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Physics 334Modern Physics

Credits Material for this PowerPoint was adopted

from Rick Trebinos lectures from Georgia Tech

which were based on the textbook Modern Physics

by Thornton and Rex. Many of the images have been

used also from Modern Physics by Tipler and

Llewellyn, others from a variety of sources

(PowerPoint clip art, Wikipedia encyclopedia

etc), and contributions are noted wherever

possible in the PowerPoint file. The PDF handouts

are intended for my Modern Physics class, as a

study aid only.

CHAPTER 4Special Theory of Relativity 1

- 4-1 Foundations of Special Relativity
- The Experimental Basis of Relativity
- Einsteins Postulates
- 4-2 Relationship between Space and Time
- The Lorentz Transformation
- Time Dilation and Length Contraction
- The Doppler Effect
- The Twin Paradox and Other Surprises

Albert Michelson(1852-1931)

It was found that there was no displacement of

the interference fringes, so that the result of

the experiment was negative and would, therefore,

show that there is still a difficulty in the

theory itself - Albert Michelson, 1907

Newtonian (Classical) Relativity

- Newtons laws of motion must be implemented with

respect to (relative to) some reference frame.

A reference frame is called an inertial frame if

Newtons laws are valid in that frame. Such a

frame is established when a body, not subjected

to net external forces, moves in rectilinear

motion at constant velocity.

Difference Between Inertial and Non-Inertial

Reference Frame

Newtonian Principle of Relativity

- If Newtons laws are valid in one reference

frame, then they are also valid in another

reference frame moving at a uniform velocity

relative to the first system. - This is referred to as the Newtonian principle of

relativity or Galilean invariance.

If the axes are also parallel, these frames are

said to be Inertial Coordinate Systems

The Galilean Transformation

- For a point P
- In one frame S P (x, y, z, t)
- In another frame S P (x, y, z, t)

The Inverse Relations

1. Parallel axes 2. S has a constant relative

velocity (here in the x-direction) with respect

to S. 3. Time (t) for all observers is a

Fundamental invariant, i.e., its the same for

all inertial observers.

A need for ether

- In Maxwells theory, the speed of light, in terms

of the permeability and permittivity of free

space, was given by - Thus the velocity of light is constant
- Aether was proposed as an absolute reference

system in which the speed of light was this

constant and from which other measurements could

be made. - Properties of Aether
- Low density
- Elasticity
- Transverse waves
- Galilean transformation

Maxwells equations are not invariant under

Galilean transformations. The Michelson-Morley

experiment was an attempt to show the existence

of aether.

Michelson-Morley experiment

Michelson and Morley realized that the earth

could not always be stationary with respect to

the aether. And light would have a different

path length and phase shift depending on whether

it propagated parallel and anti-parallel or

perpendicular to the aether.

Perpendicular propagation

Parallel and anti-parallel propagation

Supposed velocity of earth through the aether

Michelson-Morley Experimental Analysis

Exercise 4-1 Show that the time difference

between path differences after 90 rotation is

given by

Recall that the phase shift is w times this

relative delay

or

- The Earths orbital speed is v 3 104 m/s ,

and the interferometer size is L 1.2 m, So

the time difference becomes 8 10-17 s, which,

for visible light, is a phase shift of 0.2 rad

0.03 periods

The Michelson interferometer shouldve revealed a

fringe shift as it was rotated with respect to

the aether velocity. MM expected 0.4 of the width

of a fringe, and could only see 0.01 equal to the

uncertainty in the measurement.

Interference fringes showed no change as the

interferometer was rotated.

Thus, aether seems not to exist!

Einsteins Postulates

- Albert Einstein was only two years old when

Michelson and Morley reported their results. - At age 16 Einstein began thinking about

Maxwells equations in moving inertial systems. - In 1905, at the age of 26, he published his

startling proposal the Principle of

Relativity. - It nicely resolved the Michelson and Morley

experiment (although this wasnt his intention

and he maintained that in 1905 he wasnt aware of

Michelson and Morleys work)

Albert Einstein (1879-1955)

It involved a fundamental new connection between

space and time and that Newtons laws are only an

approximation.

Einsteins Two Postulates

- With the belief that Maxwells equations must be

valid in all inertial frames, Einstein proposed

the following postulates - The principle of relativity The laws of physics

are the same in all inertial reference frames. - The constancy of the speed of light The speed

of light in a vacuum is equal to the value c,

independent of the motion of the source.

Relativity of Simultaneity

- In Newtonian physics, we previously assumed that

t t. - With synchronized clocks, events in S and S can

be considered simultaneous. - Einstein realized that each system must have its

own observers with their own synchronized clocks

and meter sticks. - Events considered simultaneous in S may not be in

S. - Also, time may pass more slowly in some systems

than in others.

The constancy of the speed of light

Lorentz Transformation

- Exercise 4-2 The equations for a spherical

wavefronts in S is - x2y2z2c2t2 , Show that the equation for the

spherical wavefronts in S cannot be

x2y2z2c2t2 in the Galilean transformation. - Exercise 4-3 Show that x g (x vt) so that

x g (x vt) , yields the g factoid - and that for small velocities

Lorentz Transformation

- Exercise 4-4 Use x g (x vt) and x g

(x vt) , to find t g (t v x /c2) - Exercise 4-5 Use x g (x vt) and t g (t

v x /c2) to show that the equations for

spherical wave fronts in S and S are the same.

Lorentz Transformation Equations

A more symmetrical form

Properties of g

- Recall that b v / c lt 1 for all observers.

g equals 1 only when v 0. In general

Graph of g vs. b (note v lt c)

The complete Lorentz Transformation

If v ltlt c, i.e., ß 0 and g 1, yielding the

familiar Galilean transformation. Space and time

are now linked, and the frame velocity cannot

exceed c.

Relativistic Velocity Transformation

Exercise 4-6 Suppose a shuttle takes off quickly

from a space ship already traveling very fast

(both in the x direction). Imagine that the

space ships speed is v, and the shuttles speed

relative to the space ship is u. What will the

shuttles velocity (u) be in the rest frame?

The Inverse Lorentz Velocity Transformations

- If we know the shuttles velocity in the rest

frame, we can calculate it with respect to the

space ship. This is the Lorentz velocity

transformation for ux, uy , and uz. This is

done by switching primed and unprimed and

changing v to v

Lorentz velocity transformation

Example As the outlaws escape in their really

fast getaway ship at 3/4c, the police follow in

their pursuit car at a mere 1/2c, firing a

bullet, whose speed relative to the gun is 1/3c.

Question does the bullet reach its target a)

according to Galileo, b) according to Einstein?

vpg 1/2c

vog 3/4c

vbp 1/3c

police

outlaws

bullet

vpg velocity of police relative to ground vbp

velocity of bullet relative to police vog

velocity of outlaws relative to ground

Galileos addition of velocities

In order to find out whether justice is met, we

need to compute the bullet's velocity relative to

the ground and compare that with the outlaw's

velocity relative to the ground.

In the Galilean transformation, we simply add the

bullets velocity to that of the police car

Einsteins addition of velocities

Due to the high speeds involved, we really must

relativistically add the police ships and

bullets velocities

Gedanken (Thought) experiments

It was impossible to achieve the kinds of speeds

necessary to test his ideas (especially while

working in the patent office), so Einstein used

Gedanken experiments or Thought experiments.

Young Einstein

The complete Lorentz Transformation

If v ltlt c, i.e., ß 0 and g 1, yielding the

familiar Galilean transformation. Space and time

are now linked, and the frame velocity cannot

exceed c.

Time Dilation and Length Contraction

More very interesting consequences of the Lorentz

Transformation

- Time Dilation
- Clocks in S run slowly with respect to

stationary clocks in S. - Length Contraction
- Lengths in S contract with respect to the same

lengths in stationary S.

We must think about how we measure space and time.

In order to measure an objects length in space,

we must measure its leftmost and rightmost points

at the same time if its not at rest. If

its not at rest, we must ask someone else

to stop by and be there to help out.

In order to measure an events duration in time,

the start and stop measurements can occur at

different positions, as long as the clocks are

synchronized. If the positions are

different, we must ask someone else to stop

by and be there to help out.

Proper Time

- To measure a duration, its best to use whats

called Proper Time. - The Proper Time, t, is the time between two

events (here two explosions) occurring at the

same position (i.e., at rest) in a system as

measured by a clock at that position.

Same location

Proper time measurements are in some sense the

most fundamental measurements of a duration. But

observers in moving systems, where the

explosions positions differ, will also make such

measurements. What will they measure?

Time Dilation and Proper Time

Franks clock is stationary in S where two

explosions occur. Mary, in moving S, is there

for the first, but not the second. Fortunately,

Melinda, also in S, is there for the second.

Mary and Melinda are doing the best measurement

that can be done. Each is at the right place at

the right time.

If Mary and Melinda are careful to time and

compare their measurements, what duration will

they observe?

S

Frank

Time Dilation

- Mary and Melinda measure the times for the two

explosions in system S as t1 and t2 . By the

Lorentz transformation

This is the time interval as measured in the

frame S. This is not proper time due to the

motion of S .

Frank, on the other hand, records x2 x1 0 in

S with a (proper) time t t2 t1, so we have

Time Dilation

- 1) ?t gt ?t(? gt1) the time measured between

two events at different positions is greater

than the time between the same events at one

position this is time dilation. - 2) The events do not occur at the same space and

time coordinates in the two systems. - 3) System S requires 1 clock and S requires 2

clocks for the measurement. - 4) Because the Lorentz transformation is

symmetrical, time dilation is reciprocal

observers in S see time travel faster than for

those in S. And vice versa!

Time Dilation Example Reflection

S

S

Frank

Mary

Fred

Exercise 4-7 Show that the event in its rest

frame (S) occurs faster than in the frame thats

moving compared to it (S).

Time stops for a light wave

Because

And, when v approaches c

For anything traveling at the speed of light

In other words, any finite interval at rest

appears infinitely long at the speed of light.

Proper Length

When both endpoints of an object (at rest in a

given frame) are measured in that frame, the

resulting length is called the Proper Length.

Well find that the proper length is the largest

length observed. Observers in motion will see a

contracted object.

Length Contraction

- Frank Sr., at rest in system S, measures the

length of his somewhat bulging waist - Lp xr - xl
- Now, Mary and Melinda S, measure it, too, making

simultaneous measurements (tl tr ) of the

left, xl , and the right xr endpoints - Frank Sr.s measurement in terms of Marys and

Melindas

? Proper length

Moving objects appear thinner!

Length contraction is also reciprocal.

So Mary and Melinda see Frank Sr. as thinner than

he is in his own frame. But, since the Lorentz

transformation is symmetrical, the effect is

reciprocal Frank Sr. sees Mary and Melinda as

thinner by a factor of g also. Length

contraction is also known as Lorentz

contraction. Also, Lorentz contraction does not

occur for the transverse directions, y and z.

Lorentz Contraction

v 10 c

A fast-moving plane at different speeds.

Experimental Verification of Time Dilation

Cosmic Ray Muons Muons are produced in the

upper atmosphere in collisions between ultra-high

energy particles and air-molecule nuclei. But

they decay (lifetime 1.52 ms) on their way to

the earths surface

No relativistic correction

Top of the atmosphere

Now time dilation says that muons will live

longer in the earths frame, that is, t will

increase if v is large. And their average

velocity is 0.98c!

Detecting muons to see time dilation

- At 9000 m it takes muons (9000/0.998c 30 µs)

about 15 lifetimes to reach earth. If No 108

and t 15t, N 31 muons should reach earth.

From relativistic approach, the distance traveled

is only 600m at that speed in 1 lifetime (2 µs)

and therefore N 3.68 x 107 Experiments have

confirmed this relativistic prediction

Space-time Invariants

This is a quantity that is invariant under

Lorentz transformation. It is defined in the

following way

(?s)2 (c2?t2) - ?x2 ?y2 ?z2

- The quantity ?s2 between two events is invariant

(the same) in any inertial frame. - ?s is known as the space-time interval between

two events.

There are three possibilities for ?s2 ?s2 0

?x2 c2 ?t2, and the two events can be connected

only by a light signal. The events are said to

have a light-like separation. ?s2 gt 0 ?x2 gt c2

?t2, and no signal can travel fast enough to

connect the two events. The events are not

causally connected and are said to have a

space-like separation. ?s2 lt 0 ?x2 lt c2 ?t2,

and the two events can be causally connected. The

interval is said to be time-like.

Space-time

- When describing events in relativity, its

convenient to represent events with a space-time

diagram. - In this diagram, one spatial coordinate x,

specifies position, and instead of time t, ct is

used as the other coordinate so that both

coordinates will have dimensions of length. - Space-time diagrams were first used by H.

Minkowski in 1908 and are often called Minkowski

diagrams. Paths in Minkowski space-time are

called world-lines.

Particular Worldlines

Stationary observers live on vertical lines. A

light wave has a 45º slope.

Worldline is the record of the particles travel

through spacetime, giving its speed (1/slope) and

acceleration (1/rate of change of slope).

The Light Cone

The past, present, and future are easily

identified in space-time diagrams. And if we add

another spatial dimension, these regions become

cones.

The Doppler Effect

The Doppler effect for sound yields an increased

sound frequency as a source such as a train (with

whistle blowing) approaches a receiver and a

decreased frequency as the source recedes.

Christian Andreas Doppler (1803-1853)

- A similar change in sound frequency occurs when

the source is fixed and the receiver is moving. - But the formula depends on whether the source or

receiver is moving. - The Doppler effect in sound violates the

principle of relativity because there is in fact

a special frame for sound waves. Sound waves

depend on media such as air, water, or a steel

plate in order to propagate. Of course, light

does not!

Waves from a source at rest

Viewers at rest everywhere see the waves with

their appropriate frequency and wavelength.

Recall the Doppler Effect

A receding source yields a red-shifted wave, and

an approaching source yields a blue-shifted

wave. A source passing by emits blue- then

red-shifted waves.

The Relativistic Doppler Effect

- So what happens when we throw in Relativity?
- Exercise 4-8 Consider a source of light (for

example, a star) in system S receding from a

receiver (an astronomer) in system S with a

relative velocity v. Show that the frequency can

be obtained from - Where f0 is the proper frequency
- Exercise 4-9 What would be the frequency if the

source was approaching? - Exercise 4-10 Use the results from exercise 8

and 9 to deduce the expressions for

non-relativistic velocities.

Using the Doppler shift to sense rotation

The Doppler shift has a zillion uses.

Using the Doppler shift to sense rotation

Example The Sun rotates at the equator once in

about 25.4 days. The Suns radius is 7.0x108m.

Compute the Doppler effect that you would expect

to observe at the left and right limbs (edges) of

the Sun near the equator for the light of

wavelength l 550 nm 550x10-9m (yellow light).

Is this a redshift or a blueshift?

Aether Drag

Exercise 4-12 In 1851, Fizeau measured the

degree to which light slowed down when

propagating in flowing liquids.

Fizeau found experimentally

This so-called aether drag was considered

evidence for the aether concept. Derive this

equation from velocity addition equations.

Lorentz-FitzGerald Contraction

- Exercise 4-13 Lorentz and FitzGerald, proposed

that the null test of Michelson Morleys

experiment can be explained by using the concept

of length contraction to explain equal path

lengths and zero phase shift. Show that this

proposition can work.

The Twin Paradox

- The Set-up
- Mary and Frank are twins. Mary, an astronaut,

leaves on a trip many lightyears (ly) from the

Earth at great speed and returns Frank decides

to remain safely on Earth. - The Problem
- Frank knows that Marys clocks measuring her age

must run slow, so she will return younger than

he. However, Mary (who also knows about time

dilation) claims that Frank is also moving

relative to her, and so his clocks must run slow.

- The Paradox
- Who, in fact, is younger upon Marys return?

The Twin-Paradox Resolution

- Franks clock is in an inertial system during the

entire trip. But Marys clock is not. As long as

Mary is traveling at constant speed away from

Frank, both of them can argue that the other twin

is aging less rapidly. - But when Mary slows down to turn around, she

leaves her original inertial system and

eventually returns in a completely different

inertial system. - Marys claim is no longer valid, because she

doesnt remainin the same inertial system.

Frank does, however, and Mary ages less than

Frank.

Twin Paradox

- Exercise 4-14 A clock is placed in a satellite

that orbits Earth with a period of 108 min. (a)

By what time interval will this clock differ from

an identical clock on Earth after 1 year? (b) How

much time will have passed on Earth when the two

clocks differ by 1.0 s? (Assume special

relativity applies and neglect general

relativity.)