lec-06 pptzz

2 Prof. Shiva Prasad, Department of Physics, IIT Bombay

Recapitulate We applied the postulates of

special theory of relativity and arrived at Lorentz transformation.

We took an old example and showed that speed of light indeed remains constant in two frames under Lorentz Transformation.


Lorentz Transformation

2

= x x - vt

y y

z z

vxt t

c

2

1

1

v

c


Length Contraction

Proper length of a rod is length measured in a frame in which it is at rest.

Proper frame is a frame in which the rod is at rest.


Let us first assume that the rod is placed along x-axis.

Its proper length is largest.

Its length is shorter in other frames.


Length Measurement

Measure the co-ordinates of the two ends of the rod.

Is time of measurement important? Yes only if rod is moving.


X1 X2

Moving Rod

If the co-ordinates are measured at two different time, their difference does not give the correct length.


There may be other methods of measuring the length of a moving rod, some would be discussed later. But the co-ordinate difference of a moving rod would give length only if the two ends are measured at the same time.

Moving Rod


Let S be the frame in which the rod is at rest.

O O

z

x

y y

z

S S

A B


Events

E 1: Observer in S measures the co-ordinate of A end of the rod. (x1, t1) E 2: Observer in S measures the co-ordinate of B end of the rod. (x2, t2)


2 1 x x L in S only if 2 1 (say)t t t

22 2 2 2

11 1 1 2

,

,

vxx x vt t t

c

vxx x vt t t

c


We note that . Still This is because the rod is at rest in S. Measurement time has to be adjusted to be same in S not in S.

2 1t t

2 1x x L


2 1 2 1( )x x x x

LL


A 1 m rod AB is kept stationary in S, with end A at origin, in x-y plane making an angle of 60o with x-axis. What would be its length in S, if the relative speed between S and S is 0.6 c? Also find the angle that rod makes with x-axis in S.

Example 1


O O

z

x

y y

z

S S

x A

B


Like before, we have to define events relating to measurement of ends of the rod. And these events have to occur at same time in S. Let the co-ordinates be: E1: (x1,y1,t) E2: (x2,y2,t)


2 2

2 2

0.5 m

3 m

2

x x vt

y y

We have to carry out Lorentz Transformation like before. However, the co-ordinates of the rod are known in S.


1 1

1 1

0

0

x x vt

y y

Similarly


Like before time for the two events would be different in S, but we have not specifically written it here, as this is not of much interest in the problem.


We eventually get

2 1

2 1

0.5

3

2

x x

y y

With 21

1.251 0.6


The length of the rod in S is thus given as

2 2

2 1 2 1

220.5 3

1.25 2

0.91 0.95 m

x x y y


The angle that the rod makes with x-axis in S is given by

2 1

2 1

3 1.25tan 2.165

2 0.5

y y

x x

This gives o65.2


We note that only the x-component of the rods length is contracted in S and not the y-component.


According to an observer in S a lightening strikes at a distance of 20 km from origin at t=0. At what distance from origin, did this event occur in S, which moves with v=0.6 c in frame S?

Example 2


At t=0 and t=0 the origins of S and S were coincident. Imagine a stationary rod of 20 km extending from origin to place of lightening in S. The length of this rod would be contracted in S.

Wrong Approach


Now So the co-ordinate of lightening in S would be given by the contracted length 20/1.25=16 km.

Wrong Answer

1.25


The co-ordinate of the event in S (x=20km, t=0).Hence using Lorentz transformation the x-coordinate in S is the following

Correct Answer

1.25 20 0

25 km

x


Let us calculate the time of the event as seen in S.

Why this discrepancy?

3

2

5 8

0.6 20 101.25 0

5 10 s taking 3 10 m/s

ct

c

c


So According to S lightening had struck 5x10-5 s before the origins of the two frames coincided.


In S frame the events of two origins coinciding, and lightening striking are not simultaneous, though in S they are.


According to S, the origin of S is approaching him with a speed of 0.6c and only 5x10-5 s after the lightening striking, will the origin of S, reach his origin.


Note when the observer in S gets the information of lightening is not important. What is important is the time when the lightening took place as per his calculation.


Distance of origin of S from S in S frame, at the instant lightening strikes is

55 10 0.6 9 kmc


Assume that lightening struck at point P and a stationary rod OP of 20 km length is at rest in S. This length is proper. This length would appear to be contracted in S and would indeed be given by 16 km.


But according to S, the lightening occurred before his origin coincided with S. Hence the co-ordinate of the event does not measure the length of the rod in his frame.


P O,O

20 km

P O O

16 km 9 km

According to S at t=0

When lightening strikes

According to S at t= -5 10-5 s


Time Dilation

Proper Time interval between two events is time interval measured in a frame in which the two events occur at the same place.


In another frame the time interval between these two events would appear to be larger than the proper time interval.


Let the time interval be proper in S. Hence the two events should have the same x co-ordinates in S, say x. Let t1 and t2 be the time of the two events measured in S.


2 2 2

1 1 2

vxt t

c

vxt t

c

The time of the two events in S is then given as follows.


The time difference between these two events in S is therefore, given as follows.

2 1 2 1 t t t t

Proper time interval between two events normally is written as .


We note that to apply time dilation, only the x co-ordinates of the two events have to be same. Though for general definition of proper time interval all the co-ordinates should be same. We shall discuss this aspect in detail later.


Lorentz Transformation of x co-ordinate gives.

2 2

1 1

x x vt

x x vt

Hence the x values are different in S. But that is not so shocking. A difference is expected classically also in similar situation.


We discussed two important consequences of Lorentz Transformation: Length Contraction and Time Dilation.

We gave some examples of length contraction and warned where one can make an error in direct use of the formula.

Summary

lec-06 pptzz

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