8.286 Leture 14 d s 2 = c 2 d t 2 + a 2 ( t ) + r 2 d 2 + sin - PowerPoint PPT Presentation

Alan Guth, The Geodesic Equation, 8.286 Lecture 14, October 29, 2013, p. 1. Summary of Le ture 13: Adding Time to the Robertson{Walker Metri d r 2 � � 8.286 Le ture 14 d s 2 = − c 2 d t 2 + a 2 ( t ) + r 2 d θ 2 + sin 2 θ d φ 2 � � . 1 kr 2 − O tober 29, 2013 Meaning: If ds 2 > 0, it is the square of the spatial separation measured THE GEODESIC EQUATION by a local free-falling observer for whom the two events happen at the same time. If ds 2 < 0, it is − c 2 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 ds 2 = 0, then the two events can be joined by a light pulse. Alan Guth Massa husetts Institute of T e hnology –1– 8.286 Le ture 14, O tober 29 THE GEODESIC EQUATION Adding Time to the d s 2 = g ij ( x k ) d x i d x j . Robertson{Walker Metri Notation: Descripti on of path from A to : B x i ( λ ) , where x i (0) = x i x λ f ) = x i i , ( B . A � d r 2 � d s 2 = − c 2 d t 2 + a 2 ( t ) + r 2 d θ 2 + sin 2 θ d φ 2 from x i ( λ ) to x i ( λ + d λ ): Distance � � . 1 kr 2 x i d x j − d d s 2 = g ij ( x k d λ 2 . ( λ )) d λ d λ Distance from A to B : Application: The Geodesi Equation. � λ f d x i d x j � [ x i ( λ )] = g ij ( x k ( λ )) S d λ . d λ d λ 0 Alan Guth Alan Guth Massa husetts Institute of T e hnology Massa husetts Institute of T e hnology –2– –3– 8.286 Le ture 14, O tober 29 8.286 Le ture 14, O tober 29

Alan Guth, The Geodesic Equation, 8.286 Lecture 14, October 29, 2013, p. 2. Varying the path (calculus of variations, 1696): � λ f �� � x i ( λ ) d S ˜ 1 1 � = 2 � x i ( λ ) = x i ( λ ) + αw i ( λ ) , � ˜ d α � A ( λ, 0) 0 � α =0 w i (0) = 0 i ( � ∂g ij w k d x i d x j d w i d x j , w λ f ) = 0 , � + 2 g d λ . × ij ∂x k d λ d d d λ � λ λ � x i ( λ ) � d S ˜ � = 0 � d α � Integrate se ond term by parts! � α =0 for all w i ( λ ) , jk d x j d x k λ f d x j d S � � 1 d � 1 �� � ∂g � w i ( λ ) d λ . = d λ d λ − d λ g ij √ √ � d α A ∂x i d λ where 2 A � α =0 0 � λ f x i d˜ x j d˜ d x j d x k d x j d � 1 � 1 � ∂g � x i ( λ ) � � x k ( λ ) � ˜ = A ( λ, α ) d λ , A ( λ, α ) = g ij ˜ S . jk = g ij = . ⇒ √ √ d λ d λ A ∂x i d λ d λ d λ d λ 0 A 2 Alan Guth Alan Guth Massa husetts Institute of T e hnology Massa husetts Institute of T e hnology –4– –5– 8.286 Le ture 14, O tober 29 8.286 Le ture 14, O tober 29

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