Question

1. What basic assumptions did Einstein make in deriving his theory of Special Relativity? What experimental evidence, if any, was available at the time to support his conjectures? What are three counter-intuitive consequences of the theory?

2. [15 pts - short essay] Explain time dilation. Use an example to show how it is derived.

3. [10 pts - question] Most of the elementary particles we know about are unstable; they decay after a finite lifetime. (The muon is an example.) Only a few elementary particles are completely stable. Use time dilation to argue that a photon, which by definition travels at the speed of light, must be stable and cannot decay.

4. [10 pts - question] An ice skater is moving forward at a steady speed of 10 m/s in a straight line.

   (a) The skater tosses a coin straight (horizontally) ahead of her. The coin leaves her hand at 5 m/s (relative to the hand). What is the velocity of the coin, according to Newtonian physics, as viewed by a person standing still on the ice? (b) According to Einstein’s relativity, will the velocity of the coin obtained in question (a) be slightly larger, slightly smaller, or equal to the result you found in (a)? [You do not need to compute the actual velocity in this case]

Answer 1

Basic assumptions made of Einstien are

1. The speed of light is constant in all inertial frames.
2. The laws of physics are the same in all inertial frames of reference.

The only available experimental evidence at that time was the null result of Michelson-Morley experiment which showed the speed of light is constant in every inertial frame.

The three counter-intuitive consequences of the theory are

1. Length contraction: moving objects appear to be shorter than their proper length
2. Time Dilation: time moves slowly in a moving frame
3. No matter how much force you apply on an object it can move faster than the speed of light.

TIME DILATION: Time dilation is a phenomenon described by the theory of relativity. It can be illustrated by supposing that two observers are in motion relative to each other.

According to the theory of relativity, time dilation is a difference in the elapsed time measured by two observers, either due to a velocity difference relative to each other, or by being differently situated relative to a gravitational field. As a result of the nature of spacetime,^[2] a clock that is moving relative to an observer will be measured to tick slower than a clock that is at rest in the observer's own frame of reference. A clock that is under the influence of a stronger gravitational field than an observer's will also be measured to tick slower than the observer's own clock.

The faster the relative velocity, the more is the rate of time dilation. This case is sometimes called special relativistic time dilation. It is often interpreted as time “slowing down” for the other (moving) clock. But that is only true from the physical point of view of the local observer, and of others at relative rest (i.e. in the local observer’s frame of reference). The point of view of the other observer will be that again the local clock (this time the other clock) is correct, and it is the distant moving one that is slow. From a local perspective, time registered by clocks that are at rest with respect to the local frame of reference always appears to pass at the same rate.

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