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Lecture-XIX

pec a eory o e at v tyLorentz transformation

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The question is how (x, y, z; t) and (x0; y0; z0; t0) are related?

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If both coordinate systems are inertial (that is, no relative acceleration), then a particle moving

along a straight line in one system must move along a straight line in the other. Otherwise we

would introduce spurious forces into the system with a curved trajectory. Hence we require

linear transformations.

A reasonable assumption:

The most general, linear transformation between (x, t) and (x, t) can be written as

where a1; a2; b1; b2 are constants that can only depend on v, the velocity between thecoordinate systems, and on c.

Step 1: note that the origin of the primed frame (x0 = 0) is a point that moves with speed vas

seen in the unprimed frame. Therefore,

and hence

Step 2:

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The only way this is consistent with is if,

solving we get,

Then substituting a1, we get

Inverse Transformation

So Galilean transformations are a limiting case of the Lorentz transformations

when

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Alternate method

( )

( ) ( )

( )

, , b u t .

1,

A t 0 , t h e o r ig i n s o f t h e t w o s y s te m s c o i n c i d e .

a n d

11

11 1

x x v t y y z z t t

x x v t x v t v t t t xv

t t

x c t x c t

v

cx v t c t x x c t

v c

= = =

= + = + = +

=

= = + = + =

2

2

2

1 1H e n c e , 1 1 1 ,

1

d e p e n d s o n a n d n o t o n , t h e r e la t i v e v e l o c i t y o f S ' w i t h

r e s p e c t t o S . O n e c a n n o t d i s t i n g u i s h S a n d S ' e x c e p t f o r t h e s i g no f w h i c h d o e s n o t e f fe c t

v c

c v v

c

v v

v

+ = =

2 2

2

2 2 2 2

t h e d e p e n d e n c e o f .

1H e n c e , .

1

, , , .1 1

v c

x v t t v x c

x t y y z zv c v c

= =

= = = =

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Some consequences of Lorentz Transformations

Simultaneity:Consider a railway-man standing at the middle of a freight car of length 2L. He flicks on his

lantern and a light pulse travels out in all directions with the velocity c. Light arrives at the

two ends of the car after a time interval L/c. In this system, the freight car's rest system, the

light arrives simultaneously.

Now let us observe the same situation from a different frame, one moving to the right with

velocity v.

In the rest frame: Take the origin of coordinates at the

center of the car, and take t = 0 at the instant the lantern

as es. e two events are

In this system, the freight car's rest system, the light arrives simultaneouslyat A and B.

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Same events in the moving frame:

Message: It makes no sense to say that one event happens at the same time as another, unless

you state which frame youre talking about. Simultaneity depends on the frame in which the

observations are made.

Two events A and B have the following coordinates in the x, y system.

Event A: Event B: The distance L and time T separating the

events in the x, y system

The coordinates in the x', y' system,

with L > 0 and T > 0.

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The distance L' between the events in primed frame:

Length Contraction: Imagine that a rod is lying at rest along the x-axis in the S frame. Its

end points are measured to be at xAa ndxB, therefore its rest or proper length is xB xA= l0.

Now what is the rods length as measured by the S frame observer, for whom the rod moves

with the relative speed v? Suppose the two end points of the rod are at xA and xB as measured

at the same instant of time t from S.

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Example: A rod of length lies in the x'y' plane of its rest system and makes an angle with

the x' axis. What is the length and orientation of the rod in the lab system x, y in which the rodmoves to the right with velocity v?

The coordinates of the end points A and B in rest frame:

The coordinates of the end points A and B in Lab frame:

The angle that the rod makes with the x axis is: