First order Friedmann equation = t f. and α. Substitute ã α = α cos θ = t f = α(θ f sin θ f ) ã f = α(1 cos θ f ) = α(1 cos θ)

Transcription

1 Alan Guth, Introduction to Non-Euclidean Spaces (After finishing dynamics of homogeneous expansion), Lecture 10, October 10, 2013, p. 1. Summary of Lecture 9 First order Friedmann equation = t f ãf ãdã dt =, where a ã, t ct, 2αã ã 2 k Evolution of a closed universe: 0 0 4π Gρã3 and α. 3 c 2 Substitute ã α = α cos θ = t f = α(θ f sin θ f ) ã f = α(1 cos θ f ) = ct = α( θ sin θ) a = α(1 cos θ) k 1 2

2 Alan Guth, Introduction to Non-Euclidean Spaces (After finishing dynamics of homogeneous expansion), Lecture 10, October 10, 2013, p. 2. Age of a closed universe: (WantitintermsofH and ) 8π H 2 kc 2 a c = Gρ = 3 a 2 ã = = k H 1. 4π Gρã 3 c α = 3 c 2 α =. 2 H ( 1) 3/2 a c c = α(1 cos θ) = = (1 cos θ). k H 1 2 H ( 1) 3/2 1 = sin θ = ±. Then ct = α(θ sin θ) = t = 2 H ( 1) 3/2 { arcsin ( 2 1 ± ) } t = 2 H ( 1) 3/2 { arcsin ( 2 1 ± ) 2 1 Quadrant Phase Sign Choice θ =sin 1 () 1 Expanding 1to2 Upper 0to π 2 2 Expanding 2to Upper π 2 to π π 3 Contracting to 2 Lower π to 3 2 3π 4 Contracting 2to1 Lower 2 }. to 2π Lecture 10, October 10 4 The calculations are almost identical, except that one defines a ã, where κ k>0. κ One finds hypergeometric functions instead of trigonometric functions, with ct = α(sinh θ θ) a = α(cosh θ 1) κ instead of ct = α( θ sin θ) a = α(1 cos θ). k Age a Universe of The Matter-Dominated [ ( )] sinh 1 if < 1 2(1 ) 3/2 H t = 2/3 [ ( ) if = 1 ] 1 sin ± if > 1 2( 1) 3/2 Evolution of an Open Universe Lecture 10, October 10 5 Lecture 10, October 10 6

3 TO INTRODUCTION SPACES NON-EUCLIDEAN Alan Guth, Introduction to Non-Euclidean Spaces (After finishing dynamics of homogeneous expansion), Lecture 10, October 10, 2013, p. 3. Age of Matter-Dominated Universe Evolution of a Matter-Dominated Universe Lecture 10, October 10 7 Lecture 10, October Slides courtesy of Mustafa Amin. Used with permission.

4 Alan Guth, Introduction to Non-Euclidean Spaces (After finishing dynamics of homogeneous expansion), Lecture 10, October 10, 2013, p. 4. Slides courtesy of Mustafa Amin. Used with permission. Corrected 10/10/13 11 Corrected 10/10/13 Slides courtesy of Mustafa Amin. Used with permission. 12 Equivalent Statements of the 5th Postulate Equivalent Statements of the 5th Postulate (a) If a straight line intersects one of two parallels (i.e, lines which do not intersect however far they are extended), it will intersect the other also. (b) There is one and only one line that passes through any given point and is parallel to a given line. Lecture 10, October Lecture 10, October 10 14

5 Alan Guth, Introduction to Non-Euclidean Spaces (After finishing dynamics of homogeneous expansion), Lecture 10, October 10, 2013, p. 5. Equivalent Statements of the 5th Postulate Equivalent Statements of the 5th Postulate (c) Given any figure there exists a figure, similar to it, of any size. (Two polygons are similar if their corresponding angles are equal, and their corresponding sides are proportional.) (d) There is a triangle in which the sum of the three angles is equal to two right angles (i.e., 180 ). Lecture 10, October Lecture 10, October Giovanni Geralamo Saccheri ( ) Carl Friedrich Gauss ( ) In 1733, Saccheri, a Jesuit priest, published Euclides ab omni naevo vindicatus (Euclid Freed of Every Flaw). The book was a study of what geometry would be like if the 5th postulate were false. He hoped to find an inconsistency, but failed. Source unknown. All rights reserved. This content is excluded from our Creative Commons license. For more information, see Photo from Carl_Friedrich_Gauss.jpg (in public domain). German mathematician and physicist. Born as the son of a poor working-class parents. His mother was illiterate and never even recorded the date of his birth. His students included Richard Dedekind, Bernhard Riemann, Peter Gustav Lejeune Dirichlet, Gustav Kirchhoff, and August Ferinand Möbius. Lecture 10, October Lecture 10, October 10 18

6 Alan Guth, Introduction to Non-Euclidean Spaces (After finishing dynamics of homogeneous expansion), Lecture 10, October 10, 2013, p Slides courtesy of Mustafa Amin. Used with permission.

7 Alan Guth, Introduction to Non-Euclidean Spaces (After finishing dynamics of homogeneous expansion), Lecture 10, October 10, 2013, p Slides courtesy of Mustafa Amin. Used with permission.

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