Measurements relative to a system of reference.
(Serge Cabala )

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 Tu , the earth brought to its time zones is noted Tl .
Tu and Tl are two different referentials , relatively to which one refers events.
Imagine a moment that we have only the possibility to measure times with sundials, events would be then exclusively dated by local times, Tl would be the alone system used.

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 Tu , relatively to Tl .

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 Tl , are (New -York, 10).

The coordinates the of this same event, relatively to Tu , 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 Tl , 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 Tu , dates are everywhere get by radio receivers, the universal time of Greenwich beeing spead by a satellite for example.

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 Tl , radio receiver for Tu ).

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 Tl , 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 Tu , are: (New-York, 10).

By definition, these two events Ep and En are simultaneous realtively to Tl (Relatively to the earth brought to its time zones).

Relatively to Tu , these two events are 5 hours offsetted .
Coordinates of En , relatively to Tu , are: (New-York, 15), those of Ep , relatively to Tu , are: (Paris, 10).

One sees immediately that two explosions, one in Paris and the other in New-York, simultaneous relatively to Tu , are not simultaneous relatively to Tl .
They are offsetted of 5 hours relatively to Tl .

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 Tl is:

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

1400 is a meaning number, relatively to Tl , only.

The average speed of this plane, on the flight Paris New-York, relatively to Tu 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 Tl 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 Tu 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 Tu , in the direction Paris New-York, the speed Vl of the plane, relatively to Tl , increase, becomes infinite, then becomes negative.
Think to the Concorde plane, the local time of landing down to New-York is always inferior to the local starting time from Paris.
Tl is an inadequate system to describe coherently some phenomena, such as the motion of Concorde.

The limit speed, relatively to Tu , in the direction Paris New-York, from which one obtain paradoxes in Tl is : 1400 km / h (=7000 / 5).
A plane that displaces in the direction Paris New-York at a speed of 1400 km / h , relatively to Tu , has an infinite speed relatively to Tl .

Remark 2: In the direction New-York Paris, whatever be the speed of the plane, relatively to T , 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 Tl , 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 Tl , is the distance Paris New-York.
The plane is measuring 7000 km , relatively to Tl .

The length of this plane, relatively to Tu , cannot be known, so longer is not given the speed of this plane relatively to Tl , or relatively to Tu .

Hence, let us assume more that the speed of this plane, in the direction Paris New-York, is 1400 km / h , relatively to Tl , ( 700 km / h , relatively to Tu ) , and let us determine its length, relatively to Tu .

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 Tu , it is easier).
The length of this plane, relatively to Tu , is therefore:   7000-5 * 700 = 3500 km.

A ruler of 3500 km , relatively to Tu , moving in the direction Paris New-York at the speed of 700 km / h, relatively to Tu , is measuring 7000 km, relatively to Tl .

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

The speed of this plane , relatively to Tl , 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 Tl , is 7000 km, because the tail and the nose of the plane, are, in Tl , seen simultaneously at 10 o=clock, and because the distance (in km) in Tl , between these two events is 7000 km.

Question: what is the length of this plane , relatively to Tu ? Reply: 10500 km.
I leave to the reader the care to explain.

A ruler of 10500 km, relatively to Tu , moving in the sense New-York Paris at a speed of 700 km / h, relatively to Tu , is measuring 7000 km, relatively to Tl .

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

In the very long plane of 3500 km, relatively to Tu , moving in the sense Paris New-York, at speed of 700 km / h, relatively to Tu , times are mesured with sundials.
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 Al (plane with local times).

1. Measurement of a length, relatively to Al .

Let us determine the distance Paris New-York, relatively to Al .
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 Al : (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 Al : (nose of the plane, 10 ).

The two preceding events are simultaneous, relatively to Al .
Relatively to Al , the distance tail nose is 3500 km.
Relatively to Al , the distance Paris New-York is then 3500km.

A ruler of 7000 km fixed to the ground is perceived under the distance 3500 km in Al .

2. Measurement of the speed of Paris , relatively to Al .

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 Al : (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 Al : (tail of the plane, 10 ).

The speed of Paris, relatively to Al , is:
(distance tail-nose measured in Al ) / (10 - 5) = 3500 / 5 = 700 km / h, for a speed in absolute value.
If the plane is tail nose oriented, the speed in algebraic measure is -700 km/h.

We have therefore the next results:
Relatively to Tl , the plane is moving at 1400 km / h.
Relatively to Al the earth is moving at 700 km/h (- 700 in algebraic measure)
Relatively to Tu , the plane is moving at 700 km / h.

Remark: If one notes Au 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 Au , is again 700 km / h (- 700 km / h in algebraic measure).

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 Su (System with an universal time).
The equatorial axis on which one measures times only with sundials is noted Sl (System with local times).

Remark, Su 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 Sl , the time in each point of the axis is given by a sundial fixed on this point.
Su and Sl differ only by the type of used clocks.

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 Su , are noted (x, t).
Coordinates the of the same event, relatively Sl , are noted (x', y=)

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 Su , the motion of M is given by the equation x=vt .

a) Determine, relatively to Sl , the equation of this moving point M .
What is the speed (noted v=) of M , relatively to Sl ?

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 Sl 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 Su , and that slides on equator at v speed , always relatively to Su .
(The origin of the ruler has for equation x=vt , relatively to Su , its extremity has for equation x=vt+L , relatively to Su ).

a) What is the L= length of the ruler, relatively to Sl ?

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 Su .
The origin of this ruler is noted Or. The ruler is oriented from east to west.
The equation of Or, relatively to Su , is x=vt.
R is now infinite.
The unit of length of this ruler, relatively to Su , measure one km (it is the usual intuitive unit).
Times on R are measured with sundials.
As on earth, we have local times.
One notes Rl the system of reference thus obtained.

Let E an any event of coordinates (x, t), relatively to Su ,of coordinates (x', y=), relatively to Sl , and of coordinates (x", t") , relatively to Rl .

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

b) Let M a moving point, displacing relatively to Su , according to the equation x=kt.
Determine the equation of this moving point, relatively to Rl .
Discuss according to the value of k.
Give the equation of the motion of M in Rl , from its equation in Sl . (One will note k' the speed of M relatively to Sl )

c) Let a second ruler noted R2, moving at k speed , relatively to Su , and of length L, relatively to Su .
What is its length L", relatively to Rl ?
Express L" from L= and k=. L= being the length of R2 relatively to Sl , k' being the speed of R2 relatively to Sl .

5. One takes the ruler describes in 4. with its local times.
One changes now the unit of length on Rl .
The unit of length measures now, relatively to Su , u kilometers. u is a real positive or negative (one can suppose u positive) .

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 Rl ,one changes the unit of length as in 5. , and more, one changes the unit of time.
The unit of length on Rl , measures relatively to Su , u kilometers as in 5.
The local time unit on Rl , has for value s hours, relatively to Su .
In other words, when it flows one hour in Su , all clocks of Rl advance of 1/s units.

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.