PHZ6607 Class Notes
W. Zach Korth
9/2/2008
1
Action principle
S=
Z
L dλ
(1)
1 dxi (λ) dxj (λ)
1 L= m
= m ṙ2 + r2 θ̇2 + r2 sin2 θφ̇2
2
dλ
dλ
2
The metric of flat space in spherical polar coordinates is
ds2 = dr2 + r2 dθ2 + r2 sin2 θdφ2
(2)
(3)
On the surface of a sphere, r = a, dr = 0, so
ds2 = a2 dθ2 + sin2 θdφ2
2
(4)
Geodesic equation
The geodesic equation is given by
j
k
d 2 xi
i dx dx
+
Γ
=0
jk
dλ2
dλ dλ
(5)
where
1
Γijk = g il (glk,j + gjl,k − gjk,l )
2
is known as the affine connection.
1
(6)
2.1
Parameterized curves
If we consider a parameterized curve xi (λ) with parameter λ, the proper
length between points A and B is given by
lAB =
Z Bq
gij dxi dxj
A
(7)
But what does it mean to integrate over these infinitesimals? This is why we
choose a parameter, λ over which to integrate, as
lAB =
Z B
A
s
gij
dxi dxj
dλ
dλ dλ
(8)
We note that this form is reparameterization invariant, as any changes in the
parameter λ → f (λ) leave the physics unchanged.
3
Acceleration vector
We have what we defined as the tangent vector
dxi
u =
dλ
i
(9)
We now define an“acceleration vector”
Dui
dui
a =
≡
+ Γijk uj uk
Dλ
dλ
i
2
(10)
3.1
Relation to geodesic
How do we define a geodesic?
• A “curve of extremal length” or
•
Dui
Dλ
∝ f (λ) ui
Aside
For ui =
dxi
,
ds
ds2
dxi dxj
gij = 2 = 1
ds ds
ds
2
Du = Du · u = a · u = 0
u2 = ui uj gij =
4
(11)
(12)
Parallel transport
Suppose we have a vector field defined in 3-D space. This field would have
a value wi (xj ) along some curve xj . If we parameterize the curve as xj (λ) ,
it makes sense to ask how the vector wi changes along the curve. For this,
however, we must be sure to use an appropriate derivative.
dwi
dλ
Dwi
dλ
→
not a good vector
=
dwi
dλ
+ Γikj wk uj
(13)
proper vector
We are used to being able to move vectors around at will. This cannot
be done in curved space and indeed is not trivial even in flat space for noncartesian bases. In general, to move a vector along some curve we must
impose a condition
Dwi
dwi
=
+ Γikj wk uj = 0
(14)
dλ
dλ
which we term the condition of parallel transport.
5
Covariant derivatives
If a field wi (xj ) as above is defined on a manifold, not just along some curve
xj (λ),
i
dwi
dxj ∂wi
Dwi
i ∂w
=
=
u
=⇒
=
dλ
dλ ∂xj
∂xj
Dλ
3
∂wi
+ Γikj wk · uj
∂xj
!
(15)
Here, we define
∂wi
+ Γikj wk ≡ ∇j wk
∂xj
!
(16)
to be the covariant derivative, which is necessary to ensure that tensors
remain as tensors under differentiation. This arises because a true vector is
w = wi ei
(17)
dw = d(wi ei ) = (dwi )ei + wi (dei )
(18)
so
Hence, we have two terms in the covariant derivative: one for component
changes, and one for changes in the coordinate bases.
6
Christoffel symbols
We can now give a proper definition of the Christoffel symbols (Γijk ):
Γijk ≡ ei ∂k ej
(19)
Note that this is not a tensor, but rather a “tensor-like object”. Good tensors
can be constructed from the Christoffel symbols, however; for example, the
torsion tensor
i
Tjk
≡ Γijk − Γikj
(20)
In GR, we will be dealing with the Einstein Equations, in which manifolds
have no torsion, thus
i
Tjk
≡ Γijk − Γikj = 0 −→ Γijk = Γikj
7
(21)
Covariant derivative examples
Let us examine the form of what results from a covariant derivative of...
a scalar
∇i φ(xi ) =
a vector
∂φ
∂xi
∂V k
∇j V =
+ Γkij V i
j
∂x
k
4
(22)
(23)
a one-form
∇j Wk =
∂Wk
− Γikj Wi
∂xj
(24)
some higher-rank tensor
7.1
∇j T ik = ∂j (T ik ) + Γilj T lk + Γklj T il
(25)
∂i φ → ∇i φ
(26)
∇i V i = ∂i V i + Γiji V j
(27)
Operators
Recall some operators
gradient (scalar)
divergence (vector)
Noting the indices, we see that this must contract into a scalar. Let’s
look at the connection:
1
Γiji = g il (gli,j + gjl,i − gji,l )
2
(28)
The last two terms in the parentheses cancel, leaving
1
1 q
Γiji = g il gli,j = q ( |g|),j
2
|g|
(29)
where g is the determinant of the metric. Thus,
q
1 q
1
∇i V i = ∂i V i + q ( |g|),j V j = q ∂i ( |g|V i )
|g|
|g|
(30)
ijk ∇i Bj
(31)
curl (vector)
This equation works well for 3-D, but for higher dimensions it is useful
to define
Xij = ∇i Bj − ∇j Bi = ∂i Bj − ∂j Bi
(32)
where the last step follows from the fact that our connections Γijk are
symmetric in their lower indices.
5
the Laplacian
q
1
∇2 φ = ∇i ∇i φ = q ∂i ( |g|∇i φ)
|g|
(33)
∇i φ = g ij ∇j φ = g ij ∂j φ
(34)
with
6