Classical Fileds in Curved Spacetime

Lecture 2
Classical Fields in Curved Spacetime



1 Conventions and useful expressions
ηµν = diag(+1, −1, −1, −1) = η µν , (1)
xµ = (x0 , x1 , x2 , x3 )T = (ct, xi )T = (ct, ⃗x)T = (t, ⃗x)T , (2)
xµ = ηµν xν = (t, −⃗x)T , (3)
xµ = gµν (x)xν , (4)
ds2 = gµν (x)dxµ dxν = dxµ dxµ , (5)
1
Γσµν = g σρ (∂µ gνρ + ∂ν gρµ − ∂ρ gµν ) , (6)
2
∇µ ϕ = ∂µ ϕ , ϕ = ϕ(x), (7)
∇µ Aν = ∂µ Aν − Γλµν Aλ , Aµ = Aµ (x), (8)
∇µ Aν = ∂µ Aν + Γνµλ Aλ , (9)
1 √
∇µ Aµ = √ ∂µ −gAµ , f our − divergence, g = det(ĝ),


(10)
−g
∇ρ Tµν = ∂ρ Tµν − Γλρµ Tλν − Γλρν Tµλ , (11)
Fµν = ∇µ Aν − ∇ν Aµ = ∂µ Aν − ∂ν Aµ (due to Γλµν = Γλνµ ), (12)
Dµ := ∇µ + iAµ , gauge covariant derivative, (13)
µ
Rναβ = ∂β Γµνα − ∂α Γµνβ + Γµσβ Γσνα − Γµσα Γσνβ , (14)
α
Rµν = Rµαν = ∂ν Γαµα − ∂α Γαµν + Γβµα Γανβ − Γαµν Γβαβ , (15)
R = Rµµ = Gµν Rµν , (16)
δgµν = −gµρ gνσ δg ρσ , (17)
δg = gT r(ĝ −1 .δĝ) = g(g µν δgµν ) = −ggµν δg µν , (18)

1

, √ √ ′ δg 1√
δ −g = ( −g)g δg = − √ =− −ggµν δg µν , (19)
2 −g 2
δR
= Rαβ , (20)
δg αβ
√
1 δ( −gLM )
Tµν =√ (only in signature (+, −, −, −)). (21)
−g δg µν

2 Scalar Field in Curved Spacetime
Let us first consider scalar fields in curved spacetime:
√ √
Z   Z
4 1 µν
S [ϕ, ∂µ ϕ] = −ηd x η ∂µ ϕ∂ν ϕ − V (ϕ) = d4 x −ηLϕ , (22)
2

∂Lϕ ∂Lϕ
δS = 0 ⇒ ∂α − = 0. (23)
∂(∂α ϕ) ∂ϕ
∂Lϕ dV
=− , (24)
∂ϕ dϕ
 
∂Lϕ 1 µν ∂(∂µ ϕ) ∂(∂ν ϕ)
∂α = η ∂α ∂ν ϕ + ∂µ ϕ = (25)
∂(∂α ϕ) 2 ∂(∂α ϕ) ∂(∂α ϕ)
1  µν α
∂α η δµ ∂ν ϕ + η µν ∂µ ϕδνα =

2
1  α µ
∂α δµ ∂ ϕ + δνα ∂ ν ϕ = ∂α ∂ α ϕ ⇒

2
dV
□ϕ = − , where ∂α ∂ α ϕ = □, Klein − Gordon (26)
dϕ
Gauge invariance and coupling to the electromagnetic field:
1
LF = − Fµν F µν − Aµ J µ , (27)
4
∂LF ∂LF
∂α − =0⇒ ∂α F αβ = J β (homework). (28)
∂(∂α Aβ ) ∂Aβ
′
Global U (1) symmetry: ϕ(x) −→ ϕ (x) = eiα ϕ(x), α = const ⇒
L ϕ = L ϕ′ , ϕ(x) = ϕ1 (x) + iϕ2 (x).
′
Local U (1) gauge symmetry: ϕ (x) = eiα(x) ϕ(x) ⇒ Lϕ ̸= Lϕ′ .
To restore gauge invariance −→ minimal coupling −→ Dµ = ∂µ + iAµ ⇒

2