The Special Theory of Relativity

Abstract

This chapter gives a concise and yet rather complete introduction to the special theory of relativity. Minkowski diagrams are used to illustrate several concepts such as the relativity of simultaneity. Special relativity is a theory of flat spacetime admitting accelerated and rotating reference frames. In this chapter we also show how magnetism appears as a 2 order effect in v/c of electricity due to the Lorentz transformation.

Exercises

• 2.1. Robb’s Lorentz invariant spacetime interval formula (A. A. Robb, 1936)

Show that the spacetime interval between the emission event at the point of time $t_{A}$ and the reflection event at $t_{B}$ in Fig. 2.2 can be expressed as $\Delta s = c\sqrt {t_{A} t_{C} }$, where $t_{C}$ is the point of time when the reflected light signal arrives back at the emitter.

• 2.2. The twin paradox

On New Year’s day 2004, an astronaut (A) leaves Earth on an interstellar journey. He is travelling in a spacecraft at the speed of $v = \left( {4/5} \right)c$ heading towards Alpha Centauri. This star is at a distance of 4 ly (ly = light years) measured from the reference frame of the Earth. As A reaches the star, he immediately turns around and heads home. He reaches the Earth New Year’s day 2016 (in Earth’s time frame). The astronaut has a brother (B), who remains on Earth during the entire journey. The brothers have agreed to send each other a greeting every New Year day with the aid of radio telescope.

• (a) Show that A only sends 6 greetings (including the last day of travel), while B sends 10.

• (b) Draw a Minkowski diagram where A’s journey is depicted with respect to the Earth’s reference frame. Include all the greetings that B is sending. Show with the aid of the diagram that while A is outbound, he only receives one greeting, while on his way home he receives nine.

• (c) When does B receive signals from A?

• (d) Show how the results from (b) and (c) can be deduced from the Doppler effect.

• 2.3. Faster than the speed of light?

The quasar 3C273 emits a jet of matter that moves with the speed v₀ towards Earth making an angle ϕ to the line of sight (see Fig. 2.18).

Fig. 2.18

Light cone due to Cherenkov radiation

• (a) Assume that two signals are sent towards the Earth simultaneously, one from A and one from B. How much earlier will the signal from B reach the Earth compared to that from A?

• (b) Find an expression of the transverse distance that the emitted part has moved when it reaches B. How much time (relative to the Earth) has this part been travelling?

• (c) Calculate the velocity $v_{0}$ of the light source in terms of v and $\theta$, and find the value of $v_{0}$ if $v = 10c$ and $\theta = 10^{ \circ }$. How large must $v_{0}$ be in order that the observed transverse velocity shall be larger than c?

• 2.4. Time dilation and Lorentz contraction

• (a) At what speed does a clock move if its rate of time is 0.6 times the rate when it is at rest?

• (b) A rod moves in the x-direction. An observer following the rod measures that it makes an angle $\pi /4$ with the x-axis. What is the speed of the rod if an observer at rest on the x-axis finds that it makes an angle $\pi /3$ with the x-axis, due to the Lorentz contraction of its length component in the x-direction?

• 2.5. Atmospheric mesons reaching the surface of the Earth

Atmospheric muons are formed when molecules in the Earth’s atmosphere are hit by particles in the cosmic rays about $L_{0} = 10\;{\text{km}}$ above the surface. The half-life of a muon as measured by an observer co-moving with the muon is $t_{0} = 1.56 \times 10^{ - 6} \;{\text{s}}$. The average velocity of the muons are $v = 0.98\,c$.

• (a) According to a non-relativistic calculation, how many of ten million muons formed at 10 km height reach the Earth’s surface?

• (b) Taking the relativistic time dilation into account, how many will then reach the surface of the Earth?

• (c) How is this explained by an observer following the muons?

• 2.6. Relativistic Doppler shift

The relativistic formula for the Doppler effect of an object moving along the direction of sight with a velocity v away from the observer is

$$\lambda_{r} = \sqrt {\frac{1 + v/c}{1 - v/c}} \lambda_{e} ,$$

where $\lambda_{e}$ is the frequency measured by an observer co-moving with the object, and $\lambda_{r}$ is the frequency measured for the light received by an observer at rest. If the object moves towards the observer the signs are interchanged.

For an object moving away from an observer there is an increase in the measured wavelength—a redshift. The redshift of an object is defined as

$$z = \frac{{\lambda_{r} - \lambda_{e} }}{{\lambda_{e} }}.$$

Hence the redshift due to the Doppler effect is

$$z = \sqrt {\frac{1 + v/c}{1 - v/c}} - 1.$$

Positive value of z means redshift and negative value blueshift.

The measured value of z for the centre of our neighbour galaxy, the Andromeda Galaxy, is $z = - \,0.0004$.

Determine the velocity of the Andromeda Galaxy along the line of sight from this measurement. Which way does the Andromeda Galaxy move relative to the Milky Way?

• 2.7. The velocity of light in a moving medium

Light moves more slowly in a material medium than in empty space. The index of refraction, n, of the medium is defined as the ratio of the velocity of light in the medium when it is at rest, $u_{0}$, and the velocity of light in empty space. Then

$$n = \frac{{u_{0} }}{c}.$$

We now consider a medium moving with a velocity v in the same direction as the light. The speed of light in this medium relative to the laboratory frame is related to the speed of light in a frame co-moving with the water by the relativistic velocity addition law,

$$u = \frac{{u_{0} + v}}{{1 + \frac{{u_{0} v}}{{c^{2} }}}}.$$

Find the velocity of light in a moving medium in terms of its index of refraction and velocity.

• 2.8. Cherenkov radiation

When a particle moves through a medium with a velocity v greater than the velocity of light in the medium, it emits a cone of radiation with a half-angle $\theta$ given by $\cos \theta = c/nv$.

From Wikipedia: https://en.wikipedia.org/wiki/Cherenkov_radiation#/media/File:Cherenkov.svg.

• (a) What is the threshold kinetic energy (in MeV) of an electron moving through water in order that it shall emit Cherenkov radiation? The index of refraction of water is $n = 1.3$. The rest energy of an electron is $m_{e} c^{2} = 0.5.11\;{\text{MeV}}$.

• (b) What is the limiting half-angle of the cone for high-speed electrons moving through water?

• 2.9. Relativistic form of Newton’s 2 law

In order to simplify the calculation we shall consider motion along the x-direction only. Let the particle have rest mass $m_{0}$ and velocity v. Its momentum is $p = \gamma m_{0} v$, where $\gamma = \left( {1 - v^{2} /c^{2} } \right)^{ - 1/2}$.

The relativistic form of Newton’s 2 law is $F = {\text{d}}p/{\text{d}}t$, where F is the force acting on the particle, and t is the coordinate time.

Calculate the form of this law as expressed in terms of v and dv/dt.

• 2.10. Lorentz transformation of electric and magnetic fields

It follows from the Lorentz transformations (2.27), (2.28) and (2.30) that the partial derivatives transform as

$$\frac{\partial }{\partial t} = \gamma \left( {\frac{\partial }{{\partial t^{\prime } }} + v\frac{\partial }{{\partial x^{\prime } }}} \right),\quad \frac{\partial }{\partial x} = \gamma \left( {\frac{\partial }{{\partial x^{\prime } }} + \frac{v}{{c^{2} }}\frac{\partial }{{\partial t^{\prime } }}} \right),\quad \frac{\partial }{\partial y} = \frac{\partial }{{\partial y^{\prime } }},\quad \frac{\partial }{\partial z} = \frac{\partial }{{\partial z^{\prime } }}.$$

Deduce the transformation equations for electric and magnetic fields by using the transformation equations for the partial derivatives together with the requirement that Maxwell’s equations shall be Lorentz invariant.