An identical clock remained on Earth. What time had elapsed on the Earth clock, when the astronaut's spaceship passes the star?

She now decides to observe the two wires from a high speed vehicle travelling parallel to the wires, in the direction of the electron travel, at the same speed. Will the force between the two wires be larger, smaller, about the same, or zero? Again, marks for the explanation of the correct answer, not just for the answer. (If the electrons move at speeds that are not << c, then this question becomes much more complicated.)

#### Question 9

A train moves at constant, relativistic speed. When the train passes a pole by the track, a clock on board records the time reading. There are two poles. Each has a similar clock, which records when the front of the train passes it. The clocks on the poles have been carefully synchronised.
Call T_{train} the interval measured between the readings by the clock on the train, and T_{track} the interval between the readings of the pole clocks.

Consider the following statements:

i) According to relativity, an observer by the track sees the clock on the train running slow, so he predicts T_{train} < T_{track}.

ii) According to relativity, an observer on the train sees the clocks on the poles running slow, so she predicts T_{track} < T_{train}.

iii) The two observers can send each other their (already recorded) measurements, at their convenience. They cannot both be correct, so there must be a paradox.

Is this a paradox? If not, why not? (And if so, what shall we do with all those textbooks about relativity?)

#### Question 10

Zoe has entered her car for a new event: the Bathurst Time Trial. Her car (a '61 Holden fitted with tail fins to increase speed) is believed to be so fast that a special timing system has been devised. It works like this: at the start, a laser in the car sends a beam laterally towards an array of mirrors set up at the half-way point of the course, as shown in the two diagrams below (not to scale). The course has length L and the mirrors are at different distances w_{n} from the track, as shown in the sketch. (The mirrors are also at slightly different heights and the beam is spread in the vertical direction so that it can hit all of them, but this detail is unimportant for the following question.)
At the finish line, a reflected beam is received by one of a vertical array of detectors fixed to the side of the car. From the height of the detector that received it, one can tell which mirror reflected the beam. The further the reflector is far from the track, then the further light has travelled during the time trial, so the slower the car. The aim in this race is to have a low value of w_{n}.

i) The judges of the race determine that the pulse received by the car at the finish line was reflected by the nth mirror, at a distance w_{n} from the track. From this observation, the judges then calculate their value of the time t_{judge} taken by Zoe for the event. Derive an expression for tjudge in terms of L, w_{n} and c, the speed of light.

ii) Using your result from (i), give an expression for the speed v that the judges will record for the event. Express your answer in terms of c and the ratio w_{n}/L.

iii) Rearrange your answer to (ii) to express L/wn in terms of c and v.

iii) Zoe also observes that the light has been reflected from the nth reflector. From this observation alone, and *without* using the Lorentz transformation equations, give an expression for the time t_{Zoe} that Zoe will calculate as the time she took to complete the distance L. Your expression should not involve v.

Explain in one or two sentences how you derived your answer.

iv) Zoe and the judge determine different times: t_{judge} does not equal t_{Zoe}. Nevertheless, from independent measurements such as the Doppler shift in light, they both agree on the speed v.

Describe how Zoe (who understands relativity as well as motor mechanics) would explain the difference between the two results for t_{judge}and t_{Zoe}.

v) Describe how the judges (who also understand relativity) would explain the difference between the two results for t_{judge} and t_{Zoe}.

vi) From your results above, give an expression for the ratio t_{judge}/t_{Zoe}. Using your expression for part iii to simplify your answer, express it in terms of the ratio (v/c) and comment briefly.

vii) The laser (a gas laser) has a tube mirror at either end, with a standing light wave between the mirrors, as shown in the inset at bottom right. At what angle to the direction of the car should the laser point so that the beam will strike one of the reflectors and return to be picked up by the detector in the car at the finish line? Your answer should have a sentence or two of explanation. It should include a sketch of the situation in the frame of the judges.

#### Question 11 (A symmetrical 'twin paradox').

Two space travellers are initially at rest, each at a large distance L from a central clock, but in opposite directions. On their twentieth birthdays, each determines that the central clock records the same time. The light has taken the same time to reach each traveller, they are both at rest in the same inertial frame so, in this frame, they are equally old. Complete symmetry.
They set off to meet at the central clock. Over a distance d, which is negligible in comparison with L, they accelerate to speeds comparable with c. Everything is symmetrical.
During the unaccelerated flight, they are travelling at relativistic speeds with respect to each other. Consequently, during that part of the trip, each traveller sees that other's clock is running slow, and each sees that the other is ageing more slowly. (And remember that d<
Is this a paradox that destroys relativity? If not, why not?

(As was the case with Question 9, the argument in this question was sent to me by email with the challenge that I should put it on this site, whether or not I could answer it. So I do. However, the information you need to resolve the two 'paradoxes' is already on Einstein Light.)