Prelims 1997 Question 6

Transcription

1 Prelims 1997 Question 6 Write down the Lorentz transformations and use them to derive the expressions describing the phenomena of Lorentz contraction and time dilation. [5] The Prime Minister plans a quiet Cabinet meeting at a remote location. Accordingly, the Cabinet Secretary books a conference on a planet circling a star ten light years from Earth. State, giving reasons for your answers, whether the following statements are in principle TRUE or FALSE. Neglect the effect of acceleration. (a) The Prime Minister claims that with a fast enough spaceship the journey could be as short as one hour. (b) The Transport Secretary says that if she leaves 30 minutes after the Prime Minister, but travels at a speed such that her journey time is only 30 minutes, she will arrive at the same time as the Prime Minister. (c) The Defence Secretary says that the proper time of the Prime Minister's journey will be much longer than an hour. (d) The Health Minister claims that although the journey might only be one hour, the travellers will age more than ten years. (e) The Leader of the Opposition says that none of the group will return for at least twenty years. [20]

2 ct ct 2 S (x 2, t 2 ) Relative Velocities In S: From Lorentz Transformations: ct 1 (x 1, t 1 ) x 1 x 2 x ct v rel S ct 2 (x 2, t 2) In S : (x 1, t 1) ct 1 x 1 x 2 x

3 Spaceship Problem Spaceship A of proper length L is travelling east at speed v 1, and spaceship B of proper length 2L is travelling west at speed v 2, as seen from Earth. The pilot of spaceship A sets his clock to zero when the front of spaceship B passes him (the spaceship pilots sit in the nose cone). Use Lorentz transformations to calculate the time at which, according to the pilot of spaceship A, the tail of spaceship B passes him.

4 (a) 0 Movement of B w.r.t. A B G.B. 2L x -L U.S. 0 A (at rest) x (a) The event specified by the passing of the fronts of spaceships A and B. (b) 0 2L -L Movement of B w.r.t. A B G.B. U.S. 0 A (at rest) x x (b) The event specified by the passing of the rear of spaceship B by the front of spaceship A. Both from the point of view of an observer in A. Therefore B is shortened.

5 Fizeau s Experiment y (laboratory frame) Direction of Light u x v (water) in x u x = u x + v 1+vu c ( x/c 2 n + v 1 1 n 2 = c n + fv ) y (moves with velocity v relative to ) Direction of Light u x = c/n For transformations f = 1 Best value for f (M&M 1886) = ± for n = 1.33, f theor = 1-1/ = water at rest in x f is the

6 Relativistic Doppler Effect Stationary observer sees waves redshifted 1 2 Stationary observer sees waves blueshifted 3 S1 S2 4 u S3 S4 Source moving to right while emitting waves u

7 Relativistic Doppler Shift [ ] 1 1 β ν 2 = ν 0 1+β [ ] 1 1+β ν 2 = ν 0 1 β Observer moving away from source Observer moving towards source Useful way to remember formula: ν = ν 0 γ(1 β) - sign: moving APART + sign: moving TOGETHER

8 Hubble s Law Hubble (1929) noticed that distant galaxies always showed a RED SHIFT. They are moving AWAY from us, those further away move faster. Hubble s Law: u = DH Edwin Hubble Velocity is proportional to distance Hubble s original data Current accepted value = 67 kms -1 Mpc -1

9 Measuring Red Shifts Distance (x 10 6 l.yrs.) H = u / D H = 2.2 x s

10 Gravitational Lensing

11 Addition of Velocities from Doppler Shift Source y S y v 1 (rel. to S) S y v 2 (rel. to O 1, S ) S O 1 O 2 0 x 1 x 2 x v (rel. to S) [ ] 1 1 β1 2 ν 1 = ν 0 1+β 1 [ ] 1 1 β2 2 ν 2 = ν 1 1+β 2 β = β 1 + β 2 1+β 1 β 2 [ ] 1 [ ] 1 [ ] 1 1 β 2 1 β β 2 ν 2 = ν 0 = ν0 1+β 1+β 1 1+β 2

12 Olbers Paradox Heinrich Olbers (1926): If the Universe is uniformly filled with stars, then no matter which direction you look, your line of sight will eventually intersect a star (or other bright object). Therefore,? Average Luminosity per star: L Number density of stars: n stars / volume Brightness of star at distance r: B = L/4 r 2 Examine shell of stars at distance r. Total brightness from shell with thickness dr is: B shell = [volume of shell] x [density of stars] x [brightness per star] = 4 r 2 dr x n x L/4 r 2 = nldr. This is independent of distance, and as we add more shells increases without limit! Even with absorption etc. we still expect the sky to be as brilliant as the surface of a star! (a) The universe is of finite age - light from stars more distant than ct 0 hasn t reached us yet. (b) The universe is expanding - Doppler shift reduces energy of photons, fluxes drop faster than 1/r 2. In fact explanation (b) has only a factor of 2 influence on the brightness. Olbers paradox needs a universe of!