Frames of Reference and the Galilean Transformation – Introduction to the Special Theory of Relativity (Senior 5 Physics) | Ganzaa

Introduction to the Special Theory of Relativity

Lessons

Senior 5 · Physics

Introduction to the Special Theory of Relativity

4 lessons

Lesson 1 of 4

Frames of Reference and the Galilean Transformation

25 min

Are you moving?

Sit in a bus travelling smoothly at a steady 60 km h⁻¹ on the road to Musanze, curtains drawn. Toss a coin straight up: it falls straight back into your hand, exactly as it would at the bus stop. Pour water into a cup: it fills just as at home. No experiment done inside the bus can tell you whether the bus is moving steadily or standing still. Galileo made the same point in 1632, with a ship's cabin full of butterflies and dripping bottles. This simple idea, that steady motion cannot be detected from the inside, is the starting point of the theory of relativity.

Events and frames of reference

Physics describes events: things that happen at a particular place at a particular instant, such as a flash of lightning or a ball hitting the ground. To give an event its coordinates $(x, y, z, t)$ , an observer needs a frame of reference: a set of axes to measure position and a set of clocks to measure time.

An inertial frame is one in which Newton's first law holds: a body with no resultant force on it stays at rest or moves in a straight line at constant speed. Any frame moving at constant velocity relative to an inertial frame is also inertial. A frame that accelerates or rotates (a bus braking, a merry-go-round) is non-inertial: in it, objects appear to be pushed by forces that nothing exerts. The surface of the Earth is very nearly inertial for most experiments.

The Galilean principle of relativity states that the laws of mechanics are the same in every inertial frame. There is no experiment in mechanics that picks out one inertial frame as truly "at rest".

Definition

An event is an occurrence at a definite point in space and instant in time, with coordinates $(x, y, z, t)$ . A frame of reference is a coordinate system with clocks, used by an observer to measure where and when events happen. An inertial frame is a frame in which Newton's first law holds; it moves at constant velocity relative to any other inertial frame. Example: a bus moving steadily in a straight line is an inertial frame; the same bus going round a bend is not.

The Galilean transformation

Let frame S′ move with constant velocity $v$ along the $x$ -axis of frame S, with the origins together at $t = 0$ . An event has coordinates $(x, y, z, t)$ in S and $(x^{'}, y^{'}, z^{'}, t^{'})$ in S′.

At time $t$ the origin of S′ is a distance $v t$ along the $x$ -axis, so $x = x^{'} + v t$ . In Newton's physics, time is the same for everyone. The Galilean transformation equations are therefore

x' = x - vt, \qquad y' = y, \qquad z' = z, \qquad t' = t

Differentiating with respect to time gives the Galilean velocity addition rule and shows that accelerations are the same in both frames:

u' = u - v, \qquad a' = a

Since the acceleration is the same and masses are the same, $F = m a$ holds in both frames: Newton's laws are unchanged by a Galilean transformation, as the principle of relativity requires. Distances and time intervals are also the same in both frames.

Worked example 1: velocities in a bus

A bus moves along a straight road at 15 m s⁻¹. A passenger walks towards the front at 1.5 m s⁻¹ relative to the bus. (a) What is her velocity relative to the road? (b) A car travelling at 25 m s⁻¹ in the opposite direction passes the bus. What is its velocity relative to the bus?

Solution

Take the road as S, the bus as S′, with $v = + 15$ m s⁻¹.

(a) $u = u^{'} + v = 1.5 + 15 = 16.5$ m s⁻¹ forwards.

(b) $u = - 25$ m s⁻¹, so $u^{'} = u - v = - 25 - 15 = - 40$ m s⁻¹: the car approaches the bus at 40 m s⁻¹.

Worked example 2: transforming an event

Frame S′ moves at 20 m s⁻¹ along the $x$ -axis of S, and the origins coincide at $t = 0$ . A firework explodes at $x = 500$ m, $y = 80$ m, $t = 10$ s in S. Find its coordinates in S′.

Solution

$x^{'} = x - v t = 500 - (20) (10) = 300$ m; $y^{'} = 80$ m; $t^{'} = t = 10$ s.

The trouble with light

In 1865 James Clerk Maxwell showed that electric and magnetic fields can travel as a wave whose speed follows from two constants of electricity and magnetism:

c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}} = \frac{1}{\sqrt{(4\pi \times 10^{-7})(8.85 \times 10^{-12})}} = 3.00 \times 10^{8}\,\mathrm{m\,s^{-1}}

This was exactly the measured speed of light: light is an electromagnetic wave. But the speed relative to what? By the Galilean rule $u^{'} = u - v$ , an observer moving towards a light beam at speed $v$ should measure $c + v$ , and one moving away should measure $c - v$ . Physicists assumed that light travels at $c$ relative to an invisible medium filling space, the ether, and that the Earth, orbiting the Sun at about 30 km s⁻¹, must move through it.

In 1887 Albert Michelson and Edward Morley tried to measure the Earth's motion through the ether. Their interferometer split a beam of light into two beams that travelled along perpendicular arms and back, then recombined them to make interference fringes. A beam travelling along the "ether wind" and back should take slightly longer than one travelling across it. Turning the apparatus through 90° swaps the arms, and should shift the fringes. With arms effectively 11 m long, they expected a shift of about 0.4 of a fringe. They found less than a twentieth of that: no ether wind at all. Repeated at different times of day and year, and with ever greater precision since, the result is always the same: the speed of light is the same in every direction, whatever the motion of the Earth.

The Michelson–Morley apparatus of 1887, at what is now Case Western Reserve University in Cleveland, USA. The mirrors stand on a heavy stone slab, which floated on mercury so that it could be turned smoothly without vibration. Photo: Case Western Reserve University, public domain, via Wikimedia Commons

Worked example 3: the expected fringe shift

For a Michelson interferometer with arms of length $L$ , moving through the ether at speed $v$ , the expected fringe shift when it is rotated through 90° is approximately $\Delta N = \frac{2Lv^2}{\lambda c^2}$ . Calculate it for $L = 11$ m, $v = 3.0 \times 10^{4}$ m s⁻¹ and $\lambda = 500$ nm.

Solution

\Delta N = \frac{2(11)(3.0 \times 10^{4})^2}{(500 \times 10^{-9})(3.00 \times 10^{8})^2} = \frac{2(11)(9.0 \times 10^{8})}{(5.00 \times 10^{-7})(9.0 \times 10^{16})} = 0.44

A shift of 0.4 of a fringe was easy to see with their apparatus. Its absence was a real result, not a failure of sensitivity.

Warning

"No ether" is not "no light". The Michelson–Morley result does not mean their light was faulty. It means the Galilean rule $u^{'} = u - v$ fails for light. Either Maxwell's theory or Galileo's transformation had to change. Einstein's answer, in the next lesson, was to keep Maxwell and change our ideas of space and time.

Tip

Galileo's rule in one line: positions shift by $v t$ , velocities shift by $v$ , and time and acceleration do not change at all. Everything in relativity is a correction to this line.

Michelson–Morley Experiment introduction, Khan Academy. Watch how the two perpendicular beams were expected to take different times if the Earth moved through an ether.

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Practice

Define an inertial frame of reference and give one example of a frame that is not inertial.
State the Galilean principle of relativity.
Frame S′ moves at 12 m s⁻¹ along the $x$ -axis of S, with origins together at $t = 0$ . An event occurs at $x^{'} = 40$ m, $t^{'} = 5.0$ s in S′. Find $x$ and $t$ in S.
A moto travels east at 12 m s⁻¹. Rain falls vertically at 8.0 m s⁻¹ relative to the ground. Find the speed of the rain relative to the rider and the angle it makes with the vertical.
Show that if $x^{'} = x - v t$ and $t^{'} = t$ , the acceleration measured in S′ equals that measured in S.
Using the Galilean rule, what speed of light would an observer moving at $1.0 \times 10^{5}$ m s⁻¹ towards a lamp expect to measure? What do experiments actually find?
Explain why the Michelson–Morley experiment is called the most famous "null result" in physics, and what it implied.

Show answersHide answers

A frame in which Newton's first law holds, i.e. one moving at constant velocity (not accelerating or rotating). Not inertial: a car braking or turning, a merry-go-round.
The laws of mechanics are the same in all inertial frames; no mechanical experiment can detect uniform motion.
$x = x^{'} + v t = 40 + (12) (5.0) = 100$ m; $t = 5.0$ s.
Relative to the rider, the rain has a horizontal velocity of 12 m s⁻¹ westward and 8.0 m s⁻¹ downward: speed $\sqrt{12^2 + 8.0^2} = 14.4$ m s⁻¹, at $\tan^{-1}\frac{12}{8.0} = 56^\circ$ to the vertical, slanting towards the rider's face.
$u' = \frac{dx'}{dt'} = \frac{dx}{dt} - v = u - v$ ; since $v$ is constant, $a' = \frac{du'}{dt} = \frac{du}{dt} = a$ .
$c + v = 3.0001 \times 10^{8}$ m s⁻¹ (i.e. $3.00 \times 10^{8} + 1.0 \times 10^{5}$ ). Experiments find exactly $c = 3.00 \times 10^{8}$ m s⁻¹, regardless of the observer's motion.
It was designed to detect a definite effect (a fringe shift of about 0.4) and found none. It showed that there is no detectable ether wind and that the speed of light does not depend on the motion of the observer, contradicting the Galilean addition of velocities.

Key points

An event has coordinates $(x, y, z, t)$ in a frame of reference; inertial frames move at constant velocity.
Galilean transformation: $x^{'} = x - v t$ , $y^{'} = y$ , $z^{'} = z$ , $t^{'} = t$ ; velocities $u^{'} = u - v$ ; accelerations unchanged.
Maxwell's theory gives a fixed light speed $c = \frac{1}{\sqrt{\mu_0\varepsilon_0}}$ , which clashes with Galilean velocity addition.
Michelson–Morley (1887): no ether wind; light has the same speed in every direction.

Sources and credits

Photo: Michelson morley experiment 1887, Case Western Reserve University, Wikimedia Commons, public domain.
Video: Michelson–Morley Experiment introduction, Khan Academy, embedded from YouTube.

Written by Ganzaa. The two diagrams, the worked examples and the practice exercises are by Ganzaa.

1Are you moving?

2Events and frames of reference

3The Galilean transformation

4The trouble with light

Are you moving?

Events and frames of reference

The Galilean transformation

The trouble with light