Principles of special relativity

Albert Einstein published the special theory of relativity in 1905 based on two postulates.

1. a) State in your own words the first postulate, known as the principle of relativity.
2. b) State the second postulate, concerning the speed of light in a vacuum.
3. c) State to which type of reference frames the special theory of relativity applies.

State which of the following reference frames can be considered inertial and justify each answer:

1. a) A train braking as it enters a station.
2. b) A space probe travelling through deep space at a constant velocity.
3. c) A laboratory centrifuge rotating at a constant angular velocity.
4. d) A train station at rest relative to the ground.

The Michelson–Morley experiment (1887) aimed to measure the speed of the Earth relative to the "aether", the medium through which light was supposed to propagate, by comparing the speed of light in perpendicular directions. The diagram shows the interferometer used.

1. a) What result did they obtain?
2. b) Why is this result incompatible with the classical addition of velocities?
3. c) Which of Einstein's postulates incorporates this, and what happened to the aether hypothesis?

A passenger on a train travelling at $30\ \text{m/s}$ throws a ball forward at $10\ \text{m/s}$ relative to the train.

1. a) According to the Galilean transformation, with what velocity does an observer on the platform see the ball moving?
2. b) Write down the addition of velocities rule from classical physics that you used.

A spacecraft moves away from the Earth at $0.6c$, that is, $1.8 \times 10^{8}\ \text{m/s}$, and turns on a bow headlight (given: $c = 3 \times 10^{8}\ \text{m/s}$). The diagram summarizes the situation.

1. a) What speed would classical physics predict for the light from the headlight as measured from Earth?
2. b) What speed does an observer on Earth actually measure?
3. c) What speed would that same observer measure if the light source were at the stern, pointing towards the Earth? Justify your answer using the second postulate.

The relativistic addition of velocities is $u = \frac{u' + v}{1 + \frac{u' \times v}{c^2}}$, where $u'$ is the velocity of an object relative to a moving frame and $v$ is the velocity of the moving frame relative to the ground. A spacecraft moves away from Earth at $v = 0.8c$ and launches a probe forwards at $u' = 0.5c$ relative to the spacecraft.

1. a) Calculate the speed of the probe measured from Earth using Galileo's classical formula.
2. b) Calculate it using the relativistic formula (you may work directly in units of $c$).
3. c) Verify that the relativistic result is less than $c$ and show, by substituting $u' = c$ into the formula, that if the probe were moving at the speed of light relative to the spacecraft, it would also be measured at $c$ from Earth.

A very fast train passes a platform. At the instant when the centre of the train aligns with an observer on the platform, this observer sees two simultaneous flashes emitted from the ends of the train.

1. a) Will a passenger sitting in the centre of the train also see them as simultaneous?
2. b) Reason your answer, taking into account that the train is moving towards one of the flashes and that light from both flashes travels at $c$ for the passenger as well.

Einstein's thought experiment: two lightning bolts strike the ends of a train carriage just as its centre passes an observer on the platform, who sees both strikes at the same time. A passenger is standing in the centre of the carriage. The diagram represents the situation.

1. a) Describe what the passenger sees and explain why, using the fact that for her, the light from both strikes also travels at $c$ while she moves towards one of them.
2. b) What is this phenomenon called, where two events simultaneous for one observer are not simultaneous for another?
3. c) Draw a conclusion about the existence of a universal "now" valid for the entire galaxy.

Muons are created in the upper atmosphere by cosmic radiation and their mean lifetime at rest is $2.2\ \mu\text{s}$. If time were absolute, almost none of them would reach the Earth's surface, but ground detectors register them in large numbers.

1. a) What relativistic effect explains why so many muons reach the ground?
2. b) Which quantity ceases to be absolute in this explanation?

In a moving spacecraft, a lamp located at a point on the cabin floor flashes on and off: two events occurring at the same place inside the spacecraft. The diagram compares what the crew and an observer on Earth see.

1. a) Which observer measures the proper time interval between both events: the crew of the spacecraft or an observer on Earth? Justify your answer.
2. b) Which of the two observers measures a larger interval and what is this effect called?

Atmospheric muons are produced at an altitude of about $10\ \text{km}$, travel at $v = 0.998c$, and their mean lifetime at rest is $2.2\ \mu\text{s}$ (given: $c = 3 \times 10^{8}\ \text{m/s}$).

1. a) Calculate the average distance a muon would travel if time were absolute, that is, using its mean lifetime at rest.
2. b) Compare that distance with the $10\ \text{km}$ it descends and conclude whether, according to classical physics, it should be detected at the surface.
3. c) Name the relativistic effect that explains why they are detected, and explain the reasoning.

The GPS system uses atomic clocks installed on satellites orbiting the Earth at high speed.

1. a) Why can satellite clocks not remain synchronized with ground clocks without continuous corrections?
2. b) State the relativistic phenomenon responsible for this discrepancy.

The fastest crewed aircraft in history reached about $980\ \text{m/s}$ (given: $c = 3 \times 10^{8}\ \text{m/s}$).

1. a) Calculate the ratio of its speed to the speed of light.
2. b) In light of the result, explain why relativistic effects are negligible in everyday life.
3. c) State above what kind of speeds, compared to $c$, they begin to be noticeable.

The twin paradox: one twin remains on Earth while the other travels in a very fast spacecraft for a few years and returns. The spacetime diagram summarizes the journey.

1. a) State the apparent paradox: if motion is relative, why does it seem that each twin should see the other age more slowly?
2. b) Explain why the situation is not truly symmetrical: what happens to the travelling twin at the points of acceleration and turnaround that does not happen to the one who stays?
3. c) Conclude which of the two has aged less upon reunion.

Can an object with mass reach the speed of light in a vacuum? Answer and justify your response, taking into account the second postulate and the fact that the energy required to continue accelerating the object would grow without limit as its speed approaches $c$.

In the Large Hadron Collider (LHC), protons reach speeds of $0.9999c$ (given: $c = 3 \times 10^{8}\ \text{m/s}$).

1. a) Calculate the speed of the proton in $\text{m/s}$.
2. b) Calculate how many $\text{m/s}$ it lacks to reach $c$.
3. c) The accelerator continues to supply energy to the proton and yet it never reaches $c$. Explain why and how this extra energy manifests itself.

Supersonic aircraft exceed the speed of sound in air, but no object with mass can reach or exceed the speed of light in a vacuum.

1. a) How does the sound barrier differ physically from the speed-of-light barrier? Think about the medium through which each signal propagates and which postulate supports one of them.
2. b) Justify, in terms of energy, why reaching $c$ is impossible for an object with mass and is not merely a technical difficulty like the sound barrier was.