d 1 g g = . d d 2 x d d AND Use indices , , etc., - PowerPoint PPT Presentation

Alan Guth, Black-Body Radiation and the Early History of the Universe, 8.286 Lecture 15, October 31, 2013, p. 1. Summary of Le ture 14: 8.286 Le ture 15 THE SPACETIME GEODESIC EQUATION O tober 31, 2013 λσ d x λ d x σ BLACK-BODY RADIATION d x ν d � � 1 ∂g g = . µν d τ d τ 2 ∂x µ d τ d τ AND Use indices µ , ν , etc., which are summmed from 0 to 3, where THE EARLY HISTORY x 0 ≡ t . OF THE UNIVERSE Use τ to parameterize the path, where τ = proper time measured along the path. Alan Guth Massa husetts Institute of T e hnology –1– 8.286 Le ture 15, O tober 31 THE SCHWARZSCHILD METRIC RADIAL GEODESICS For µ = r , the geodesic equation gives − 1 � 2 GM � � 2 GM � d s 2 = − c 2 d τ 2 = − c 2 d t 2 + r 2 1 − 1 d − rc 2 rc 2 2 2 � � � � � � d d r 1 d r 1 d t + r 2 d θ 2 + r 2 sin 2 θ d φ 2 . g = ∂ r g rr + ∂ r g tt . rr d d τ 2 d τ 2 d τ τ Describes the metric for any spherically symmetric mass distribution, for the region outside the mass distribution. M is the mass of the object, G is which simplifies to Newton’s constant, and c is (of course) the speed of light. At r = R S , where d 2 r GM = . 2 GM − d τ 2 r 2 R S = c 2 It looks like Newton, but r is not really the distance from the is called the Schwarzschild horizon, the metric is singular. But the singularity is not physical, and can be removed by a different choice of origin, and τ is the proper time measured along the trajectory. coordinates. R S is, however, a horizon: anything with r < R S can never Alan Guth get out. Massa husetts Institute of T e hnology –2– –3– 8.286 Le ture 15, O tober 31

Alan Guth, Black-Body Radiation and the Early History of the Universe, 8.286 Lecture 15, October 31, 2013, p. 2. Solving the Radial Infall Equation The proper time τ needed to reach radial variable r is � �� � � � r r 0 − r 0 � r 0 tan − 1 τ ( r ) = + r ( r 0 − r ) . 2 GM r The infalling object will be ripped apart by the singularity at r = 0 in a finite amount of the object’s proper time. But from the outside, it will take an infinite amount of coordinate time t before the object reaches the horizon. Alan Guth Massa husetts Institute of T e hnology –4– 8.286 Le ture 15, O tober 31

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