with k < 0 ! Open Universe: Robertson-Walker metri. Name: - PowerPoint PPT Presentation

Alan Guth, Non-Euclidean Spaces: Spacetime Metric and Geodesic Equation, 8.286 Lecture 13, October 24, 2013, p. 1. Summary of Le ture 12: Open Universe Metri 8.286 Le ture 13 O tober 24, 2013 Closed Universe: NON-EUCLIDEAN SPACES: 2 � d r � + r 2 d θ 2 + sin 2 θ d φ 2 d s = a 2 ( t ) 2 � � , kr 2 1 SPACETIME METRIC − AND THE where k > 0. GEODESIC EQUATION Same thing, but with k < 0 ! Open Universe: Robertson-Walker metri . Name: Alan Guth Massa husetts Institute of T e hnology –1– 8.286 Le ture 13, O tober 24 Summary, Cont: Summary, Cont: Why is This Homogeneous? Why is This the Open Universe Metri ? Open Universe: d r 2 � � Open Universe: d s 2 = a 2 ( t ) d θ 2 + sin 2 θ d φ 2 + r 2 � � with k < 0 . , 1 − kr 2 r 2 � d � d s 2 = a 2 ( t ) + r 2 d θ 2 + sin 2 θ d φ 2 � � , For closed universe ( k > 0), show homogeneity explicitly by 1 + κr 2 showing that any point ( r 0 , θ 0 , φ 0 ) is equivalent to the origin: Construct a map ( r, θ, φ ) → ( r ′ , θ ′ , φ ′ ) which preserves metric, where κ = − k > 0. and maps ( r 0 , θ 0 , φ 0 ) to the origin, ( r ′ = 0 , θ ′ , φ ′ ). Requirements: Isotropy and Homogeneity Construct map in three steps: Isotropy about the origin is obvious: θ and φ appear as ( r, θ, φ ) → ( x, y, z, w ) ( x ′ , y ′ , z ′ , w ′ ) → ( r ′ , θ ′ , φ ′ ) . − → rotation a 2 ( t ) r 2 (d θ 2 + sin 2 θ d φ 2 ) , The same map works for k < 0, showing that any point can be mapped to the origin by a metric-preserving mapping.(We will exactly as on a sphere of radius a ( t ) r . not show it.) Alan Guth Massa husetts Institute of T e hnology –2– –3– 8.286 Le ture 13, O tober 24

Alan Guth, Non-Euclidean Spaces: Spacetime Metric and Geodesic Equation, 8.286 Lecture 13, October 24, 2013, p. 2. Summary, Cont: From Spa e to Spa etime In special relativity, We will not show it, but any 3D homogeneous isotropic y B ) 2 + ( t B ) 2 . space can be described by the Robertson-Walker metric, for c 2 ( t A s 2 2 2 ( x A x B ) + ( y A z A z B ) ≡ − − − − − AB k positive, negative, or zero (flat universe). For k > 0, the universe is finite. For k < = 0, the universe is s 2 AB is Lorentz-invariant — it has the same value for all inertial reference frames. infinite. Meaning of s 2 AB : The Gauss–Bolyai–Lobachevsky geometry is the 2-dimensional If positive, it is the distance between the two events in the frame in which open universe. they are simultaneous. (Spacelike.) If negative, it is the time interval between the two events in the frame in which they occur at the same place. (Timelike.) If zero, it implies that a light pulse could travel from A to B (or from B to A ). Alan Guth Alan Guth Massa husetts Institute of T e hnology Massa husetts Institute of T e hnology –4– –5– 8.286 Le ture 13, O tober 24 8.286 Le ture 13, O tober 24

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