Kiefer C. Quantum gravity

28 COVARIANT APPROACHES TO QUANTUM GRAVITY
For electrodynamics, right and left polarized states are given by the vectors
e R = √ 1 (e y +ie z ) , e L = √ 1 (e y − ie z ) . (2.16)
2 2
Under the above rotation, they transform as
e ′ R =e−iθ e R , e ′ L =eiθ e L . (2.17)
The left (right) circular polarized electromagnetic wave thus has helicity 1 (−1).
Instead of (2.8), one has here
A µ = e µ e ikx + e ∗ µe −ikx (2.18)
with k µ k µ =0andk ν e ν = 0. It is possible to perform a gauge transformation
without leaving the Lorenz gauge, A µ → A ′ µ = A µ + ∂ µ Λwith✷Λ =0.WithΛ
being a plane-wave solution, Λ = 2Re[iλe ikx ], one has instead of (2.9),
e µ → e µ − λk µ . (2.19)
The field equations of linearized gravity can be obtained from the Lagrangian
(Fierz and Pauli 1939),
L = 1
64πG (f µν,σ f µν,σ − f µν,σ f σν,µ − f νµ,σ f σµ,ν
−f µ µ,ν f ρ
ρ, ν +2f ρν ,ν f σ ) 1
σ,ρ +
2 T µνf µν . (2.20)
The Euler–Lagrange field equations yield (writing f ≡ f µ µ )
f
σ
µν,σ
− f
σ
σµ,ν
(
σ
− fσν,µ + f ,µν + η µν f αβ ,αβ − f
,σ
σ
)
= −16πGT µν . (2.21)
The left-hand side of this equation is −2 times the Einstein tensor, −2G µν ,in
the linear approximation. It obeys the linearized Bianchi identity ∂ ν G µν =0,
which is consistent with ∂ ν T µν = 0. The Bianchi identity is a consequence of
the gauge invariance (modulo a total divergence) of the Lagrangian (2.20) with
respect to (2.4). 3
Performing in (2.21) the trace and substituting the η µν -term yields
✷f µν − f
σ
σµ,ν
σ
− fσν,µ + f ,µν = −16πG ( T µν − 1 2 η µνT ) . (2.22)
Using the harmonic condition (2.3), one finds the linearized Einstein equations
(2.2). 4
3 The Bianchi identity in electrodynamics reads ∂ µ(∂ νF µν ) = 0, consistent with the chargeconservation
law ∂ νj ν =0.
4 It is often useful to make a redefinition f µν → √ 32πGf µν; cf. Section 2.2.2.