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PH604 Special Relativity (8 lectures)

Books: “Special Relativity, a first encounter”, Domenico Giulini, Oxford “Introduction to the Relativity Principle”, G.Barton, Wiley + many others in Section QC.6

• Newtonian Mechanics and the Aether• Einstein’s special relativity and Lorentz transformation

and its consequences• Causality and the interval• Relativistic Mechanics• Optics and apparent effects

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PH604 Special Relativity

Books: “Special Relativity, a first encounter”, Domenico Giulini, Oxford “Introduction to the Relativity Principle”, G.Barton, Wiley + many others.

Notes: http://galileo.phys.virginia.edu/classes/252/lecture1.htm

Lecture: Susskind http://www.youtube.com/watch?v=toGH5BdgRZ4

• 7 lectures + class test• Weeks 1-3 tuesdays x 2• Week 4: Class test monday (2/3 50%)

• Newtonian Mechanics and the Aether• Einstein’s special relativity and Lorentz transformation• Consequences• Causality and the interval• Relativistic Mechanics• Optics and apparent effects

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Newtonian Mechanics and the Aether

1. Newtonian Mechanics and Newton’s law of Inertia

2. The relativity principle of Galileo and Newtonian

3. Questions with regard to Newtonian Mechanics

4. The “Aether” – does it exist?

5. Michelson – Morley Experiment

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1.Newtonian Mechanics and Newton’s law of Inertia--Newton’s Law: m a = F: Predict the motions of the planets, moons, comets, cannon balls, etc

--This law is actually not always correct! (surprised?)

http://www.phys.vt.edu/~takeuchi/relativity/notes/section02.html

--Inertial Frame: A frame in which the Newton’s law is correct.

--Any frame that is moving at a constant relative velocity to the first inertial frame is also an inertial frame.

--The frames in which Newton’s law does NOT hold that are accelerating with respect to inertial frames and are called non-inertial frames.

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zz

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ZZ’’

YY’’

XX’’OO’’

SSSS’’

uuTwo inertial reference frames S and S’ moving with a constant velocity u relative to each other

S: as (x,y,z,t) and in

Common sense shows the two measurements are related by:

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Or in vector form:

v ’ = v - u v ’ = v - u

a’ = aa’ = a

( u is // to x and x’)

A moving object is described in

S’: as S’: as (x’, y’, z’, t’)(x’, y’, z’, t’)

--This is the Galilean transformation. Note the universal time, t=t’

2. The relativity principle of Galileo and Newtonian

--They would assert that Mechanics only deals with relative motion and that ‘absolute’ motion can never be measured.

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zz

…….but spatial coordinates must be transformed.but spatial coordinates must be transformed

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2a. Universal Time

The difference between the -coordinate in the moving frame and the x -coordinate in the stationary frame is exactly the distance travelled by the frame in time t.

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A fundamental example of an invariant quantity, in all forms of relativity, is an event in space-time.

Suppose two events, E1,E2, have the same space-time coordinates in a particular inertial frame of reference,.. then they will have the same space-time coordinates in every inertial frame of reference.

2b. Invariance & Simultaneity

In Galilean relativity, spatial separation (length) is invariant.

In Galilean relativity, time intervals are invariant.

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2b. Invariance & Simultaneity

……. and so we can consider simultaneous events, which are events that occur at the same time, but not necessarily at the same location. Then, if two events are simultaneous in one frame, they will be simultaneous in any other frame.

This, along with time-invariance and spatial separation itself, gets dropped in special relativity.

Consider a particle trajectory x(t).

Neither position nor velocity are invariant……..but acceleration is! Why? zz

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Distance Force Mass, all invariants m a = F

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3. The finite speed of light

The speed of light was measured from astronomical phenomena

Io around jupiter, Roemer 1672, HuygensStellar aberration: Annual: maximum angle of approximately 20 arcseconds

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4. Questions with regard to Newtonian Mechanicsi) phenomena on a very small scale we need Quantum Mechanics;

ii) Phenomena where the speed of motion is near the speed of light “c”

we need relativity

Modern experiment that shows the limitation of Newtonian mechanics:

We shall be concerned with case ii) in this course.

T Kinetic Energy of electrons (between 0.5-15 MeV)

A relation between V2 vs K.E of the electrons can be plotted.

Van de Graaf accelerator Experiment: [American Journal of Physics, Volume 32, Issue 7, pp. 551-555 (1964). ]

Accelerator Pulsed electrons beam

Measure the rise in temperature

Target B

D

The V. of electrons can be determined by: V = D / time

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Newtonian Mechanics: K.E. = ½ mv2

Newtonian Mechanics

v2

K.E.

Experiment

--N-M prediction is valid at low energy (velocities).

--Experiment: Vmax 3108(m s-1)C

C2

O

--The Vmax of the electrons appear to equal the speed of light in Vacuum.

--Other ‘massless particles’ such as neutrinos appear only to move at C as well

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5. Speed of Light: existence of Aether ?Maxwell’s electromagnetic theory predicted that light should travel with a constant speed in vacuum, irrespective of reference frames:

00

1

C

How light propagates through a vacuum ?

--All other wave motions known, needed some form of ‘medium’

-- Wave velocity would be relative to the ‘medium’

Suggestion:Perharps, even a vacuum contains a very tenuous ‘medium’ --- the ‘Aether’, then the constant velocity of light is relative to this absolute frame, and thespeed of light in other ‘inertial’ systems would not be C.

if so, can we detect it?

Direct measurement of the relative motion to aether is difficult, but If it existed in space, we should be able to measure the motion of the Earth relative to aether -- Michelson-Morley (1887).

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Michelson-Morley Experiment –Detect the Earth moving through the Aether??

--In 1887 Michelson and Morley built an interferometer

To measure the movement of the Earth through the Aether.

beam splitter

Light source

Mirror 1

Mirror 2

Detector

Even though this instrument can be a few meters in size, it can detect changes in distance of hundreds of nanometers

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Interferometer, stationary in the Aether

Interferometer Moving Through the Aether

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v

Aether wind speedl2

l1

The time for light to travel along l1 arm and back: (downstream)

)1(

2

2

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cvc

lvc

lvc

lt

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222 )( lvct

The time for light to travel along l2 arm (cross stream)

V t

C t

travel along l2

arm and back:

If the light has frequency of f, the number of fringes that corresponds with differences, t1-t2 of the light travel in the two arms is:

2

22

2

21

21

1

2

1

2)(

cvc

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Since the test was to see if any fringes moved as the whole apparatus was turned through 90o.Then the roles of l1 and l2 would be exchanged, and the new number of fringes would be

So the observed number of fringe shift on rotation through 90o should be:

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Michelson & Morley made apparatus long enough to detect 1/3 of a fringe, with =500nm, so that l1 + l2 =17m, Nfringe = 108v2 /(3c2)

--But they could detect no shift at all (at any time of year!)

(Michelson wondered if the aether was somehow getting stuck to the earth, like the air in a below-decks cabin on a ship.)

Measured speed: 186,350 miles per second with a likely error of around 30 miles per second.

Meanwhile, Maxwell’s equations predicted a definite speed for electrmagnetic waves, and it is the speed found by measurements.  But what is the speed to be measured relative to?

--The only possible conclusion from this series of very difficult experiments was that the whole concept of an all-pervading aether was wrong from the start.