December 5, 2008
Tetrads
The problem: How to put gravity into a Lagrangian density?
The problem: How to put gravity into a Lagrangian density? The solution: The Principle of General Covariance
The problem: How to put gravity into a Lagrangian density? The solution: The Principle of General Covariance 1. Ignoring gravity, find the equations of motion
The problem: How to put gravity into a Lagrangian density? The solution: The Principle of General Covariance 1. Ignoring gravity, find the equations of motion 2. Make the following substitutions:
The problem: How to put gravity into a Lagrangian density? The solution: The Principle of General Covariance 1. Ignoring gravity, find the equations of motion 2. Make the following substitutions: Lorentz Tensors become Tensor-like objects
The problem: How to put gravity into a Lagrangian density? The solution: The Principle of General Covariance 1. Ignoring gravity, find the equations of motion 2. Make the following substitutions: Lorentz Tensors become Tensor-like objects Derivatives become
The problem: How to put gravity into a Lagrangian density? The solution: The Principle of General Covariance 1. Ignoring gravity, find the equations of motion 2. Make the following substitutions: Lorentz Tensors become Tensor-like objects Derivatives become Minkowski tensors (η matrices) become the metric tensor for curved spacetime g µν
A major issue: this method does not work for spinors
A major issue: this method does not work for spinors No representations of GL(4) act like spinors under an infinitesimal Lorentz transformation
Tetrads g µν = ξα ξ β x µ x ν η αβ (1) where ξ represents a coordinate system under the influence of gravity.
Tetrads g µν = ξα ξ β x µ x ν η αβ (1) where ξ represents a coordinate system under the influence of gravity. Let zx α be normal local coordinates to each point in space-time X
Tetrads g µν = ξα ξ β x µ x ν η αβ (1) where ξ represents a coordinate system under the influence of gravity. Let zx α be normal local coordinates to each point in space-time X g µν (x) = zα X (x) z β X (x) x µ x ν η αβ (2)
Tetrads Let: ( z α ) X (x) x µ ( z β X (x) ) x ν x=x x=x = V α µ (X ) (3) = V β ν (X ) (4)
Tetrads Let: ( z α ) X (x) x µ ( z β X (x) ) x ν x=x x=x = V α µ (X ) (3) = V β ν (X ) (4) V α µ (X ) is a tetrad.
Tetrads Let: ( z α ) X (x) x µ ( z β X (x) ) x ν x=x x=x = V α µ (X ) (3) = V β ν (X ) (4) V α µ (X ) is a tetrad. Fix z α X, change x µ to x µ
Tetrads Let: ( z α ) X (x) x µ ( z β X (x) ) x ν x=x x=x = V α µ (X ) (3) = V β ν (X ) (4) V α µ (X ) is a tetrad. Fix z α X, change x µ to x µ V α µ V α µ (5)
Tetrads V α µ = zα X x µ (6) V µ α = zα X x ν x µ x ν (7) V µ α = x ν zx α x µ x ν (8)
Tetrads V α µ = zα X x µ (6) V µ α = zα X x ν x µ x ν (7) V µ α = x ν zx α x µ x ν (8) V α µ = x ν x µ V α ν (9)
Tetrads Taking the Lorentz transform of z α X
Tetrads Taking the Lorentz transform of z α X z α X z α X = Λα β (X )zβ X (10)
Tetrads Taking the Lorentz transform of z α X z α X z α X = Λα β (X )zβ X (10) V α µ (X ) = ( ) x µ Λ α β zβ x (11) V α µ (X ) = Λ α β z β X x µ (12) V α µ (X ) Λ α β V β µ (X ) (13)
Tetrads Let B µ be a generally covariant vector.
Tetrads Let B µ be a generally covariant vector. V µ α B µ = B α (14)
Tetrads Let B µ be a generally covariant vector. V µ α B µ = B α (14) (1) Under a local Lorentz transformation it will behave as a vector.
Tetrads Let B µ be a generally covariant vector. V µ α B µ = B α (14) (1) Under a local Lorentz transformation it will behave as a vector. (2) Under a general coordinate transformation it will transform as four scalars.
Tetrads Let α be a covariant derivative and ψ be a field
Tetrads Let α be a covariant derivative and ψ be a field α ψ Λ β α(x)d ( Λ(x) ) β ψ(x) (15)
Tetrads Let α be a covariant derivative and ψ be a field α ψ Λ β α(x)d ( Λ(x) ) β ψ(x) (15) D(Λ) is the matrix representation of the infinitesimal Lorentz group
Tetrads Let α be a covariant derivative and ψ be a field α ψ Λ β α(x)d ( Λ(x) ) β ψ(x) (15) D(Λ) is the matrix representation of the infinitesimal Lorentz group Γ µ (x) = 1 2 Σαβ V ν α ( µ V βν (x) ) (16)
Tetrads Where Σ αβ is the group generator for the Lorentz group
Tetrads Where Σ αβ is the group generator for the Lorentz group V βν = g µν V µ β (17)
L(x) = 1 2 i(ψγα α ψ α ψγ α ψ) mψψ (18)
L(x) = 1 2 i(ψγα α ψ α ψγ α ψ) mψψ (18) Σ αβ = 1 4 [γ α, γ β ] (19)
L(x) = 1 2 i(ψγα α ψ α ψγ α ψ) mψψ (18) Σ αβ = 1 4 [γ α, γ β ] (19) where the γ terms are the Dirac matrices
Γ µ (x) = 1 8 [γ α, γ β ]V ν α (x) ( µ g µν (x)v µ β (x)) (20)
Γ µ (x) = 1 8 [γ α, γ β ]V ν α (x) ( µ g µν (x)v µ β (x)) (20) { 1 L(x) = det V 2 i( ψγ µ µ ψ ( µ ψ)γ µ ψ ) } mψψ (21)
Γ µ (x) = 1 8 [γ α, γ β ]V ν α (x) ( µ g µν (x)v µ β (x)) (20) { 1 L(x) = det V 2 i( ψγ µ µ ψ ( µ ψ)γ µ ψ ) } mψψ (21) where γ µ = V µ α γ α
{γ µ, γ ν } = γ µ γ ν + γ ν γ µ (22) V α µ γ α Vβ ν γβ + Vβ ν γβ V α µ γ α (23) V α µ Vβ ν {γα, γ β } (24)
{γ µ, γ ν } = γ µ γ ν + γ ν γ µ (22) V α µ γ α Vβ ν γβ + Vβ ν γβ V α µ γ α (23) V α µ Vβ ν {γα, γ β } (24) {γ α, γ β } = 2η αβ (25)
{γ µ, γ ν } = γ µ γ ν + γ ν γ µ (22) V α µ γ α Vβ ν γβ + Vβ ν γβ V α µ γ α (23) V α µ Vβ ν {γα, γ β } (24) {γ α, γ β } = 2η αβ (25) {γ µ, γ ν } = 2V µ α V ν β η αβ (26)
{γ µ, γ ν } = γ µ γ ν + γ ν γ µ (22) V α µ γ α Vβ ν γβ + Vβ ν γβ V α µ γ α (23) V α µ Vβ ν {γα, γ β } (24) {γ α, γ β } = 2η αβ (25) {γ µ, γ ν } = 2V µ α V ν β η αβ (26) {γ µ, γ ν } = 2g µν (27)
Spinors do not work with the Principle of General Covariance
Spinors do not work with the Principle of General Covariance Contracting the spinor into the tetrad solves this dilemma
Spinors do not work with the Principle of General Covariance Contracting the spinor into the tetrad solves this dilemma All an approximation; quantum effects neglected
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.