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JF Theoretical Physics PY1T10Special Relativity

12 Lectures (plus problem classes)Prof. James Lunney

Room: SMIAM 1.23, jlunney@tcd.ie

BooksSpecial Relativity FrenchUniversity Physics Young and Freedman - Chapter 37Berkeley Physics Course, Vol 1, Mechanics

A biography of Einstein provided by the American Institute of Physicshttp://www.aip.org/history/einstein/

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In 1905 Albert Einstein published 3 remarkable papers:

•Special theory of relativity (STR)•Photoelectric effect•Brownian motion

STR implies that measurements in different inertial frames of reference having relative velocities give different values of length and time.

Very strange, non-intuitive, contrary to normal experience where v<<cExpect to relativistic effects when v~c

In the limit of low v (v<<c) STR → Newtonian mechanics

Introduction

Albert Einstein , Zurich (1879-1955)

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Course outline•Inertial frames and Galilean transformations

•Michelson-Morley experiment

•Einstein’s postulates, relativity of simultaneity

•Lorentz transformation

•Length contraction and time dilation

•Relativistic Doppler effect, moving clocks

•Transformation of velocities

•Relativistic dynamics: energy, mass, momentum

•Pair creation, fission, fusion

•Collisions, Compton effect

•Energy-momentum invariant

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Classical Mechanics

Newton I: a = 0 (v =const) when F = 0Newton II: F = ma – valid for non-accelerated (inertial) reference frameInertial ref. frame? - frame defined by Newton IFrame fixed to surface of earth not inertial – gravity, acceleration due to rotation of earth (small)Frame fixed to star is inertial frame to good very good approx.

Are the laws of physics the same in different inertial frames (moving with relative v)?

Hypothesis of Galilean invariance: Basic laws of physics are the same in all inertial frames (central idea in STR)Formal way of saying: If I am in a train with no windows I cannot say if I am moving or not. No trouble keeping my balance , coin falls straight down…….

Galilean invariance accepted by 19 th century physicists, but with light we should be able to detect uniform motion.

Galilean Transformation: Transforms coordinates of an event between inertial frames moving at a speed relative to each other.

eg. A boat firing a canon from its deck.

2 inertial frames: Boat, and Water.

Boat moves at 15 m/s, canon fired at 100 m/s relative to the boat. How fast relative to the water?

Galilean transformation: Speed = 100 + 15 = 115 m/s relative to the water.

15 m/s

115 m/s

Galilean Transformation - 1

Inertial frame S’ moves with velocity v relative to inertial frame S, along x-axis. Origins coincide at t = 0 and t’ = 0.

𝑥 = 𝑥′ + 𝑣𝑡 𝑦 = 𝑦′ 𝑡 = 𝑡′ 𝑧 = 𝑧′

𝑑𝑥 = 𝑑𝑥′ + 𝑣𝑑𝑡′ 𝑑𝑦 = 𝑑𝑦′ 𝑑𝑡 = 𝑑𝑡′

𝑑𝑥

𝑑𝑡=

𝑑𝑥′

𝑑𝑡+ 𝑣

𝑑𝑡′

𝑑𝑡𝑢𝑥 = 𝑢𝑥′ + 𝑣, 𝑢𝑦 = 𝑢𝑦′, 𝑢𝑧 = 𝑢𝑧′

𝑢 = 𝑢′ + 𝑣

vt

y S

zx, x’

event 𝑥, 𝑦, 𝑧, 𝑡(𝑥′, 𝑦′, 𝑧′, 𝑡′)

o

y’ S’

z’

v

o’

Galilean Transformation - 2

𝑢 = 𝑢′ + 𝑣

𝑎 =𝑑𝑢

𝑑𝑡=

𝑑𝑢′

𝑑𝑡= 𝑎′

𝐹 = 𝑚 𝑎, 𝐹′ = 𝑚′ 𝑎′

But 𝑚 = 𝑚′, 𝑎 = 𝑎′

𝐹 = 𝐹′

Observers in both frames agree on magnitude and direction of Force.

Galilean Transformation - 3

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Theories of Light

Early theories (Newton): stream of particles

1667 Robert Hooke: vibration – wave theory

By beginning of 19 cent. there was strong evidence for wave theory • Polarisation• Interference• Diffraction

Maxwell (1861): light is an electromagnetic wave

Sound can be transmitted in solid, liquid or gas, but not vacuum

Expect to need a medium to carry light wave – ether (aether)

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The ether

Expect light to propagate at constant velocity wrt ether

The velocity of light measured by an observer will depend on his motion wrt ether

If we measure velocity of light in different directions we should be able to detect our motion through the ether:

•Same in both directions – stationary•Not same in both direction – moving

Michelson and Morley tried to measure this effect

Earth moves around Sun at 3×104 m s-1

Velocity of light, c = 3×108 m s-1 - need to measure very small effect

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Michelson–Morley experiment - 1

Albert Michelson,Ohio (1852-1931)

Edward Morley,Ohio (1838-1923)

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PSource

Telescope

Mirror M1

Mirror M2

Compensating plate C

Half-silvered plate P

‘Ether wind’

Aim: to measure the velocity of our motion through some absolute space as defined by ether (or check to see if c is the same in all directions)

•Used multiple reflections to extend optical path length

•For rotation the whole interferometer floated in bath of mercury

Michelson–Morley experiment - 2

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Michelson–Morley experiment - 3

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See interference fringes in telescopeFringes are parallel lines if M1 not exactly perpendicular to M2

Suppose equipment moves at speed v in direction PM1 wrt frame defined by ether, ie. ether wind is blowing in direction M1P

Rotate interferometer by 900. Then PM2 points into ether wind

Expect to see fringe shift : Why?

What value δ? v≈3×104 m s-1, c = 3×108 m s-1, therefore v/c = 10-4 , λ = 6×10-7m1881 Michelson used l =1.2 m giving δ = 0.04, observed δ = 0.02 (max)1887 Michelson used l =11 m giving δ = 0.4, observed δ = 0.01 (max)Conclusion

• No fringe shift when apparatus is rotated• We cannot detect any motion relative to ether (absolute space)

2

22

lv

c

Michelson–Morley experiment - 4

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So what about the ether?

No ether – 1890’s not ready to accept this conclusion OREther exists, motion through it is real , but compensating effects at work

In M-M expt; along ether wind; , perp. to ether wind:

Lorentz-Fitzgerald contraction: If a body contracts along its direction of motionthrough the ether by factor (1-v2/c2)1/2 then t1 = t2 and there is no fringe shift.

Not just an ad hoc idea.

“We know that electric forces are affected by the motion of electrified bodies relative to the ether and it seems a not improbable supposition that the molecular forces are affected by the motion and that the size of the body alters consequently” G F Fitzgerald, 1889

1 2

2

2

1

lct

vc

2 12 2

2

2

1

lct

vc

Lorentz-Fitzgerald contraction

George Francis Fitzgerald: Professor and Fellow of Trinity College Dublin, 1877 – 1901

Nephew of George Johnston Stoney, who coined the term “electron”. Fitzgerald proposed using it to describe particles found by JJ Thomson.

Was one of the first to suggest one cannot surpass the speed of light. Proposed “Fitzgerald Contractions”, which state that bodies contract along direction of motion the faster they travel. ("The Ether and the Earth's Atmosphere" 1889)

The contraction hypothesis was correct, but for the wrong reason.

Contraction now called “Lorentz-Fitzgerald Contraction” after Fitzgerald and Hendrik Lorentz, who independently derived them.

George Francis Fitzgerald, Dublin (1851-1901)

Hendrik Lorentz, Leiden (1853-1928)

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Measurement of length and time

Einstein: Analysis of motions based on an abstraction - existence of universal absolute time. We should not rely on a metaphysical notions about time.Rather, we make observations with physical devices – clocks ( pendulum clock, watch with vibrating quartz crystal , rotating earth, vibrating molecule…

To measure velocity of a body:

𝑣 =𝑟2−𝑟1

𝑡2−𝑡1𝑟1, 𝑡1 (𝑟2, 𝑡2)

Clock at 𝑟1 reads 𝑡1 when body arrives at 𝑟1Another clock at 𝑟2 reads t2 when body arrives there.

Need to define what we mean by same time at two locations

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Einstein’s Postulates

Postulate 1: All inertial frames are equivalent wrt the laws of physics

Postulate 2: The speed of light in empty space always has the same value, ie. the speed of light is independent of the motion of the source or receiver

P2 explains the null result of the M-M expt. M-M does not prove P2. Rather we can use P1 and P2 to make predictions which can be tested by experiment.

Use Galilean transformation (GT) to describe Newtonian mechanics in different inertial frames.But P2 is not consistent with GT – need to find new set of transformations.

Later we will see that Newtonian mechanics does not work for very high velocity.

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Synchronising clocks

If we could transmit signals at infinite speed, no problem. Use large, but finite speed of light.

Observation stations A and B at rest in same frame of reference.Clock at A can record time differences between events that occur in immediate vicinity of A. Similarly for B.We have: A time and B time

To establish common time: By definition time for light to travel A→B equals time for B→ASend out light signal from A at t =0Reflect light at B and return to A at time t = t0

Time when light arrives at B is t = t0/2

We have synchronised the clocks at A and B