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Special Theory of Relativity

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IntroductionIntroduction

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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Einstein's TheoryEinstein'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.

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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

Galilean Transformation

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Given two frames, one moving at constant velocity with respect to the other, how do we transform the coordinates?

zyx x'

y'z' v

Galilean Transformation

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Given two frames, one moving at constant velocity with respect to the other, how do we transform the coordinates?

zyx x'

y'z' v

Transformation Equations For velocity

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The velocity is the time derivative of the position

uu

uu

uu

z'

''

z

y'

''

y

x'

''

x

dt

dz

dt

dz

dt

dy

dt

dy

vvdt

dx

dt

dx

Transformation for acceleration and force

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To obtain the acceleration transformation equations, we differentiate the velocity equations with respect to time

Since the mass is not affected by the motion of the reference frame: invariant in all frames:

aa

aa

aa

'

''

'

''

'

'' )(

zzz

z

y

yy

y

xxxx

x

dt

du

dt

du

dt

du

dt

du

dt

du

dt

vud

dt

du

We have: aa '

'' a mFa mF

Newton’s laws of motion are exactly the same in all inertial frames

Consequence of Galilean Transformation

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Invariance of the length

ionransformatGalilean tby invariant islength so

xxxx

vtx xandvt x xsince

frame) (moving frame S in the

object an of extremes theof scoordinate theare xand x

frame) (unmoving frame S in the

object an of extremes theof scoordinate theare xand x

121'

2'

2'

11'

'

2'

1'

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Consequence of Galilean Transformation

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Consequence of Galilean Transformation

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Using his own postulates, and nothing else, Einstein imagines an experiment with light rays, and demonstrates that Special Relativity gives a different answer:

Lorentz transformation

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Velocity addition theorem

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Consequences of Lorentz transformation

Time dilationTime dilation

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Relativity of the massRelativity of the mass

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Mass -energy equivalence

"The mass of a body is a measure of its energy content"

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Energy momentum relation

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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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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– 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 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!