The history of
Lorentz transformation
In physics, the Lorentz transformations are a six-parameter family of Linear transformation, linear coordinate transformation, transformations from a Frame of Reference, coordinate frame in spacetime to another frame that moves at a constant velo ...
s comprises the development of
linear transformations
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
forming the
Lorentz group
In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicis ...
or
Poincaré group
The Poincaré group, named after Henri Poincaré (1906), was first defined by Hermann Minkowski (1908) as the group of Minkowski spacetime isometries. It is a ten-dimensional non-abelian Lie group that is of importance as a model in our und ...
preserving the
Lorentz interval and the
Minkowski inner product .
In
mathematics, transformations equivalent to what was later known as Lorentz transformations in various dimensions were discussed in the 19th century in relation to the theory of
quadratic forms,
hyperbolic geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ''P ...
,
Möbius geometry, and
sphere geometry, which is connected to the fact that the group of
motions in hyperbolic space, the
Möbius group or
projective special linear group, and the
Laguerre group are
isomorphic to the
Lorentz group
In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicis ...
.
In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, Lorentz transformations became known at the beginning of the 20th century, when it was discovered that they exhibit the symmetry of
Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.
...
. Subsequently, they became fundamental to all of physics, because they formed the basis of
special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The laws ...
in which they exhibit the symmetry of
Minkowski spacetime
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of Three-dimensional space, three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two Event (rel ...
, making the
speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
invariant between different inertial frames. They relate the spacetime coordinates of two arbitrary
inertial frames of reference with constant relative speed ''v''. In one frame, the position of an event is given by ''x,y,z'' and time ''t'', while in the other frame the same event has coordinates ''x′,y′,z′'' and ''t′''.
Mathematical prehistory
Using the coefficients of a
symmetric matrix
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally,
Because equal matrices have equal dimensions, only square matrices can be symmetric.
The entries of a symmetric matrix are symmetric with ...
A, the associated
bilinear form, and a
linear transformation
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
s in terms of
transformation matrix
In linear algebra, linear transformations can be represented by matrices. If T is a linear transformation mapping \mathbb^n to \mathbb^m and \mathbf x is a column vector with n entries, then
T( \mathbf x ) = A \mathbf x
for some m \times n matrix ...
g, the Lorentz transformation is given if the following conditions are satisfied:
:
It forms an
indefinite orthogonal group
In mathematics, the indefinite orthogonal group, is the Lie group of all linear transformations of an ''n''-dimensional real vector space that leave invariant a nondegenerate, symmetric bilinear form of signature , where . It is also called the p ...
called the
Lorentz group
In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicis ...
O(1,n), while the case det g=+1 forms the restricted
Lorentz group
In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicis ...
SO(1,n). The quadratic form becomes the
Lorentz interval in terms of an
indefinite quadratic form of
Minkowski space
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the iner ...
(being a special case of
pseudo-Euclidean space In mathematics and theoretical physics, a pseudo-Euclidean space is a finite-dimensional real -space together with a non- degenerate quadratic form . Such a quadratic form can, given a suitable choice of basis , be applied to a vector , giving
q(x ...
), and the associated bilinear form becomes the
Minkowski inner product.
[Ratcliffe (1994), 3.1 and Theorem 3.1.4 and Exercise 3.1] Long before the advent of special relativity it was used in topics such as the
Cayley–Klein metric
In mathematics, a Cayley–Klein metric is a metric on the complement of a fixed quadric in a projective space which is defined using a cross-ratio. The construction originated with Arthur Cayley's essay "On the theory of distance"Cayley (1859), ...
,
hyperboloid model
In geometry, the hyperboloid model, also known as the Minkowski model after Hermann Minkowski, is a model of ''n''-dimensional hyperbolic geometry in which points are represented by points on the forward sheet ''S''+ of a two-sheeted hyperbolo ...
and other models of
hyperbolic geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ''P ...
, computations of
elliptic functions and integrals, transformation of
indefinite quadratic forms,
squeeze mapping
In linear algebra, a squeeze mapping, also called a squeeze transformation, is a type of linear map that preserves Euclidean area of regions in the Cartesian plane, but is ''not'' a rotation or shear mapping.
For a fixed positive real number , th ...
s of the hyperbola,
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
,
Möbius transformations,
spherical wave transformation, transformation of the
Sine-Gordon equation,
Biquaternion
In abstract algebra, the biquaternions are the numbers , where , and are complex numbers, or variants thereof, and the elements of multiply as in the quaternion group and commute with their coefficients. There are three types of biquaternions co ...
algebra,
split-complex numbers,
Clifford algebra, and others.
Electrodynamics and special relativity
Overview
In the
special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The laws ...
, Lorentz transformations exhibit the symmetry of
Minkowski spacetime
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of Three-dimensional space, three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two Event (rel ...
by using a constant ''c'' as the
speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
, and a parameter ''v'' as the relative
velocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
between two
inertial reference frames. Using the above conditions, the Lorentz transformation in 3+1 dimensions assume the form:
:
In physics, analogous transformations have been introduced by
Voigt (1887) related to an incompressible medium, and by
Heaviside (1888), Thomson (1889), Searle (1896) and
Lorentz (1892, 1895) who analyzed
Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.
...
. They were completed by
Larmor (1897, 1900) and
Lorentz (1899, 1904), and brought into their modern form by
Poincaré (1905) who gave the transformation the name of Lorentz. Eventually,
Einstein (1905) showed in his development of
special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The laws ...
that the transformations follow from the
principle of relativity and constant light speed alone by modifying the traditional concepts of space and time, without requiring a
mechanical aether in contradistinction to Lorentz and Poincaré.
Minkowski (1907–1908) used them to argue that space and time are inseparably connected as
spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
.
Regarding special representations of the Lorentz transformations:
Minkowski (1907–1908) and
Sommerfeld (1909) used imaginary trigonometric functions,
Frank (1909) and
Varićak (1910) used
hyperbolic function
In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the u ...
s,
Bateman and Cunningham (1909–1910) used
spherical wave transformations,
Herglotz (1909–10) used Möbius transformations,
Plummer (1910) and
Gruner (1921) used trigonometric Lorentz boosts,
Ignatowski (1910) derived the transformations without light speed postulate,
Noether (1910) and Klein (1910) as well
Conway (1911) and Silberstein (1911) used Biquaternions,
Ignatowski (1910/11), Herglotz (1911), and others used vector transformations valid in arbitrary directions,
Borel (1913–14) used Cayley–Hermite parameter,
Voigt (1887)
Woldemar Voigt
Woldemar Voigt (; 2 September 1850 – 13 December 1919) 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 ...
(1887)
[Voigt (1887), p. 45] developed a transformation in connection with the
Doppler effect and an incompressible medium, being in modern notation:
[Pais (1982), Kap. 6b]
:
If the right-hand sides of his equations are multiplied by γ 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. Optical phenomena in free space are
scale,
conformal, and
Lorentz invariant
In a relativistic theory of physics, a Lorentz scalar is an expression, formed from items of the theory, which evaluates to a scalar, invariant under any Lorentz transformation. A Lorentz scalar may be generated from e.g., the scalar product of ...
, so the combination is invariant too.
For instance, Lorentz transformations can be extended by using factor
:
[Lorentz (1915/16), p. 197]
:
.
''l''=1/γ gives the Voigt transformation, ''l''=1 the Lorentz transformation. 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. It was demonstrated by Poincaré and Einstein that one has to set ''l''=1 in order to make the above transformation symmetric and to form a group as required by the relativity principle, therefore the Lorentz transformation is the only viable choice.
Voigt sent his 1887 paper to Lorentz in 1908, and that was acknowledged in 1909:
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 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.
[Bucherer (1908), p. 762]
Heaviside (1888), Thomson (1889), Searle (1896)
In 1888,
Oliver Heaviside
Oliver Heaviside FRS (; 18 May 1850 – 3 February 1925) was an English self-taught mathematician and physicist who invented a new technique for solving differential equations (equivalent to the Laplace transform), independently developed ...
[Heaviside (1888), p. 324] investigated the properties of
charges in motion according to Maxwell's electrodynamics. He calculated, among other things, anisotropies in the electric field of moving bodies represented by this formula:
:
.
Consequently,
Joseph John Thomson
Sir Joseph John Thomson (18 December 1856 – 30 August 1940) was a British physicist and Nobel Laureate in Physics, credited with the discovery of the electron, the first subatomic particle to be discovered.
In 1897, Thomson showed that ...
(1889)
[Thomson (1889), p. 12] found a way to substantially simplify calculations concerning moving charges by using the following mathematical transformation (like other authors such as Lorentz or Larmor, also Thomson implicitly used the
Galilean transformation ''z-vt'' in his equation
):
:
Thereby,
inhomogeneous electromagnetic wave equation
In electromagnetism and applications, an inhomogeneous electromagnetic wave equation, or nonhomogeneous electromagnetic wave equation, is one of a set of wave equations describing the propagation of electromagnetic waves generated by nonzero sour ...
s are transformed into a
Poisson equation
Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with t ...
.
[Miller (1981), 98–99] Eventually,
George Frederick Charles Searle[Searle (1886), p. 333] noted in (1896) that Heaviside's expression leads to a deformation of electric fields which he called "Heaviside-Ellipsoid" of
axial ratio
Axial ratio, for any structure or shape with two or more axes, is the ratio of the length (or magnitude) of those axes to each other - the longer axis divided by the shorter.
In ''chemistry'' or ''materials science'', the axial ratio (symbol P) i ...
:
Lorentz (1892, 1895)
In order to explain the
aberration of light
In astronomy, aberration (also referred to as astronomical aberration, stellar aberration, or velocity aberration) is a phenomenon which produces an apparent motion of celestial objects about their true positions, dependent on the velocity of t ...
and the result of the
Fizeau experiment
The Fizeau experiment was carried out by Hippolyte Fizeau in 1851 to measure the relative speeds of light in moving water. Fizeau used a special interferometer arrangement to measure the effect of movement of a medium upon the speed of light.
...
in accordance with
Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.
...
, Lorentz in 1892 developed a model ("
Lorentz ether theory
What is now often called Lorentz ether theory (LET) 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 cen ...
") in which the aether is completely motionless, and the speed of light in the aether is constant in all directions. In order to calculate the optics of moving bodies, Lorentz introduced the following quantities to transform from the aether system into a moving system (it's unknown whether he was influenced by Voigt, Heaviside, and Thomson)
[Lorentz (1892a), p. 141][Miller (1982), 1.4 & 1.5]
:
where ''x
*'' is the
Galilean transformation ''x-vt''. Except the additional γ in the time transformation, this is the complete Lorentz transformation.
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 two steps. At first an implicit Galilean transformation, and later the expansion into the "fictitious" electromagnetic system with the aid of the Lorentz transformation. In order to explain the negative result of the
Michelson–Morley experiment
The Michelson–Morley experiment was an attempt to detect the existence of the luminiferous aether, a supposed medium permeating space that was thought to be the carrier of light waves. The experiment was performed between April and July 188 ...
, he (1892b)
[Lorentz (1892b), p. 141] introduced the additional hypothesis that also intermolecular forces are affected in a similar way and introduced
length contraction
Length contraction is the phenomenon that a moving object's length is measured to be shorter than its proper length, which is the length as measured in the object's own rest frame. It is also known as Lorentz contraction or Lorentz–FitzGerald ...
in his theory (without proof as he admitted). The same hypothesis was already made by
George FitzGerald in 1889 based on Heaviside's work. While length contraction was a real physical effect for Lorentz, 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:
[Lorentz (1895), p. 37]
:
For solving optical problems Lorentz used the following transformation, in which the modified time variable was called "local time" (german: Ortszeit) by him:
[Lorentz (1895), p. 49 for local time and p. 56 for spatial coordinates.]
:
With this concept Lorentz could explain the
Doppler effect, the
aberration of light
In astronomy, aberration (also referred to as astronomical aberration, stellar aberration, or velocity aberration) is a phenomenon which produces an apparent motion of celestial objects about their true positions, dependent on the velocity of t ...
, and the
Fizeau experiment
The Fizeau experiment was carried out by Hippolyte Fizeau in 1851 to measure the relative speeds of light in moving water. Fizeau used a special interferometer arrangement to measure the effect of movement of a medium upon the speed of light.
...
.
Larmor (1897, 1900)
In 1897, Larmor extended the work of Lorentz and derived the following transformation
[Larmor (1897), p. 229]
:
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, explaining the
Michelson–Morley experiment
The Michelson–Morley experiment was an attempt to detect the existence of the luminiferous aether, a supposed medium permeating space that was thought to be the carrier of light waves. The experiment was performed between April and July 188 ...
. It's notable that Larmor was the first who recognized that some sort of
time dilation
In physics and relativity, time dilation is the difference in the elapsed time as measured by two clocks. It is either due to a relative velocity between them ( special relativistic "kinetic" time dilation) or to a difference in gravitational ...
is a consequence of this transformation as well, because "individual electrons describe corresponding parts of their orbits in times shorter for the
estsystem in the ratio 1/γ". Larmor wrote his electrodynamical equations and transformations neglecting terms of higher order than ''(v/c)''
2 – when his 1897 paper was reprinted in 1929, Larmor added the following comment in which he described how they can be made valid to all orders of ''v/c'':
[Larmor (1897/1929), p. 39]
In line with that comment, in his book Aether and Matter published in 1900, Larmor used a modified local time ''t″=t′-εvx′/c
2'' instead of the 1897 expression ''t′=t-vx/c
2'' by replacing ''v/c''
2 with ''εv/c''
2, so that ''t″'' is now identical to the one given by Lorentz in 1892, which he combined with a Galilean transformation for the ''x′, y′, z′, t′'' coordinates:
[Larmor (1900), p. 168]
:
Larmor knew that the Michelson–Morley experiment was accurate enough to detect an effect of motion depending on the factor ''(v/c)''
2, 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'' and ''t″'' as given above) as:
[Larmor (1900), p. 174]
:
by which he arrived at the complete Lorentz transformation. Larmor showed that Maxwell's equations were invariant under this two-step transformation, "to second order in ''v/c''" – it was later shown by Lorentz (1904) and Poincaré (1905) that they are indeed invariant under this transformation to all orders in ''v/c''.
Larmor gave credit to Lorentz in two papers published in 1904, in which he used the term "Lorentz transformation" for Lorentz's first order transformations of coordinates and field configurations:
Lorentz (1899, 1904)
Also Lorentz extended his theorem of corresponding states in 1899. First he wrote a transformation equivalent to the one from 1892 (again, ''x''* must be replaced by ''x-vt''):
[Lorentz (1899), p. 429]
:
Then he introduced a factor ε of which he said he has no means of determining it, and modified his transformation as follows (where the above value of ''t′'' has to be inserted):
[Lorentz (1899), p. 439]
:
This is equivalent to the complete Lorentz transformation when solved for ''x″'' and ''t″'' and with ε=1. Like Larmor, Lorentz noticed in 1899
[Lorentz (1899), p. 442] also some sort of time dilation effect in relation to the frequency of oscillating electrons ''"that in ''S'' the time of vibrations be ''kε'' times as great as in ''S
0''"'', where ''S
0'' is the aether frame.
In 1904 he rewrote the equations in the following form by setting ''l''=1/ε (again, ''x''* must be replaced by ''x-vt''):
[Lorentz (1904), p. 812]
:
Under the assumption that ''l=1'' when ''v''=0, he demonstrated that ''l=1'' must be the case at all velocities, therefore length contraction can only arise in the line of motion. So by setting the factor ''l'' to unity, Lorentz's transformations now assumed the same form as Larmor's and are now completed. Unlike Larmor, who restricted himself to show the covariance of Maxwell's equations to second order, Lorentz tried to widen its covariance to all orders in ''v/c''. He also derived the correct formulas for the velocity dependence of
electromagnetic mass Electromagnetic mass was initially a concept of classical mechanics, denoting as to how much the electromagnetic field, or the self-energy, is contributing to the mass of charged particles. It was first derived by J. J. Thomson in 1881 and was for ...
, and concluded that the transformation formulas must apply to all forces of nature, not only electrical ones.
[Lorentz (1904), p. 826] However, he didn't achieve full covariance of the transformation equations for charge density and velocity. When the 1904 paper was reprinted in 1913, Lorentz therefore added the following remark:
Lorentz's 1904 transformation was cited and used by
Alfred Bucherer in July 1904:
[Bucherer, p. 129; Definition of s on p. 32]
:
or by
Wilhelm Wien
Wilhelm Carl Werner Otto Fritz Franz Wien (; 13 January 1864 – 30 August 1928) was a German physicist who, in 1893, used theories about heat and electromagnetism to deduce Wien's displacement law, which calculates the emission of a blackbody ...
in July 1904:
[Wien (1904), p. 394]
:
or by
Emil Cohn
Emil Georg Cohn (28 September 1854 – 28 January 1944), 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 t ...
in November 1904 (setting the speed of light to unity):
[Cohn (1904a), pp. 1296-1297]
:
or by
Richard Gans
Richard is a male given name. It originates, via Old French, from Old Frankish and is a compound of the words descending from Proto-Germanic ''*rīk-'' 'ruler, leader, king' and ''*hardu-'' 'strong, brave, hardy', and it therefore means 'stron ...
in February 1905:
[Gans (1905), p. 169]
:
Poincaré (1900, 1905)
Local time
Neither Lorentz or Larmor gave a clear physical interpretation of the origin of local time. However,
Henri Poincaré 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
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.
[Poincaré (1900), pp. 272–273] In order to synchronise the clocks here on Earth (the ''x*, t''* frame) a light signal from one clock (at the origin) is sent to another (at ''x''*), and is sent back. It's supposed 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 or ether ("luminiferous", meaning "light-bearing") was the postulated medium for the propagation of light. It was invoked to explain the ability of the apparently wave-based light to propagate through empty space (a vacuum), so ...
system for Lorentz and Larmor). 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 ''δt
a+δt
b'' and the time ''t*=(δt
a+δt
b)/2'' is ascribed to the moment when the light signal reached the distant clock. In the rest frame the time ''t=δt
a'' is ascribed to that same instant. Some algebra gives the relation between the different time coordinates ascribed to the moment of reflection. Thus
:
identical to Lorentz (1892). By dropping the factor γ
2 under the assumption that
, Poincaré gave the result ''t*=t-vx*/c
2'', which is the form used by Lorentz in 1895.
Similar physical interpretations of local time were later given by
Emil Cohn
Emil Georg Cohn (28 September 1854 – 28 January 1944), 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 t ...
(1904)
[Cohn (1904b), p. 1408] and
Max Abraham
Max Abraham (; 26 March 1875 – 16 November 1922) was a German physicist known for his work on electromagnetism and his opposition to the theory of relativity.
Biography
Abraham was born in Danzig, Imperial Germany (now Gdańsk in Poland) t ...
(1905).
[Abraham (1905), § 42]
Lorentz transformation
On June 5, 1905 (published June 9) Poincaré formulated transformation equations which are algebraically equivalent to those of Larmor and Lorentz and gave them the modern form:
[Poincaré (1905), p. 1505]
:
.
Apparently Poincaré was unaware of Larmor's contributions, because he only mentioned Lorentz and therefore used for the first time the name "Lorentz transformation". Poincaré set the speed of light to unity, pointed out the group characteristics of the transformation by setting ''l''=1, and modified/corrected Lorentz's derivation of the equations of electrodynamics in some details in order to fully satisfy the principle of relativity, ''i.e.'' making them fully Lorentz covariant.
In July 1905 (published in January 1906)
[Poincaré (1905/06), pp. 129ff] Poincaré showed in detail how the transformations and electrodynamic equations are a consequence of the
principle of least action
The stationary-action principle – also known as the principle of least action – is a variational principle that, when applied to the '' action'' of a mechanical system, yields the equations of motion for that system. The principle states tha ...
; he demonstrated in more detail the group characteristics of the transformation, which he called
Lorentz group
In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicis ...
, and he showed that the combination ''x
2+y
2+z
2-t
2'' is invariant. He noticed that the Lorentz transformation is merely a rotation in four-dimensional space about the origin by introducing
as a fourth imaginary coordinate, and he used an early form of
four-vector
In special relativity, a four-vector (or 4-vector) is an object with four components, which transform in a specific way under Lorentz transformations. Specifically, a four-vector is an element of a four-dimensional vector space considered as a ...
s. He also formulated the velocity addition formula, which he had already derived in unpublished letters to Lorentz from May 1905:
[Poincaré (1905/06), p. 144]
:
.
Einstein (1905) – Special relativity
On June 30, 1905 (published September 1905) Einstein published what is now called
special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The laws ...
and 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. While Lorentz considered "local time" to be a mathematical stipulation device for explaining the Michelson-Morley experiment, Einstein showed that the 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. 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.
The notation for this transformation is equivalent to Poincaré's of 1905, except that Einstein didn't set the speed of light to unity:
[Einstein (1905), p. 902]
:
Einstein also defined the velocity addition formula:
[Einstein (1905), § 5 and § 9]
:
and the light aberration formula:
[Einstein (1905), § 7]
:
Minkowski (1907–1908) – Spacetime
The work on the principle of relativity by Lorentz, Einstein,
Planck
Max Karl Ernst Ludwig Planck (, ; 23 April 1858 – 4 October 1947) was a German theoretical physicist whose discovery of energy quanta won him the Nobel Prize in Physics in 1918.
Planck made many substantial contributions to theoretical p ...
, together with Poincaré's four-dimensional approach, were further elaborated and combined with the
hyperboloid model
In geometry, the hyperboloid model, also known as the Minkowski model after Hermann Minkowski, is a model of ''n''-dimensional hyperbolic geometry in which points are represented by points on the forward sheet ''S''+ of a two-sheeted hyperbolo ...
by
Hermann Minkowski in 1907 and 1908.
[Minkowski (1907/15), pp. 927ff][Minkowski (1907/08), pp. 53ff] Minkowski particularly reformulated electrodynamics in a four-dimensional way (
Minkowski spacetime
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of Three-dimensional space, three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two Event (rel ...
). For instance, he wrote ''x, y, z, it'' in the form ''x
1, x
2, x
3, x
4''. By defining ψ as the angle of rotation around the ''z''-axis, the Lorentz transformation assumes the form (with ''c''=1):
[Minkowski (1907/08), p. 59]
:
Even though Minkowski used the imaginary number iψ, he for once
directly used the
tangens hyperbolicus in the equation for velocity
:
with
.
Minkowski's expression can also by written as ψ=atanh(q) and was later called
rapidity. He also wrote the Lorentz transformation in matrix form:
[Minkowski (1907/08), pp. 65–66, 81–82]
:
As a graphical representation of the Lorentz transformation he introduced the
Minkowski diagram
A spacetime diagram is a graphical illustration of the properties of space and time in the special theory of relativity. Spacetime diagrams allow a qualitative understanding of the corresponding phenomena like time dilation and length contractio ...
, which became a standard tool in textbooks and research articles on relativity:
[Minkowski (1908/09), p. 77]
Sommerfeld (1909) – Spherical trigonometry
Using an imaginary rapidity such as Minkowski,
Arnold Sommerfeld (1909) formulated the Lorentz boost and the relativistic velocity addition in terms of trigonometric functions and the
spherical law of cosines In spherical trigonometry, the law of cosines (also called the cosine rule for sides) is a theorem relating the sides and angles of spherical triangles, analogous to the ordinary law of cosines from plane trigonometry.
Given a unit sphere, a "sph ...
:
[Sommerfeld (1909), p. 826ff.]
:
Frank (1909) – Hyperbolic functions
Hyperbolic functions were used by
Philipp Frank (1909), who derived the Lorentz transformation using ''ψ'' as
rapidity:
[Frank (1909), pp. 423-425]
:
Bateman and Cunningham (1909–1910) – Spherical wave transformation
In line with
Sophus Lie
Marius Sophus Lie ( ; ; 17 December 1842 – 18 February 1899) was a Norwegian mathematician. He largely created the theory of continuous symmetry and applied it to the study of geometry and differential equations.
Life and career
Marius Soph ...
's (1871) research on the relation between sphere transformations with an imaginary radius coordinate and 4D conformal transformations, it was pointed out by
Bateman and
Cunningham
Cunningham is a surname of Scottish origin, see Clan Cunningham.
Notable people sharing this surname
A–C
* Aaron Cunningham (born 1986), American baseball player
*Abe Cunningham, American drummer
* Adrian Cunningham (born 1960), Australian ...
(1909–1910), that by setting ''u=ict'' as the imaginary fourth coordinates one can produce spacetime conformal transformations. Not only the quadratic form
, but also
Maxwells equations are covariant with respect to these transformations, irrespective of the choice of λ. These variants of conformal or Lie sphere transformations were called
spherical wave transformations by Bateman.
[Bateman (1909/10), pp. 223ff][Cunningham (1909/10), pp. 77ff] However, this covariance is restricted to certain areas such as electrodynamics, whereas the totality of natural laws in inertial frames is covariant under the
Lorentz group
In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicis ...
.
[Klein (1910)] In particular, by setting λ=1 the Lorentz group can be seen as a 10-parameter subgroup of the 15-parameter spacetime conformal group .
Bateman (1910–12) also alluded to the identity between the
Laguerre inversion and the Lorentz transformations. In general, the isomorphism between the Laguerre group and the Lorentz group was pointed out by
Élie Cartan
Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometr ...
(1912, 1915–55),
[Cartan (1912), p. 23] Henri Poincaré (1912–21)
[Poincaré (1912/21), p. 145] and others.
Herglotz (1909/10) – Möbius transformation
Following
Felix Klein
Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and grou ...
(1889–1897) and Fricke & Klein (1897) concerning the Cayley absolute, hyperbolic motion and its transformation,
Gustav Herglotz (1909–10) classified the one-parameter Lorentz transformations as loxodromic, hyperbolic, parabolic and elliptic. The general case (on the left) and the hyperbolic case equivalent to Lorentz transformations or squeeze mappings are as follows:
[Herglotz (1909/10), pp. 404-408]
:
Varićak (1910) – Hyperbolic functions
Following
Sommerfeld (1909), hyperbolic functions were used by
Vladimir Varićak in several papers starting from 1910, who represented the equations of special relativity on the basis of
hyperbolic geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ''P ...
in terms of Weierstrass coordinates. For instance, by setting ''l=ct'' and ''v/c=tanh(u)'' with ''u'' as rapidity he wrote the Lorentz transformation:
[Varićak (1910), p. 93]
:
and showed the relation of rapidity to the
Gudermannian function
In mathematics, the Gudermannian function relates a hyperbolic angle measure \psi to a circular angle measure \phi called the ''gudermannian'' of \psi and denoted \operatorname\psi. The Gudermannian function reveals a close relationship betwee ...
and the
angle of parallelism
In hyperbolic geometry, the angle of parallelism \Pi(a) , is the angle at the non-right angle vertex of a right hyperbolic triangle having two asymptotic parallel sides. The angle depends on the segment length ''a'' between the right angle an ...
:
:
He also related the velocity addition to the
hyperbolic law of cosines:
[Varićak (1910), p. 94]
:
Subsequently, other authors such as
E. T. Whittaker
Sir Edmund Taylor Whittaker (24 October 1873 – 24 March 1956) was a British mathematician, physicist, and historian of science. Whittaker was a leading mathematical scholar of the early 20th-century who contributed widely to applied mathema ...
(1910) or
Alfred Robb (1911, who coined the name rapidity) used similar expressions, which are still used in modern textbooks.
Plummer (1910) – Trigonometric Lorentz boosts
w:Henry Crozier Keating Plummer (1910) defined the Lorentz boost in terms of trigonometric functions
[Plummer (1910), p. 256]
:
Ignatowski (1910)
While earlier derivations and formulations of the Lorentz transformation relied from the outset on optics, electrodynamics, or the invariance of the speed of light,
Vladimir Ignatowski (1910) showed that it is possible to use the principle of relativity (and related
group theoretical principles) alone, in order to derive the following transformation between two inertial frames:
[Ignatowski (1910), pp. 973–974][Ignatowski (1910/11), p. 13]
:
The variable ''n'' can be seen as a space-time constant whose value has to be determined by experiment or taken from a known physical law such as electrodynamics. For that purpose, Ignatowski used the above-mentioned Heaviside ellipsoid representing a contraction of electrostatic fields by ''x''/γ in the direction of motion. It can be seen that this is only consistent with Ignatowski's transformation when ''n=1/c''
2, resulting in ''p''=γ and the Lorentz transformation. With ''n''=0, no length changes arise and the Galilean transformation follows. Ignatowski's method was further developed and improved by
Philipp Frank and
Hermann Rothe (1911, 1912),
[Frank & Rothe (1911), pp. 825ff; (1912), p. 750ff.] with various authors developing similar methods in subsequent years.
[Baccetti (2011), see references 1–25 therein.]
Noether (1910), Klein (1910) – Quaternions
Felix Klein
Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and grou ...
(1908) described Cayley's (1854) 4D quaternion multiplications as "Drehstreckungen" (orthogonal substitutions in terms of rotations leaving invariant a quadratic form up to a factor), and pointed out that the modern principle of relativity as provided by Minkowski is essentially only the consequent application of such Drehstreckungen, even though he didn't provide details.
[Klein (1908), p. 165]
In an appendix to Klein's and Sommerfeld's "Theory of the top" (1910),
Fritz Noether
Fritz Alexander Ernst Noether (7 October 1884 – 10 September 1941) was a Jewish German mathematician who emigrated from Nazi Germany to the Soviet Union. He was later executed by the NKVD.
Biography
Fritz Noether's father Max Noethe ...
showed how to formulate hyperbolic rotations using biquaternions with
, which he also related to the speed of light by setting ω
2=-''c''
2. He concluded that this is the principal ingredient for a rational representation of the group of Lorentz transformations:
[Noether (1910), pp. 939–943]
:
Besides citing quaternion related standard works by
Arthur Cayley (1854), Noether referred to the entries in Klein's encyclopedia by
Eduard Study
Eduard Study ( ), more properly Christian Hugo Eduard Study (March 23, 1862 – January 6, 1930), was a German mathematician known for work on invariant theory of ternary forms (1889) and for the study of spherical trigonometry. He is also known f ...
(1899) and the French version by
Élie Cartan
Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometr ...
(1908). Cartan's version contains a description of Study's
dual number
In algebra, the dual numbers are a hypercomplex number system first introduced in the 19th century. They are expressions of the form , where and are real numbers, and is a symbol taken to satisfy \varepsilon^2 = 0 with \varepsilon\neq 0.
Du ...
s, Clifford's biquaternions (including the choice
for hyperbolic geometry), and Clifford algebra, with references to Stephanos (1883), Buchheim (1884–85), Vahlen (1901–02) and others.
Citing Noether, Klein himself published in August 1910 the following quaternion substitutions forming the group of Lorentz transformations:
[Klein (1910), p. 300]
:
or in March 1911
[Klein (1911), pp. 602ff.]
:
Conway (1911), Silberstein (1911) – Quaternions
Arthur W. Conway in February 1911 explicitly formulated quaternionic Lorentz transformations of various electromagnetic quantities in terms of velocity λ:
[Conway (1911), p. 8]
:
Also
Ludwik Silberstein
Ludwik Silberstein (1872 – 1948) was a Polish-American physicist who helped make special relativity and general relativity staples of university coursework. His textbook '' The Theory of Relativity'' was published by Macmillan in 1914 with a se ...
in November 1911
[Silberstein (1911/12), p. 793] as well as in 1914, formulated the Lorentz transformation in terms of velocity ''v'':
:
Silberstein cites Cayley (1854, 1855) and Study's encyclopedia entry (in the extended French version of Cartan in 1908), as well as the appendix of Klein's and Sommerfeld's book.
Ignatowski (1910/11), Herglotz (1911), and others – Vector transformation
Vladimir Ignatowski (1910, published 1911) showed how to reformulate the Lorentz transformation in order to allow for arbitrary velocities and coordinates:
[Ignatowski (1910/11a), p. 23; (1910/11b), p. 22]
:
Gustav Herglotz (1911)
[Herglotz (1911), p. 497] also showed how to formulate the transformation in order to allow for arbitrary velocities and coordinates v=''(v
x, v
y, v
z)'' and r=''(x, y, z)'':
:
This was simplified using vector notation by
Ludwik Silberstein
Ludwik Silberstein (1872 – 1948) was a Polish-American physicist who helped make special relativity and general relativity staples of university coursework. His textbook '' The Theory of Relativity'' was published by Macmillan in 1914 with a se ...
(1911 on the left, 1914 on the right):
[Silberstein (1911/12), p. 792; (1914), p. 123]
:
Equivalent formulas were also given by
Wolfgang Pauli
Wolfgang Ernst Pauli (; ; 25 April 1900 – 15 December 1958) was an Austrian theoretical physicist and one of the pioneers of quantum physics. In 1945, after having been nominated by Albert Einstein, Pauli received the Nobel Prize in Physics ...
(1921), with
Erwin Madelung
Erwin Madelung (18 May 1881 – 1 August 1972) was a German physicist.
He was born in 1881 in Bonn. His father was the surgeon Otto Wilhelm Madelung. He earned a doctorate in 1905 from the University of Göttingen, specializing in crystal structu ...
(1922) providing the matrix form
:
These formulas were called "general Lorentz transformation without rotation" by
Christian Møller (1952),
[Møller (1952/55), pp. 41–43] who in addition gave an even more general Lorentz transformation in which the Cartesian axes have different orientations, using a
rotation operator . In this case, v′=''(v′
x, v′
y, v′
z)'' is not equal to -v=''(-v
x, -v
y, -v
z)'', but the relation
holds instead, with the result
:
Borel (1913–14) – Cayley–Hermite parameter
Émile Borel
Félix Édouard Justin Émile Borel (; 7 January 1871 – 3 February 1956) was a French mathematician and politician. As a mathematician, he was known for his founding work in the areas of measure theory and probability.
Biography
Borel was ...
(1913) started by demonstrating Euclidean motions using Euler-Rodrigues parameter in three dimensions, and Cayley's (1846) parameter in four dimensions. Then he demonstrated the connection to indefinite quadratic forms expressing hyperbolic motions and Lorentz transformations. In three dimensions:
[Borel (1913/14), p. 39]
:
In four dimensions:
[Borel (1913/14), p. 41]
:
Gruner (1921) – Trigonometric Lorentz boosts
In order to simplify the graphical representation of Minkowski space,
Paul Gruner (1921) (with the aid of Josef Sauter) developed what is now called
Loedel diagrams, using the following relations:
[Gruner (1921a),]
:
In another paper Gruner used the alternative relations:
[Gruner (1921b)]
:
See also
*
History of special relativity
The history of special relativity consists of many theoretical results and empirical findings obtained by Albert A. Michelson, Hendrik Lorentz, Henri Poincaré and others. It culminated in the theory of special relativity proposed by Albert Eins ...
References
Historical mathematical sources
Historical relativity sources
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*. For Minkowski's and Voigt's statements see p. 762.
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English translation
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*; English translation by David Delphenich
On the mechanics of deformable bodies from the standpoint of relativity theory
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* (Reprint of Larmor (1897) with new annotations by Larmor.)
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English translation
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* Written by Poincaré in 1912, printed in Acta Mathematica in 1914 though belatedly published in 1921.
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Secondary sources
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*Barrett, J.F. (2006), The hyperbolic theory of relativity,
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* See also "Michelson, FitzGerald and Lorentz: the origins of relativity revisited"
Online
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* (Only pages 1–21 were published in 1915, the entire article including pp. 39–43 concerning the groups of Laguerre and Lorentz was posthumously published in 1955 in Cartan's collected papers, and was reprinted in the Encyclopédie in 1991.)
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*; First edition 1911, second expanded edition 1913, third expanded edition 1919.
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In English:
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External links
*Mathpages
{{DEFAULTSORT:Lorentz transformations
Equations
History of physics
Hendrik Lorentz
Historical treatment of quaternions