in Curved Space I β ∇αψ → Λα(x)D Λ(x) ∇βψ(x) (15)

I D(Λ) is the matrix representation of the inﬁnitesimal Lorentz group

I 1 Γ (x) = ΣαβV ν∇ V (x) (16) µ 2 α µ βν

Outline Introduction: A Useful Method Tetrad Formalism Tetrads 1 The Spin 2 Field Covariant Derivatives Conclusion Bibliography Covariant Derivatives

I Let ∇α be a covariant derivative and ψ be a ﬁeld

Erik Olsen Spinors in Curved Space I D(Λ) is the matrix representation of the inﬁnitesimal Lorentz group

I 1 Γ (x) = ΣαβV ν∇ V (x) (16) µ 2 α µ βν

Outline Introduction: A Useful Method Tetrad Formalism Tetrads 1 The Spin 2 Field Covariant Derivatives Conclusion Bibliography Covariant Derivatives

I Let ∇α be a covariant derivative and ψ be a ﬁeld

I β ∇αψ → Λα(x)D Λ(x) ∇βψ(x) (15)

Erik Olsen Spinors in Curved Space I 1 Γ (x) = ΣαβV ν∇ V (x) (16) µ 2 α µ βν

Outline Introduction: A Useful Method Tetrad Formalism Tetrads 1 The Spin 2 Field Covariant Derivatives Conclusion Bibliography Covariant Derivatives

I Let ∇α be a covariant derivative and ψ be a ﬁeld

I β ∇αψ → Λα(x)D Λ(x) ∇βψ(x) (15)

I D(Λ) is the matrix representation of the inﬁnitesimal Lorentz group

Erik Olsen Spinors in Curved Space Outline Introduction: A Useful Method Tetrad Formalism Tetrads 1 The Spin 2 Field Covariant Derivatives Conclusion Bibliography Covariant Derivatives

I Let ∇α be a covariant derivative and ψ be a ﬁeld

I β ∇αψ → Λα(x)D Λ(x) ∇βψ(x) (15)

I D(Λ) is the matrix representation of the inﬁnitesimal Lorentz group

I 1 Γ (x) = ΣαβV ν∇ V (x) (16) µ 2 α µ βν

Erik Olsen Spinors in Curved Space I µ Vβν = gµνVβ (17)

Outline Introduction: A Useful Method Tetrad Formalism Tetrads 1 The Spin 2 Field Covariant Derivatives Conclusion Bibliography Covariant Derivatives

αβ I Where Σ is the group generator for the Lorentz group

Erik Olsen Spinors in Curved Space Outline Introduction: A Useful Method Tetrad Formalism Tetrads 1 The Spin 2 Field Covariant Derivatives Conclusion Bibliography Covariant Derivatives

αβ I Where Σ is the group generator for the Lorentz group

I µ Vβν = gµνVβ (17)

Erik Olsen Spinors in Curved Space I 1 Σ = [γ , γ ] (19) αβ 4 α β

I where the γ terms are the Dirac matrices

Outline Introduction: A Useful Method Tetrad Formalism 1 The Spin 2 Field Conclusion Bibliography 1 The Spin 2 Field

I 1 L(x) = i(ψγα∂ ψ − ∂ ψγαψ) − mψψ (18) 2 α α

Erik Olsen Spinors in Curved Space I where the γ terms are the Dirac matrices

Outline Introduction: A Useful Method Tetrad Formalism 1 The Spin 2 Field Conclusion Bibliography 1 The Spin 2 Field

I 1 L(x) = i(ψγα∂ ψ − ∂ ψγαψ) − mψψ (18) 2 α α I 1 Σ = [γ , γ ] (19) αβ 4 α β

Erik Olsen Spinors in Curved Space Outline Introduction: A Useful Method Tetrad Formalism 1 The Spin 2 Field Conclusion Bibliography 1 The Spin 2 Field

I 1 L(x) = i(ψγα∂ ψ − ∂ ψγαψ) − mψψ (18) 2 α α I 1 Σ = [γ , γ ] (19) αβ 4 α β

I where the γ terms are the Dirac matrices

Erik Olsen Spinors in Curved Space I n1 o L(x) = det V iψγµ∇ ψ − (∇ ψ)γµψ − mψψ (21) 2 µ µ

µ α I where γµ = Vα γ

Outline Introduction: A Useful Method Tetrad Formalism 1 The Spin 2 Field Conclusion Bibliography 1 The Spin 2 Field

I 1 Γ (x) = [γ , γ ]V ν(x)∇ g (x)V µ(x) (20) µ 8 α β α µ µν β

Erik Olsen Spinors in Curved Space µ α I where γµ = Vα γ

Outline Introduction: A Useful Method Tetrad Formalism 1 The Spin 2 Field Conclusion Bibliography 1 The Spin 2 Field

I 1 Γ (x) = [γ , γ ]V ν(x)∇ g (x)V µ(x) (20) µ 8 α β α µ µν β I n1 o L(x) = det V iψγµ∇ ψ − (∇ ψ)γµψ − mψψ (21) 2 µ µ

Erik Olsen Spinors in Curved Space Outline Introduction: A Useful Method Tetrad Formalism 1 The Spin 2 Field Conclusion Bibliography 1 The Spin 2 Field

I 1 Γ (x) = [γ , γ ]V ν(x)∇ g (x)V µ(x) (20) µ 8 α β α µ µν β I n1 o L(x) = det V iψγµ∇ ψ − (∇ ψ)γµψ − mψψ (21) 2 µ µ

µ α I where γµ = Vα γ

Erik Olsen Spinors in Curved Space I α β {γ , γ } = 2ηαβ (25)

I µ ν µ ν {γ , γ } = 2Vα Vβ ηαβ (26)

I µ ν {γ , γ } = 2gµν (27)

Outline Introduction: A Useful Method Tetrad Formalism 1 The Spin 2 Field Conclusion Bibliography 1 The Spin 2 Field

I {γµ, γν} = γµγν + γνγµ → (22) µ α ν β ν β µ α Vα γ Vβ γ + Vβ γ Vα γ → (23) µ ν α β Vα Vβ {γ , γ } (24)

Erik Olsen Spinors in Curved Space I µ ν µ ν {γ , γ } = 2Vα Vβ ηαβ (26)

I µ ν {γ , γ } = 2gµν (27)

Outline Introduction: A Useful Method Tetrad Formalism 1 The Spin 2 Field Conclusion Bibliography 1 The Spin 2 Field

I {γµ, γν} = γµγν + γνγµ → (22) µ α ν β ν β µ α Vα γ Vβ γ + Vβ γ Vα γ → (23) µ ν α β Vα Vβ {γ , γ } (24)

I α β {γ , γ } = 2ηαβ (25)

Erik Olsen Spinors in Curved Space I µ ν {γ , γ } = 2gµν (27)

Outline Introduction: A Useful Method Tetrad Formalism 1 The Spin 2 Field Conclusion Bibliography 1 The Spin 2 Field

I {γµ, γν} = γµγν + γνγµ → (22) µ α ν β ν β µ α Vα γ Vβ γ + Vβ γ Vα γ → (23) µ ν α β Vα Vβ {γ , γ } (24)

I α β {γ , γ } = 2ηαβ (25)

I µ ν µ ν {γ , γ } = 2Vα Vβ ηαβ (26)

Erik Olsen Spinors in Curved Space Outline Introduction: A Useful Method Tetrad Formalism 1 The Spin 2 Field Conclusion Bibliography 1 The Spin 2 Field

I {γµ, γν} = γµγν + γνγµ → (22) µ α ν β ν β µ α Vα γ Vβ γ + Vβ γ Vα γ → (23) µ ν α β Vα Vβ {γ , γ } (24)

I α β {γ , γ } = 2ηαβ (25)

I µ ν µ ν {γ , γ } = 2Vα Vβ ηαβ (26)

I µ ν {γ , γ } = 2gµν (27)

Erik Olsen Spinors in Curved Space I Contracting the spinor into the tetrad solves this dilemma

I All an approximation; quantum eﬀects neglected

Outline Introduction: A Useful Method Tetrad Formalism 1 The Spin 2 Field Conclusion Bibliography Conclusion

I Spinors do not work with the Principle of General Covariance

Erik Olsen Spinors in Curved Space I All an approximation; quantum eﬀects neglected

Outline Introduction: A Useful Method Tetrad Formalism 1 The Spin 2 Field Conclusion Bibliography Conclusion

I Spinors do not work with the Principle of General Covariance

I Contracting the spinor into the tetrad solves this dilemma

Erik Olsen Spinors in Curved Space Outline Introduction: A Useful Method Tetrad Formalism 1 The Spin 2 Field Conclusion Bibliography Conclusion

I Spinors do not work with the Principle of General Covariance

I Contracting the spinor into the tetrad solves this dilemma

I All an approximation; quantum eﬀects neglected

Erik Olsen Spinors in Curved Space Outline Introduction: A Useful Method Tetrad Formalism 1 The Spin 2 Field Conclusion Bibliography Bibliography

I G. Arfken and H. Weber, Mathematical Methods for Physicists Elsevier Academic Press, 2005, sixth ed. N.D. Birrell and P.C.W. Davies. Quantum Fields in Curved Space Cambridge University Press, 1982. Steven Weinberg. Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. John Wiley and Sons, 1972.

Erik Olsen Spinors in Curved Space