Alan Guth, Non-Euclidean Spaces: Open Universes and the Spacetime Metric, 8.286 Lecture 12, October 22, 2013, p. 1.
Summary of Lecture 11:
Surface of a Sphere
8.286 Lecture 12
October 22, 2013
NON-EUCLIDEAN SPACES:
OPEN UNIVERSES AND
THE SPACETIME METRIC
x2 + y 2 + z 2 = R 2 .
Alan Guth
Massachusetts Institute of Technology
8.286 Lecture 12, October 22
–1–
Polar Coordinates:
Varying
x = R sin θ cos φ
y = R sin θ sin φ
θ:
ds = R dθ
z = R cos θ ,
Alan Guth
Massachusetts Institute of Technology
8.286 Lecture 12, October 22
–2–
Alan Guth
Massachusetts Institute of Technology
8.286 Lecture 12, October 22
–3–
Alan Guth, Non-Euclidean Spaces: Open Universes and the Spacetime Metric, 8.286 Lecture 12, October 22, 2013, p. 2.
Varying θ and φ
Varying
φ:
Varying
ds = R sin θ dφ
Varying
θ:
ds = R dθ
ds = R sin θ dφ
φ:
ds2 = R2 dθ 2 + sin2 θ dφ2
Alan Guth
Massachusetts Institute of Technology
8.286 Lecture 12, October 22
Alan Guth
–4–
Review of Lecture 11:
A Closed Three-Dimensional Space
–5–
Massachusetts Institute of Technology
8.286 Lecture 12, October 22
Metric for the Closed 3D Space
x 2 + y 2 + z 2 + w 2 = R2
Varying
x = R sin ψ sin θ cos φ
Varying
y = R sin ψ sin θ sin φ
z = R sin ψ cos θ
θ or φ:
ψ:
ds = R dψ
ds2 = R2 sin2 ψ(dθ 2 + sin2 θ dφ2 )
If the variations are orthogonal to each other, then
ds2 = R2 dψ 2 + sin2 ψ dθ 2 + sin2 θ dφ2
w = R cos ψ ,
ds = R dψ
Alan Guth
Massachusetts Institute of Technology
8.286 Lecture 12, October 22
–6–
Alan Guth
Massachusetts Institute of Technology
8.286 Lecture 12, October 22
–7–
Alan Guth, Non-Euclidean Spaces: Open Universes and the Spacetime Metric, 8.286 Lecture 12, October 22, 2013, p. 3.
Review of Lecture 11:
Implications of General Relativity
Proof of Orthogonality of Variations
ψ = displacement of point when ψ is changed to ψ + dψ.
Let dR
θ = displacement of point when θ is changed to θ + dθ.
Let dR
θ has no w-component =⇒ dR
ψ · dR
θ = dR
(3) · dR
(3) ,
dR
ψ
θ
where (3) denotes the projection into the x-y-z subspace.
(3) is radial; dR
(3) is tangential
dR
ψ
θ
=⇒
Alan Guth
Massachusetts Institute of Technology
8.286 Lecture 12, October 22
ds2 = R2 dψ 2 + sin2 ψ dθ2 + sin2 θ dφ2 , where R is radius
of curvature.
According to GR, matter causes space to curve.
a2 (t)
R cannot be arbitrary. Instead, R2 (t) =
.
k
Finally,
ds2 = a2 (t)
(3) = 0
(3) · dR
dR
ψ
θ
–8–
2
dr 2
2
2
2
,
+
r
+
sin
θ
dφ
dθ
1 − kr 2
sin ψ
where r = √ . Called the Robertson-Walker metric.
k
–9–
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8.286 The Early Universe
Fall 2013
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