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1. Einstein’s special relativity 2. Events and space-time in Relativity3. Proper time and the invariant interval4. Lorentz transformation5. Consequences of the Lorentz transformation6. Velocity transformation

Einstein’s special relativity and Lorentz transformation and its consequences

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1. Principle of Relativity by Einstein (1905)

2) The speed of light in a vacuum is the same for all observers, regardless of their relative motion or of the motion of the source of the light. http://en.wikipedia.org/wiki/Theory_of_relativity

It is based on the following two postulates:1) The laws of physics are the same for all observers in uniform motion relative to one another (principle of relativity),

- need a transformation of coordinates which preserves the laws of physics

A B

V

Observer in the car: the light pulse reaches A and B at the same time

A B

V

Observer to whom the car is moving with relative V: the light pulse reaches A before B

Simultaneity breaks down time cannot be regarded as a universal entity

- need a different transformation from Galileo’s but will converge to it for V<<C

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2. Events and space-time in Relativity

When and where is the object under our interest An Event in Relativity.

--An event is a point defined by (t, x, y, z), which describes the precise location of a “happening” which occurs at a precise point in space and at a precise time.

--“Space-time” is often depicted as a “Minkowski diagram”.

Space rSpace r

constantconstant

acceleratedaccelerated

decelerateddeceleratedTime (ct)Time (ct)

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World lines

The world line of an object is the unique path of that object as it travels through 4-dimensional space-time.

. World lines of particles/objects at constant speed are called geodesics.

Space rSpace r

constantconstant

acceleratedaccelerated

decelerateddeceleratedTime (ct)Time (ct)

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Events and space-time in Relativity

What is simultaneous in a moving frame is not simultaneous in the stationary frame.

Here the signal has to be sent later ( t > 0) from A …….

3 Proper time and the invariant interval

In 3-dimensional EUCLIDIAN space: P1

P2

In coordinate system O: ),,(),,,( 22221111 zyxPzyxP

In coordinate system O’: )',','(),',','( 22221111 zyxPzyxP

2222222 ''' zyxzyxr

In relativity, we would like to find a similar quantity for pairs of events, that is frame-independent, or the same for all observers, that is

invariant interval

2222222 )()()()()()()( zyxtcrtcs

∆t is the difference in time between the events

∆r is the difference between the places of occurrence of the events.

3.1 invariant interval (i.e. c is constant)

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3.2 Events, INTERVAL AND THE METRIC

A metric specifies the interval between two events

3.3 Proper time (length) the invariant interval

The proper time between two events is the time experienced by an observer in whose frame the events take place at the same point.

According to the definition of the interval between two events:

rc || --there always is such a frame since positive interval means:

0)()()() 222 rtcsIfii

0)()()() 222 rtcsIfi

2)( sc

so a frame moving at vector v = (∆r) /(∆t), in which the events take place at the same point, is moving at a speed < c

--It is still invariant even though there is no frame in which both events take place at the same point. (or (c∆t)2 < 0).

, the interval is said to be “timelike”

, the interval is said to be “spacelike”

--There is no such frame because necessarily it would have to move faster than the speed of light.

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0)()()() 222 rtcsifiii ,the interval is said to be “light-like” or null…defining a null geodesic

22 )()(.,. rtcei This is the case in which

Or, in which the two events lie on the worldline of a photon.

Because the speed of light is the same in all frames……. …. an interval equal to zero in one frame must equal zero in all frames.

The three cases have different causal properties, which will be discussed later.

Sometimes the proper distance is defined to be the distance separating two events in the frame in which they occur at the same time. It only makes sense if the interval is negative, and it is related to the interval by 2S

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We assume that relative transformation equation for x is the same as the Galileo Trans. except for a constant multiplier on the right side, i.e,

)2()''(

)1()('

utxx

utxx

where is a constant which can depend on u and c but not on the coordinates. (based on Postulate 1)

4. A transformation formula – Lorentz Transformation

How to find the factor ?

By tracing the propagation of a light wave front in two different frames, one of which is moving with a velocity of V along x-axis w.r.t. the other.

4.1 The formula fits into Einstein’s two postulates

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Assume a light pulse that starts at the origins of S and S’ at t =t’=0

zz

YY

XXOO

ZZ’’

YY’’

XX’’OO’’

After a time interval the front of the wave moves

ZZ’’

YY’’

XX’’OO’’

zz

YY

XXOO

uu(X, t) in S

(X’, t’) in S’

and

It is recorded as:

By Einstein’s postulates 2: x = ct

x’=ct’

SSSS’’

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Substituting ct for x and ct’ for x’ in eqs. (1) and (2)

)'3()('

)3(c

uc

t

t

)2()''(

)1()('

utxx

utxx

)3()()(' tucutctct

)4(')()''( tucutctct

)'4()(

')4(

uc

c

t

t

22

2

222

22

/1

1

1

1

cuc

uuc

c

Let (3Let (3’’) = (4) = (4’’))u < c so u < c so is is always > 1always > 1

When u << cWhen u << c

~ 1 ~ 1

If u~c, If u~c,

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)2()''(

)1()('

utxx

utxx

The relativistic transformation for x and and x’ is The relativistic transformation for x and and x’ is

If u << c If u << c

~1 ~1 utxx

utxx

'

;'

Lorentz transf.Lorentz transf. Galileo transf.Galileo transf.

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The transformation between t and t’ can be derived:

)1()()('' tucutxctx

)(2c

uctt ][

2c

uxt )(' utct

ct

Divide c into Eq.(1)

For the wave front of light, x=ct, x’=ct’

Divide c into Eq(2) )2(')( tucctx

]'

'[2c

uxt )''( utct

ct

)

''(

2c

uctt

The complete relativistic transformation (L.T.) is

)5(]'

'[,','),''(2c

uxttzzyyutxx

)6(][',','),('2c

uxttzzyyutxx

4. 2 The interval of two events under Lorentz transformation.

For two events, (t1, x1,y1,z1) and (t2,x2,y2,z2), we define:

(T, X, Y, Z) = (t1-t2,x1-x2,y1-y2,z1-z2)

)('

)('

cTc

vXX

Xc

vcTcT

then Lorentz transformation becomes

]2

2['

222

22

222222'2

vXTTvX

vXTXc

vTcXTc

22' )()(.,. SSei The interval of two events is an invariant under Lorentz Transformation.For short: the interval is a Lorentz scalar.

2222'2'2

2

22222 .,.),1)((

XTcXTc

eic

vXTc

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4.3 Lorentz transfermation in 4-dimensional formula

The L-T could be formally defined as a genernal linear transformation that leaves all intervals between any pair of events unaltered.

Introduce 4-D vector

1

1

1

1

z

y

x

ct

x

)xx()xx( 2121122 gS

1111 ,,,ct x zyx

Here we have introduced:

1000

0100

0010

0001

g

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L-T can be expressed as

1000

0100

00

00

c

vc

v

z

y

x

ct

'

'

'

'

z

y

x

ct

LXX'