Special Theory of Relativity

10th Lecture

Transformations Equation

Transformation equations are used to transform between the

coordinates of two reference frames. There are two types of

transformation equations.

1. Galilean Transformation Equation (without relativistic effect!)

2. Lorentz Transformation Equation (with relativistic effect!)

Galileo Galilei Hendric Lorentz

Galilean Transformation

The Galilean transformation is used to transform

between the coordinates of two reference frames

which differ only by constant relative motion

within the constructs of Newtonian physics. This is

the passive transformation point of view. The

equations below, although apparently obvious,

break down at speeds that approach the speed of

light owing to physics described by relativity theory.

Galileo formulated these concepts in his description of uniform motion.

The topic was motivated by Galileo's description of the motion of a

ball rolling down a ramp, by which he measured the numerical value

for the acceleration of gravity near the surface of the Earth.

P. T. O

Galilean Transformation

The notation below describes the relationship

under the Galilean transformation between

the coordinates ( x, y, z, t ) and ( x′, y′, z′, t′ )

of a single arbitrary event, as measured in two

coordinate systems S and S', in uniform

relative motion ( velocity v ) in their common

x and x ′ directions, with their spatial origins

coinciding at time t = t ′ = 0:

Galilean Inverse Transformation

S S’

S S’

v

v

Galilean Transformation Equations

They enable us to relate a measurement in one inertial reference

frane to another. For example, suppose we measure the velocity of a

vehicle moving in the in x-direction in system S, and we want to

know what would be the velocity of the vehicle in S’.

Galilean Velocity Transformation Equations

dt

dzU

dt

dyU

dt

dxU

z

y

x

'

''

'

''

'

''

dt

dzU

dt

dyU

dt

dxU

z

y

x

Galilean Velocity Transformation Equations

S S’

S S’

v

v

Galilean Acceleration Transformation Equations

dt

dUa

dt

dUa

dt

dUa

zz

y

y

xx

'

''

'

''

'

''

dt

dUa

dt

dUa

dt

dUa

zz

y

y

xx

Galilean Acceleration Transformation Equations

S S’

S S’

v

v

Lorentz Transformation

In physics (1904), the Lorentz transformation

or Lorentz-Fitzgerald transformation

describes how, according to the theory of special

relativity, different measurements of space and

time by two observers can be converted into the

measurements observed in either frame of

reference.

The Lorentz transformation was originally the result of attempts by Lorentz

and others to explain how the speed of light was observed to be independent

of the reference frame, and to understand the symmetries of the laws of

electromagnetism. Albert Einstein later re-derived the transformation from

his postulates of special relativity. The Lorentz transformation supersedes the

Galilean transformation of Newtonian physics, which assumes an absolute

space and time. According to special relativity, the Galilean transformation is a

good approximation only at relative speeds much smaller than the speed of

light.

Lorentz Transformation EquationsConsider two observers O and O' , each using their own Cartesian coordinate

system to measure space and time intervals. O uses (t, x, y, z) and O ' uses (t' , x' , y'

, z' ). Assume further that the coordinate systems are oriented so that, in 3

dimensions, the x-axis and the x' -axis are collinear, the y-axis is parallel to the y' -

axis, and the z-axis parallel to the z' -axis. The relative velocity between the two

observers is v along the common x-axis. Also assume that the origins of both

coordinate systems are the same, that is, coincident times and positions.

If all these hold, then the coordinate systems are said to be in standard

configuration. A between the forward Lorentz Transformation and the inverse

Lorentz Transformation can be achieved if coordinate systems are in . The

symmetric form highlights that all physical laws should remain unchanged under a

Lorentz transformation.

Lorentz Transformation Equations

These are the simplest forms. The Lorentz

transformation for frames in standard

configuration can be shown to be:

where:

v is the relative velocity between frames in the

x - direction,

1/(1-v²/c²)^½ , is the Lorentz factor,

Lorentz Transformation Equations

S S’

S S’Lorentz Inverse

Transformation

Equations

Lorentz Transformation

Equations

v

v

Lorentz Velocity Transformation Equations

S S’

S S’Lorentz Inverse Velocity Transformation Equations

Lorentz Velocity Transformation Equations

v

v

Lorentz Acceleration Transformation Equations

S S’

S S’

Lorentz Inverse Acceleration Transformation Equations

Lorentz Acceleration Transformation Equations

v

v

Length Contraction (using Lorentz Transformation)

Length contraction is the observation that a moving object appears

shorter than a stationary object.

Let us assume there is a rod of length L with respect to the stationary

frame S’, is moving with constant velocity v.

Length with respect to the S frame = L =

Length with respect to the S’ frame = Lo =1

1

1

2 xx

12 xx

Length Contraction (using Lorentz Transformation)

Therefore,

Using Lorentz transformation equations;

11

1

1 vtxx

vtxx 1

and 22

1

2 vtxx

1122

1

1

1

2 vtxvtxxx

1212

1

1

1

2   ttvxxxx

Where,21 tt

Therefore, 12

1

1

1

2 xxxx

LLo  

LLo        Length measured by an

observer in the Frame S

Length measured by an

observer in the Frame S’

Time Dilation (using Lorentz Transformation)

Let us assume two events that takes place at a certain point, one after the

other. We can consider, an artist painting a picture as an example. The

first event could be the commencement of painting the picture and the

second event could be the completion of painting the picture.

Time interval with respect to the S frame = to =

Time interval with respect to the S’ frame = t =1

1

1

2 tt

12 tt

Time Dilation (using Lorentz Transformation)

Therefore,

Using Lorentz transformation equations;

xxxc

vtt

1121

1

1 ;    

x

c

vtt

2

1

and1

2222

1

2 ;     xxxc

vtt

121222

1

1

1

2 xc

vtx

c

vttt

12212

1

1

1

2 xxc

vtttt

Where,

21 xx

Therefore, 12

1

1

1

2 tttt

0 tt

ott       Time interval measured by an

observer in the Frame S

Time interval measured by an

observer in the Frame S’

Relative Motion for the two bodies in Relativity

Let us assume two objects are moving in an opposite direction to

each other,

A

VA

B

VB

Using Lorentz transformation equations;

2

1

1c

vU

vUU

x

xx

For this example;Ax VU 1 ,

Bx VU         and),( BAVv

Direction of B is opposite to the A

21

c

vV

vVV

B

BA

21

c

VV

VVv

BA

BA

2

),(

1c

VV

VVVv

BA

BABA

Let us assume two objects are moving in a same direction,

A

VA

B

VB2

1c

VV

VVv

BA

BA

This v denotes V(A,B). V(A,B) has a negative value. :. The

direction of V(A,B) should be the opposite direction.

A

VA

B

VB

For this example;Ax VU 1 ,

Bx VU    and),( BAVv

2

),(

1c

VV

VVVv

BA

BABA

Find the Energy & Momentum in S’ frame w.r.t S’ frame

vPEvE x )(1

A quantity that remains unchanged by a Lorentz transformation is said

to be Lorentz invariant. Such quantities play on especially important role

in special theory of relativity. The norm of any four vector is Lorentz

Invariant.

,

2

1 )(c

EvPvP xx

S S’

andyy PP 1

zz PP 1

Where,

2

2

1

1)(

c

vv

, 22

2

2

2

1

cmvc

c

v

mmcE o

o

and xox

oxx UmvU

c

v

mmUP

2

2

1

v

Lorentz Invariant

22222121 cpEcpE

A quantity that remains unchanged by a Lorentz transformation is said

to be Lorentz invariant. Such quantities play on especially important role

in special theory of relativity. The norm of any four vector is Lorentz

Invariant.

and

22221212 rtcrtc S S’

v

Four Vectors / Four

Cdts Systems

In the theory of relativity, a four-vector is a vector in a four-dimensional

real vector space, called Minkowski Space. It differs from a vector in that

it can be transformed by Lorentz Transformation. The usage of the

four-vector name tacitly assumes that its components refer to a standard

basis. The components transform between these bases as the space and

time coordinates differences, ( Δx, Δy, Δz, Δt ) under spatial

translations, rotations and boosts [ a change by a constant velocity to

another inertial reference frame]

Minkowski Four-Dimensional Space (“World”)

We can characterize the Lorentz Transformation still more simply if we

introduce the imaginary Sqrt[-1] = i ct in place of t, as time variable.

If in accordance with this, we insert,

tcix

zx

yx

xx

       

     

     

     

4

3

2

1

Where, 1i

and similarly for the accented system K’, then the condition which is

identically satisfied by the transformation can be expressed, thus;

2222 1

4

1

3

1

2

1

1

2

4

2

3

2

2

2

1       xxxxxxxx

That is, by the afore-mentioned choice of “co-ordinates” is

transformation into this equation.

Minkowski Four-Dimensional Space (“World”)

We see from the above equation that the imaginary time co-ordinate X4

enters into the condition of transformation in exactly the same way as

the space co-ordinates X1, X2 and X3. It is due to this fact that,

according to the theory of relativity, the “ time ” X4 enters into natural

laws in the same form as the space co-ordinates X1, X2 and X3.

A four-dimensional continuum described

by the co-ordinates X1, X2 and X3 was

called “World” by Minkowski, who also

termed a point-event a “World-point”.

From a “happening” in three-dimensional

space, Physics becomes, as it were, an

“existence” in the four-dimensional

“World”.

Thank You !