Schwarzschild - HW#9Supplement Phys410Fall 2011

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HW#9—Supplement —Phys410—Fall 2011 Prof. Ted Jacobson Room 4115, (301)405-6020 www.physics.umd.edu/grt/taj/410a/ jacobson@umd.edu S9.3 Motion in the Schwarzschild spacetime Consider the action for a test particle in a gravitational field, S = - mc 2 R ds , where the field is that of a central mass. The line element in this field is ds 2 = F ( r ) dt 2 - 1 F ( r ) dr 2 /c 2 - ( r 2 /c 2 )( 2 + sin 2 θdφ 2 ) , (1) where F ( r ) = 1 - r g /r , and r g is the Schwarzschild radius , r g = 2 GM/c 2 . This is the unique spherically symmetric vacuum solution to the Einstein gravitational field equations, up to coordinate changes. The parameter M is the mass of the central body. [If there is no central body, as would happen if a star collapsed to form a black hole, then this line element applies all the way down to the radius r = r g , where F = 0. This is the event horizon of the black hole. The blow up of the line element at r = r g does not signify any physical divergence. Rather, this (Schwarzschild) coordinate system is not well-behaved at
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Schwarzschild - HW#9Supplement Phys410Fall 2011

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