Exercises: 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.

A lighthouse on the coast emits a light signal. It is measured by two observers: the lighthouse keeper, who is at rest, and the captain of a boat approaching the lighthouse at $20\ \text{m/s}$ (given: $c = 3 \cdot 10^{8}\ \text{m/s}$).

1. a) According to classical physics, what value would each observe?
2. b) According to special relativity, what value does each observe? Justify your answer using the corresponding postulate.

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.

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 space probe moving at a speed of $v$ relative to the Earth turns on its bow headlight. According to classical physics, the light from that headlight should be measured from the Earth at $c + v$.

1. a) Explain why this prediction contradicts the second postulate of special relativity.
2. b) What speed does an observer on Earth actually measure for the light from the headlight?

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?

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.

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 $c$.

A spacecraft moves away from the Earth at $0,6c$, that is, $1,8 \cdot 10^{8}\ \text{m/s}$, and turns on a bow headlight (given: $c = 3 \cdot 10^{8}\ \text{m/s}$). 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 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?

In the Large Hadron Collider (LHC), protons reach speeds of $0,9999c$ (given: $c = 3 \cdot 10^{8}\ \text{m/s}$).

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.

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?

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.

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

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.

Light propagates in water at $2,25 \cdot 10^{8}\ \text{m/s}$, which is less than in a vacuum ($c = 3 \cdot 10^{8}\ \text{m/s}$).

1. a) Does this fact contradict the second postulate of relativity? Justify your answer by specifying exactly what the postulate refers to.
2. b) In nuclear reactor pools, there are particles moving through water faster than $2,25 \cdot 10^{8}\ \text{m/s}$ (although always below $c$) and emitting a bluish glow called Cherenkov radiation. Does this contradict relativity? Reason the difference with the previous section.

In classical physics, time, length, and mass are absolute quantities: their value does not depend on the observer measuring them.

1. a) Of these three quantities, state which become relative in the special theory of relativity.
2. b) Which quantity remains invariant for all inertial observers according to the second postulate?
3. c) If a universal time no longer exists, what happens to the idea that two distant events occur "at the same time" for everyone?

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: $c = 3 \cdot 10^{8}\ \text{m/s}$).

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 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.

The relativistic addition of velocities is $u = \frac{u' + v}{1 + \frac{u' \cdot v}{c^2}}$, 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.

If the clocks on GPS satellites were not corrected, they would accumulate a time shift of about $38\ \mu\text{s}$ per day. Positioning relies on measuring the time it takes for the signal from each satellite, travelling at $c = 3 \cdot 10^{8}\ \text{m/s}$, to arrive.

1. a) Calculate the error in distance introduced by a time shift of $38\ \mu\text{s}$ in the arrival time measurement.
2. b) In light of the result, how long would it take for the system to become useless for navigation if it were not corrected? Justify your answer.
3. c) Explain why the operation of GPS is considered an everyday verification of relativity.

The Lorentz factor $\gamma$ measures the magnitude of relativistic effects: it equals $1,005$ at $0,1c$; $1,15$ at $0,5c$; $2,29$ at $0,9c$; and $7,09$ at $0,99c$. The graph plots $\gamma$ against velocity.

1. a) At which of the specified speeds does the relativistic correction first exceed $5\ \%$?
2. b) Based on these values, explain why Newtonian mechanics works so well in civil and aeronautical engineering.
3. c) Predict, without using a calculator, whether the value of $\gamma$ at $0,999c$ will be slightly greater than $7$ or much greater, reasoning from the trend in the data and what must happen as $v$ approaches $c$.

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.

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.

A proton ($m = 1,67 \cdot 10^{-27}\ \text{kg}$) moves at $0,99c$; at this speed, the Lorentz factor is $\gamma = 7,09$. The classical kinetic energy is $E_c = \frac{1}{2} \cdot m \cdot v^2$ and the relativistic kinetic energy is $E_c = (\gamma - 1) \cdot m \cdot c^2$ (given: $c = 3 \cdot 10^{8}\ \text{m/s}$).

1. a) Calculate the kinetic energy predicted by classical physics.
2. b) Calculate the kinetic energy predicted by special relativity.
3. c) Calculate the ratio of the two predictions and conclude why the classical formula for kinetic energy is not valid near $c$.