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test2 - PHYS5523 Theory of Relativity Test 2 The...

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PHYS5523 Theory of Relativity Test 2: The Gravitational Field of a Single Body November 2, 2011 General Instructions: This is a three hour, open book test with three problems. Complete all three problems. Problem 1: A Radially Infalling Observer. a) Demonstrate that a radially infalling observer always measures the speed of radially moving photons (whether they are traveling inwards or outward) to be c relative to his own velocity. Accomplish this by finding the velocity of both the photons and the observer himself in the reference frame of a radially infalling observer as given by the Gullstrand-Painleve coordinates. The Schwarschild metric in Gullstrand-Painleve coordinates is ds 2 = - (1 - 2 M r ) dt 2 + 2 radicalbigg 2 M r dt dr + dr 2 + r 2 2 + r 2 sin 2 θdφ 2 (1) where t is the Gullstrand-Painleve coordinate time, which is the same as the proper time for an observer freely-falling from infinity, and r , θ and φ are the Schwarzschild coordinates of the same name. b) Now consider the viewpoint of a radially infalling observer (observer 1) compared to that of an observer stationary at infinity in the Schwarzschild radial coordinate (observer 2). Each of them is observing photons emitted by
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