Original exercises are proposed at the end of this lecture.

I) Introduction

A notion, often badly developed in physics, is the measurement, relatively
to a system of reference.

The measurement of times, the measurement of speeds, the measurement
of dimensions of a moving body, relatively to a referential, require precise
definitions.

Common sense is not sufficient, it induce to errors and paradoxes.

Some aspects of these measurements are never approached, and never illustrated by simple examples.

Have you in the past, heard speaking about:

The speed of a plane, relatively to the earth brought to its system of time zones?

The length of a mobile ruler, relatively to the earth on which times are everywhere measured with sundials?

Two simultaneous events, relatively to the earth and its time zones?

Do you know what are coordinates of a terrestrial event, relatively to the earth and its system of time zones? (This last point is better known. )

II) Beginning of the study

You will find hereafter illustrations of these different measures.

On earth exist two measurement systems of times.

1. The universal time given by the meridian of Greenwich.

This time is used to date astronomical events, and
serves as reference to the calculation of times given by time zones.

2. Local times given by terrestrial time zones. They are close to times given by sundials.

When one told to someone: I will meet you tomorrow at 10 o=clock
in New-York, it is always implied that this number 10 will be given by
a New-York clock (and not by a Parisian clock).

In our terrestrial activities, one refers always to local times.

All along this text, the measurement of lengths on earth is the classical
measurement, the taken unit being one kilometer.

Take care: this measurement is the measurement of motionless bodies,
of bodies fixed to the ground.

I remember that the distance Paris New-York is approximately 7000 km,
and that local times are offsetted of 5 hours, relatively to the universal
time.

When it is 10 o=clock in
Paris, time given by a Parisian clock, it is only 5 o=clock
in New-York, time given by a New-York clock .

And more, I suppose that the Parisian time is the time of Greenwich
meridian .

Hence, Parisian clocks indicate the universal time.

These data are not matching exactly geographical data, but I take these data to illustrate my purposes.

In following, the earth on which times are given by the universal time
is noted **T _{u}** , the earth brought to its time zones is
noted

Imagine a moment that we have only the possibility to measure times with sundials, events would be then exclusively dated by local times,

One supposes all along this text, that the local time given by a sundial,
always coincides with the time of its time zone.

III) Notion of simultaneity, speed, length, relatively
to **T _{u}** , relatively to

1. __Coordinates of an event.__

Let an explosion, **En ,** which happens in New-York at 10 o=clock,
local time.

The coordinates of this event, relatively to **T _{l}** ,
are (New -York, 10).

The coordinates the of this same event, relatively to **T _{u}**
, are (New-York, 15).

(In the couples, the word New-York, can be replaced by the terrestrial coordinates of this city.)

Every event on earth is thus located by its place and by a date.

The date being always given by the terrestrial clock found on the place
of the event at the moment it produces.

If the system of reference is **T _{l}** , clocks are sundials
(the time given by a sundial is supposed coinciding with the time of its
time zone) .

If the system of reference is

Retain thus that the date of an event is always given by a machine (a
clock) that is found in the center of the event at the moment it happens
(sundial for **T _{l}** , radio receiver for

2. __Simultaneity__.

Let an explosion **Ep**, produced in Paris at 10 o=clock,
local time given by a Parisian clock.

Coordinates of **Ep** , relatively to **T _{l}** , are:
(Paris, 10).

Let an other explosion **En,** produced in New-York at 10 o=clock,
local time given by a New-York clock.

Coordinates of **En** , relatively to **T _{u}** , are:
(New-York, 10).

__By definition__, these two events **Ep** and **En** are simultaneous
realtively to **T _{l}** (Relatively to the earth brought to
its time zones).

Relatively to

Coordinates of

One sees immediately that two explosions, one in Paris and the other
in New-York, simultaneous relatively to **T _{u}** , are not
simultaneous relatively to

They are offsetted of 5 hours relatively to

3. __Measurement of speeds.__

Let a plane that leaves Paris at 10 o=clock,
local time given by a Parisian clock (or Parisian sundial).

One supposes that this plane arrives to New-York at 15 o=clock,
local time given by a New-York clock .

__By definition__, the average speed of this plane, on the flight
Paris New-York, relatively to **T _{l}** is:

Vl = 7000 / (15-10) =1400 km / h.

1400 is a meaning number, relatively to **T _{l}** , only.

The average speed of this plane, on the flight Paris New-York, relatively
to **T _{u}** is :

Vu = 7000 / ((15+5) -10) =700 km / h

A same plane leaves now New-York at 8 o=clock, local time given by a New-York clock, and arrives to Paris at 23 o=clock, local time given by a Parisian clock.

__By definition__, the average speed of this plane, on the flight
New-York Paris, relatively to **T _{l}** is:

V'l = 7000 / (23 - 8) = 7000 / 15 = 466,66.. km / h

The average speed of this plane, on the flight New-York Paris, relatively
to **T _{u}** is :

V'u = 7000 / (23 - (8+5)) = 7000 / 10 = 700km / h = Vu

Let us suppose, as previously, that times are measured only by sundials.

The alone manner to measure a speed, would be this given by the above
definition.

And one would ask many questions about the origin of the difference
between Vl and V'l.

Remark 1: When one increases the Vu speed of the plane , relatively
to **T _{u}** , in the direction Paris New-York, the speed Vl
of the plane, relatively to

Think to the Concorde plane, the local time of landing down to New-York is always inferior to the local starting time from Paris.

The limit speed, relatively to **T _{u}** , in the direction
Paris New-York, from which one obtain paradoxes in

A plane that displaces in the direction Paris New-York at a speed of 1400 km / h , relatively to

Remark 2: In the direction New-York Paris, whatever be the speed of
the plane, relatively to **T _{u }** , the local time of
landing is always greater than the local time of starting.

V'l never becomes infinite, and has always same sign.

And more, one notices that V'l (in absolute value) is always strictly inferior to 7000 / 5 = 1400 km / h.

4. __Length of a moving body__ .

Let imagine now a very long rigid plane (or a file of airplanes).

The tail of this plane is seen above Paris at 10 o=clock exactly, local time given by a Parisian clock.

The nose of this plane is seen above New-York at 10 o=clock exactly, local time given by a New-York clock .

Relatively to **T _{l}** , one sees simultaneously , the tail
of the plane above Paris, and the nose of this same plane above New-York.
.

__By definition__, the length of this plane, relatively to **T _{l}**
, is the distance Paris New-York.

The plane is measuring 7000 km , relatively to

The length of this plane, relatively to **T _{u}** , cannot
be known, so longer is not given the speed of this plane relatively to

Hence, let us assume more that the speed of this plane, in the direction
Paris New-York, is 1400 km / h , relatively to **T _{l}** , (
700 km / h , relatively to

When the tail of the plane is above Paris at 10 o=clock,
local time, its nose is above the sea, between Paris and New-York, and
the plane has still to fly during 5 hours, before its nose will be found
above New-York (I reason relatively to **T _{u}** , it is easier).

The length of this plane, relatively to

A ruler of 3500 km , relatively to **T _{u}** , moving in
the direction Paris New-York at the speed of 700 km / h, relatively to

Let now an other very long rigid plane, moving in the sense New-York Paris.

The speed of this plane , relatively to **T _{l}** , is assumed
to be 7000 / 15 = 466.66 km / h.

Its nose is seen above Paris at 10 o=clock, local time, its tail is seen above New-York at10 o=clock, local time.

As in the preceding case, the length of this plane, relatively to **T _{l}**
, is 7000 km, because the tail and the nose of the plane, are, in

Question: what is the length of this plane , relatively to **T _{u}**
? Reply: 10500 km.

I leave to the reader the care to explain.

A ruler of 10500 km, relatively to **T _{u}** , moving in
the sense New-York Paris at a speed of 700 km / h, relatively to

IV) Measurements, in the plane brought to local times.

In the very long plane of 3500 km, relatively to **T _{u}**
, moving in the sense Paris New-York, at speed of 700 km / h, relatively
to

In the plane, the lenght of a body fixed to the plane, is measured of usual manner, the unit being one km.

This system is noted

1. __Measurement of a length, relatively to__
**A _{l}** .

Let us determine the distance Paris New-York, relatively to **A _{l}**
.

We have seen above, that, if the tail of the plane is seen above Paris at 10 o=clock, Parisian local time, then the nose of the plane is seen above New-York at 10 o=clock, local time of New-York.

Times are measured in the plane with sundials.

When the tail of the plane is above Paris, the sundial of the tail
indicates 10 o=clock, just as
sundial situated on the ground of Paris.

Coordinates the of the event "to see Paris since the tail of the plane"
are, relatively to **A _{l}** : (tail of the plane, 10 ).

The nose of the same plane is above New-York at 10 o=clock, local time of New-York, the sundial of the nose of the plane indicates equally 10 o=clock as the sundial situated on the ground of New-York. .

Coordinates the of the event "to see New-York since the nose of the
plane" are, relatively to **A _{l}** : (nose of the plane, 10
).

The two preceding events are simultaneous, relatively to **A _{l}**
.

Relatively to

Relatively to

A ruler of 7000 km fixed to the ground is perceived under the distance
3500 km in **A _{l}** .

2. __Measurement of the speed of Paris , relatively
to __**A _{l}** .

One supposes the nose of the plane above Paris at 5 o=clock,
local time of Paris.

The sundial situated in the nose of the plane indicates also 5 o=clock.

Coordinates the of the event "one sees Paris since the nose of the
plane" are, relatively to **A _{l}** : (nose of the plane, 5
hours).

The tail of the plane is found above Paris at 10 o=clock,
local time, as in 1.

The coordinates of the event "to see Paris since the tail of the plane"
are, relatively to **A _{l}** : (tail of the plane, 10 ).

The speed of Paris, relatively to **A _{l}** , is:

(distance tail-nose measured in

If the plane is tail nose oriented, the speed in algebraic measure is -700 km/h.

We have therefore the next results:

Relatively to **T _{l}** , the plane is moving
at 1400 km / h.

Relatively to

Relatively to

Remark: If one notes **A _{u}** the plane in which times are
measured by universal clocks (time sends by radio from Greenwich meridian
), one notices that the speed of the earth, relatively to

End of lecture

If you have well understood, you are now ready to solve following exercices, and to understand relativity in a new and clearer manner.

Small exercises on time zones

The proposed exercises are original, simple and progressive.

The aim is to familiarize yourself with measurements, relatively to a system of reference, and to well perceive the preference of the framework of the classical mechanics, because it is a framework in which it is easier to reason.

The framework of the classical mechanics being formed by an Euclidian space and a universal time.

Installation of studied systems.

The equator is supposed to be an infinite straight axis, the origin
being the Greenwich meridian.

This axis is oriented form east to west.

The sun motion above equator is supposed perfectly regular, always
straight above the equator.

The equator is represented by an axis Ox, and the sun is a luminous
point in translation above this axis, in the direction of crescent x.

The unit of length on the equator is one km.

The unit of time is one hour.

On the equator, times are measured by sundials, which give local times, or by the universal time given by the sundial of the Greenwich meridian, universal time being sent by radio for example.

On the equator origin, local time and universal time are coinciding.

The equatorial axis brought to the universal time is noted **S _{u}**
(System with an universal time).

The equatorial axis on which one measures times only with sundials is noted

Remark, **S _{u}** is a pecular local times system, the time
is given, in each point of the axis, by a radio receiver situated on this
point (with a possible correction), while for

One admits, for local times, a gap of 12 hours every 20000 km, relatively to the universal time.

Coordinates the of an event on the equator (explosion for example) ,
relatively to **S _{u}** , are noted (x, t).

Coordinates the of the same event, relatively

**Exercises**

**1.** Determine formulae allowing to
calculate (x', y=) knowing (x,
t). Then determine (x, y) from (x=,
t=).

**2.** Let a moving point M on the equator
(Ox axis).

Relatively to **S _{u}** , the motion
of M is given by the equation x=vt .

a) Determine, relatively to **S _{l}** , the equation of this
moving point M .

What is the speed (noted v=) of M , relatively to

b) Determine the limit of v' when v tends toward minus infinite ?

c) What is the limit speed (noted w) of v from which the description
of the motion of M in **S _{l}** becomes incoherent?

d) Determine the limit of v' when v tends toward plus infinite.

What do you observe?

**3.** Let a ruler (noted R) of length
L , relatively to **S _{u}** , and that slides on equator at
v speed , always relatively to

(The origin of the ruler has for equation x=vt , relatively to

a) What is the L= length of
the ruler, relatively to **S _{l }**?

b) What is the limit of L= when v tends toward minus infinite?

c) How behaves L= when v tends toward w (by inferior values), reaches it, then exceeds w? What are paradoxes?

**4.** One takes the precedent R ruler
moving at v speed, relatively to **S _{u}** .

The origin of this ruler is noted Or. The ruler is oriented from east to west.

The equation of Or, relatively to

R is now infinite.

The unit of length of this ruler, relatively to

Times on R are measured with sundials.

As on earth, we have local times.

One notes

Let E an any event of coordinates (x, t), relatively to **S _{u}**
,of coordinates (x', y=), relatively
to

a) Express (x", t" ) from (x, t) then from (x', y=).

b) Let M a moving point, displacing relatively to **S _{u}**
, according to the equation x=kt.

Determine the equation of this moving point, relatively to

Discuss according to the value of k.

Give the equation of the motion of M in

c) Let a second ruler noted R2, moving at k speed , relatively to **S _{u}**
, and of length L, relatively to

What is its length L", relatively to

Express L" from L= and k=. L= being the length of R2 relatively to

**5.** One takes the ruler describes
in 4. with its local times.

One changes now the unit of length on **R _{l}**
.

The unit of length measures now, relatively to

a) Do with u , precedent questions 4. a), b), c) .

b) What value is necessary to give to u , so that, formulae giving (x", t" ) from (x', y=), can be obtained with formulae giving (x', y=) from (x", t" ), by replacing simply in these last v by -v ?

**6.** One takes the ruler describes
in 4. with its local times.

On **R _{l}** ,one changes the unit of
length as in 5. , and more, one changes the unit of time.

The unit of length on

The local time unit on

In other words, when it flows one hour in

a) Do precedent questions 4. a), b), c), with now, constants u and s.

b) What values can be given to u and s, so that formulae giving (x@,t@)
from (x, t), are same type that those used in Lorentz=transformations.

Are these values unique ?

What is then the value of the constant c? (c, constant
that appears in Lorentz= formulae
).

Answers are gived on the french pages.