ch37 young freedman2

8/12/2019 Ch37 Young Freedman2

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Chapter 37: Relativity

Events and Inertial

Reference Frames

Principles of Einsteins

Special Relativity

Relativity of Simultaneity,Time Intervals, Length

Lorentz Transformation

Relativistic Momentum &Energy

General Relativity


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Albert Einstein and the Special Theoryof RelativityThe Special Theory of Relativity was published by Albert Einstein in 1905

when he was still working in a Swiss patent office.

It stands as one of the greatest intellectual achievement of the 20th century.

(In the same year, two groundbreaking works on the theory of Brownian motion

and the Photoelectric Effect were also published.)

Special Relativity led to a fundamental

rethinking of the concepts of space and time:

The measurement of a time interval The measurement of length

The concept of simultaneity/Causality


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Summary of ChangesBefore Special Relativity: space and time are thought to be as absolute

properties of the background or stage where objects act upon.

Different observers from different inertial reference frames will agree

on all the measurements on the previous slide.

After Special Relativity: These space and time measurements are relative

to the observers frame of reference. A stationary observer with respect to a

moving observer will see:

a moving clock runs slower

a moving meter-stick gets shorter

two events being simultaneous for one inertial observer will not necessary

be simultaneous for the other.


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Reality Check

One should note that these relativistic effects are only dramatic for objects

moving at large relative speeds near c!

For everyday objects moving with ordinary slowerspeeds, these effects are small.

In fact, one can show that Einsteins theory reduces to Newtonian mechanicsin the limit u


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Events

Since relativity deals with the fundamental concepts of space and time,

we need to have a concrete basis for our analysis of these quantities:

Event: An occurrence in the physical universe characterized by its position

andtime. We label each event by (x,y,z,t)

Example: a car crash (the event) occurs at a

particular location and time.

(x,y,z)

tan event

space-time


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Inertial Reference Frames

Inertial Reference Frames:

A coordinate system (x,y,z,t) for labeling events attached to an observer

who is not accelerating and there is no net force acting on it.

e.g., A space ship far away from any stars in deep space stopped or in

constant velocity motion.

An effective Inertial Reference Frames:

g

u

S

SBoth Sand S are not truly

inertial ref. frames because

gravity acts on them!

But since acts on them equally

and is (their relative

velocity), we can treat them as

effective inert. ref. frames.

g

to u


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Relative Motion

u

S

S

u

S

S

Sand S are in relative motion

Swould say that S

moves to the rightwithrespect to him/her at a

constant speed u.

S would equivalentlysay that Smoves to the

leftwith respect to

him/her with the same

constant speed u.


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Galilean Relativity (before Einstein)

Principle of Galilean Relativity: TheLaws of Mechanics must be the same in

all inertial reference frames, i.e., Newtons Laws of Motion apply equally to

all inertial observers in relative motion with constant velocity.

u

S

S

Example: observer in S throw a ball straight up.


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Galilean Relativity (before Einstein)

S

In S inertial ref. frame:

In S inertial ref. frame:

u

Su

Although the observed trajectoriesin S& S are not the same, the same

Newtons Equation

describes the observations in both

situations!

mF a


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Galilean Coordinate Transformation

How can one translate physical quantities from one inertialreference frame to another?

u is the relative speed between S& S

(an event)


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Galilean Coordinate Transformation

At a later time t, (x,y,z,t) & (x,y,z,t) for event P are related by:

'

'

'

'

x x ut

y y

z z

t t

Galilean space-time

transformation equations

Notice that in this classical viewpoint,

't tso that clocks runs at the same rates in all inertial reference frames !

One can show that Newtons 2nd law is invariant

under this coordinate transformation !

Galilean

Relativity


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Galilean Velocity Transformation

Let say there is a particle at point P moving in thex-direction with its

instantaneous velocity vx as measured by an observer in S-frame.

How is the velocity vx measured by an observer in S-frame related to vx?

In S-frame, the particle moves a distance of dx in a time dt, so that

x

dxv

dt

From the Galilean Coordinate Transformation, we have in differential form,

'dx dx udt


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Galilean Velocity Transformation

'dx dx

udt dt Dividing dton both sides of the equation gives,

'x x

v v u Galilean Velocity Transformation

relative velocity

between framesvelocity of

particle as

measured by S

velocity of

particle as

measured by S

'

'

'dx dx dxu u

dt dt dt

Since dt=dt, we have,


14/27

The Constancy of the Speed of Light

Recall theMichelson-Morley experiment: Similar to a boat (light) traveling in a

flowing river (ether), the speed of light was expected to depends on its relativemotion with respect to the ether.

Presumed ether wind direction

u

c-u

c+u

u is the relative speed between

the frames (water & shore)

Result: Speed of light c does not follow the Galilean Velocity Transformation

and c is the same for all inertial observers in relative motion.


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The Constancy of the Speed of Light

Recall fromMaxwells Equations in electromagnetic theory: EM waves can be

shown to travel according to the plane wave equation,

2 2

2 2 2

, 1 ,E B E B

x c t

at the same speed c.

If we believeMaxwells Equations to be correct in all inertial reference frames,then we must accept that EM waves travel at the same speed c in all inertial

reference frames.

(Recall Sec. 32.2)


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Einsteins Postulates for SpecialRelativity

1. All laws of physics must be the same in all inertial reference frames.

Specific observations might be different but the same phenomena mustbe described by the same physical law.

Not just the laws of mechanics (as in the Galilean viewpoint). All laws

of physics include mechanics, EM, thermodynamics, QM, etc.

Same emf is induced

in the coil !


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Einsteins Postulates

2. The speed of light c in vacuum is the same in all inertial reference frames

and is independent of the observer or the source.

These two postulates form the basis of Einsteins Special Theory of Relativity.

(old vel addition rule applies to (slowly moving)

missile and it will be generalized for objects

moving close to c.)

(old vel addition rule does not apply to light.

c is the SAME in all frames ! )


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Stating the Results First

u

S

S

flash a flash b

Time Dilation: (moving clock runs slow)

2 2

0

, 1 1t t u c

Measured

by S

Measured

by S

Length Contraction: (moving ruler get shorter)

0L L

Simultaneity:

Two flashes

simultaneous in S but

not in S.


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Notes on Relative Motion

u

S

S

flash a flash b

flash a flash b

- Both observers in Sand S have their own measurement devices and they

can also measure his/her partners devices and compare with his/her own.

- Although time duration and length might depend on the observers inertial

frame, they will agree on the following three items:

c is the same in all frames

their relative speed u is the same all physical laws apply equally


20/27

Relativity of Time Intervals

Measuring Time Intervals with a light clock:

One time unit is measured by the duration

of two events: laser light leaving (tick)

laser light return (tock)

mirror

laser

Consider a boxcar moving with respect to the ground and we are

interested in the measurement of an interval of time by both Sand S

from a clock placed in the boxcar.

u

S

S


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Relativity of Time IntervalsIn the S frame: Mavis O is moving with the boxcar

the clock is stationary with Mavis O

distance travelled by light = 2d

speed of light = c

According to Mavis O,

0

2d

t c (measured in S)


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Now, consider the observation from Stanleys S frame (stationary frame),

Note: speed of light is still c in this frame but it now travels further !


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Thus, if tis the timebetween the bounces of the

laser light in S frame, then

it must be longerthan t0.

From the given geometry, we can explicitly calculate t:

2 2

2 2

2 2 22 2 2

22

2 2

2 2

2 2 4

4

u t c t d l

c t u t t d c u

dt

c u

2 2

2

1

dt

c u c

where

2 2

1 1 u c

0t t

2t l c

(light travels atthe same speed

c in S!)

1


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Time Dilation

0

2 21ttu c

This is the time dilation formula in SR.

Since u is strictly less than c,

and

2 21 1 1u c

0t t always !

(Note: both observers Sand S will agree on this

relationship between time intervals as long as they are

both looking at the same clock in S.)


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Proper Time

0

t t

is called the proper time and it is a special (or proper) time interval

since it is the time interval of the clock measured by an observer stationary

with respect to that clock, i.e., the two events (tick & tock) occur at the same

location.

0t

All other time intervals, , are measured by observers in relative motion

with respect to the clock.

t

All observers have his/her own proper time and all other observersmeasuring other observers clocks will notnecessary beproper.

The proper time will always be the shortesttime interval among all observers.


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Unreality Check

2 22 ut l c

If speed of light changes

according to Galilean VelocityTransformation, then

u

c2 2u c

Then,

Following the same calculation as previously, we have

2 2

2 2 2 2

2 2

2 2

u t td l u c

d u

2 2

2 2

2 2

t tc u

2 2

4

c t

22 2

02

4dt t

c

(no time dilation: all inertial

observers measure the same

time interval.)


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Relativity of Length

Distance between two points on a rigid body P & Q can be measured by a

light signals round trip time.

PQ

mirror

laser

l

l can be measured by the time interval: t2t1, 2 1 2 12 ( ) ( )2

cl c t t l t t

As we have seen, t will be different for different inertial observers, l willalso !

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