Unveiling the Special Theory of Relativity

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Unveiling the Special Theory of Relativity

• Sunil Mukhi
• Tata Institute of Fundamental Research, Mumbai

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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.

Ann.Physik 17 (1905), 891-921.
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• What was the background to this work?
• What was the new idea that he proposed?
• How was this experimentally confirmed?
• How does this influence our thinking today?

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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 as
  well as 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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• 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
  back-and-forth travel time of light, parallel and
  perpendicular to the supposed ether

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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,
  Fitzgerald, Larmor, Lorentz, Poincare) had 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

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, in 1900, asked the 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 how to
hide your sources.
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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 ad-hoc
  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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• Given two frames, one moving at constant velocity
  with respect to the other, how do we transform
  the coordinates?
• The traditional answer 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 answer

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• It is easily checked that this equation, unlike
  the traditional one, preserves
  .

• 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
• Time dilation and "twin paradox"
• Addition law for velocities
• Lorentz transformation of Maxwell equations
• Doppler shift
• Radiation pressure on perfect mirrors
• 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/S
  R/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 Sources Alvaeger, Sadeh,
• 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 Einsteins 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 Einsteins
  life and work are perhaps the following
• Think clearly
• Follow your intuition
• 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
  Einsteins 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"