Lecture 10
Parallel transport
Objectives:
• Parallel transport
• Geodesics
• Equations of motion
Reading: Schutz 6; Hobson 3; Rindler 10.
In this lecture we are finally going to see how the metric determines the
motion of particles. First we discuss the concept of “parallel transport”.
10.1
Parallel transport
In SR, the equation for force-free motion of a particle is
~
~ = dU = 0,
A
dτ
~
i.e a straight line through spacetime as well as 3D space with the vector U
remaining constant along the line parameterised by τ .
This is extended to the curved spacetime of GR by the notion of parallel
“transport” in which a vector is moved along a curve staying parallel to
itself and of constant magnitude.
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LECTURE 10. PARALLEL TRANSPORT
40
Figure: Parallel transport of a vector from A to B, keeping
it parallel to itself and of constant length at all points.
Consider the change of a vector V~ along a line parameterised by λ
dV~
dV α
d~eα
=
~eα + V α
.
dλ
dλ
dλ
We can write
∂~eα dxβ
d~eα
=
.
dλ
∂xβ dλ
Using this and the definition of the connection
∂~eα
= Γγ αβ ~eγ ,
β
∂x
gives
dV~
dV α
dxβ
α γ
=
~eα + V Γ α β
~eγ .
dλ
dλ
dλ
Swapping dummy indices α and γ in the second term finally leads to
dV~
=
dλ
µ
¶
dxβ γ
dV α
α
~eα .
+ Γ γβ
V
dλ
dλ
This is a vector with components
DV α
dV α
dxβ γ
=
+ Γα γβ
V ,
Dλ
dλ
dλ
and is known variously as the “intrinsic”, “absolute” or “total” derivative.
One also sometimes sees the vector written as
dV~
= ∇U~ V~ ,
dλ
where U α = dxα /dλ is the “tangent vector” pointing along the line (= fourvelocity if λ = τ ).
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LECTURE 10. PARALLEL TRANSPORT
The components are very similar to the covariant derivative
V α ;β = V α ,β + Γα γβ V γ .
In fact if we write
(a cheat: V α
α
α
β
α
∂V dx
∂V
dV
=
=
U β,
β
dλ
∂x dλ
∂xβ
might only be defined on the line) then we can write
DV α /Dλ is to
dV α /dλ as V α ;β is
to V α ,β .
DV α
= V α ;β U β .
Dλ
Parallel transport: if a vector V~ is “parallel transported” along a line then
dV~
= 0,
∇U~ V~ =
dλ
or in component form:
dV α
dxβ γ
DV α
=
+ Γα γβ
V = 0.
Dλ
dλ
dλ
10.2
Straight lines or “geodesics”
With parallel transport we can extend the idea of “straight” lines to curved
spaces:
Definition: a line is “straight” if it parallel transports its own
tangent vector.
~ = 0 or,
In other words straight lines in curved spaces are defined by ∇U~ U
α
α
α
setting V = U = dx /dλ
dxβ dxγ
d2 xα
α
+
Γ
= 0.
γβ
dλ2
dλ dλ
More compactly
ẍα + Γα γβ ẋβ ẋγ = 0,
using the “dot” notation for derivatives wrt λ.
• These are force-free equations of motion
~ = dU
~ /dτ = 0 to GR.
• Extends SR A
Shows how V α
must change for
V~ to remain
constant.
LECTURE 10. PARALLEL TRANSPORT
42
• In GR, gravity is not a force but a distortion of spacetime
• Metric gαβ → Γγ αβ → particle motion.
• Straight lines are often called geodesics. “Great circles” are geodesics
on spheres.
10.2.1
Affine parameters
~ = kU
~ , i.e. the tangent vector
We could have defined “straight” by ∇U~ U
changes by a vector parallel to itself. However in such cases one can always
transform to a new parameter, say µ = µ(λ), such that ∇U~ ′ U~ ′ = 0, where
U~ ′ is the new tangent vector. µ is then called an affine parameter. Proper
I will always
time τ is affine for massive particles.
assume affine
parameters.
10.3
Example: motion under a central force
Consider motion under Newtonian gravity
dV~
GM
= − 2 ~rˆ.
dt
r
In general coordinates the left-hand side is
dV α
+ Γα βγ V β V γ .
dt
In polar coordinates V~ = (ṙ, θ̇).
From last time Γr θθ = −r, Γθ rθ = Γθ θr = 1/r with all others = 0. Therefore:
GM
dV r
+ Γr θθ V θ V θ = − 2 ,
dt
r
and
dV θ
+ Γθ rθ V r V θ + Γθ θr V θ V r = 0.
dt
These give
r̈ − rθ̇2 = −
and
GM
,
r2
2
θ̈ + ṙθ̇ = 0.
r
The second can be integrated to give the well known conservation of angular
momentum r2 θ̇ = h.
LECTURE 10. PARALLEL TRANSPORT
43
These two equations are the equations of planetary motion which lead to
ellipses and Kepler’s laws. The point here is how the connection allows one
to cope with familiar equations in awkward coordinates. In much of physics
such coordinates can be avoided, but not in GR where there is no sidestepping
the connection. Note here how the centrifugal term, rθ̇2 , appears via the
connection.