Alan Guth, The Geodesic Equation, 8.286 Lecture 14, October 29, 2013, p. 1.
Summary of Lecture 13: Adding Time
to the Robertson{Walker Metric
8.286 Lecture 14
2
2
2
2
ds = −c dt + a (t)
October 29, 2013
THE GEODESIC EQUATION
dr 2
+ r 2 dθ2 + sin2 θ dφ2
2
1 − kr
.
Meaning:
If ds2 > 0, it is the square of the spatial separation measured
by a local free-falling observer for whom the two events happen
at the same time.
If ds2 < 0, it is −c2 times the square of the time separation
measured by a local free-falling observer for whom the two
events happen at the same location.
If ds2 = 0, then the two events can be joined by a light pulse.
Alan Guth
–1–
Massachusetts Institute of Technology
8.286 Lecture 14, October 29
THE GEODESIC EQUATION
Adding Time to the
Robertson{Walker Metric
ds2 = −c2 dt2 + a2 (t)
Application:
dr 2
+ r 2 dθ2 + sin2 θ dφ2
2
1 − kr
Notation: ds2 = gij (xk ) dxi dxj .
Description of path from A to B :
xi (λ) , where xi (0) = xiA ,
Distance from xi (λ) to xi (λ + dλ):
.
ds2 = gij (xk (λ))
Distance from A to B:
The Geodesic Equation.
i
S [x (λ)] =
0
Alan Guth
Massachusetts Institute of Technology
8.286 Lecture 14, October 29
xi (λf ) = xiB .
–2–
Alan Guth
Massachusetts Institute of Technology
8.286 Lecture 14, October 29
λf
dxi dxj 2
dλ .
dλ dλ
gij (xk (λ))
dxi dxj
dλ .
dλ dλ
–3–
Alan Guth, The Geodesic Equation, 8.286 Lecture 14, October 29, 2013, p. 2.
Varying the path (calculus of variations, 1696):
dS x̃i (λ) dα
x̃i (λ) = xi (λ) + αwi (λ) ,
wi (0) = 0 ,
d S x̃i (λ) dα
wi (λf ) = 0 ,
=0
Integrate second term by parts!
α=0
for all wi (λ) ,
λf 1 ∂gjk dxj dxk
d
1
dxj
dS √
√
−
=
g
wi (λ) dλ .
ij
i dλ dλ
dλ
∂x
dλ
dα α=0
A
2 A
0
where
S x̃i (λ) =
λf
1
1 λf
=
2 0
A(λ, 0)
α=0
dwi dxj
∂gij k dxi dxj
dλ .
w
+ 2gij
×
∂xk
dλ dλ
dλ dλ
A(λ, α) dλ ,
0
Alan Guth
Massachusetts Institute of Technology
8.286 Lecture 14, October 29
dx̃i dx̃j
A(λ, α) = gij x̃k (λ)
.
dλ dλ
–4–
=⇒
d
1
dxj
1 ∂gjk dxj dxk
√ gij
= √
.
dλ
dλ
2 A ∂xi dλ dλ
A
Alan Guth
Massachusetts Institute of Technology
8.286 Lecture 14, October 29
–5–
MIT OpenCourseWare
http://ocw.mit.edu
8.286 The Early Universe
Fall 2013
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