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# Relativity theory of Lorentz

*For the Einsteinism that is often confused with the Relativity Theory, see Einstein's general and special theory of relativity.*

The **relativity theory of Lorentz** is the comprehensive theory that is obtained after supplementing
the Maxwell theory with the equations of the Lorentz transformation
in order to make the Maxwell theory consistent with the relativity principle.
The relativity principle implies that any experiment with material systems, carried out on
bodies moving uniformly with respect to one another, would give exactly the same result, so that it would be
inherently impossible to tell, from an experiment confined to a body, whether that body was at rest or moving
uniformly through the aether.

The name "relativity theory of Lorentz" is used because this theory is often ascribed to Lorentz
under reference to the 1904 paper by Lorentz
^{[1]}.
This was done for example by Dingle
^{[2]}.
Another factor in the choice of the name "relativity theory of Lorentz" is that the
"Lorentz transformation" which plays a central role in "relativity theory of Lorentz" is also called after Lorentz
^{[3]}
^{[4]}.

The relativity theory of Lorentz is based on the following three assumptions:

- A procedure for the synchronization of clocks (to be given below).
- A proposal that motion of a material body through the aether produces a contraction in the direction of motion by the factor (1 – v
^{2}/c^{2})^{1/2}. - A proposal that motion of a material body through the aether produces a slowing down of all rhythmical processes by the factor (1 – v
^{2}/c^{2})^{1/2}.

Where v is the velocity of the body and c the velocity of light. Using these starting points, the relation between the coordinates of an event referred to a non-moving system, and the
coordinates of the same event referred to a system moving uniformly is given by the Lorentz transformation^{[2]}.

The relativity theory of Lorentz provides an explanation for numerous key physical experiments, including the Michelson-Morley experiment
^{[2]}
^{[5]}
and the Fizeau experiment.

## Contents

## Assumptions

### Synchronization of clocks

The clock synchronization procedure that is used in this section corresponds to the procedure used by Henri Poincaré in a publication about the theory of Lorentz published in 1900^{[6]} and described by Henri Poincaré in 1904 in more detail^{[7]}.

Clock A and clock B are two equally fast running clocks at a distance L apart. The task is to synchronize them, which simply means that at the same time, both clocks will indicate the same time.
All that is needed for the synchronization is to make a single adjustment to clock B in which clock B is set to the time of clock A at the moment of the adjustment. Because clock A and B run equally fast, this single adjustment will then ensure that both clocks will remain synchronized also after the adjustment.
The adjustment of clock B is performed as follows: Clock A sends out a signal containing its reading at the moment of emitting the signal (this reading is denoted as T_{A}).
It is assumed the signal travels with the speed of light (c) towards clock B.
When clock B receives the signal, it sets its time to T_{A}, plus the calculated time it took for the signal to reach clock B (which is L/c). Therefore, at the moment clock B receives the signal, clock B is set to T_{A} + L/c. In this ways clock B is synchronized with clock A if the calculated time L/c indeed corresponds to the actual time it took for the signal to reach clock B.

### Contraction of lengths

Lorentz proposed that motion of a material body through the aether produces a contraction in the direction of motion by the factor (1 – v^{2}/c^{2})^{1/2}, where v is the velocity of the material body relative to the eather and c is the speed of light relative to the aether. This proposal was made in order to explain the outcome of the Michelson-Morley experiment.

### Slowing down of rhythmical processes

Lorentz proposed that motion of a material body through the aether produces a slowing down of all rhythmical processes by the factor (1 – v^{2}/c^{2})^{1/2}, where v is the velocity of the material body relative to the eather and c is the speed of light relative to the aether. This additional proposal was needed to also explain the Fizeau experiment for the speed of light (moving medium).

## Derivation of the Lorentz transformation

The length between 0 and X’ in the moving system is L = X’(1-v^{2}/c^{2})^{1/2} (because of the second starting point: the assumed contraction in the direction of movement). The length between the 0 of the non-moving frame and the 0 of the moving frame is vt (because it is assumed that both 0’s coincide at t=0). Therefore:

X = X’ (1-v^{2}/c^{2})^{1/2} + vt

It follows that:

X’ = (X-vt)/ (1-v^{2}/c^{2})^{1/2}

which is the equation for X’ of the Lorentz transformation.

The other equation of the Lorentz transformation is an equation that expresses t’ as a function of x and t. t' is the reading of a clock in the moving system that happens to be at x at time t (hence t' is the reading of clock B in the figure on the right). This equation is derived as follows:
Clock A is set such that at time t=0 its reading is zero as well. The reading of clock A at arbitrary time t is denoted by T_{A}(t). Because of the assumed slowing down of all rhythmic processes in the moving system (third starting point), it follows that T_{A}((t) = t (1-v^{2}/c^{2})^{1/2} .
To find the reading of clock B the above-described synchronization procedure is followed
(the synchronization procedure makes no error when both clocks are not moving. Because the procedure is applied to moving clocks, a synchronization error is made, which results in both clocks not indicating the same reading at the same time; this synchronization error cannot be avoided as because of the principle of relativity absolute velocities cannot be detected.).
A light beam that moves from clock A towards clock B will move with speed c-v relative to and away from clock A and it has the same speed relative to (but towards) clock B. Therefore, the signal that clock B receives at time t was sent out from clock A at time t- L/(c-v).
Hence, the reading that is received by clock B is T_{A}((t- L/(c-v)).
Because of the synchronization procedure, the reading of clock B at time t (i.e. when it receives the signal) is set to T_{A}((t- L/(c-v)) + X’/c.
Hence,

t' = T_{B}((t) = T_{A}((t- L/(c-v)) + X’/c

= (t- (x-vt) /(c-v)) (1-v^{2}/c^{2})^{1/2} + ((X-vt)/ (1-v^{2}/c^{2})^{1/2}

=[ t - vx/c^{2} ] / (1-v^{2}/c^{2})^{1/2}

The above result corresponds to the equation of the Lorentz transformation that expresses t’ as a function of x and t. The equation has a straightforward interpretation: the term containing x is needed because of the synchronization error; the factor (1-v^{2}/c^{2})^{1/2} is needed because of the slowing down of all rhythmical processes in the moving system. This can be seen as follows:
only for the clock located at x=vt no synchronization error is made. For this clock application of the above equation results in t' = t (1-v^{2}/c^{2})^{1/2}. Hence t' < t. This shows that the factor (1-v^{2}/c^{2})^{1/2} is indeed because of the slowing down of all rhythmical processes in the moving system.
Summarizing the above, the following system of equations was derived that expresses x' and t' as a function of x and t:

- x’ = (x-vt)/ (1-v
^{2}/c^{2})^{1/2} - t' =[ t - vx/c
^{2}] / (1-v^{2}/c^{2})^{1/2}

This system of equations is known as the Lorentz transformation.
An interesting consequence of the Lorentz transformation that two events, e.g. one event with coordinates (x'_{1}, t'_{1}) and another event with coordinates (x'_{2}, t'_{2}) are not simultaneous if t'_{1} = t'_{2}. In the same way, if the (x,t) coordinates of two events are given by (x_{1}, t_{1}) and (x_{2}, t_{2}) and if t_{1} = t_{2} (in other words both events are simultaneous), the Lorentz transformation results in t'_{1} ≠ t'_{2}.
It is noted that lengths measured using the x-coordinate represent true (uncontracted) lengths, whereas lengths measured using the x'-coordinate do not represent true lengths. Also, time differences measured using the t-coordinate represent true time differences, whereas time differences using the t'-coordinate are affected by both the contraction and the slowing down.

## Physical phenomena explained by the theory

Some key scientific experiments, of which the outcomes are explained by the relativity theory of Lorentz, are discussed.

### Michelson-Morley experiment

*For a more detailed overview see the main article Michelson-Morley experiment.*

If the speed of light is constant relative to a luminiferous aether, different speeds of light would be measured if the laboratory in which the speed is measured moves through the aether itself. For example, if the laboratory moves at a speed v in the same direction of the pulse of light of which the speed is measured, and the speed of the pulse of light relative to the aether is c, the apparent speed at which the pulse propagates in the laboratory is c-v; in other words the pulse would appear to be moving slower. The essence of the Michelson-Morley experiment is to measure this change of speed of light in the moving laboratory.

In this experiment, a pulse of light is sent from point S (the symbol S stands for "source") towards point X.
where a beam splitter splits the light pulse into two. One light pulse travels from X towards A, where it is reflected using a mirror towards D (D=light detector).
The other light pulse continues from X towards B, where it is reflected using a mirror back to X, where the pulse
is reflected by the beam splitter towards D as well.
The pulses of light both end up at D, where the difference in their arrival times is measured
^{[8]}.
The first pulse of light travels along the path S-X-A-X-D. The second pulse along the path S-X-B-X-D.
Because both pulses travel along S-X and X-D, any time difference must be caused by a difference in the
time it takes for the first puls to traverse the path X-A-X and the time it takes for the second
pulse to traverse the path X-B-X.

The experimental setup is such that both arms, AX and BX have the same length and are placed orthogonal to each other. Therefore, if the laboratory is not moving, the arrival times of both beams at D should be the same. However, if the whole setup is moving at a constant speed, this may cause a difference in the arrival times, if the speed of the laboratory is either more in the direction of the line AX or more in the direction of the line BX.

However, no difference in arrival times was detected by Michelson and Morley,
even though the experiment was designed to detect this difference if the speed of light is indeed constant relative to the aether.
The result, therefore, seemed to indicate that the speed of light is not constant relative to a fixed aether.
However, that conclusion would imply that Maxwell's equations (which imply a constant speed of light)
would need to be revised or altered. But Maxwell's equations have been used to explain a wide range of physical phenomena.
Therefore, Lorentz and Fitzgerald explained the negative result of the Michelson-Morley experiment
by proposing that the movement of a material body through the aether leads to its contraction (Lorentz) in the direction of
movement or
its elongation in the direction perpendicular to the movement (Fitzgerald),
but leaving the Maxwell equations unchanged.
Lorentz calculated that the contraction coefficient needed for explaining the null-result of the Michelson-Morley experiment was (1-v^{2}/c^{2})^{1/2}.

### Fizeau experiment

*For a detailed overview see the main article Fizeau experiment for the speed of light (moving medium).*

## Historical background

Since Maxwell developed his equations in the 1860s it has been known that light is a wave-phenomenon, which propagates at a speed that is independent on
the movements of its source. Light consist of vibrations in what was called the "luminiferous aether" or simply "aether". The speed of light, from Maxwell's equations, is given by 1/√(ε μ) where ε is the dielectric constant of the vacuum (or aether) and μ is the magnetic permeability of the vacuum (or aether). Waves require a medium to propagate. For example, sound is a wave of pressure variations in the air. Light, although it can propagate through air, glass, etc. almost undisturbed, does not require a "material" like air or glass, because it can travel undisturbed through the vacuum of outer space (the light of the stars would otherwise not reach us).
It was therefore thought that there is an all-pervading "aether" that is the medium in which light manifests itself. This was also the interpretation of Maxwell himself ^{[9]}:

But interestingly, the existence of a luminiferous aether contradicts another principle, namely the principle of relativity. To demonstrate this, both principles are repeated here:

1. **The principle of relativity.** This scientific principle, which states that the laws of physical phenomena should be the same, whether for an observer fixed, or for an observed carried along in a uniform movement of translation, was first formulated in it modern form by the scientist Poincaré before the inception of Lorentz theory of relativity. The principle of relativity entails that "absolute velocities" are meaningless. In other words, there is no aether with respect to which absolute velocities can be defined.

2. **The velocity of transmission of light in vacuum is constant.** It is noted that the constancy of the speed of light is a scientific consequence of Maxwell's equations that were published in the 1860s before the inception of the relativity theory of Lorentz. The constancy means that the speed light is independent on the velocity of the source of the light, in the same way as the speed of sound is independent on the velocity of its source.

Now the contradiction is revealed: The contradiction is that if the velocity of light is constant and not relative to its source (as implied by the second statement), the only apparent alternative is that each beam of light has that velocity with respect to the aether, but precisely this is denied by the first statement. The Michelson-Morley experiment performed in 1881 and 1887 was designed to resolve the contradiction by proving that the first statement (the relativity principle) is false. The experiment attempted to detect the movement of the earth through the aether by looking at the speed of light in different directions in an earth-based laboratory. However, no differences in speed were observed, and the contradiction remained.

Because Maxwell's equations had been very successful in describing a wide range of physical phenomena, physicists were looking for a way of saving the equations of Maxwell. Hence, because Maxwell's equations predict a constant speed of light, this means that physicists were looking at finding a theory that explains the observed phenomena (including the result of the Michelson-Morley experiment). This desired theory should then maintain the validity of the Maxwell equations, at the price of giving up the absolute validity of the relativity principle.
The "relativity theory of Lorentz" did exactly this. In this theory, it was proposed that objects that move at a velocity v through the aether contract in the direction of movement by a factor √(1-v^{2}/c^{2}) and at the same time experience a slowing down of all rhythmical processes by the same factor. These proposals reconciled the Maxwell equations with the null result of the Michelson-Morley experiment; the relativity principle was given up because absolute motion produces a real physical effect (slowing down and contraction) on objects in the relativity theory of Lorentz.

Lorentz acknowledged that this proposal was ad hoc ^{[10]}.

## See also

## References

- ↑ Hendrik Lorentz (1904) Electromagnetic phenomena in a system moving with any velocity smaller than that of light. Proceedings of the Royal Netherlands Academy of Arts and Sciences, 1904, 6: 809–831.
- ↑
^{2.0}^{2.1}^{2.2}Dingle, H (1972) Science at the crossroads, 109 pp. - ↑ Poincaré published in 1905 an article in which the Lorentz tranformation is given. Strangely, he calls it the "Lorentz transformation" under reference to the 1904 article by Lorentz, which contains a transformation that
*looks like*the Lorentz transformation, but that is not mathematically identical with it. Hence, it can be said that Poincaré erroneously attached the name of Lorentz to the transformation. Poincaré, Henri (1905), "On the Dynamics of the Electron", Comptes rendus hebdomadaires des séances de l'Académie des Sciences 140: 1504–1508 - ↑ Einstein later (for example in his 1924 book, but not in his 1905 paper) used the name "Lorentz transformation". Why did Einstein uncritically repeat Poincaré's erroneous naming? Was this an attempt to associate the undoubtedly more scientific work of Lorentz with his own pseudoscientific work?
- ↑ The explanation of the relativity theory of Lorentz uses a contraction in the direction of motion; the slowing down is not essential for the explanation (it is included to account for other physical phenomena). In fact, the contraction is not even necessary to explain the null result of the Michelson-Morley experiment; Fitzgerald for example used an expansion of the experimental equipment in the directions perpendicular to the direction of motion. No physical experiment has to date been performed to decide experimentally which one of these two options (contraction in the direction of motion or expansion in the directions perpendicular to the direction of motion) corresponds to reality.
- ↑
Poincaré, Henri (1900), "La théorie de Lorentz et le principe de réaction", Archives néerlandaises des sciences exactes et naturelles 5: 252–278.
The English translation of this publication has the title "
*The Theory of Lorentz and The Principle of Reaction*" and can be downloaded from http://www.physicsinsights.org/poincare-1900.pdf (accessed December 2014) - ↑
In his publication,
*Poincaré, Henri (1904), “L'état actuel et l'avenir de la physique mathématique”, Bulletin des sciences mathématiques 28 (2): 302-324*, Poincaré writes (here translated into English): "*Imagine two observers who wish to adjust their timepieces by optical signals; they exchange signals, but as they know that the transmission of light is not instantaneous, they are careful to cross them. When station B perceives the signal from station A, its clock should not mark the same hour as that of station A at the moment of sending the signal, but this hour augmented by a constant representing the duration of the transmission.*" - ↑ It has been discussed by Dingle that no real time differences were measured, because the time difference was obtained as a result of a calculation using Maxwell's equations. See H. Dingle (1972) Science at the Crossroads, Chapter 8, page 75.
- ↑ On page 382 of the book Hecht E. (1987) Optics, Addison-Wesley Publishing Company, 676 pp, Maxwell is quoted as follows: "Aethers were invented for the planets to swim in, to constitute electric atmospheres and magnetic effluvia, to convey sensations from one part of our bodies to another, and so on, until all space had been filled three of four times over with aethers.... The only aether which has survived is that which was invented by Huygens to explain the propagation of light."
- ↑ Herbert Dingle, in his 1972 textbook
*Science at the Crossroads*, quotes Lorentz: "It need hardly be said that the present theory is put forward with all due reserve."