x 2 + y 2 + z 2 = R 2 . OPEN UNIVERSES AND THE SPACETIME METRIC - - PowerPoint PPT Presentation

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Alan Guth, Non-Euclidean Spaces: Open Universes and the Spacetime Metric, 8.286 Lecture 12, October 22, 2013, p. 1. Summary of Leture 11: Surfae of a Sphere 8.286 Leture 12 Otober 22, 2013 NON-EUCLIDEAN SPACES: x 2 + y 2 + z 2 =

SLIDE 1

Alan Guth, Non-Euclidean Spaces: Open Universes and the Spacetime Metric, 8.286 Lecture 12, October 22, 2013, p. 1.

8.286 Le ture 12 O tober 22, 2013 NON-EUCLIDEAN SPACES: OPEN UNIVERSES AND THE SPACETIME METRIC Summary

Le ture 11: Surfa e

a Sphere

x2 + y2 + z2 = R2 .

Alan Guth Massa husetts Institute

T e hnology 8.286 Le ture 12, O tober 22

–1–

Polar Coordinates:

x = R sin θ cos φ y = R sin θ sin φ z = R cos θ ,

Alan Guth Massa husetts Institute

T e hnology 8.286 Le ture 12, O tober 22

–2–

Varying θ:

ds = R dθ

Alan Guth Massa husetts Institute

T e hnology 8.286 Le ture 12, O tober 22

–3–

SLIDE 2

Alan Guth, Non-Euclidean Spaces: Open Universes and the Spacetime Metric, 8.286 Lecture 12, October 22, 2013, p. 2.

Varying φ:

ds = R sin θ dφ

Alan Guth Massa husetts Institute

T e hnology 8.286 Le ture 12, O tober 22

–4–

Varying θ and φ Varying θ:

ds = R dθ

Varying φ:

ds = R sin θ dφ ds2 = R2 dθ2 + sin2 θ dφ2

Alan Guth Massa husetts Institute

T e hnology 8.286 Le ture 12, O tober 22

–5–

Review

Le ture 11: A Closed Three-Dimensional Spa e

x2 + y2 + z2 + w2 = R2 x = R sin ψ sin θ cos φ y = R sin ψ sin θ sin φ z = R sin ψ cos θ w = R cos ψ , ds = R dψ

Alan Guth Massa husetts Institute

T e hnology 8.286 Le ture 12, O tober 22

–6–

Metri for the Closed 3D Spa e Varying ψ:

ds = R dψ

Varying θ or φ:

ds2 = R2 sin2 ψ(dθ2 + sin2 θ dφ2) If the variations are orthogonal to each other, then ds2 = R2 dψ2 + sin2 ψ

dθ2 + sin2 θ dφ2

Alan Guth Massa husetts Institute

T e hnology 8.286 Le ture 12, O tober 22

–7–

SLIDE 3

Alan Guth, Non-Euclidean Spaces: Open Universes and the Spacetime Metric, 8.286 Lecture 12, October 22, 2013, p. 3.

Proof

Orthogonality

Variations

Let

dRψ = displacement of point when ψ is changed to ψ + dψ.

Let

dRθ = displacement of point when θ is changed to θ + dθ.

(3)

d Rθ has no w-component = ⇒ d Rψ · d Rθ = d Rψ ·

(3)

d Rθ , where (3) denotes the projection into the x-y-z subspace. (3) dRψ is radial; (3) dRθ is tangential = ⇒ (3) dRψ · (3) dRθ = 0

Alan Guth Massa husetts Institute

T e hnology 8.286 Le ture 12, O tober 22

–8–

Review

Le ture 11: Impli ations

General Relativity

ds2 = R2 dψ2 + sin2 ψ

dθ2 + sin2 θ dφ2

, where R is radius

f curvature.

According to GR, matter causes space to curve. a2 R cannot be arbitrary. Instead, R2 (t) (t) = k . Finally, d ds2 = a2(t)

+ r2 1 − kr2

dθ2 + sin2 θ dφ2

, sin ψ where r = √ . Called the Robertson-Walker metric. k

–9–

SLIDE 4

x 2 + y 2 + z 2 = R 2 . OPEN UNIVERSES AND THE SPACETIME METRIC - - PowerPoint PPT Presentation

Alan Guth, Non-Euclidean Spaces: Open Universes and the Spacetime Metric, 8.286 Lecture 12, October 22, 2013, p. 1.

x2 + y2 + z2 = R2 .

–1–

x = R sin θ cos φ y = R sin θ sin φ z = R cos θ ,

–2–

ds = R dθ

–3–

Alan Guth, Non-Euclidean Spaces: Open Universes and the Spacetime Metric, 8.286 Lecture 12, October 22, 2013, p. 2.

ds = R sin θ dφ

–4–

ds = R dθ

ds = R sin θ dφ ds2 = R2 dθ2 + sin2 θ dφ2

–5–

x2 + y2 + z2 + w2 = R2 x = R sin ψ sin θ cos φ y = R sin ψ sin θ sin φ z = R sin ψ cos θ w = R cos ψ , ds = R dψ

–6–

ds = R dψ

ds2 = R2 sin2 ψ(dθ2 + sin2 θ dφ2) If the variations are orthogonal to each other, then ds2 = R2 dψ2 + sin2 ψ

–7–

Alan Guth, Non-Euclidean Spaces: Open Universes and the Spacetime Metric, 8.286 Lecture 12, October 22, 2013, p. 3.

Let

Let

(3)

d Rθ has no w-component = ⇒ d Rψ · d Rθ = d Rψ ·

(3)

d Rθ , where (3) denotes the projection into the x-y-z subspace. (3) dRψ is radial; (3) dRθ is tangential = ⇒ (3) dRψ · (3) dRθ = 0

–8–

ds2 = R2 dψ2 + sin2 ψ

, where R is radius

According to GR, matter causes space to curve. a2 R cannot be arbitrary. Instead, R2 (t) (t) = k . Finally, d ds2 = a2(t)

+ r2 1 − kr2

, sin ψ where r = √ . Called the Robertson-Walker metric. k

–9–

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8.286 The Early Universe

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