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Unformatted text preview: HW#4 —Phys776—Spring2005 Prof. Ted Jacobson Due Thursday, March 17, before class. Room 4115, (301)4056020 [email protected] 1. The past light cone of a point in Minkowski spacetime is a null hypersurface whose generators form a null geodesic congruence. Consider a point on the cone where the radius of a spherical cross section is r . The expansion θ at this point is negative, hence according to the focusing theorem θ will diverge to minus infinity in a finite affine parameter. (a) Show that the radial coordinate r is an affine parameter along the null generators of the cone. (b) Using r as the affine parameter, what is the value of θ in terms of r ? (c) At what value of the affine parameter range Δ r does the focusing theorem predict θ will diverge? Compare your answer to the value of Δ r that you infer directly from the definition of the cone, and show that the two answers agree. 2. Consider any static spherically symmetric metric in Schwarzschildlike coordinates, ds 2 = f ( r ) dt 2 g ( r ) dr 2 r 2 d Ω 2 . (1) (a) Show that the ingoing and outgoing radial light rays are always geodesics.(a) Show that the ingoing and outgoing radial light rays are always geodesics....
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