Special Relativity

Special Relativiy 1

Special Relativity(Translated from Relativités et quanta

clarifiés)

Bernard Schaeffer PhD

Special Relativiy2

1. INTRODUCTION

Special relativity originated one century ago from unsolved problems and

various observations incompatible with the ideas of that epoch. Maxwell

predicted the existence of radiation pressure, already imagined by Newton and

observable with the Crookes radiometer. The Maxwell equations have been

criticized because they were not conserved in the Galilean transformation. With

the newtonian absolute movement, speed and time one predicted that light

should be dragged by the Earth’s movement. Michelson-Morley experiment had

to prove the existence of the Ether. The negative result of the experiment led to

light speed invariance.

Special relativity is special because it is limited to uniform translation, without

any acceleration. Its fundamental postulate is the invariance of light speed in a

change of Galilean reference frame.

The Galilean transformation had to be replaced by the Lorentz transformation in

order to take into account this experimental result, already known from the

Maxwell equations. Einstein deduced directly the Lorentz transformation

without using the Maxwell equations. From the Lorentz transformation one

deduce various transformations : time, length, speed, acceleration, mass…

Acceleration ought to be incompatible with Galilean frames but Einstein took

the precaution of saying that special relativity should be applied to the "slowly

accelerated electron". Using time as a fourth spatial dimension, one obtains the

pseudo-euclidean space of Minkowski, euclidean by using an imaginary fourth

dimension.

Completed by Newton’s laws, special relativity became the relativistic dynamics

whose principal application is the formula giving the energy contained in a mass

at rest or in movement. The diagram below shows the logical process from the

Lorentz transformation to E = mc2.

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Relativité restreinte

Linéarité c = cste Réciprocité

Transformation de Lorentz

� = 1

1 - v2

c2t' = � t - vx

c2

x = � x' + vt't = � t' + vx'c2

x' = � x - vt

Directe

Réciproque

Dynamique relativiste

E c = m - m0 c2

Energie cinétique

Energie proportionnelleà la masse

x = x'�

Dilatation du temps:Immobilité de la règle dans leréférentiel R' en mouvement

x' = 0

Contraction des longueurs:Instantané depuis le référentiel R

de l'observateurt = 0

t = �t'

Accélération d �vdt

= dv'dt'

Vitesse limite = cvx =

v'x + v

1 +v v'xc2

Loi de Newtonrelativiste F =

d mvdt

E = m c2

Equations deMaxwell

Théorême de Pythagoredans l'espace à quatredimensions deMinkowski

s2 = x2 + y2 + z2 + ict 2

ds2 = dr2 + r2 d�2 + sin2� d�2 + d ict 2

Masse relativiste m = � m0

Flow chart of special relativity

Special Relativiy4

2. MICHELSON-MORLEY EXPERIMENT

2.1. THE AETHER

The Michelson-Morley experiment consists to compare light speed in the

directions parallel and perpendicular to the motion of the Earth on its axis. If

Aether exists, still in an absolute reference frame, light speed should be constant

in this reference frame, like sound speed in the air in the absence of wind.

According to the Galilean principle of superposition, the speed of light is

increased or decreased, depending on the direction and amplitude of the wind as

may be shown with a ultrasound anemometer measuring the wind velocity based

on the transit time of ultrasonic acoustic signals.

There are two possiblilities, either the Aether is still relative to the Universe and

the Earth is in motion, the speed of light will vary with the orientation, or the

Aether is stuck to the Earth and light speed is independant of direction.

Let us take a closer look to this experiment with the swimmer analogy.

2.2. THE MICHELSON SWIMMER

Crossing a lake

Let us consider a swimmer crossing a lake of width L0. The time t0 of a round-

trip crossing at speed c is given by

t0 =2L 0

c

Crossing a river

To cross a river, the swimmer has to swim obliquely upstream in order not to be

dragged downstream. The relative speed c of the swimmer has to be larger than

the driving speed v of the current. If c = v, the swimmer stays on the spot and

the duration of the crossing is infinite. If c < v his route seems oblique to an

observer staying on a boat dragged by the current and perpendicular to an

observer staying on the bank. In any case the duration of the crossing is larger

with a current than without : this is a kind of time dilation ! Nevertheless, the

time is absolute in classical mechanics and the swimmer has the same time as an

observer on the bank or on a boat. In a given time, the distance covered by the

swimmer is the vector sum of the distances covered along and across the river.

Special Relativiy 5

The same is valid for the speeds obtained by dividing by the corresponding time

increment.

The absolute speed v1 is the swimmer speed for an observer on the bank. The

absolute speed is perpendicular to the bank and given by the Pythagorean

theorem :

c2 = v21 + v2

The duration of the round-trip crossing is :

t1 =2L 0

v1

=2L 0

c2 – v2

That is to say :

t1=

t0

1 –v2

c2

The crossing time increases with the speed of the current and becomes infinite

when the speed of the current attains that of the swimmer in still water. Time

seems to be dilated for an observer on the bank.

Swimming along the river

The velocities add the way down and subtract the way up. The durations add.

Therefore, the time t2 necessary for a round-trip along the river with the same

distance L0 parallel to the current is :

t2 =L 0

c - v+

L 0

c + v

that is

t2=

t0

1 -v2

c2

This time dilation, with a slightly different formula, is larger.

Comparing the travel times

For the same distance, it takes a longer time to swim along the river bed than to

cross it. Both times are larger than in still water. The time difference between

swimming perpendicular and parallel to the stream is :

Special Relativiy6

t2 – t1 =t0

1 –v2

c2

–t0

1 –v2

c2

�v2

2c2t0

This is the formula that Michelson proposed himself to transpose to light. The

swimmer velocity c is that of light (nowadays a photon). The current speed v is

the velocity of the Aether wind.

2.3. MICHELSON INTERFEROMETER

The Michelson interferometer is a very sentitive equipment made of two

perpendicular mirrors M1 et M2 and a half transparent mirror, inclined at an

angle of 45 degrees, so that half the light pulse goes on through the glass, half is

reflected. The two arms of the apparatus have equal lengths, are perpendicular

and may rotate. One has two beams from the same light source, reflected

parallel to the incident ray and coming again together through the semi-

reflecting mirror. Equal optical paths may be adjusted very precisely in order to

obtain interference fringes. The fringes should move with the orientation of the

interferometer if the speed of light depends on that of the solar system

(400 km/s). The shift should be maximum in the direction of the constellation

Virgo. According to Michelson, the precision of the apparatus is even enough to

detect the Earth’s Aether wind due to the rotation of the Earth around the Sun

(30 km/s or 0.04 fringe). A later improvement with an eleven meters optical

path, should even detect the Aether wind due to the rotation of the Earth on its

axis (400 m/s or 0.005 fringe. One should note that there are already three

absolute reference systems. The newtonian notion of absolute space is thus

physically incorrect.

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The formula derived for the swimmer remains valid for the Aether wind :

�t = t2 – t1 �v2

2c2t0

In order to minimize errors, the shift of the interference fringes, being measured

at 0° and 90°, has to be multiplied by two :

�t �v2

c2t0 =

2 Lv2

c3

The variation of the optical path is then :

n � = c �t �2Lv2

c2

The number n of fringes shifted by the translation at 30 km/s of the Earth around

the Sun is, for a 600 nm wavelength, with c = 300.000 km/s and with arms of

1.2 m length

n �2Lv

2

� c2=2 � 1.2 � 10

-4 2

6 � 10- 7= 0.04

To their amazement, Michelson and Morley found that the velocity of light was

independent of its direction of travel through space. There was no observable

fringe shift although the expected effect was twice the experimental error.

Michelson and Morley carried out, in 1887, a new experiment ten times more

sensitive with the same null result : there is no Aether wind. The speed of light

seems to be constant, even in single trip as shown by measuring the speed of

light emitted by the �-ions (or pions or pi-mesons). However, there still exist

people who believe in the Aether, like the Nobe Prize winner Maurice Allais.

2.4. CONTRACTION AND DILATATION

Going back to the formulae giving the crossing times :

t1 =t0

1 –v2

c2

et t2 =t0

1 -v2

c2

t0 is the round-trip time in still water (v = 0)

t1 is the round-trip time for crossing the river with a current of speed v

t2 is the round-trip time along the river bed with a current of speed v

The Michelson-Morley experiment having shown that, for light, these two times

were equal. One has :

Special Relativiy8

2L 1

c 1 –v2

c2

=2L 2

c 1 –v2

c2

L0 is replaced by L1, parallel to the movement, and L2, perpendicular to the

movement. L1 and L2 are, indeed, the only adjustable parameters. In order to

verify the preceding equality, one needs to have :

L 2 = L 1 1 -v2

c2

Fitzgerald had read a paper from Heavyside showing that the electric field of the

moving charge distribution undergoes a distortion, with the longitudinal

components of the field being affected by the motion but the transverse ones

not. Then, if we assume that intermolecular forces are electrical, then we have

L1 = L0 and

L2= L

01 -

v2

c2

which is the Fitzgerald-Lorentz contraction, a consequence of the Maxwell

equations. Then

t1 =2L 1

c 1 –v2

c2

=2L 0

c 1 –v2

c2

=t0

1 –v2

c2

where t0 is the round-trip time in the absence of Aether. It shows that the time

dilates. But the time is independant of the direction of movement. A clock,

laser, for example will have a period increasing with speed, but without change

under a slow rotation, even if its dimensions vary with speed. This should be

true for any type of clock, mechanical, optical or electronical. Then

t1= t

2=

t0

1 -v2

c2

The period of the pendulum of a clock

A clock with one beat per second will have one beat in two seconds at a speed of

261,000 km/s for a fixed observer.

Special Relativiy 9

3. RELATIVISTIC KINEMATICS

3.1. GALILEAN TRANSFORMATION

According to newtonian mechanics, the the absolute velocity is the sum of the

relative and the transferred (may also be called dragging or entrainment, in fluid

mechanics) velocities :

va = vt + vr

For a constant speed, the abscissa is a linear function of the time. We may then

write :

xa = vt t + xr

or with other notations :

x = x' + v t'

where t = t’ : the time does not depend on the reference frame. In fact, it does :

time is different in New York and in Paris but the difference is a constant.

We may also write it

x' = x � v t

It is exactly the same, except for the sign of v with t = t’. It is the reciprocal

Galilean transformation.

3.2. DERIVATION OF THE LORENTZTRANSFORMATION

The Galilean transformation needs to be generalized to take into account the

constancy of the speed of light. The frame of reference R of the observer,

generally considered as motionless, corresponds to the absolute reference frame

of newtonian mechanics. The frame of reference R’ is the relative reference

frame. The speed v is the classical transferred velocity, assumed to be constant.

The Lorentz transformation, in its simplest form, is usually written in two

dimensions (space and time), with the x axis coinciding with the velocity vector

of R’ relatively to R.

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Linearity

In special relativity t � t’. The simplest linear relationship between spacetime in

the R and R’ reference frames is, with three independent coefficients, �, �, �,

function of the velocity v of the particle and v of the light is:

x' = � x � vt

t' = � t + �x

This is the Lorentz transformation de Lorentz that becomes the Galilean

transformation for � = � = 1 and � = 0.

Constancy of light speed

In order to have a speed of light c independent of the reference frame, one needs

to have x = ct and x' = ct'. The first of the preceeding equations then becomes :

ct' = � ct - vt

Using the second one

t' = � t + �ct

we get :

c� t + �ct = � ct - vt

Simplifying by t et dividing by c�, one obtains the relation

1 + �c = �

�1 � v

c

Relativity principle

According to the principle of relativity, there is no privileged reference frame.

One has to find the same relationship when passing from R to R’ or, inversely

from R’ to R. The relative speed v of the frames needs however a change of sign

for the same reason as for the Galilean transformation. The direct transformation

is

ct' = � c � v t

The reciprocal transformation of the abscissa isct = � ct' + vt'then

ct' = � c � v t = � c � v�

cct' + vt'

or

Special Relativiy 11

c= � c � v�

cc + v

After transformation this expression gives the Lorentz factor :

� = 1

1 � v2

c2

The second Lorentz relation, using x = ct and

1 + �c =�

�1 � v

c

becomes :

t' = � t + �x = � 1 + �c t = ��

�1 � v

ct = � 1 � v

ct =

t � vtc

1 � v2

c2

Replacing t = x/c, we get the Lorentz transformation of the time :

t' = � t –vx

c2

Algebraic form

The constants � et � being determined, one obtains the direct Lorentz la

transformation :

x' = x � vt

1 � v2

c2

t' =t � xv

c2

1 � v2

c2

and the reciprocal

x = x' + vt'

1 � v2

c2

t =

t' + vx'

c2

1 � v2

c2

When light speed c tends to infinity, the Lorentz factor � tends towards one. The

preceding formulae become the Galilean transformation :

x' = x - vt et t' = t

or

x = x' + vt et t = t'

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where the resultant displacement is the sum of the relative and of the transferred

displacements.

Matrix form

The Lorentz transformation

x' = � x – vt

t' = � t –xv

c2

may be written in matrix form :

x' t' = �

1

–v

c2

– v

1xt

Using i = � 1 , y = ict and y' = ict', we obtain :

x' y' = �

1

–iv

c

iv

c

1

xy

Multipliantthe transformation matrix by its transpose, on obtains the unit

matrix :

�

1

–iv

c

iv

c

1�

1

iv

c

–iv

c

1= �2

1 +iv

c

2

–iv

c+

iv

c

–iv

c+

iv

c

–iv

c

2

+ 1

= �2

1 –v2

c2

0

0

1 –v2

c2

=10

01

Its transposed is also its inverse. There is conservation of the lengths in the

space x, y = ict which is then euclidian. The Lorentz transformation is then a

rotation of an imaginary angle.

In the littérature one fins a matrix presentation of the four-dimensional Lorentz

transformation représented by the capital lambda (the L of Lorentz) :

�i j

= �

1

– �00

– �1

00

0

0

10

0

0

01

where � = v/c. In an arbitrary direction, sans explicit it, on may write the general

form of a linear transformation in the four-dimensional spacetime as :


�i j

=

a11

a12

a13

a14

a12

a22

a23

a24

a13

a23

a33

a34

a14

a24

a34

a44

where the aij are of the form

aij =�x'i

�xj

= ai,j

where the partial derivative is indicated by a comma.

The comma, representing a partial derivation, abridges considerably the

formulae in relativity. In two dimensions the a linear transformation is :

dx' =�x'�x

dx +�x'�y

dy = x', xdx + x', ydy

dy' =�y'�x

dx +�y'�y

dy = y', xdx + y', ydy

Rotation in spacetime

By putting ict = y, ict' = y', tg (i) = iv/c, one has

� =1

1 -v2

c2

=1

1 +iv

c

2=

1

1 + tg2 i�= cos i�

The Lorentz factor � is real ; indeed

� = cos i� =exp(i2�) + exp(– i2�)

2=

exp(– �) + exp(�)

2= ch �

The transformation becomes, in matrix form, a rotation of an imaginary angle

i :

x' y' = �

1

–iv

c

iv

c

1

xy

= cos i�1

– tg i�

tg i�

1

xy

=cos i�

– sin i�

sin i�

cos i�xy

Using the hyperbolic functions, one eliminates the imaginary quantities by

replacing y = ct and y’ = ct' :

x = x' ch � – y' sh �

y = – x' sh � + y' ch �

These are formulae analogous to those of rotation, where the trigonometric

functions are replaced by hyperbolic functions. The terms in hyperbolic sines

are preceded each with a minus sign. In the rotation, only one sine is preceded

Special Relativiy14

with a minus sign. The Lorentz transformation matrix is symmetrical while the

rotation matrix is antisymmetrical.

The Lorentz transformation is then a hyperbolic rotation in a pseudo-euclidean

space or in a true rotation in a euclidean space, but with an imaginary angle. In

this euclidean space, the time is an imaginary distance, ict. The Lorentz

transformation may be generalised in vectorial form for some rare practical

applications.

3.3. TIME AND LENGTH

Time dilation

Let us consider a motionless observer in a reference frame R. He looks at a

clock (not a pendulum clock, depending on the Earth gravity) moving at a

velocity v. He measures a time interval t between two beats of this moving

clock.

An observer moving with the clock (x’ = 0) in R’ measures a time interval t’

between two beats of his clock.

The second equation of the Lorentz transformation is :

t =

t' + vx'

c2

1 � v2

c2

With x’ = 0, we get

t = t'

1 – v2

c2

= �t'

The time interval between two beats looks larger for a moving clock. It becomes

infinite when the speed approaches that of light. A photon is immortal. A

meson has a limited life that can be measured practically motionless in the

laboratory and at high speed in the atmosphere. A longer life was found at high

speed than at rest in accord with the preceding formula.

The twin paradox is something similar but usually misinterpreted. One compares

two twins, one staying on Earth and the other flying with a rocket near the speed

of light. The twin staying on earth will see the other aging slower. Now let us

apply the principle of relativity : there is no preferred frame. Then the twin on

the rocket will see the twin on earth also aging slower. Both of the twins will see

the other one aging the same way, with or without acceleration and when they

will meet again they will have the same age. Indeed, acceleration, being a


differential of space to time, is relative between the twins like time, space and

velocity. Within the scope of special relativity the acceleration is not absolute. It

is often assumed without proof that there is a stationary and a travelling twin

(relative to what absolute frame?).

Time, space and their derivatives depend on the relative velocity between the

frames. No reference frame is preferred.

Length contraction

With the same kind of reasoning, let us consider an observer in a frame R

measuring a length x of a ruler moving at a relative speed v in a reference frame

R’. The observer in the moving frame R’measures a length x’. He has to take an

instantaneous photograph, that is, t = 0. We use the first Lorentz equation

x' =x � vt

1 � v2

c2

where we put t = 0 :

x' = x

1 � v2

c2

The length apparent to the motionless observer being x’, we have :

x = x' 1 � v2

c2

which is the Lorentz-Fitzgerald contraction. A direct measure does not seem to

exist, but it is taken into account in the calculation of the synchrotron radiation,

the diameter of the accelerator being different in the frame of the high speed

electron and in the frame of the laboratory.

Like the time, the lenth of a ruler parallel to the speed depends on the relative

speeds of the ruler and the observer. A ruler contracts at high speeds while the

time dilates.

3.4. COMPOSITION OF VELOCITIES

In classical kinematics, velocities simply add vectorially according to the

Galilean transformation.

Special Relativiy16

Colinear velocities

In relativistic kinematics, near the speed of light things are more complicated.

We shall limit ourselves first to a single spatial dimension, with colinear

velocities.

The Lorentz transformation is valid, in principle, only for Galilean reference

frames, that is, for constant transferred speeds. The relative speed v between two

Galilean frames R and R’ and the Lorentz factor � are constants. The Lorentz

transformation may then be written in différential form :

dx = � dx' + v dt'

dt = � dt' + v dx'

c2

with

� = 1

1 - v2

c2

Using vx = dx/dt and v’x = dx’/dt’, we get the relativistic composition of

velocities :

vx = dxdt

=� dx' + v dt'

� dt' + v dx'

c2

=

dx'dt'

+ v

1 + v dx'

c2 dt'

that is :

vx =v'x + v

1 +v v'xc2

For an infinite light speed, the denominator is equal to one. We then recover the

classical formula of speed addition where the absolute velocity vx = va is the

sum ot the transferred velocity v = vt and of the relative velocity v’x = vr :

va = vt + vr

In einsteinian relativity, velocities add as in Galilean relativity except that a

factor prevents to reach the speed of light. Let us chek it. If v’x = c, as for a

photon in a frame moving at speed v, then we have :

vx =c + v

1 +v c

c2

= cc + v

c + v= c

The velocity of a photon does not depend on the speed of the reference frame.


The relativistic composition of the velocities is no more the Galilean addition of

velocities. There is a factor preventing to overpass light speed. For low

velocities one recover the classical principle of superposition of velocities.

Non colinear velocities

Let us consider two frames R and R’ whose axes Ox and O’x’ coincide, their

origins O and O’ moving away from each other with velocity v.

Considérons des référentiels R et R’ dont les axes Ox et O’x’ coïncident, les

origines O et O’ s’éloignant l’une de l’autre à la vitesse v. Les composantes des

vitesses seront donc

The components of the velocity are vx and vy in R, v’x and v’y in R’. The

relation between vx and v’x is the same as for colinear velocities. Using the

differential form of the Lorentz transformation, one has dy = dy’, in the absence

of transverse contraction :

dt = � dt' + v dx'

c2= � 1 + v

c2

dx'dt'

dt'

which gives

vy = dydt

= dy'

� dt' + v dx'

c2

=

dy'dt'

� 1 + v

c2

dx'dt'

and lastly

vy =v'y 1 � v2

c2

1 +vv'xc2

This formul is used to calculate the relativistic aberration.

3.5. LONGITUDINAL DOPPLER

The Doppler effect is observed when a vibrating source whose frame is R’

emitting sound or light waves approaches or moves away from the observer, as

for example, a noisy motorcycle.

When the source is moving towards the observer whose frame is R, the center of

each new wavefront is slightly displaced towards him. The wavefronts begin to

bunch up towards the observer and spread further apart behind the source. An

observer in front of the source will hear a higher frequency, and an observer

behind the source will hear a lower frequency.

Special Relativiy18

The same happens for electromagnetic waves from radars or lasers that are used

to measure speeds. Il is also observed for redshifted spectral lines emitted by

galaxies at the origin of the expanding universe and Big-Bang theories. There is

also a Doppler effect due to matter emission in supernovae. A similar effect is

the redshift due to gravitation at the surface of the stars that may be considered

as a Doppler effect only through the principle of equivalence of general

relativity.

In classical physics, the relative frequency shift, as seen by the observer is :

��

�s

=�r – �s

�s

= – vc

where vr is the velocity of the receptor and vs the velocity of the source. v is the

relative velocity between the source and the receptor, positive when the observer

(the receptor) goes away from the source. The frequency decreases when v > 0.

This formula needs only to be multiplied by the Lorentz factor � to remain valid

when v approaches the speed of light as we will show. The Lorentz

transformation of the time :

t =

t' +vx'

c2

1 -v2

c2

becomes, for a light ray of velocity c, with x’ = ct’,

t = � t' + v c t'

c2= � 1 + v

ct'

For one période, that is t = T in R and t’ = T’ in R’ :

T = � 1 + vc

T' =1 + v

c

1 – vc

T'

The frequency being the inverse of the period, one has :

�

�'=

1 – vc

1 + vc

When the velocity is positive, that is when the source and the receptor move

away from each other, the frequency perceived by the observer is lower. At low

speeds, � � 1, we may develop the formula up to the second order :

�r – �s

�s

� – vc

+ v2

2 c2= –

vr – vs

c+ 1

2

vr – vs

c

2

It differs from the classical formula by the second order term :


�r – �s

�s

� –vr – vs

c+

v2r

c2

For velocities near the speed of light, with a negative velocity, v � � c, we

obtain the ultra-relativistic formula :

�r

�s

= � 1 – vc

� � 1 + cc

= 2 �

The frequency increases indefinitely with the velocity. This formula is used in

the theory of the synchrotron. With a positive velocity, the frequency tends to

zero.

3.6. RELATIVISTIC STELLAR ABERRATION

The stellar aberration is similar to rain falling along the window of a train. The

rain is falling vertically when the train is at rest and inclined when the train is

moving. When the speed of the train is much larger than the velocity of the

falling rain, the rain appears to move horizontally.

Let us make a a simple thought experiment with a vertical tube standing up

under the rain. The rain falls in it to its bottom without touching the inner wall

of the tube. Now let us move : the rain will no more attain directly the bottom

of the tube. In order to do it, we have to incline the tube from an angle � such

that tg � = v/c, ratio of the velocity v of the falling rain and your speed c. This

formula, purely geometrical, has nothing to do with relativity. By replacing rain

by light from a star at infinity, one may do the same experiment with a

telescope. With v = 30 km/s, the velocity of the Earth around the sun and

c = 300.000 km/s, that of light, the angle is � = 21" = 10-4

radian, for the annual

aberration of stars. This calculated value is in accord with the numerous

observations made since the 18th century by Bradley. In order to show that the

phenomenon does not occur inside the telescope, Airy showed, by filling the

telescope with water, that the refractive index had no influence.

The stellar aberration should not be confused neither with the optical aberration

of optical instruments nor with the parallax of stars near the Earth. The stellar

aberration is a phenomenon similar to the Doppler effect but concerns the

direction of propagation instead of its frequency.

Let us now calculate the relativistic aberration. One may consider, according to

the principle of relativity that the star moves along the x’ axis of the reference

frame R’ at the velocity v of the Earth, motionless relative to the terrestrial

observer in the frame R. The axis x and x’ coincide. The light ray, with velocity

c, is inclined at an angle relative to x. The projections on the axis x’ and y’

are

x’ = ct’ cos ’ and y’ = ct’ sin ’

Special Relativiy20

The Lorentz transformation equations write, with � = v/c :

x = � x' + vt' = � ct' cos � + �

y = y' = ct' sin �

By making the ratio y/x, one eliminates t’ to obtain the relativistic formula of the

aberration of light :

tg � = yx

=sin �'

� cos �' + �

For a star at the zénith, the angle is ’ � 90°, that gives

tg � = 1� �

= cv

1 – v2

c2

When the velocity is low, the angle with the vertical line being also low,

� = ’ – is near 1/tg and one finds again the classical aberration of stars :

�� vc

An electron moving at a velocity low relatively to that of light emits

electromagnetic radiation in a wide range of directions. At a relativistic velocity,

near the speed of light, the light emission is concentrated towards the front of

the electron, the angle tends to zero :

� � 1 – v2

c2

The stellar aberration should prove that the Earth moves relatively to a

referential frame bound to the Aether (invented by Maxwell !). The Michelson

experiment had shown that the velocity of light was not influenced by that of the

Earth. To explain this, it was imagined that the Earth’s gravitational field

somehow “dragged” the aether around with it in such a way as locally to

eliminate its effect. If the velocity of the Aether were local to the Earth, the

stellar aberration would vary with the altitude, which is not the case. Therefore

stellar aberation is incompatible with the absence of Aether wind.

The stellar aberration is explained geometrically in classical kinematics. For

relative velocities approaching the speed of light, a relativistic correction is

needed. For example, the synchrotron radiation is concentrated towards the front

of the electron beam.

3.8. TRANSFORMATION OF ACCELERATIONS

Changement of reference frames is more complicated for accelerations than for

velocities. We shall restrict ourselves to rectilinear motion and to uniform

circular motion.


Classical kinematics

In classical rational mechanics, the term "Galilean transformation" was not in

use. One said only that kinematics differed from geometry by the introduction of

time. It seemed natural that displacements add vectorially. Velocity was simply

the vector derivative of displacement and acceleration the vector derivative of

velocity. For the sake of simplicity let us stay in only one space coordinate.

Let us consider the acceleration of an electron in a electric field. Let R be the

reference frame of the laboratory and R’ a Galilean reference frame. Let x and

x’, vx and ax, v’x and a’x, respectively, abscissas, velocities and accelerations of

the electron in frames R and R’.

According to the Galilean transformation, the velocity is the derivative of the

abscissa. For a constant velocity, we have

x = x’ + vt

By dérivation, we get

dx/dt = dx’/dt + v

or

v’x = v’x + v

This formula remains valid even if v varies. After a subsequent derivation we

obtain the acceleration :

ax = d2 x

dt2= d

dtdx'dt

+ v = d2 x'

dt2+ dv

dt= a'x + dv

dt

If the frame R’ coincides with the electron v’x = 0 and a’x = 0. Then :

ax = dvdt

We don’t need Galilean reference frames to know the acceleration. Let us see

what happens when using the Lorentz transformation.

Acceleration parallel to velocity

Simple me thod

In relativity, when changing from a frame R to a frame R’ of relative velocity v,

time dilates with speed and length parallel to the velocity contracts according to

the formulae :

dt = � dt' et dx = dx'/�

Let vx and ax, v’x and a’x, respectively, velocities and accelerations of a particle

with abscissas x et x’ in frames R and R’ of the motionless and mobile

observers. The acceleration is the second derivative of space relative to time.

Special Relativiy22

Therefore, length being divided by � and time multiplied by �, the acceleration

has to be multiplied by ��3. The acceleration is then :

ax = d2 x

dt2=

d2 x'�

d �t' 2= �–3 d2 x'

dt'2= 1 �

v2

c2

3

2 a'x

The acceleration is smaller for the observer than for the particle. We may also

write :

�3 dvx

dt=

dv'xdt'

If the electron is motionless in frame R’, then vx = v. In relativistic kinematics,

we need to take into account the Lorentz factor �(v) with v variable. Let us

compute

�3 = 1

1 – v2

c2

3= d

dvv

1 – v2

c2

=d �vdv

which is a formula found by Lorentz in his "Theory of the electron". Using this

result, we may rewrite the proper acceleration :

dv'dt'

=d �vdt

In the frame of the electron, the proper velocity is v’ = 0 but the proper

acceleration dv’/dt’ is not zero. This true also in classical mechanics. A

passenger at rest in the frame R’ of a lift will feel an acceleration with no

motion. In a relativistic speed, the acceleration measured in the lift, with an

accelerometer, is different from the acceleration measured from the ground, with

an optical method.

The proper acceleration of a particle is simply obtained by deriving, relatively to

the apparent time t, the apparent velocity v multiplied by the Lorentz factor �.

The relativistic acceleration measured by the observer differs from the proper

acceleration while they are equal in classical mechanics. The acceleration is zero

at the speed of light.

Bette r method

We have seen that the Lorentz transformation of the velovity is:

vx =v'x + v

1 +v v'x

c2

The acceleration being the derivative of the velocity, we have, with v = constant

for Galilean reference frames R and R':


dvx

dt=

v'x + v

1 +v v'xc2

= 1

1 +v v'xc2

dv'xdt

-v'x + v

1 +v v'xc2

2

v

c2

dv'xdt

Then, after simplification:

dvx

dt=

1 - v2

c2

1 +v v'x

c2

2

dv'xdt

The Lorentz transformation of the time increment is:

dt =

dt' + v dx'

c2

1 - v2

c2

=

1 + v

c2

dx'dt'

1 - v2

c2

dt' =

1 +v v'xc2

1 - v2

c2

dt'

Remplacing dt on the right of the above expression of the acceleration we obtain

the acceleration ax in the observer's frame as a function of the proper

acceleration a'x:

ax =dvx

dt=

1 - v2

c2

1 +v v'x

c2

3

dv'xdt'

=

1 - v2

c2

1 +v v'xc2

3

a'x

We may now put v'x = 0. If we had done it before derivation,we would have

gotten a null acceleration. The same thing would happen in classical kinematics.

We have then the above formula:

ax =dvx

dt= 1 - v2

c2

3

2dv'xdt'

= � - 3 a'x

Constant proper acceleration

An example of a constant proper acceleration g may be an electron accelerated

in a constant electric field or a mass in the constant gravity near the surface of

the Earth :

a'x =dv'xdt'

= g

We have then a differential equation :

Special Relativiy24

d �vdt

= g

or

d �v = d gt

that integrates into �v = gt. The integration constant is zero if the initial velocity

is v = 0 at t = 0. With

� = 1

1 � vc

2

One may write after integration :

v = c

1 + cgt

2

For slow speeds, that is for c = � and t = 0, the formula becomes v = gt. When t

increases indefinitely, the velocity approaches asymptotically the speed of light.

The apparent acceleration, for the observer in the R frame, decreases continually

toward zero but remains constant in the mobile reference frame R’. This

formula, used in particle accelerators, may be written :

v =dx

dt=

d

dt

1

2gt2

1 +gt

c

2

When t is small, the denominator is equal to one, giving the classical law of

falling bodies. When time increases, the velocity continues to increase, but at a

decreasing rate. The infinitesimal displacement

dx =c2

2g

dgt

c

2

1 +gt

c

2

may be integrated as

x =c2

g1 +

gt

c

2

After some algebra, one gets the equation of a hyperbola :

x2 � c2 t2 = c2

g2

This is the reason why the relativistic uniformly accelerated movement is called

hyperbolic.


Variable proper acceleration

We had obtained above the formula giving the relativistic acceleration:

d �vdt

= g

This formula remains valid for a variable acceleration like gravitation:

d �vdt

= �GM

r2= d

drGM

r

For a radial velocity, we may write v=dr/dt, which gives

vd �v

dr= d

drGM

r

and thus:

vd �v = d GMr

Now, we have the identity

d� = d 1

1 – v2

c2

= �12

1 – v2

c2

�3

2 �2v

c2dv =

�3

2d v

c

2= v

c2d �v

which gives

d� = d GM

c2 r

and integrates in

1

1 – v2

c2

= GM

c2 r+ constant

The gravitational potential energy of a proper mass m0 is:

V = � m0GM

r

Multiplying both sides by the proper mass m0 of the particle and by c2, we

obtain the relativistic conservation of energy:

m0 c2 1

1 – v2

c2

� 1 + V = constant

This is indeed T + V = constant.

Special Relativiy26

v = dxdt

= c 1 – 1

1 + V

m0 c2

2

By using the lorentz transformation of the acceleration and assuming that the

acceleration derives from a potential, we have obtained the relativistic

conservation of energy. From it, we deduced the relativistic velocity of a particle

in function of the potential. This approach is not valid for a photon in a

gravitational field.

Acceleration perpendicular to velocity

Acceleration is the second time derivative of the abscissa y, dy/dt, now

perpendicular to the velocity dx/dt. According to the Lorentz transformation,

there is no transverse contraction ; then y is not affected by the frame change :

y = y’. Only the time is dilatated. We have :

ay = d2 y

dt2= d2 y'

d �t' 2= �–2 d2 y'

dt'2= �–2a'y = 1 �

v2

c2a'y

This formula may be applied to electrons accelerated in a synchrotron where the

speed is practically v � c. The acceleration is centripetal and perpendicular to the

velocity, the trajectory being circular with radius r. The acceleration a’y in the

frame R’ of the electron determines the radiation :

a'y = �2 ay = �2 c2

rwhere r is the bending radius of the synchrotron as seen in the frame R’ of the

electron. c is the velocity of the electron, almost equal to the speed of light,

equal in R and R’. The classical Larmor formula gives the radiation power

emitted by the electron :

P = 14� �

0

2e2 a2

3c3

By replacing the acceleration in the frame of the electron we get :

P = q2

6� �0

c3�2 c2

r

2

=q2 c�4

6� �0

r2

It is also important to know the frequency of the radiation. At low speeds, the

frequency is the Larmor frequency �L, obtained by equating the centrifugal

force and the Lorentz force m0 v �L = evB in SI units as everywhere in this

book. The Larmor frequency is also �L = v/r :


�L = evBm0 v

= eBm0

= vr

It is no more necessary to know the magnetic induction, replaced by the radius

of the synchrotron, much easier to grasp. At the speed of light, v = c and the

radius r is contracted according to the Lorentz factor. The frequency of the

fundamental mode is then :

� 0 = �cr

There is both a relativistic Doppler and a relativistic aberration. Both multiplie

the frequency by �. The so-called critical frequency of the synchrotron is then :

�C � �3 cr

The spectrum produced by the synchrotron extends in a practically continuous

manner from the fundamental frequency �0 to the critical frequency �c. The use

of the special relativity theory avoids the use of retarded potentials and

simplifies greatly the calculation of the Larmor formula at relativistic speeds.

3.9. DIFFERENTIAL OPERATORS

Wave propagation is obtained by solving partial differential equations where

differential operators appear. � instead d are used in the presence of more than

one independant variable. We shall see how these total and partial derivative

operators

ddt

,�

�x,�

�t,�2

�x2et

�2

�t2

transform in the Lorentz transformation. The total differential of a function

f(x, t)is the same in the "motionless" frame R and in the "mobile" frame R’ :

df =�f�x

dx +�f�t

dt =�f�x'

dx' +�f�t'

dt'

Let us express the partial derivatives in R’ with the help of the Lorentz

transformation in differential form :

dx' = � dx – v dt

dt' = � dt – v dx

c2

Replacing dx’ and dt’ in the above total differential of f in R’ :

df = � dx – v dt�f�x'

+ � dt – v dx

c2

�f�t'

By grouping the terms in dx and dt one gets :

Special Relativiy28

df = ��f�x'

–�f�t'

v

c2dx + �

�f�t'

–�f�x'

v dt

By equaling both expressions of df, we have :

�f�x

dx +�f�t

dt = ��f�x'

–�f�t'

v

c2dx + �

�f�t'

–�f�x'

v dt

Identifying the dx and dt terms and suppressing the f we obtain the partial

derivative operators :

�

�x= � �

�x'– vc2

�

�t'

�

�t= � �

�t'– v �

�x'

It is to be pointed out that the minus signs are here on the side of the primed

variables. They are on the unprimed side in the original Lorentz transformation

above. We have also the reciprocal expressions:

��x'

= � ��x+ vc2

��t

��t'

= � ��t+ v �

�x

These formulas will be used to show the invariance of the electromagnetic wave

equation in the Lorentz transformation. When c = �, � = 1, one obtains the

Galilean transformation of the operators. The particle being motionless in its

proper reference frame R' with velocity v relative to R, we have dx’/dt’ = 0. The

derivative of f with respect to t’ is then:

dfdt'

= �f�t'

+ dx'dt'

�f�x'

= �f�t'

Using the preceding expression of �/�t’ one obtains:

dfdt'

= ��f�t

+ v�f�x

= �dfdt

Putting � = 1, on gets the formula of the material derivative of newtonian fluid

mechanics. Using the proper time = t', we may write the total derivative

operator:

dd�= � d

dtThe partial derivatives are different in both classical and relativistic kinematics.

The total derivatives are equal in classical kinematics but different in relativistic

kinematics. Anyway, putting � = 1 gives always the classical formula to which

one may refer in case of doubt about signs. In case of doubt about the position of

�, it suffices to remind that the classical formulas are valid in the proper frame.


These formulas will be used to check the conservation of the wave equations in

the Galilean and Lorentz transformations.

3.10. WAVE EQUATIONS

d’Alembert equation

From the Maxwell equations one may obtain the d'Alembert equation where the

celerity is that of light:

�2�

�x2+�2�

�y2+�2�

�z2–

1

c2�2�

�t2= 0

where � is the function representing the amplitude of the wave in the frame R.

We shall show that this equation is conserved in a Lorentz transformation. The

d’Alembertian operator

�2

�x2+

�2

�y2+�2

�z2�

1

c2�2

�t2= � +

�2

� ict 2

may be written, for the sake of simplification, in a two dimensional spacetime R,

x for space and t for time:

�2

�x2– 1

c2

�2

�t2=

�

�x+

�

c �t�

�x–

�

c �t

We have seen that, in the Lorentz transformation, the differential operators

transform as:

�

�x= �

�

�x'– v

c2

�

�t'

�

�t= �

�

�t'– v

�

�x'

where v is the velocity of R’ relative to R and

� = 1

1 – v2

c2

is the Lorentz factor. Replacing these operators by their expression in the wave

equation, one obtains:

�

�x+

�

c �t= � 1 –- v

c�

�x'+

�

c �t'

and also, by changing c in – c :

Special Relativiy30

�

�x–

�

c �t= � 1 + v

c�

�x'–

�

c �t'

which gives the wave equation:

� 1 – vc

�

�x'+

�

c �t'� 1 + v

c�

�x'–

�

c �t'� = 0

or

�2 1 – v2

c2

�

�x'+

�

c �t'�

�x'–

�

c �t'� = 0

which is the original equation since

�2 1 – v2

c2= 1

The wave equation is the same in R and R'. The variables are primed in R' and

unprimed in R. The wave function � and its celerity c are unchanged. The

electromagnetic wave equation is invariant in a Lorentz transformation but the

celerity has to be that of light in the vacuum. A sound wave equation has the

same form in the absence of entrainment but is not invariant under a Lorentz

transformation.

Hertz equation

The so-called Hertz equation is the equation of mechanical waves, valid with

entrainment, for example in a wind of velocity v:

�2�

�x2– 1

c2

d2�

dt2= 0

� is the wave function that may be the density, pressure, stress, strain, volume,

displacement… The main difference with the d’Alembert equation is the

presence of straight d's for a total derivative operator instead of round �'s for a

partial derivative operator in the time derivative. The celerity is not the celerity

of light but that of mechanical waves. One may often find this equation with

round �'s in the literature but it is correct only in the absence of entrainment. We

shall check that it is invariant in the Galilean transformation. We will write

explicitely the convective term:

�2

�x2�

1

c2�

�t+ v

�

�x2� = 0

where v is the entrainment velocity. In a Galilean transformation with velocity u

the "absolute" velocity in frame R becomes v = v’ + u where v' is the velocity in


the moving frame R' and u the velocity of R' relative to R. The derivation

operators may be obtained by putting c = �, � = 1 in the Lorentz transformation:

�

�x=

�

�x'�

�t=

�

�t'� u

�

�x'

Using v = v’ + u and these expressions, the total derivative operator becomes:

�

�t+ v

�

�x=

�

�t'– u

�

�x+ v

�

�x=

�

�t'+ v – u

�

�x'=

�

�t'+ v'

�

�x'

The velocity u of R' is eliminated. The total derivative operator is the same in

the R' frame except for the primes. The Hertz equation of waves, also called

non-linear, is therefore conserved in the Galilean transformation:

�

�x'

2– 1

c2

�

�t'+ v'

�

�x'

2� = 0

It may also be deduced, without calculation that it is also conserved in the

Lorentz transformation. This is true only if the velocity of light is used in the

Lorentz transformation and the celerity of the mechanical waves in the Hertz

equation. The mechanical wave equation is not conserved in a pseudo-Lorentz

transformation where the same c is used in both the wave equation and the

Lorentz transformation. A d'Alembert equation will not work in the wind.

3.11. MINKOWSKI SPACE-TIME

Minkowski metric

The Pythagorean theorem is conserved in a rotation since lengths are conserved.

We have seen that the Lorentz transformation is equivalent to a rotation of an

imaginary angle i such that tg (i) = iv/c :

x' y' =cos i�– sin i�

sin i�cos i�

xy

The length s of a segment has to be conserved in a rotation,in vertue of the

Pythagorean theorem :

s2 = x'2 + y'2 = x2 + y2

which gives, when y et y’ are replaced by ict and ict’ :

s2 = x'2 + ict' 2 = x2 + ict 2 = x'2 - c2 t'2 = x2 - c2 t2

A minus sign, due to the square of i appears. The euclidean planar space is

transformed in a flat pseudo-euclidean space called Minkowski space-time. It

Special Relativiy32

may directly checked, by replacing t' and x' by their expressions issued from the

Lorentz transformation:

x' = � x – vt

t' = � t – vx

c2

that the métric

s'2 = � c2 t'2 + x'2 = � c2

t � vx

c2

1 � v2

c2

2

+x � vt

1 � v2

c2

2

is conserved after developing and simplifying:

s'2 = 1

1 � v2

c2

x2 1 � v2

c2+ v2 � c2 t2 = � c2 t2 + x2 = s2

The Minkowski metric is conserved by a Lorentz transformation and is easier to

use than the Lorentz transformation. In general relativity, there is no practical

transformation.

Cartesian coordinates

The three-dimensional physical space is no more absolute since Einstein.

Minkowski has shown that the phenomena discovered by Lorentz and clarified

by Einstein could be described with a four-dimensional space. If the fourth

dimension is defined as the distance travelled during the time t multiplied by i

where i the square root of – 1 one obtains a four-dimensional euclidean space:

s2 = x2 + y2 + z2 + w2

where

w = � c2 t2 = ct � 1 = ict

Without i, it is the Minkowski pseudo-euclidean space. The proper distance is

then given by the metric:

s2 = x2 + y2 + z2 � c2 t2

It is recognized by the minus sign before the t2

term. Some authors put c = 1.

The Minkowski space-time is a euclidean space deformed by the combination of

uniform dilatation and shear. It is therefore without curvature and hence a "flat"

space with constant coefficients of the metric. It is not euclidean since the

coefficients are not equal to one. In the ordinary euclidean space, lengths are

conserved by translation or rotation. In the Minkowski space-time, the rotation

is replaced by the Lorentz transformation.


We shall now define more precisely the notion of metric. In euclidean three-

dimensional analytic geometry, the spatial distance dl between two near points

is:

dl2 = dx2

+ dy 2 + dz 2 =i

dxi dxi = dxi dxi

One uses the differential notation although it is not necessary in special

relativity where the movements are uniform, without acceleration. The sign �

may suppressed thanks to the Einstein convention where repeated indices denote

summation (not always) over their range. The generalized distance (or space-

time interval) ds between two events becomes in the Minkowski space-time:

ds2 = � c2 dt2 + dx2 + dy2 + dz2 = – c2 dt2 + dl2

dl is the displacement in the physical space during the time dt at velocity v.

There are no more parentheses here. This simplified writing is not really correct

but it is commonly in use. Indeed dx2

= 2x dx � (dx)2. Some authers use

parentheses but their formulas are not very readable.

The metric may also be written as:

ds2 = � c2 dt2 + dldt

2dt2 = � c2 dt2 + v2 dt2 = � 1 � v2

c2c2 dt2 < 0

In relativity, the velocity v being less than the speed of light c, ds2

is negative

therefore ds is an imaginary number. For that reason, one prefer often to use the

proper time :

d�2 = – ds2

c2= dt2 – dx2 + dy2 + dz2

c2> 0

Using the physical velocity

v =dx2

dt2+

dy2

dt2+

dz2

dt2

the metric writes

d�2 = 1 – v2

c2dt2 > 0

and is real. The velocity v of a photon is equal to the speed of light c, then

d = ds = 0. The trajectory of a photon is a staight line The length of the

trajectory is zero since all ist elements have a zero length. A massive particle has

always a velocity less than the speed of light. When its velocity is zero, that is

when the particle is motionless in its proper frame, we have d = dt. At low

velocities, d � dt with d < dt. The proper time is always smaller than the

physical time.

In four-dimensional Riemannian geometry, the metric is generalized as follows:

ds2 = gwwdw2 + gxxdx2 + gyydy2 + gzzdz2 = gijdxi dxj

Special Relativiy34

where w = ict. The gij are called coefficients ou components of the metric. A

four-dimensional metric tensor may be represented by a matrix:

gww

gwx

gwy

gwz

gwx

gxx

gxy

gxz

gwy

gxy

gyy

gyz

gwz

gxz

gyz

gzz

ou

g00

g01

g02

g03

g01

g11

g12

g13

g02

g12

g22

g23

g03

g13

g23

g33

Generally the indexes w, 4 or 0 correspond to the time. The matrix is symmetric,

in the diagonal terms like gxy dx dy, dx and dy may be commuted without

changing the value of ds2. Therefore gxy = gyx. Practically, for the sake of

simplicity, we shall use almost always diagonal matrices, without gxy, gxt… as in

the Minkowski metric:

ds2 = � c2 dt2 + dx2 + dy2 + dz2 = d ict 2 + dx2 + dy2 + dz2

that may be written, as in a Riemannian space:

ds2 = gttd ict 2 + gxxdx2 + gyydy2 + gzzdz2

where gtt = gxx = gyy = gzz = 1. gtt, gxx, gyy and gzz are the only non-zero

components of the metric Minkowski tensor. They are all equal to one, the

minus sign appearing only when the square of i is carried out. The signs of the

coefficients may vary according the conventions used. The sign of gtt is usually

opposed to the others but it seems preferable to use (ict)2

instead of � c2t2

or

even ± t2

with c = 1 which forbiddens any checking with dimensional analysis.

The Minkowski metric is represented by a 4 � 4 diagonal matrix :

gij =

-1000

0100

0010

0001

= �i j

or

1000

0-100

00-10

000-1

if the metric of type ds2 or d 2 (d 2 is sometimes called ds2, in a so-called West-

Coast or Lorentz metric). The �ij désignate the gij of the Minkowski metric. All

the diagonal �ij are equal to ± 1, using a physical or geometrical unit system.

The determinant is g = � 1 or g = 1 if the fourth dimension is ict. In this latter

case, the diagonal terms are all equal to one.

In general relativity, the coefficients of the metric are function of the coordinates

trough the gravitational potential and the Minkowski space becomes tangent to

the curved pseudo-Riemannian space-time. We shall consider the space-time of

general relativity as a four-dimensional Riemannian (not pseudo-Riemannian)


space with w = ict (Einstein uses x0

= ct) in order to avoid the minus sign

problem.

Spherical coordinates

Spherical coordinates are defined as the position vector r, the colatitude and

the longitude � :

Let us consider the small spherical rectangle on the sphere. Its width is

r sin d� and its height is r d. The Pythagorean theorem may be applied to

this rectangle to obtain its diagonal:

r d� = r d�2 + sin � d� 2

Simplifying by r, d� gives the metric on the sphere. We may similarly

increment r with dr to obtain a new rectangular triangle

dl = dr2 + r2 d�2

Special Relativiy36

A last step gives the length element in a four-dimensional euclidean space with

the fourth dimension w = ict:

ds = d ict 2 + dl2 = d ict 2 + dr2 + r2 d�2

Replacing d� we obtain the full metric:

ds2 = d ict 2 + dr2 + r2 d�2 + sin2� d�2

which is the pseudo-euclidean Minkowski metric:

ds2 = – c2 dt2 + dr2 + r2 d�2 + sin2� d�2

or, in matrix form:

gij =

gtt

00

0

0grr

0

0

00

g��

0

000

g��

=

± 1000

0100

00r2

0

000

r2 sin2�

In radial symmetry the metric simplifies:

ds2 = � c2 dt2 + dr2

3.12. RELATIVISTIC LAGRANGIANS

Variational calculus

The calculus of variations is issued from the principles expressed by Heron of

Alexandria, Huygens, Fermat, Hamilton, d’Alembert, Maupertuis and also from

the works of Lagrange, Euler and others. The Lagrange equations may be

obtained either from variational principles or Newton's laws. These ideas may

be resumed by the principles of the shortest way (geometric aspect) or of the

least effort (mechanical aspect).

The effective trajectory is the one corresponding to the extremal way or time.

The derivative of the way has to be zero all along the way.

The shortest way in a plane

In order to find the shortest way from one point to the other, for example on a

surface, one has to know the metric giving the shortest distance between two

nearby points. It is important to define the metric in terms of differential

changes in the coordinates since not all coordinate systems are linear like the

Euclidean ones. The Pythagorean theorem defines the metric of the plane where

the line element is given by:

ds2 = dx2 + dy2


On a surface, the Pythagorean theorem is generalized by the formula invented

by Gauss:

ds2 = gxx dx2 + 2 gxy dx dy + gyy dy2

The shortest way between two points A and B in the plane is:

S =A

B

ds =A

B

dx2 + dy2 =A

B

1 + y 2 dx =A

B

L y dx =

A

B

L dydx

dx

where the Lagrangian is

L y = 1 + y 2

and

y = y' =dy

dx

is the slope of the curved way.

The symbol � (lower case delta) instead of d (straight d for total differential), or

� (curly d for partial derivative), shows a virtual infinitesimal variation.

Developing the Lagrangian L in the first order, we may write the virtual

variation of the way ds:

� ds = �L y dx =�

�yL y �y dx =

�L y

�y

ddx

�y dx

The differentiation being commutative, one may write:

� ds =�L y

�y�

dydx

dx =�L y

�y�y dx

Expliciting the Lagrangian, we get:

� ds =�

�y1 + y x 2

�y dx

Let us write:

A =�

�y1 + y 2 et B = �y

Let us integrate by parts this expression :

1

2�

�y1 + y 2 d

dx�y =

1

2

A dBdx

= dAdx

B2

1–

1

2

B dAdx

= 0 –

1

2

�y ddx

�

�y1 + y 2

The integrant has to be null whatever �y:

ddx

�

�y1 + y 2 = 0

Special Relativiy38

or, using L:

ddx

�L

�y= 0

Carrying the partial derivative relative to y', the Lagrange equation becomes:

d

dx

2 y x

2 1 + y x2

= 0

It integrates in

y x

1 + y x 2= constant

or y’ = dy/dx = constant. The trajectory y(x) is a straight line.


4. RELATIVISTIC DYNAMICS

4.1. INTRODUCTION

The Lorentz transformation, like the Galilean transformation is supposed to be

valid only between Galilean reference frames. We have seen above that the

observed acceleration is a function of the proper acceleration through the

Lorentz formula of the accelerated electron. Accelerated motion is therefore not

out of the scope of special relativity.

Relativistic dynamics is the special relativity with addition of Newton' laws.

Mass dilatation results simply from the application of the Lorentz

transformation to the acceleration as we shall see. If the acceleration is defined

as the derivative of the velocity, Newton's second law must be written with the

variable mass included in the derivand:

F = d mvdt

Energy being the product of the force and the displacement, as in classical

mechanics, one obtains an expression that reduces to the newtonian formula at

low speeds. When the effort F acts on a body and make it move of an increment

dx, the work done by F is transformed into kinetic energy dT = F dx. By

integration of this equation, one obtains the kinetic energy � mv2

in newtonian

mechanics where the mass is constant. In relativistic dynamics, we have to take

account of the variable mass, function of the velocity. We will show that the

kinetic energy depends only on the mass via the velocity and a universal

constant proportionality factor.

4.2. RELATIVISTIC MASS

The Lorentz transformation of colinear accelerations is given by the formula:

d �vdt

= dvd�

where t is the time in the frame R of the observer ; t’ is the proper time in

frame R’ of the particle ; v is the velocity of frame R’ relative to R as defined

for the Lorentz transformation. The accelerations are the same in both frames

when � = 1 e.g. for velocities low relative to the speed of light. Let us multiply

both sides of the preceding equation by the proper (or intrinsic or rest), mass of

the particle, constant and independant of the observer, m0:

Special Relativiy40

d �m0 v

dt=

d m0 v

d�

The relativistic (or inertial or apparent) mass m, depending on the reference

frame, is defined by the relation

m = � m0 =m0

1 – v2

c2

If one does not want to use the relativistic mass, one has to use the relativistic

acceleration as defined earlier or to always replace the mass by �m0. We will use

here this French et Feynman notation, except, eventually, at low speeds where

they are equal. According to this formula, the relativistic mass dilates with the

same law as the time and increases indefinitely when the velocity tends to the

speed of light.

The Young’s double slits experiment with single photons shows the double

nature, undulatory and corpuscular, of light. It is often asserted that the photon

has no mass. This is of course true but only for the rest mass since, for v = c, the

denominator of the above formula being zero, the numerator has also to be zero.

The relativistic mass of the photon may be determined only from the

equivalence of mass and energy and the quanta hypothesis.

The variable mass is useful to allow a generalization of Newton' laws in the

domain of relativistic velocities, near the speed of light.

4.3. RELATIVISTIC NEWTON'S SECOND LAW AREVOIR

The Lorentz transformation of the accelerations,

d �vdt

= dvd�

becomes, when multiplied by the rest mass m0 :

d �m0 v

dt=

d m0 v

d�

where is the proper time, in the mobile frame where the particle is at rest. v is

the relative velocity of the observer to the particle reference frames. In the

proper reference frame, the second Newton's law applies classically since the

velocity is zero, thus low relative to the speed of light. One may then write :

F =d m0 v

d�=

d �m0 v

dt=

d mvdt

The force having the same value in the observer's and in the particle reference

frames, is therefore conserved in a change of frame, in one dimension of space


at least. In relativistic dynamics, the force, according to the relativistic Newton's

second law is:

F =dp

dt

where

p =m0 v

1 � v2

c2

is the momentum, product of relativistic mass and velocity. The proper time

does not appear here any more since everything happens in the observer's frame.

Let us take the example of a voyager moving away in a rocket and an observer

remaining on the Earth. Both will be able to measure their relative acceleration

with the help of an optical instrument like a laser velocimeter. The voyager will

measure his acceleration with a mechanical accelerometer made of a load

attached to a spring. With an identical instrument, the observer will measure the

acceleration of gravity. Only the optical method will give the same relative

acceleration for the observer and the voyager. In order to get the same result

with the optical and mechanical measures, the voyager will have to subtract the

acceleration of gravity, varying with the distance from the Earth. He will know

the force from the ballistic caracteristics of the rocket.

Now, what happens at relativistic speeds? As the rocket reaches the speed of

light, the relative acceleration tends to zero but the proper acceleration may

remain constant if the proper force is constant.

In newtonian mechanics, will both measures give the same result at relativistic

velocities? The optical method will give the constant velocity of light and

therefore a null acceleration. The mechanical method will give the the assumed

constant proper acceleration. The applied force may be known from the ballistic

caracteristics of the rocket. The observer on Earth has no means to know the

thrust.

Anyway how to measure independently acceleration and force?

The voyager measures his proper acceleration with the accelerometer and the

relative acceleration with the laser. At the speed of light, he will be unable to

measure anything. At a slightly lower speed, he will measure a

Of course, Newton's law is valid in every frame but, in the proper frame, the

acceleration relative to the observer is measurable with a mechanical

accelerometer and an optical instrument. The observer on Earth is able to

measure the acceleration with an optical instrument only. He is unable to

measure the force.

Another example is that of a lienarly accelerated electron. Only the accelerating

potential (or the electrostatic field) and the velocity may be known.

Special Relativiy42

est la quantité de mouvement fonction de la masse au repos m0 et de la vitesse v.

Le temps propre n’apparaît plus ici car tout se passe dans le référentiel de

l’observateur. One uses the letter a = dv/dt rather than � to designate the

acceleration in order to avoid confusion with the Lorentz factor. Knowing that

p = �m0v, when the force derives from a potential V, one may write

F = m0ddt

v

1 – v2

c2

= –�V�x

This is the same as using the relativistic mass

m = � m0

with the classical acceleration

dv

dt

or the rest mass m0, invariable, with the relativistic acceleration

d �vdt

Newton's second law of motion is relativity compatible if one takes into account

the mass variation with velocity. It needs only to derive momentum instead of

the velocity alone. Another method would be to consider the relativistic

acceleration, not used.

4.4. ENERGIE CINÉTIQUE

In classical mechanics, the kinetic energy is T = �mv2. The velocity v,

according to relativity, is limited by the speed of light. The maximum kinetic

energy would be �mc2 if the mass were independent of the velocity. It is a first

approach of the relativistic energy.

A second approach is to calculate the classical kinetic energy T with the

relativistic mass:

m =m0

1 – v2

c2

or, for m � m0:

T = mv2

2= m c2

21 –

m0

m1 +

m0

m� m � m0 c2

The kinetic energy is proportional to the mass variation. We shall show that it is

the relativistic formula. Let us apply the relativistic newtonian law. The

variation dT of the kinetic energy being equal to the work of the applied force F


during the displacement dx, we have, by applying the relativistic second

newton's law:

dT = F dx = F v dt = m0

d �vdt

v dt = m0 v d �v

Having the identity:

v d �v = v d v

1 – v2

c2

= v dv

1 – v2

c2

+

v2 v dv

c2

1 – v2

c2

3

2

= v dv

1 – v2

c2

1 +

v2

c2

1 – v2

c2

= dv2

2 1 – v2

c2

3

2

= d 1

1 – v2

c2

= d�

The incremental kinetic energy dT = m0 d� may be integrated:

T = m0� c2

+ constant

The constant is obtained by noticing that the kinetic energy must be zero at rest

when � = 1. The constant is therefore - m0c2

and the kinetic energy:

T = m – m0 c2

The relativistic kinetic energy is proportional to the mass difference between

rest and motion.

4.5. E = MC2

The conversion of mass in energy had already being considered by Newton.

Formulas like Einstein's had been proposed by Thomson, Heaviside et Poincaré.

Lise Meitner used Einstein's theory to show that the mass lost during the fission

of uranium was changed changed to energy.

We shall derive, using the expression of the kinetic energy, T = (m – m0) c2, the

most famous formula of modern physics. c and m0 being constant, the increase

of the kinetic energy is due only to the increase of the relativistic mass m. In

classical mechanics, the energy E is undetermined to an arbitrary additive

constant E0. We may choose it such that E = T + E0 = m c2. The total energy in

motion is then E = �m0c2. At rest, v = 0, then � = 1 and E0 = m0c

2. The rest

energy is a constant for a particle at rest.

Rather than choosing arbitrarily E0, one may call a evident principle. Indeed, the

proportionality between mass and energy is well known in practice, for example

by the car drivers. The energy contained in a given mass of fuel is proportional

to it according to a coefficient K depending on its heat content. There should

exist a maximum value of K corresponding to the maximum energy available

when all the matter is transformed into pure energy. K should be a universal

constant independent of the reference frame and from the velocity if mass and

energy are equivalent. For a given object, the total energy will be:

E = K m

in the frame of the observer and

Special Relativiy44

E0 = K m0

in the prper frame of the object. The difference in these two energies is due only

to the velocity: it is the kinetic energy:

T = E � E 0 = K m – m0

K being a universal constant by assumption, only the mass depends on the

speed. Now, the application of the second law of Newton combined with the

definition of energy had shown that the kinetic energy was:

T = m � m0 c2

Identifying these two las expressions, one finds K = c2 and, therefore, the total

energy in motion or at rest is:

E = m c2

The Lorentz factor

� = mm0

= m c2

m0 c2= E

E 0

represents the ratio of the total energy in motion to the total energy at rest as

well as the ratio of the corresponding masses. The available energy in a particle

depends on the observer e.g. if the particle is in motion or not relatively to the

observer. This is not only true for relativistic velocities but also in classical

mechanics. A car driver is often only aware of the damage he can cause at the

time of a shock. The kinetic energy, even newtonian, is relative since it exists

only relatively to an obstacle, that is, depends on the reference frame.

All the derivations leading to E = mc2, need additive hypothesis

In a few words we shall resume the reasoning conducting to this formula. Its

origin is in the velocity of any material object limited to that of light. If a

constant force is applied to the object to accelerate it, the velocity being limited

and the force constant, it is necessary that the mass increases to avoid

overcoming the velocity of light. The simplest formula giving an infinite mass

for v = c is the dilatation of mass given by relativity:

m =m0

1 – v2

c2

From this formula one gets the newtonian kinetic energy as a function of mass,

approximated at low velocity but also valid for relativistic velocities:

T = m – m0 c2

By assuming proportionality between mas and energy, one finds that the

proportionality constant is c2.

All the demonstrations using the transformation of matter into light or collisions

need one or two supplementary assumptions. The hypothesis of proportionality


of energy and matter with a universal constant seems better. The kinetic energy

is thus proportional to the mass variation. Using the relativistic formula for the

kinetic energy one obtains the value of the coefficient K = c2.

4.6. POTENTIAL ENERGY

The variation dV of the potential energy is the product of force F and

displacement dx with opposite sign. In the international system (SI), the

potential energy is expressed in joules (J or N.m.). The energy units

The second Newton's law gives the relationship between potental and kinetic

energy.

dV = � F x dx = �d mv

dtv dt = � d mv2

2= � dT

V is the potential energy, not to be confused with the potential like the

gravitational potential equal to the potential energy divided by the mass. The

gravitational potential energy is always negative except eventually near the

Earth's surface. The electrostatic potential is the potential energy divided by the

electric charge. The sign of the electrostatic potential depends on the sign of the

electric charge. The kinetic energy T is always positive. An adimensional

potential is represented by the letter �. The field is the derivative relative to

space of the potential. The force is the space derivative of the potential energy.

The potential disappears in special relativity, when switching from Newton to

Einstein, reappears in relativistic dynamics as the Lagrangian "à la Landau" and

in general relativity in the metrics, disappears again in the Einstein equations in

the same way as in the gravitational or electrostatic Laplace equation.

In classical mechanics, the total mechanical energy is the sum of the kinetic and

potential energies. The conservation of energy is a consequence of Newton's

laws and of the definition of energy. Conservation of energy is expressed by the

relation T + V = constant expressing the relation between kinetic and potential

energies. In special relativity, the total energy is E = mc2, without any reference

to any potential energy. In relativistic dynamics, the conservation of energy

could be written as

T + V = m � m0 c2 + V = constant

Using the definitions of the classical total mechanical energy and of theLagrangian L:

m0 c2

1 � v2

c2

� m0 c2 + V = V0

We will encounter below the Lagrangians "à la Landau":

Special Relativiy46

L = � m0 c2 1 � v2

c2� V(x)

and in Newtonian limit of general relativity:

L =d�d�

= dtd�

1 – v2

c2+ 2V

m0 c2= 1

both differing from the first one. This problem seems to be the clue of the

incompatibility between special and general relativity.

4.7. ELECTRON ACCELERATION

Energy

The total mechanical energy, sum of kinetic T and potential energy V, is an

arbitrary constant in the absence of dematerialization. In relativistic dynamics

the kinetic energy being T = (m - m0) c2, the conservation of energy writes:

m0 c2

1 � v2

c2

– m0 c2 + V =m0 c2

1 �v2

0

c2

� m0 c2 + V0

where V and v, V0 and v0, are respectively the potential and velocity at two

different places in the physical space. We may write v0 = 0:

m0 c2

1 � v2

c2

� m0 c2 = V0 � V = �V

or

v = c 1 � 1

1 + �V

m0 c2

2

�V must be positive in order to have a real value of the velocity v. Therefore,

the formula is not applicable to gravitation nor to an attractive electrostatic

Coulomb force. The velocity tends asymptotically to the speed of light c when

the potential difference increases indefinitely as is observed in particle

accelerators.

To check experimentally the formula, the velocity of the particle is measured as

a function of the applied potential. The first measures were made in 1915 by

Guye and Lavanchy measured in 1915 the ratio e/m in function of the velocity.

Bertozzi, in 1964, measured the speeds of electrons with kinetic energies in the


range 0.5–15 MeV. The kinetic energy, determined by calorimetry,verifies that

an electric field exerts a force on a moving electron in its direction of motion

that is independent of its speed. Four experimental points seem to be

insufficient. More precise measurements should be made.

The Stanford linear accelerator (SLAC) is three kilometers long to accelerate

electrons to 20 GeV with 82.650 one inch long accelerating structures divided in

three cells. The accelerating voltage is thus less than 100 kV per stage, clearly

less than 0.5 MeV, the total rest energy of the electron.

The circular trajectory of cyclotrons and synchrotrons is obtained thanks to the

magnetic part of the Lorentz force, perpendicular to the trajectory. The Lorentz

and cetrifugal forces are in equilibrium (SI units):

q v B = m0v2

Rwhere B is the magnetic induction, R the radius of the ring, me the rest mass and

e the electric charge of the electron. In practice B and R have to be adjusted in

function of the speed desired:

B R =mev

e 1 � v2

c2

The magnetic field being limited by the power of the electro-magnets, the

accelerators have an increasing size, like that of the CERN with a radius of

4 km.

Time

Electrically charged particles are accelerated by an electrostatic field. We use

here the word acceleration in the sense of increase of velocity, while it is

increase in energy for accelerator specialists. It may be understood since a

particle reaches the speed of light for relatively low energies, of the order of one

MeV for an electron and one Gev for a proton.

Let us apply the relativistic second Newton's law to an electron with a constant

eletrostatic acceleration:

ddt

v

1 � v2

c2

= �dVdx

= Fme

= eEme

= g

where e is the electric charge, E the electric field, me the mass of the electron

and g the constant proper acceleration. The calculation, already seen, gives

Special Relativiy48

v =c

1 +mec

e E t

2

The velocity of the electron tends asymptotically to the speed of light c.

4.8. RELATIVISTIC LAGRANGIAN "À LALANDAU"

In Minkowski space the motion is rectilinear and with constant speed. We shall

determine the lagrangian of a particle subjected to a force deriving from a

potential V :

F = –�V�x

The second Newton law is :

ddt

mv = –�V�x

where m may vary with speed or some other variables. The time is the observer

time. In relativistic dynamics we have

ddt

m0 v

1 – v2

c2

= –�V�x

where m0 is the proper masse, constant. The expression in parentheses may be

integrated :

ddt

�

�v– m0 c2 1 – v2

c2= –

�V�x

If the potential is independant of the velocity v, one may subtract it, on the left.

On the right side one may add the derivative of the radical, independant on the

abscissa x :

ddt

�

�v- m0 c2 1 - v2

c2– V –

�

�x– m0 c2 1 – v2

c2– V = 0

Let us define the lagrangian "à la Landau" as

L = � m0 c2 1 � v

2

c2� V(x)

The preceding equation becomes the Lagrange equation :

�L�x

– ddt

�L�v

= 0


If V = 0, one gets the relativistic lagrangian of a free particle :

L = 1 – v2

c2

The lagrangian "à la Landau" differs from the relativistic T – V :

T � V = m � m0 c2 � V = m0 c2 1

1 � v2

c2

� 1 � V

There seems to be a problem, even if both lagrangians give the newtonian

lagrangian at low velocity :

L �12

m0 v2 � V x + constant

The lagrangian "à la Landau" is used in particle accelerators taking into account

the electrostatic and magnetic potentials :

L = – m0 c2 1 – v2

c2– q �(x) + q v •A

The potential V is replaced by q� where q is the electrostatic charge and � the

electrostatic potential. A is the vector potential.

The lagrangian "à la Landau" works for acceleration energies larger than the

total rest energy of the accelerated particle. From the fundamental law of the

relativistic dynamics we have obtained a "relativistic" lagrangian where the

distinction between proper time and absolute time does not appear. This

lagrangian is incompatible with Minkowski space and seems unable to predict

any light deviation by the sun, contrarily to newtonian mechanics as we shall see

in the following chapter dedicated to general relativity.

4.9. ANTIMATTER

The total energy E = mc2

may be writen :

E 2 = m2 c4 =m2

0 c4

1 -v2

c2

-m2

0 c2 v2

1 -v2

c2

+m2

0 c2 v2

1 -v2

c2

=m2

0 c4

1 -v2

c2

1 -v2

c2+ mc2 v2

By replacing mv by the linear momentum p one obtains a useful relation, called

dispersion relation between energy E and relativistic momentum p

E2= m

2

0c4+ p

2c2

This formula works for a zero proper mass particle like a photon. The mass

being squared, by taking its square root, there are two solutions with positive

and negative masses :

Special Relativiy50

E = ± m20 c4 + p2 c2

According to quantum mechanics also, there should exist negative masses called

antimatter but the existence of negative masses has never been proved. When

one speaks of antiparticles, it is about particles of the same mass but of opposite

electrical charges. A photon and an antiphoton cannot be distinguished. The

antineutron has been discovered in 1956 through its annihilation, but has not

been observed directly.


5. CONCLUSION ON SPECIAL RELATIVITY

The unsolvable problems encoutered at the end of the 19th centuryhave been

clarified by Einstein with his special relativity. He has rederived the Lorentz

transformation with a different basis. He modified the classical mechanics by

taking again the Galilean principle of relativity abused by Newton with his

absolute time and space. The Galilean transformation is replaced by that of

Lorentz, so that speed and acceleration are not any more simple derivatives of

space with respect to time.

The speed of electromagnetic waves is that of light and depends only of electric

and magnetic properties of matter measured in the laboratory. The

electromagnetic wave equation does not depend on any absolute reference

frame, contrarily to mechanical waves. The light wave is insensitive to the wind

even of Aether.

The Michelson experiment did not give the result predicted by the Newtonian

mechanics, even with the use of extra-terrestrial light. Lorentz and Fitzgerald

invented time dilatation and length contraction. Stellar aberration, pi-ion

experiment, double star Algol, none of them contradicts the constancy of light

speed, at least in the absence of gravitation. Superluminal velocities of so-called

tachyons would have been observed but have been explained by a perspective

effect. The measure of mesons lifetimes, the Fizeau experiment and the Doppler

effect are quantitatives verifications of the Lorentz factor and of the slowing

down of the time.

The relativistic dynamics, useful in practice, is a generalization of the newtonian

dynamics. Adding the hypothesis of proportionality between energy and mass

leads to the well known formula E = mc2. The relativistic lagrangian "à la

Landau" is equivalent to the relativistic Newton's second law, useful in the

particle accelerators, but ineffective for gravitation.

Therefore, the theory is incomplete, as compared to rational mechanics valid in

electrostatics and gravitation although not at speeds near that of light.

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Special Relativity

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