Homeworks General Relativity - Physics General Relativity Homework Due Wednesday th September Jacob Lewis Bourjaily Problem 1 a We are to use the

Homeworks General Relativity - Physics General Relativity...

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Physics  , General Relativity Homework Due Wednesday,  th September  Jacob Lewis Bourjaily Problem 1 a) We are to use the spacetime diagram of an observer O to describe an ‘experiment’ specified by the problem 1.5 in Schutz’ text. We have shown the spacetime diagram in Figure 1 below. t x Figure 1. A spacetime diagram representing the experiment which was required to be described in Problem 1.a. b) The experimenter observes that the two particles arrive back at the same point in spacetime after leaving from equidistant sources. The experimenter argues that this implies that they were released ‘simultaneously;’ comment. In his frame, his reasoning is just, and implies that his t -coordinates of the two events have the same value. However, there is no absolute simultaneity in spacetime, so a different observer would be free to say that in her frame , the two events were not simultaneous. c) A second observer O moves with speed v = 3 c/ 4 in the negative x -direction relative to O . We are asked to draw the corresponding spacetime diagram of the experiment in this frame and comment on simultaneity. Calculating the transformation by hand (so the image is accurate), the experiment ob- served in frame O is shown in Figure 2. Notice that observer O does not see the two emission events as occurring simultaneously. t x Figure 2. A spacetime diagram representing the experiment in two different frames. The worldlines in blue represent those recorded by observer O and those in green rep- resent the event as recorded by an observer in frame O . Notice that there is obvious ‘length contraction’ in the negative x -direction and time dilation as well. 1
2 JACOB LEWIS BOURJAILY d) We are to show that the invariant interval between the two emission events is invariant. We can proceed directly. It is necessary to know that in frame O the events have coordinates p 1 = (5 / 2 , - 2) and p 2 = (5 / 2 , 2) while in frame O they have coordinates p 1 = γ (1 , - 1 / 8) and p 2 = γ (4 , 31 / 8) where γ 2 = 16 7 . Δ s 2 = ( p 1 - p 2 ) 2 = 16; Δ s 2 = ( p 1 - p 2 ) 2 = γ 2 ( - 9 + 16) = 16 . We see that the invariant interval is indeed invariant in this pointless example. Problem 2. a) We are to show that rapidity is additive upon successive boosts in the same direction. We may as well introduce the notation used in the problem: let v = tanh α and w = tanh β ; this allows us to write γ = 1 1 - tanh 2 α = cosh α and = sinh α , and similar expressions apply for β . We see that using this language, the boost transformations are realized by the matrices 1 γ - - γ 7→ cosh α - sinh a - sinh α cosh α , (a.1) and similarly for the boost with velocity w . Two successive boosts are then composed by 2 : cosh β - sinh β - sinh β cosh β cosh α - sinh α - sinh α cosh α = cosh α cosh β + sinh α sinh β - sinh α cosh β - cosh α sinh β - sinh α cosh β - cosh α sinh β cosh α cosh β + sinh α sinh β = cosh( α + β ) - sinh( α + β ) - sinh( α + β ) cosh( α + β ) This matrix is itself a boost matrix, now parameterized by a rapidity parameter ( α + β ). Therefore, successive boosts are additive for rapidity.

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