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Lecture 4

Relativity of Simultaneity Revisited

ct

x

O

0=v

A B C

x

ct

O

0v

A B C

vtxxi+=

ct

x

Two events simultaneous in one inertial frame are not simultaneous

in any other inertial frame moving relative to the first.

OR

Clocks synchronized in one inertial frame are not synchronizedin any other inertial frame moving relative to the first.

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Lecture 4

Lorentz Transformation

Requirements:

Transformation must be linear in both space and timecoordinates

Transformation must reduce to Galilean transformationfor sufficiently small values of the relative speed between

two inertial frames

Mathematically, we have

btaxx

tbxax

=

+=

abv =

0=x

0=x

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Lecture 4

Lorentz Transformation-Cont d

Now, consider the motion of a light pulse originating at the

origin of both inertial frames

tcx

ctx

=

=

tbacbtctatc

tbactbtcact

)()(

)()(

==

+=+=

bac

c

t

t

c

bac

=

=

+

)( 222

2222

vca

bcac

=

=

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Lecture 4

Lorentz Transformation-Cont d

2

21

1)(

c

v

av

=

abtxaabtxax

btaxaxxaxtb

+=+=

==

)1(

)(

22atx

b

at +

=

21

2

22

22

22222/1

/1

/1

11

1

c

v

v

cv

cv

cv

v

cv

b

a=

=

=

Lorentz factor

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Lecture 4

Lorentz Transformation-Cont d

To summarize

)(

)(

2c

xvtt

zzyy

tvxx

+=

=

=

+=

)(

)(

2c

vxtt

zzyy

vtxx

=

=

=

=

=+=

22c

vxttx

c

vt

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Lecture 4

x

ct ct

x

OO

The time axis in S follows:

))(/(

0)(

ctcvvtx

vtxx

==

=

xct )/1( =

The spatial axis in S follows:

2

2

/

0)/(

cvxt

cvxtt

=

=

xct =

Relationship between Inertial Frames

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Lecture 4

Coupling of Space and Time

)(

)(

2

c

vxtt

vtxx

=

=

(x1, t1) (x2, t2)

[ ]

=

=

212

1212

121212

)()(

)()(

c

xxvtttt

ttvxxxx

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Lecture 4

Length Contraction

A rod is at rest in S, and lies along thex-axis. Let its two ends be

denoted by space-time coordinates (x1, t1) and (x2, t2),then thelength of the rod is always measured as

Now, what is the length of the rod as measured by an observer in

S?

x1 = (x1 vt)

x2 = (x2 vt)

L = x2(t) x

1(t)

x2 x1 = (x2 x1)

L = Lp 1v

2

c2

Lp= x2

x1 Proper Length

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Lecture 4

Time Dilation

A clock is at rest in S. Consider two events corresponding to

different readings of the clock, (x, t1)and(x, t2).The timeinterval between the two events in Sis denoted by

p=

t2

t1

what is the time interval between the same two events as

measured by an observer in S?

Proper Time

t1= ( t1 +

v xc2 )

t2= ( t2 +

v xc

2 )

t2 t1 = ( t2 t1)

=

p

1 v2

c2

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Lecture 4

Relativistic Doppler Effect

+= xc

vtt

2

+= xc

vtt

2

+= )(2

tc

c

vtt

lightIf the time interval represents

the period of the light wave,

t=1

f, t =

1

fp

f =1 v

c

1+ vc

fp

fp :proper frequency, if the

source of light is at rest

in S.

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Lecture 4

Redshift and Blueshift

beta valuec

v= 21

1

=Lorentz factor

We define z ( fp

f) f

1)1(

1)1(2

2

++

+=

z

z

f =1

1+ fp

z If

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Lecture 4

General formula for Doppler shift: f =fp

1

1 cos

Transverse Doppler Effect

t= t +v

c2 x

x = 0

t = ( t )

t=1

f, t =

1

fpf =

fp

Transverse Doppler shift always represents

redshift!

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Lecture 4

Reading Assignments

Chapter 1, 1-6