History of lorentz transformations
Encyclopedia
The Lorentz transformation
s relate the space-time coordinates, (which specify the position x, y, z and time t of an event) relative to a particular inertial frame of reference (the "rest system"), and the coordinates of the same event relative to another coordinate system moving in the positive x-direction at a constant speed v, relative to the rest system. It was devised as a theoretical transformation which makes the velocity of light invariant between different inertial frames. The coordinates of the event in this "moving system" are denoted x′, y′, z′ and t′. Before 1905, the rest system was identified with the "aether", the supposed medium which transmitted electro-magnetic waves, and the moving system as commonly identified with the earth as it moved through this medium. Early approximations of the transformation were published by Voigt
(1887) and Lorentz
(1895). They were completed by Larmor
(1897, 1900) and Lorentz (1899, 1904) and were brought into their modern form by Poincaré
(1905), who gave the transformation the name of Lorentz. Eventually, Einstein
(1905) showed in the course of his development of special relativity
, that this transformation concerns the nature of space and time.
In this article the historical notations are replaced with modern notations, where
is the Lorentz factor
, v is the relative velocity of the bodies, and c is the speed of light
.
and an incompressible medium, Voigt (1887) developed a transformation, which was in modern notation:
If the right-hand sides of his equations are multiplied by
they are the modern Lorentz transformation
. In Voigt's theory the speed of light is invariant, but his transformations mix up a relativistic boost together with a rescaling of space-time. Maxwell's electrodynamics is both scale invariant and Lorentz invariant, so the combination is invariant too. But scale transformations are not a symmetry of all the laws of nature, only of electromagnetism, so these transformations cannot be used to formulate a principle of relativity
in general. Lorentz acknowledged Voigt's work in 1909 by saying:
Also Hermann Minkowski
said in 1908 that the transformations which play the main role in the principle of relativity were first examined by Voigt in the 1887. Voigt responded in the same paper by saying, that his theory was based on an elastic theory of light, not an electromagnetic one. However, he concluded that some results were actually the same.
") in which the aether is completely motionless, and the speed of light in the aether is constant in all directions. To calculate the optics of moving bodies, Lorentz (independently of Voigt) introduced the following quantities to transform from the aether system into a moving system.
where x* is the Galilean transformation
x-vt. While t is the "true" time for observers resting in the aether, t' is an auxiliary variable only for calculating processes for moving systems. It is also important that Lorentz and later also Larmor formulated this transformation in 2 steps. At first the Galilean transformation - and later the expansion into the "fictitious" electromagnetic system with the aid of the Lorentz transformation. He also (1892b) introduced the additional hypothesis that also intermolecular forces are affected in a similar way and introduced length contraction
in his theory (without proof as he admitted). While for Lorentz length contraction was a real physical effect, he considered the time transformation only as a heuristic working hypothesis and a mathematical stipulation.
In 1895, Lorentz further elaborated on his theory and introduced the "theorem of corresponding states". This theorem states that a moving observer (relative to the ether) in his „fictitious“ field makes the same observations as a resting observers in his „real“ field for velocities to first order in v/c. Lorentz showed that the dimensions of electrostatic systems in the ether and a moving frame are connected by this transformation:
For solving optical problems Lorentz used the following transformation, whereby for the time variable he used the expression "local time" (Ortszeit):
With this concept Lorentz could explain the Doppler effect
, the aberration of light
, and the Fizeau experiment
.
This transformation is just the Galilean transformation for the x, y, z coordinates but contains Lorentz’s "local time". Larmor knew that the Michelson–Morley experiment was accurate enough to detect an effect of motion depending on the factor v²/c², and so he sought the transformations which were "accurate to second order" (as he put it). Thus he wrote the final transformations (where x* = x − vt) as:
Larmor showed that Maxwell's equations were invariant under this two-step transformation, "to second order in v/c", as he put it. Larmor noted that if it is assumed that the constitution of molecules is electrical then the Fitzgerald-Lorentz contraction is a consequence of this transformation. It's notable that Larmor was the first who recognized that some sort of time dilation
is a consequence of this transformation as well, because individual electrons describe corresponding parts of their orbits in times shorter for the [rest] system in the ratio 1/γ.
and l is an undetermined factor. This factor was set to unity by him in 1904, so Lorentz's equations now assumed the same form as Larmor's (as mentioned above x* must be replaced by x − vt):
In connection with this he also derived the correct formulas for the velocity dependence of mass. He concluded, that this transformation must apply to all forces of nature, not only electrical ones and therefore length contraction is a consequence of this transformation.
in 1900 commented on the origin of Lorentz’s “wonderful invention” of local time.
He remarked that it arose when clocks in a moving reference frame are synchronised by exchanging signals which are assumed to travel with the same speed c in both directions, which lead to what is nowadays called relativity of simultaneity
, although Poincaré's calculation does not involve length contraction or time dilation. In order to synchronise the clocks here on Earth (the x*, t* frame) we send a light signal from one clock (at the origin) to another (at x*), and bounce it back. We suppose that the Earth is moving with speed v in the x-direction (= x*-direction) in some rest system (x,t) (i.e. the luminiferous aether
system for Lorentz and Larmor). We calculate that the time of flight outwards is
and the time of flight back is
The elapsed time on the clock when the signal is returned is δto + δtb and we ascribed the time t* = (δto + δtb)/2 to the moment when the light signal reached the distant clock. In the rest frame, of course, the time t = δto is ascribed to that same instant. Some algebra gives the relation between the different time coordinates ascribed to the moment of reflection. Thus
Poincaré gave the result t* = t − vx*/c2, which is the form used by Lorentz in 1895. Poincaré dropped the factor ε ≅ 1 under the assumption that
Similar physical interpretations of local time were later given by Emil Cohn
(1904) and Max Abraham
(1905).
Apparently Poincaré was unaware of Larmor's contributions, because he only mentioned Lorentz and therefore used for the first time the name "Lorentz transformation". He showed that Lorentz's application of the transformation on the equations of electrodynamics didn't fully satisfy the principle of relativity
. So by pointing out the group characteristics of the transformation Poincaré demonstrated the Lorentz covariance of the Maxwell-Lorentz equations.
In July 1905 (published in January 1906) Poincaré showed that the transformations are a consequence of the principle of least action
; he demonstrated in more detail the group characteristics of the transformation, which he called Lorentz group
, and he showed that the combination x2 + y2 + z2 − c2t2 is invariant. He noticed that the Lorentz transformation is merely a rotation in four-dimensional space about the origin by introducing ct√−1 as a fourth imaginary coordinate, and he used an early form of four-vector
s.
And unlike Lorentz and Poincaré who still distinguished between "real" time in the aether and "apparent" time for moving observers, Einstein showed that the transformations concern the nature of space and time. Einstein's views on space and time, together with Poincaré's four-dimensional approach, were further elaborated by Hermann Minkowski
who gave, besides other things, a geometric representation of the Lorentz transformation by using Minkowski diagram
s.
Lorentz transformation
In physics, the Lorentz transformation or Lorentz-Fitzgerald transformation describes how, according to the theory of special relativity, two observers' varying measurements of space and time can be converted into each other's frames of reference. It is named after the Dutch physicist Hendrik...
s relate the space-time coordinates, (which specify the position x, y, z and time t of an event) relative to a particular inertial frame of reference (the "rest system"), and the coordinates of the same event relative to another coordinate system moving in the positive x-direction at a constant speed v, relative to the rest system. It was devised as a theoretical transformation which makes the velocity of light invariant between different inertial frames. The coordinates of the event in this "moving system" are denoted x′, y′, z′ and t′. Before 1905, the rest system was identified with the "aether", the supposed medium which transmitted electro-magnetic waves, and the moving system as commonly identified with the earth as it moved through this medium. Early approximations of the transformation were published by Voigt
Woldemar Voigt
Woldemar Voigt was a German physicist, who taught at the Georg August University of Göttingen. Voigt eventually went on to head the Mathematical Physics Department at Göttingen and was succeeded in 1914 by Peter Debye, who took charge of the theoretical department of the Physical Institute...
(1887) and Lorentz
Hendrik Lorentz
Hendrik Antoon Lorentz was a Dutch physicist who shared the 1902 Nobel Prize in Physics with Pieter Zeeman for the discovery and theoretical explanation of the Zeeman effect...
(1895). They were completed by Larmor
Joseph Larmor
Sir Joseph Larmor , a physicist and mathematician who made innovations in the understanding of electricity, dynamics, thermodynamics, and the electron theory of matter...
(1897, 1900) and Lorentz (1899, 1904) and were brought into their modern form by Poincaré
Henri Poincaré
Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...
(1905), who gave the transformation the name of Lorentz. Eventually, Einstein
Albert Einstein
Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...
(1905) showed in the course of his development of special relativity
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
, that this transformation concerns the nature of space and time.
In this article the historical notations are replaced with modern notations, where
is the Lorentz factor
Lorentz factor
The Lorentz factor or Lorentz term appears in several equations in special relativity, including time dilation, length contraction, and the relativistic mass formula. Because of its ubiquity, physicists generally represent it with the shorthand symbol γ . It gets its name from its earlier...
, v is the relative velocity of the bodies, and c is the speed of light
Speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
.
Voigt (1887)
In connection with the Doppler effectDoppler effect
The Doppler effect , named after Austrian physicist Christian Doppler who proposed it in 1842 in Prague, is the change in frequency of a wave for an observer moving relative to the source of the wave. It is commonly heard when a vehicle sounding a siren or horn approaches, passes, and recedes from...
and an incompressible medium, Voigt (1887) developed a transformation, which was in modern notation:
If the right-hand sides of his equations are multiplied by
they are the modern Lorentz transformationLorentz transformation
In physics, the Lorentz transformation or Lorentz-Fitzgerald transformation describes how, according to the theory of special relativity, two observers' varying measurements of space and time can be converted into each other's frames of reference. It is named after the Dutch physicist Hendrik...
. In Voigt's theory the speed of light is invariant, but his transformations mix up a relativistic boost together with a rescaling of space-time. Maxwell's electrodynamics is both scale invariant and Lorentz invariant, so the combination is invariant too. But scale transformations are not a symmetry of all the laws of nature, only of electromagnetism, so these transformations cannot be used to formulate a principle of relativity
Principle of relativity
In physics, the principle of relativity is the requirement that the equations describing the laws of physics have the same form in all admissible frames of reference....
in general. Lorentz acknowledged Voigt's work in 1909 by saying:
Also Hermann Minkowski
Hermann Minkowski
Hermann Minkowski was a German mathematician of Ashkenazi Jewish descent, who created and developed the geometry of numbers and who used geometrical methods to solve difficult problems in number theory, mathematical physics, and the theory of relativity.- Life and work :Hermann Minkowski was born...
said in 1908 that the transformations which play the main role in the principle of relativity were first examined by Voigt in the 1887. Voigt responded in the same paper by saying, that his theory was based on an elastic theory of light, not an electromagnetic one. However, he concluded that some results were actually the same.
Lorentz (1892, 1895)
In 1892 Lorentz developed a model ("Lorentz ether theoryLorentz ether theory
What is now often called Lorentz Ether theory has its roots in Hendrik Lorentz's "Theory of electrons", which was the final point in the development of the classical aether theories at the end of the 19th and at the beginning of the 20th century....
") in which the aether is completely motionless, and the speed of light in the aether is constant in all directions. To calculate the optics of moving bodies, Lorentz (independently of Voigt) introduced the following quantities to transform from the aether system into a moving system.
where x* is the Galilean transformation
Galilean transformation
The Galilean transformation is used to transform between the coordinates of two reference frames which differ only by constant relative motion within the constructs of Newtonian physics. This is the passive transformation point of view...
x-vt. While t is the "true" time for observers resting in the aether, t' is an auxiliary variable only for calculating processes for moving systems. It is also important that Lorentz and later also Larmor formulated this transformation in 2 steps. At first the Galilean transformation - and later the expansion into the "fictitious" electromagnetic system with the aid of the Lorentz transformation. He also (1892b) introduced the additional hypothesis that also intermolecular forces are affected in a similar way and introduced length contraction
Length contraction
In physics, length contraction – according to Hendrik Lorentz – is the physical phenomenon of a decrease in length detected by an observer of objects that travel at any non-zero velocity relative to that observer...
in his theory (without proof as he admitted). While for Lorentz length contraction was a real physical effect, he considered the time transformation only as a heuristic working hypothesis and a mathematical stipulation.
In 1895, Lorentz further elaborated on his theory and introduced the "theorem of corresponding states". This theorem states that a moving observer (relative to the ether) in his „fictitious“ field makes the same observations as a resting observers in his „real“ field for velocities to first order in v/c. Lorentz showed that the dimensions of electrostatic systems in the ether and a moving frame are connected by this transformation:
For solving optical problems Lorentz used the following transformation, whereby for the time variable he used the expression "local time" (Ortszeit):
With this concept Lorentz could explain the Doppler effect
Doppler effect
The Doppler effect , named after Austrian physicist Christian Doppler who proposed it in 1842 in Prague, is the change in frequency of a wave for an observer moving relative to the source of the wave. It is commonly heard when a vehicle sounding a siren or horn approaches, passes, and recedes from...
, the aberration of light
Aberration of light
The aberration of light is an astronomical phenomenon which produces an apparent motion of celestial objects about their real locations...
, and the Fizeau experiment
Fizeau experiment
The Fizeau experiment was carried out by Hippolyte Fizeau in 1851 to measure the relative speeds of light in moving water. Albert Einstein later pointed out the importance of the experiment for special relativity...
.
Larmor (1897, 1900)
Larmor in 1897 and 1900 presented the transformations in two parts. Similar to Lorentz, he considered first the transformation from a rest system (x, y, z, t) to a moving system (x′, y′, z′, t′)
This transformation is just the Galilean transformation for the x, y, z coordinates but contains Lorentz’s "local time". Larmor knew that the Michelson–Morley experiment was accurate enough to detect an effect of motion depending on the factor v²/c², and so he sought the transformations which were "accurate to second order" (as he put it). Thus he wrote the final transformations (where x* = x − vt) as:
Larmor showed that Maxwell's equations were invariant under this two-step transformation, "to second order in v/c", as he put it. Larmor noted that if it is assumed that the constitution of molecules is electrical then the Fitzgerald-Lorentz contraction is a consequence of this transformation. It's notable that Larmor was the first who recognized that some sort of time dilation
Time dilation
In the theory of relativity, time dilation is an observed difference of elapsed time between two events as measured by observers either moving relative to each other or differently situated from gravitational masses. An accurate clock at rest with respect to one observer may be measured to tick at...
is a consequence of this transformation as well, because individual electrons describe corresponding parts of their orbits in times shorter for the [rest] system in the ratio 1/γ.
Lorentz (1899, 1904)
Also Lorentz, by extending his theorem of corresponding states, derived in 1899 the complete transformations. However, he used the undetermined factor l as an arbitrary function of v. Like Larmor, in 1899 also Lorentz noticed some sort of time dilation effect, and he wrote that for the frequency of oscillating electrons "that in S the time of vibrations be kl times as great as in S0", where S0 is the ether frame,and l is an undetermined factor. This factor was set to unity by him in 1904, so Lorentz's equations now assumed the same form as Larmor's (as mentioned above x* must be replaced by x − vt):
In connection with this he also derived the correct formulas for the velocity dependence of mass. He concluded, that this transformation must apply to all forces of nature, not only electrical ones and therefore length contraction is a consequence of this transformation.
Local time
Neither Lorentz or Larmor gave a clear physical interpretation of the origin of local time. However, Henri PoincaréHenri Poincaré
Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...
in 1900 commented on the origin of Lorentz’s “wonderful invention” of local time.
He remarked that it arose when clocks in a moving reference frame are synchronised by exchanging signals which are assumed to travel with the same speed c in both directions, which lead to what is nowadays called relativity of simultaneity
Relativity of simultaneity
In physics, the relativity of simultaneity is the concept that simultaneity–whether two events occur at the same time–is not absolute, but depends on the observer's reference frame. According to the special theory of relativity, it is impossible to say in an absolute sense whether two events occur...
, although Poincaré's calculation does not involve length contraction or time dilation. In order to synchronise the clocks here on Earth (the x*, t* frame) we send a light signal from one clock (at the origin) to another (at x*), and bounce it back. We suppose that the Earth is moving with speed v in the x-direction (= x*-direction) in some rest system (x,t) (i.e. the luminiferous aether
Luminiferous aether
In the late 19th century, luminiferous aether or ether, meaning light-bearing aether, was the term used to describe a medium for the propagation of light....
system for Lorentz and Larmor). We calculate that the time of flight outwards is
and the time of flight back is
The elapsed time on the clock when the signal is returned is δto + δtb and we ascribed the time t* = (δto + δtb)/2 to the moment when the light signal reached the distant clock. In the rest frame, of course, the time t = δto is ascribed to that same instant. Some algebra gives the relation between the different time coordinates ascribed to the moment of reflection. Thus
Poincaré gave the result t* = t − vx*/c2, which is the form used by Lorentz in 1895. Poincaré dropped the factor ε ≅ 1 under the assumption that
Similar physical interpretations of local time were later given by Emil Cohn
Emil Cohn
Emil Georg Cohn , was a German physicist.-Life:Cohn was born in Neustrelitz, Mecklenburg on 28 September 1854. He was the son of August Cohn, a lawyer, and Charlotte Cohn. At the age of 17, Cohn began to study jurisprudence at the University of Leipzig...
(1904) and Max Abraham
Max Abraham
Max Abraham was a German physicist.Abraham was born in Danzig, Imperial Germany to a family of Jewish merchants. His father was Moritz Abraham and his mother was Selma Moritzsohn. Attending the University of Berlin, he studied under Max Planck. He graduated in 1897...
(1905).
Lorentz transformation
On June 5, 1905 (published June 9) Poincaré simplified the equations (which are algebraically equivalent to those of Larmor and Lorentz) and gave them the modern form (Poincaré set the speed of light to unity):Apparently Poincaré was unaware of Larmor's contributions, because he only mentioned Lorentz and therefore used for the first time the name "Lorentz transformation". He showed that Lorentz's application of the transformation on the equations of electrodynamics didn't fully satisfy the principle of relativity
Principle of relativity
In physics, the principle of relativity is the requirement that the equations describing the laws of physics have the same form in all admissible frames of reference....
. So by pointing out the group characteristics of the transformation Poincaré demonstrated the Lorentz covariance of the Maxwell-Lorentz equations.
In July 1905 (published in January 1906) Poincaré showed that the transformations are a consequence of the principle of least action
Principle of least action
In physics, the principle of least action – or, more accurately, the principle of stationary action – is a variational principle that, when applied to the action of a mechanical system, can be used to obtain the equations of motion for that system...
; he demonstrated in more detail the group characteristics of the transformation, which he called Lorentz group
Lorentz group
In physics , the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical setting for all physical phenomena...
, and he showed that the combination x2 + y2 + z2 − c2t2 is invariant. He noticed that the Lorentz transformation is merely a rotation in four-dimensional space about the origin by introducing ct√−1 as a fourth imaginary coordinate, and he used an early form of four-vector
Four-vector
In the theory of relativity, a four-vector is a vector in a four-dimensional real vector space, called Minkowski space. It differs from a vector in that it can be transformed by Lorentz transformations. The usage of the four-vector name tacitly assumes that its components refer to a standard basis...
s.
Einstein (1905)
On June 30, 1905 (published September 1905) Einstein gave a new derivation of the transformation, which was based only on the principle on relativity and the principle of the constancy of the speed of light. Contrary to Lorentz, who considered "local time" only as a mathematical stipulation, Einstein showed that the "effective" coordinates given by the Lorentz transformation were in fact the inertial coordinates of relatively moving frames of reference. For quantities of first order in v/c this was also done by Poincaré in 1900, while Einstein derived the complete transformation by this method. The notation for this transformation is identical to Poincaré's of 1905, except that Einstein didn't set the speed of light to unity:And unlike Lorentz and Poincaré who still distinguished between "real" time in the aether and "apparent" time for moving observers, Einstein showed that the transformations concern the nature of space and time. Einstein's views on space and time, together with Poincaré's four-dimensional approach, were further elaborated by Hermann Minkowski
Hermann Minkowski
Hermann Minkowski was a German mathematician of Ashkenazi Jewish descent, who created and developed the geometry of numbers and who used geometrical methods to solve difficult problems in number theory, mathematical physics, and the theory of relativity.- Life and work :Hermann Minkowski was born...
who gave, besides other things, a geometric representation of the Lorentz transformation by using Minkowski diagram
Minkowski diagram
The Minkowski diagram was developed in 1908 by Hermann Minkowski and provides an illustration of the properties of space and time in the special theory of relativity. It allows a quantitative understanding of the corresponding phenomena like time dilation and length contraction without mathematical...
s.