The Theory of Special Relativity
Learning Objectives

Electromagnetism and Electromagnetic Waves.

Nature of Light.

Do Electromagnetic Waves propagate in a Medium (Luminiferous Ether)?

Galilean transformation in Newtonian physics.

The Michelson-Morley experiment.

Invariance of the speed of light under the Lorentz transformation.

Einstein’s interpretation of the Lorentz transformation.
Learning Objectives

Electromagnetism and Electromagnetic Waves.

Nature of Light.

Do Electromagnetic Waves propagate in a Medium (Luminiferous Ether)?

Galilean transformation in Newtonian physics.

The Michelson-Morley experiment.

Invariance of the speed of light under the Lorentz transformation.

Einstein’s interpretation of the Lorentz transformation.
Electromagnetism

James Clerk Maxwell united the phenomena of electricity
and magnetism under the theory of electromagnetism, which
is expressed through Maxwell’s equations as published in
1861/1862.
Gauss’s Law
for Magnetism
James Clerk Maxwell,
1831-1879
Single integral with
circle: closed loop
Double integral with
circle: closed surface
Electromagnetism

James Clerk Maxwell united the phenomena of electricity
and magnetism under the theory of electromagnetism, which
is expressed through Maxwell’s equations as published in
1861/1862.
The net outward normal electric flux through any
closed surface is proportional to the total electric
charge enclosed within that closed surface.
James Clerk Maxwell,
1831-1879
Magnetic field has a divergence equal to 0.
Or, equivalently, no monopoles.
The magnetic field in space
around an electric current is
proportional to the electric
current which serves as its source.
The induced electromotive force
in any closed circuit is equal to
the negative of the time rate of
change of the magnetic flux
through the circuit.
Single integral with
circle: closed loop
Double integral with
circle: closed surface
Electromagnetic Waves

In this course, you do not need to know how to derive or apply Maxwell’s
equations.

You only need to know one thing: an accelerating charge produces an
electromagnetic wave.
Electromagnetic Waves

In this course, you do not need to know how to derive or apply Maxwell’s
equations.

You only need to know one thing: an accelerating charge produces an
electromagnetic wave.
Speed of Electromagnetic Waves

Suppose you have two charges (e.g., two electrons) separated by a distance.

If you move one charge, will the other charge instantaneously feel a change in
force from the first charge?
ee-
Speed of Electromagnetic Waves

Maxwell’s equations predict the speed of electromagnetic waves to be*
(Although I use the symbol c, which stands for the speed of light, it is important to
understand that when Maxwell’s published his equations it had not yet been
proven that light comprises electromagnetic waves!)
* μ0 is the permeability and ε0 the permittivity of free space (a theoretically perfect vacuum).
Permeability/Permittivity is a measure of the amount of resistance encountered when
forming an magnetic/electric field in a medium. Vacuum permeability and permittivity are
both fundamental constants of nature.
Learning Objectives

Electromagnetism and Electromagnetic Waves.

Nature of Light.

Do Electromagnetic Waves propagate in a Medium (Luminiferous Ether)?

Galilean transformation in Newtonian physics.

The Michelson-Morley experiment.

Invariance of the speed of light under the Lorentz transformation.

Einstein’s interpretation of the Lorentz transformation.
Nature of Light

By mid-1800’s, there had been numerous experiments of increasing accuracy to
measure the speed of light. For example, in the 1850’s, the French physicist Léon
Foucault measured a value for the speed of light that is only 5% higher than the
modern value.

Because the predicted speed of electromagnetic waves appeared to match the
measured speed for light, Maxwell postulated (but did not prove) that light
comprises electromagnetic waves.
Nature of Light

Foucault’s apparatus involves light reflecting off a rotating mirror towards a
stationary mirror. As the rotating mirror will have moved slightly in the time it
takes for the light to bounce off the stationary mirror (and return to the rotating
mirror), it will thus be deflected away from the original source, by a small angle.
Nature of Light

In a series of experiments beginning in 1865, the German physicist Heinrich Hertz
showed that electromagnetic waves have the same properties (e.g., can be
transmitted and received, undergoes reflection and refraction, has the same speed)
as light, and therefore that light comprises electromagnetic waves.
Learning Objectives

Electromagnetism and Electromagnetic Waves.

Nature of Light.

Do Electromagnetic Waves propagate in a Medium (Luminiferous Ether)?

Galilean transformation in Newtonian physics.

The Michelson-Morley experiment.

Invariance of the speed of light under the Lorentz transformation.

Einstein’s interpretation of the Lorentz transformation.
Luminiferous Ether

Waves that we are familiar with propagate in a medium:
- sound waves propagate through air (travelling pressure fluctuations in air)
- water waves propagate through water (travelling pressure fluctuations in
water)

What kind of medium does electromagnetic waves (light) propagate in?
Luminiferous Ether

In the 1800s, it was thought that light (electromagnetic waves) propagate in
luminiferous ether. (Luminiferous = light bearing. Ether = substance which
permeates space but does not provide any mechanical resistance.)

Even Maxwell himself thought so, writing that:

If this is the case, then the speed
of light should depend on our
motion relative to the ether.
Which situation then, in this
example, would we measure the
speed of light to be higher?
Luminiferous Ether

In the 1800s, it was thought that light (electromagnetic waves) propagate in
luminiferous ether. (Luminiferous = light bearing. Ether = substance which
permeates space but does not provide any mechanical resistance.)

Even Maxwell himself thought so, writing that:

If this is the case, then the speed
of light should depend on our
motion relative to the ether.

This “commonsense” treatment
of space and time (i.e., vectorial
addition of velocities) in
Newtonian physics is based
mathematically on the Galilean
transformations.
Learning Objectives

Electromagnetism and Electromagnetic Waves.

Nature of Light.

Do Electromagnetic Waves propagate in a Medium (Luminiferous Ether)?

Galilean transformation in Newtonian physics.

The Michelson-Morley experiment.

Invariance of the speed of light under the Lorentz transformation.

Einstein’s interpretation of the Lorentz transformation.
The Galilean Transformations

The Galilean transformations treats space and time separately and as absolutes.

Consider a laboratory where there are meter sticks and synchronized clocks
everywhere so that the position and time of any event that occurs in this
laboratory can be measured at the location of that event; i.e., the purely
geometrical effect of time delay inherent for light to propagate from a given
location to an observer can be safely ignored.

Consider two inertial reference frames,
S and S´, constituting two such
laboratories where the frame S´ is
moving in the positive x-direction with
constant velocity u. The clocks in the
two reference frames are started when
the origins of the coordinate systems,
O and O´, coincide at time t = t´ = 0.
The Galilean Transformations

Consider, for simplicity, a stationary object (apple) located at position (x, y, z) in
the S frame. After a time t has elapsed, this same object will be located at a
position as measured in the S´ frame of
at a time in the S/ frame of
the same transformations apply
object.
Obviously,
for a moving
The Galilean Transformations

How does the velocity v of an object as measured in the S frame relate to the
velocity v´ of the same object in the S´ frame?

We simply take the time derivatives of Eqs. 4.1-4.3 to get
or in vector form
The Galilean Transformations

How does the acceleration a of an object as measured in the S frame relate to the
acceleration a´ of the same object in the S´ frame?

We simply take the time derivatives of Eq. 4.5 (keeping in mind that u is constant)
to get
Thus, the same acceleration is measured in both
reference frames.

In summary, the Galilean transformations are
simply the “commonsense” treatment of space
and time in Newtonian physics.
Learning Objectives

Electromagnetism and Electromagnetic Waves.

Nature of Light.

Do Electromagnetic Waves propagate in a Medium (Luminiferous Ether)?

Galilean transformation in Newtonian physics.

The Michelson-Morley experiment.

Invariance of the speed of light under the Lorentz transformation.

Einstein’s interpretation of the Lorentz transformation.
Does Light Propagate in a Luminifereous Ether?

In 1887, Albert A. Michelson and Edward W. Morley began an
experiment (which they gradually improved over time) to
determine whether light propagates in a luminifereous ether.
Luminifereous ether
Albert A. Michelson,
1852-1931
Motion of Earth
through ether
Edward W. Morley,
1838-1923
Does Light Propagate in a Luminifereous Ether?

To understand their experiment, consider the following:
Suppose we arrange a competition between two swimmers in a river that is
100-m wide and flowing at 3 m/s. Both swimmers are equally strong and swim
at a constant speed of 5 m/s. Swimmer 1 is asked to swim 100 m upstream and
back. Swimmer 2 is asked to swim directly across the river and back to the
same point. Who wins?
Swimmer 2
100 m
Swimmer 1
100 m
Does Light Propagate in a Luminifereous Ether?

Swimmer 1 swims upstream at a net speed of 5-3=2 m/s, taking 50 s to swim
100 m. Swimmer 1 then swims downstream at a net speed of 5+3 = 8 m/s, taking
12.5 s to swim another 100 m. Total time = 50 + 12.5 = 62.5 s.

In 1 s, swimmer 2 swims 5 m at an angle θ to the bank and is swept 3 m
downstream, and therefore experiences a net speed of 4 m/s directly across.
Swimmer 2 therefore takes 25 s to swim 100 m directly across to the other band,
and another 25 s to swim 100 m directly back. Total time = 25 + 25 = 50 s.

If we stage the competition at an even
wider river, would their time difference
be smaller or larger?
θ
The Michelson-Morley Experiment

Light from a source partially passes through a half-silvered mirror, is reflected
back, and reflected by the half-silvered mirror to observer.

Light from the same source is partially reflected by the half-silvered mirror, is
reflected back, and passes through the half-silvered mirror to observer.
The original Michelson-Morley experimental setup
Half-silvered mirror
Light source
Direction of Earth’s motion through ether
The Michelson-Morley Experiment

Light from a source partially passes through a half-silvered mirror, is reflected
back, and reflected by the half-silvered mirror to observer.

Light from the same source is partially reflected by the half-silvered mirror, is
reflected back, and passes through the half-silvered mirror to observer.

Imagine that you align the blue arm to be in
the direction through which Earth is moving
through the ether. For light to take equal
times to travel along the two arms and hence
interfere constructively to produce
maximum light intensity at the screen, you
would need to make the green arm longer
than the blue arm. (Remember: cross-stream
swimmer wins!)
Half-silvered mirror
Light source
Direction of Earth’s motion through ether
The Michelson-Morley Experiment
Light from a source partially passes through a half-silvered mirror, is reflected
back, and reflected by the half-silvered mirror to observer.

Light from the same source is partially reflected by the half-silvered mirror, is
reflected back, and passes through the half-silvered mirror to observer.

Now rotate the entire experimental setup by 90.
The green arm is now aligned with the direction
through which Earth is moving through the ether.
Light takes longer to travel along the green (longer)
arm than the blue (shorter) arm, and so the two
light rays do not (in general) arrive at the screen in
phase and do not interfere constructively.

In practice, do not know which direction Earth is
moving through ether, so have to repeat experiment
at various orientations. Also, on a given day, Earth
may happen to moving at the same speed and
direction as the ether, so have to repeat the
experiment over different months.
Light source

Half-silvered mirror
Direction of Earth’s motion through ether
The Michelson-Morley Experiment

Michelson and Morley used multiple mirrors to increase the arm lengths so as to
attain a measurable time delay in light travel time (if indeed light propagates in
luminiferous ether) between the two arms. (Recall that if we stage the
competition between the two
swimmers in a wider river,
their time difference would be
correspondingly larger.)
The original Michelson-Morley
experimental setup
The Michelson-Morley Experiment

In practice, see fringes at the screen because the two light rays are not exactly
parallel and the wavefront not exactly planar. This is an advantage as it is easier
to see changes in the fringe pattern rather than changes in brightness at the screen.

On rotating the equipment, expect the fringe pattern to change (i.e., fringes move
radially in our out).

Despite repeating the experiment in different
orientations and at different times, Michelson
and Morley could not detect any changes, thus
demonstrating that light does not propagate in
luminiferous ether (and obey the Galilean
Light source
transformations)!
a mirror
Half-silvered
Learning Objectives

Electromagnetism and Electromagnetic Waves.

Nature of Light.

Do Electromagnetic Waves propagate in a Medium (Luminiferous Ether)?

Galilean transformation in Newtonian physics.

The Michelson-Morley experiment.

Invariance of the speed of light under the Lorentz transformation.

Einstein’s interpretation of the Lorentz transformation.
How to Interpret the Michelson-Morley Experiment?

The Dutch physicist Hendrik Antoon Lorentz, along with George FitzGerald,
Joseph Larmor, and Woldemar Voigt, wished to preserve the idea that light
propagates in a luminiferous ether. In that case, the Galilean transformation had
to be abandoned, otherwise light would have different speeds depending on our
motion relation to the ether.

They proposed a mathematical transformation – which Poincaré in 1905 named
the Lorentz transformations – under which Maxwell’s equations were invariant
when transformed from the ether to a moving reference frame. Thus, under the
Lorentz transformation, electromagnetic waves – and therefore light – have the
same speed in all reference frames even though light propagates in a luminiferous
ether.

Lorentz and his collaborators understood that this transformation predicted the
effects of length contraction and time dilation, and indeed used these effects to
explain the results of the Michelson-Morley experiment under the notion that light
propagates in a luminiferous ether.
How to Interpret the Michelson-Morley Experiment?

Lorentz proposed that all matter comprises electrical charges in the empty space
of ether, held together by electric and magnetic forces. When matter moves with
respect to the ether, moving electrical charges create magnetic fields (according to
Maxwell’s equations) such that the electrical charges settle into a new
configuration with the body suffering contraction along the direction of motion.

In other words, Lorentz believed that space
is an absolute, and that objects contract
along the direction of motion.
Half-silvered mirror
Light source
Matter at rest in ether
Matter moving to right
with respect to ether
Direction of Earth’s motion through ether
Learning Objectives

Electromagnetism and Electromagnetic Waves.

Nature of Light.

Do Electromagnetic Waves propagate in a Medium (Luminiferous Ether)?

Galilean transformation in Newtonian physics.

The Michelson-Morley experiment.

Invariance of the speed of light under the Lorentz transformation.

Einstein’s interpretation of the Lorentz transformation.
How to Interpret the Michelson-Morley Experiment?

Instead of matter contracting in such a way so that we measure the same speed for
light as we move through the ether, Albert Einstein discarded the notion that light
propagates in a luminifereous ether and postulated that light travels at the same
speed in all reference frames.

In other words, Einstein discarded the idea that space is an absolute and that
objects contract along the direction of motion. Instead, space is not an absolute,
and has dimensions that depend on our relative motions.

Einstein assumed that all other postulates in Newtonian physics remained valid.
Thus, his postulate that light travels at the same speed in all reference frames was
the only difference compared with, but marked a crucial break from, Newtonian
physics.

Based on this single postulate (preserving all other postulates in Newtonian
physics), Einstein derived the transformation for space (length) and time as
measured from different reference frames. As we shall see, not surprisingly, the
transformations Einstein derived were identical to the Lorentz transformations,
but with an entirely different interpretation.