University of California at San Diego – Department of Physics – Prof. John McGreevy
General Relativity (225A) Fall 2013
Assignment 7 – Solutions
Posted November 4, 2013
Due Wednesday, November 20, 2013
1. Infinitesimal coordinate transformations. Under a coordinate transformation,
xµ → x̃µ (x), the metric tensor transforms as
gµν (x) 7→ g̃µν (x̃) = ∂˜µ xρ ∂˜ν xσ gρσ (x) .
Show that for an infinitesimal transformation x̃µ = xµ + µ (x), this takes the form
δgµν ≡ g̃µν (x) − gµν (x) = − (∇µ ν + ∇ν µ ) .
(This will be useful for the following problem.)
x̃µ = xµ + µ (x), ∂˜µ xρ = δµρ − ∂˜µ ρ = δµρ − ∂˜µ xα ∂α ρ = δµρ − ∂µ ρ + O(2 )
In the following the +O(2 ) is understood, that is, we treat 2 as zero.
δgµν ≡
=
=
=
=
=
=
=
g̃µν (x) − gµν
(x) = ∂˜µ xρ |x− ∂ν xσ |x− gρσ (x − ) − gµν (x)
δµρ − ∂µ ρ (δνσ − ∂ν σ ) (gρσ (x) − α ∂α gρσ (x)) − gµν (x)
−(∂µ ρ )gρν − (∂ν σ )gµσ − α ∂α gµν
− (∂µ (ρ gρν ) − ρ (∂µ gρν ) + ∂ν (σ gµσ ) − σ (∂ν gµσ ) + α ∂α gµν )
− (∂µ ν + ∂ν µ − α g ρα (∂µ gρν + ∂ν gµρ − ∂ρ gµν ))
− ∂µ ν + ∂ν µ − 2α Γαµν
− ∂µ ν − α Γαµν + ∂ν µ − α Γανµ
− (∇µ ν + ∇ν µ ) .
(1)
2. Conservation of the improved stress tensor. Our improved stress tensor is defined
as
2 δS
T µν ≡ √
g δgµν
where S is an action for a matter field (such as a scalar field φ or a Maxwell field Aµ , or
a particle trajectory). We would like to show that T µν defined this way is covariantly
conserved when evaluated on solutions of the equations of motion of the matter fields.
For simplicity consider a scalar field φ. Its EoM is 0 =
1
δS
.
δφ
(a) Show that under an infinitesimal coordinate transformation xµ → xµ + µ (x), the
action changes as
Z
1 √ µν
δS µ
4
δ S = − d x
gT (∇µ ν + ∇ν µ ) +
∂µ φ .
2
δφ
Using the chain rule,

Z
δ S =


 δS
d x
 δgµν (x)
| {z }
4
=
√
g µν
T
2
δgµν (x)
| {z }
=−∇µ ν −∇ν µ


δS
+
δφ(x) 
δφ(x) | {z } 

µ
=− ∂µ φ
Note the sign in the second term: a scalar field transforms as φ̃(x̃) = φ(x), which
means
δφ(x) = φ̃(x) − φ(x) = φ(x − ) − φ(x) = −µ ∂µ φ(x).
(b) By using the invariance of the action under coordinate transformations, show that
the equation of motion for φ implies
∇µ T µν = 0.
Coordinate invariance of the action implies that the preceding expression is zero;
δS
on its own, so
the scalar EoM imply that 0 = δφ(x)
Z
Z
√
IBP
4 1√
µν
0 = δ S = − d x
gT (∇µ ν + ∇ν µ ) = + d4 x g∇µ T µν ν (x)
2
for all means that 0 = ∇µ T µν .
3. Not so much GR in D = 1 + 1.
Show that in two spacetime dimensions, the left hand side of the Einstein equation
1
Rµν − gµν R = 8πGTµν
2
vanishes identically. This means that the metric does not have any dynamics in D =
1 + 1, and the Einstein equations impose the constraint Tµν = 0 on the matter fields.
Recall from the previous problem set that in two dimensions, the Riemann tensor is
1
Rijkl = R (gik gjl − gil gjk )
2
which means the Ricci tensor is


1
1
Rik = g jl Rijkl = R gij δll −gij  = Rgik
|{z}
2
2
=n=2
2
and the Ricci scalar is
1
R = g ik Rik = R2 = R
2
So the einstein tensor is
1
1
1
Gik = Rik − Rgik = Rgik − Rgik = 0.
2
2
2
3