GR Notes for Oct. 8th
Yaqi Han
Symmetries and Killing vectors
For a metric gµν , if
∂σ∗ gµν = 0,
(1)
the transformation xσ∗ → xσ∗ + aσ∗ is a symmetry of the metric, since the
metric is invariant under this transformation.
Geodesic equation
V µ ∇µ V ν = 0 =⇒ P µ ∇µ P ν = 0
dPν
− Γσµν Pσ P µ = 0
dτ
!
dPν
1 σρ ∂gρν ∂gµρ ∂gµν
=m
− g
+
−
Pσ P µ
µ
ν
ρ
dτ
2
∂x
∂x
∂x
(2)
P µ ∇ µ Pν = m
(3)
Since g σρ Pσ = P ρ , the first term and the third term in the parenthesis
cancel.
1
dPν
= ∂ν gµρ P ρ P µ
(4)
m
dτ
2
If ν = σ∗ ,
dPσ∗
= 0.
(5)
dτ
Pσ∗ is a constant.
Let K be a vector ∂σ∗ , (K)µ = δσµ∗ . K is called a Killing vector. We have
Pσ∗ = K µ Pµ
(6)
0 = P µ ∇µ Pσ∗ = P µ ∇µ (Kν P ν )
= K ν P µ ∇ µ P ν + P µ P ν ∇ µ Kν
= P µ P ν ∇(µ Kν) ∀ geodesics
(7)
=⇒ ∇(µ Kν) = 0
(8)
∇(µ Kν) = 0 =⇒ K is a Killing vector.
(9)
Killing’s equation
1
For a flat space
ds2 = dx2 + dy 2 + dz 2
(10)
there are three Killing vectors,
xµ = (1, 0, 0),
y µ = (0, 1, 0),
z µ = (0, 0, 1).
(11)
In polar coordinates,
ds2 = dr2 + r2 dθ2 + r2 sin2 θdφ2
(12)
the three Killing vectors are,
Rµ = (0, 0, 1) = ∂φ , → Lz
S µ = cos φ∂θ − cot θ sin φ∂φ = (0, cos φ, − cot θ sin φ),
T µ = − sin φ∂θ − cot θ cos φ∂φ = (0, − sin φ, − cot θ cos φ).
(13)
Converted into cartesian coordinates, they are
Rµ = (−y, x, 0) = x∂y − y∂x ,
S µ = (z, 0, −x),
T µ = (0, −z, y).
(14)
Check Killing’s equation,
K µ = Rµ = (−y, x, 0), Kν = (−y, x, 0)
∇(µ Kν) = 0
∂x Kx + ∂x Kx = 0
∂x Ky + ∂y Kx = 1 − 1 = 0
∂y Ky + ∂y Ky = 0
∂y Kz + ∂z Ky = 0
∂z Kz + ∂z Kz = 0
∂x Kz + ∂z Kx = 0
(15)
In 3 dimension, there are at most 6 Killing vectors. In flat space, there’s
a maximum number of Killing vectors. Flat space is maximally symmetric.
Examples:
1. ds2 = dx2 + dy 2 (flat space, curvature R = 0)
There are 3 Killing vectors: (1, 0), (0, 1), (−y, x).
2
2. ds2 = a2 dθ2 + a2 sin2 θdφ2 (spherical shell, curvature R =
There are 3 Killing vectors: R, S, T .
This can be rewritten as,
 "
!
θ
dθ2
+ dφ2 = a2 sin2 θ d ln tan
ds2 = a2 sin2 θ
2
sin θ
2
2
)
a2

!#2
+ dφ2 
(16)
3. ds2 =
a2
(dx2
y2
+ dy 2 ) (Poincaré half-plane, curvature R = − a22 )
In n-dimensional space, the maximum number of Killing vectors is n +
= n(n+1)
. In flat spacetime, there are 10 Killing vectors. Apart from
2
the 6 from flat space, there are 4 more,
n(n−1)
2
W µ = ∂t , Bxµ = (x, t, 0, 0), Byµ = (y, 0, t, 0), Bzµ = (z, 0, 0, t)
(17)
Suppose we have a stress-energy tensor T µν , ∇µ T µν = 0.
µ
νµ
J(T
) = Kν T
µ
∇µ J(T
) = 0,
since T νµ = T µν
(18)
J µ =total energy if Kν = ∂t .
Some relations for Riemann tensors, Ricci tensors, and Ricci scalar:
l
∇µ ∇νK l = Rνµσ
Kσ
∇µ ∇νK ν = Rµσ K σ
K µ ∇µ R = 0
(19)
Geodesic Deviation
For a familiy of geodesics (Figure 3.9 from Carroll’s textbook, next page),
choose s, t as coordinates, we have the tangent vector T µ and the deviation
vector S µ ,
∂xµ
∂xµ
, Sµ =
.
(20)
Tµ =
∂t
∂s
[S, T ] = 0 =⇒ S µ ∇µ T ν = T µ ∇µ S ν
(21)
We can define “relative velocity of geodesics” and “relative acceleration
of geodesics” as
V ν = T µ ∇µ S ν , Aν = T µ ∇µ V µ
(22)
3
Aµ = T ρ ∇ρ (T σ ∇σ S µ ) = T ρ ∇ρ (S σ ∇σ T µ )
= (T ρ ∇ρ S σ )∇σ T µ + S σ T ρ ∇ρ ∇σ T µ
µ
= (T ρ ∇ρ S σ )∇σ T µ + S σ T ρ (∇σ ∇ρ T µ + Rνρσ
T ν)
µ
= S ρ ∇ρ T σ ∇σ T µ + S σ [∇σ (T ρ ∇ρ T µ ) − (∇σ T ρ )∇ρ T µ ] + Rνρσ
T ν T ρS σ
µ
= Rνρσ
T ν T ρS σ
(23)
The equation
µ
Aµ = Rνρσ
T ν T ρS σ
is called the geodesic deviation equation.
4
(24)