### Document

```Unveiling the Special
Theory of Relativity
Sunil Mukhi
Tata Institute of Fundamental Research,
Mumbai
Pune, 1 October 2005
1
Introduction

In 1905, Albert Einstein
changed our perception of the
world forever.
 He published the paper "On
the Electrodynamics of Moving
Bodies".
 In this, he presented what is
now called the Special Theory
of Relativity.
Pune, 1 October 2005
Ann.Physik 17
(1905), 891-921.
2

What was the background to this work?
 What was the new idea that he proposed?
 How does this influence our thinking today?
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3
The Special Theory of Relativity

The laws of Physics are known to be unchanged
("invariant") under rotations..

A rotation mixes the space coordinates
but does not change the length of any object.

So it is a linear transformation:
that preserves
,
.
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
Special Relativity extends this invariance to
certain transformations of space and time
together.

Collect the space coordinates
time t into a four-component vector:

Here c is the speed of light. According to
Relativity, it is the same in every reference frame.
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as well as
5

Relativity states that all laws of physics are
invariant under those linear transformations:
which leave

unchanged.
This quantity is like a "length" in spacetime,
rather than just space.
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
We will now examine the physical meaning of
this statement, as well as how it came to be
proposed by Einstein.
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Electrodynamics

The crisis that motivated
Einstein's work was related
to the laws of electricity
and magnetism, or
electrodynamics.
 These laws were known,
thanks to Maxwell, and
embodied in his famous
equations.
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
These equations depend on the speed of light, c.

In what frame is this speed to be measured?
 It was thought that light propagates via a medium
called "ether", much as sound waves propagate
via air or water.
 In that case, the speed of light should change
when we move with respect to the ether - just as
for sound in air.

So c would be the speed of light as measured
while one is at rest relative to the ether.
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Michelson-Morley experiment:
the design

Experiments
were performed
to compare the
speed of light
when moving
along or
against the
ether.
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
The original experiment compared the backand-forth travel time of light, parallel and
perpendicular to the supposed ether:
v
L
c
c+v
ether, v
c-v
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
Using traditional mechanics, it follows that the
transit times are:

So there should be an observed discrepancy:
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
However, the experiment did not find this
result! In fact it found no discrepancy in the
transit time.
Michelson-Morley
experiment: the
actual apparatus
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The Fitzgerald-Lorentz Contraction

Before 1905, various attempts (by Voigt,
been made to explain this strange result.
 It turns out that all these authors discovered
some important aspects of the truth.
 In his short 1895 paper "Michelson's
Interference Experiment", Lorentz presented
a point of view directly related to the
experiment.
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
Lorentz noted that the
excess transit time in the
parallel direction could
be compensated if the
apparatus shrinks when
oriented along the ether.
 For this we must assume
that the contracted
length L' is related to the
original one by:
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Hendrik Antoon Lorentz
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
Lorentz and Fitzgerald never denied the existence of
ether. They postulated an independent effect
("contraction") that masked its visible consequences.

However Poincare,
question:
"Does the ether
really exist?"
Henri Poincare'
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
Did Einstein know of these earlier works?
 His 1905 paper has no references!
 And he is once supposed to have said:
“The secret to creativity is knowing
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Einstein's Theory

In 1905, at the age of 26, Einstein unveiled
his own ideas on the issue.
 Like Poincare, he questioned the existence of
ether, and like Lorentz, he ended up
postulating a length contraction.
 But what was really striking was that he laid
down a foundational principle, from which all
the desired results flowed naturally and
elegantly.
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
Einstein started with a simple observation
involving a magnet and a conductor in relative
motion.
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
He noted that in both cases, an identical
electric current is induced on the conductor.
 It is not the case that the moving object
always induces a current on the stationary
one (that would be "reciprocity" rather than
"relativity").
 From this, he argued that only relative motion
is physically meaningful: hence the laws of
physics are the same in all (inertial) frames of
reference.
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
Next he added a startling corollary. The
speed of light, being of fundamental
importance in physics, must be the same in
all reference frames.
 He realised that this was "apparently
irreconcilable" with requiring that the laws of
physics are the same in all frames, but then
showed that it was perfectly consistent.
 And as a consequence, the concept of ether
would turn out to be "superfluous".
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“The laws of physics are the same in all
inertial frames.“
“The speed of light is constant in all frames."
The Postulates of the Special Theory of Relativity
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Clocks, Rigid Bodies,
Electromagnetism

In a rather stern tone for a 26-year-old, Einstein
stressed the need to understand:
"the relationships between rigid bodies (systems of
coordinates), clocks, and electromagnetic processes.
Insufficient consideration of this circumstance lies at
the root of the difficulties which the electrodynamics of
moving bodies at present encounters."

This opened the way for him to question a lot of
common-sense notions.
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
The rest of the paper is derived from the
postulates with masterly confidence and no adhoc assumptions.
 He starts by questioning simultaneity and the
absolute nature of time.
 He stresses the importance of physical
interpretation:
"a mathematical description of this kind has no
physical meaning unless we are quite clear as to
what we understand by `time'."
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
Einstein then proposes a definition of
simultaneity based on synchronizing clocks
using a light ray.
 It follows that two events which are
simultaneous in one frame need not be
simultaneous in another.
 Within this simple framework, he then derives
the Lorentz contraction of a moving rod.
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z
z'
y
v
y'
x
x'

Given two frames, one
moving at constant
velocity with respect to
the other, how do we
transform the
coordinates?
would have been:
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Lorentz transformation

Using his own
postulates, and
nothing else,
Einstein imagines
an experiment with
light rays, and
demonstrates that
Special Relativity
gives a different
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
It is easily checked that this equation, unlike the
.
 In fact, this had to be the case. A light ray from the
origin reaches
at time:

Requiring this equation to hold in both systems
immediately tells us that
is
equal in both frames.
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
It is reassuring to notice that all the formulae of
Relativity reduce to those of traditional mechanics
if we take
.

This is the limit of velocities v that are small
compared to the speed of light c.
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
In the rest of the paper, Einstein worked out most
of the consequences of the Relativity axioms that
we are familiar with today:
–
–
–
–
–
–
Lorentz transformation of Maxwell equations
Doppler shift
Relativistic dynamics of accelerated electrons
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Inertia and Energy

One final consequence of his ideas remained to
be worked out.
 In a subsequent paper in the same year: "Does
the Inertia of a Body Depend on Its Energy
Content", Einstein presented his most famous
equation.
 Combining energy conservation with Relativity, he
showed that if a body emits an energy E in the
form of radiation, its mass decreases by E/c2.
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
This turned out to
be one of the most
far-reaching
conclusions from
Relativity.
"The mass of a body
is a measure of its
energy content"
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Experimental Tests

An excellent source of information on experimental tests of Special
Relativity is the webpage:
http://math.ucr.edu/home/baez/physics/Relativity/SR/experiments.html
Early experiments (pre-1905): Roentgen, Eichenwald, Wilson, Rayleigh, Arago, Fizeau, Hoek, Bradley, Airy.
Round-trip tests of light speed isotropy: Michelson and Morley,
Kennedy and Thorndike, Modern
Laser/Maser Tests,
One-way tests of light speed isotropy: Cialdea, Krisher, Champeny, Turner & Hill.
Tests of light speed from moving sources: Cosmological Sources: DeSitter, Brecher;
Terrestrial
Measurements of the speed of light and other limits on it: NBS Measurements, 1983 Redefinition
of the Meter, Limits on Variations with Frequency, Limits on Photon Mass.
Tests of the principle of relativity and Lorentz invariance: Trouton Noble, Other.
Tests of the isotropy of space: Hughes-Drever, Prestage, Lamoreaux, Chupp, Phillips, Brillet and Hall.
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Tests of time dilation and transverse Doppler effect: Ives and Stilwell; Particle Lifetimes, Doppler
Shift Measurements.
Tests of the twin paradox: Haefle and Keating, Vessot et al, Alley, Bailey et al., The Clock Hypothesis.
Tests of relativistic kinematics: Elastic Scattering, Limiting Velocity c, Relativistic Mass Variations,
Calorimetric Test of SR.
Other experiments: Fizeau, Sagnac, Michelson and Gale, g-2 Tests of SR, The Global Positioning System (GPS),
Lunar Laser Ranging, Cosmic Background Radiation (CMBR), Constancy of Physical Constants, Other.
Experiments which apparently are NOT consistent with SR/GR
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Influence on Modern Physics

Today, fundamental physics is formulated in
the language of Relativistic Quantum Field
Theory.
 This (difficult!) subject combines the
postulates of Special Relativity with those of
Quantum Mechanics.
 The result is the "Standard Model" of particle
physics, that in principle explains every
interaction in nature not involving gravity.
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
The Standard Model has been subjected to
extremely sophisticated precision tests.
 Each of these, among other things, is a test of
Special Relativity!
 In the realm of elementary particle physics,
we have learned to think relativistically.
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What can we learn from Einstein’s style of
research?
 He was motivated by logic, clarity and
physical meaning. And he had no great love
for mathematics.
 But it would be wrong to deduce that he was
strongly experiment-driven. Indeed, he said:

"A theory can be proved by experiment; but no path
leads from experiment to the birth of a theory.”
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
The true lessons to be derived from Einstein’s life
and work are perhaps the following:
–
–
–
–
–
–

Think clearly
Do not be discouraged by others
Work hard
Learn all you can – but use only what you need
And above all, have a goal that you care about.
There are also lessons to be learned from
Einstein’s critics:
– Criticism if right will be forgotten, if wrong then remembered
– Each new idea looks jarring. That neither makes it right nor wrong.
– Progress usually comes from the least expected direction. But for
this reason, we cannot guess where to expect it!
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"On the Electrodynamics of Moving Bodies"
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```