Lorentz transformation - AbsoluteAstronomy.com

In physics

Physics

Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, the Lorentz transformation or Lorentz-Fitzgerald
George FitzGerald
George Francis FitzGerald was an Irish professor of "natural and experimental philosophy" at Trinity College in Dublin, Ireland, during the last quarter of the 19th century....

transformation describes how, according to the theory 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...

, 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

Physicist

A physicist is a scientist who studies or practices physics. Physicists study a wide range of physical phenomena in many branches of physics spanning all length scales: from sub-atomic particles of which all ordinary matter is made to the behavior of the material Universe as a whole...

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...

. It reflects the surprising fact that observers moving at different velocities

Velocity

In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...

may measure different distances

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...

, elapsed times

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...

, and even different orderings of events

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...

.

The Lorentz transformation was originally the result of attempts by Lorentz and others to explain how the speed of light

Light

Light or visible light is electromagnetic radiation that is visible to the human eye, and is responsible for the sense of sight. Visible light has wavelength in a range from about 380 nanometres to about 740 nm, with a frequency range of about 405 THz to 790 THz...

was observed to be independent of the reference frame, and to understand the symmetries of the laws of electromagnetism. Albert 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...

later re-derived the transformation from his postulates of special relativity

Special relativity

. The Lorentz transformation supersedes 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...

of Newtonian physics, which assumes an absolute space and time (see Galilean relativity). According to special relativity, this is a good approximation only at relative speeds much smaller than the speed of light.

If space is homogeneous

Homogeneity (physics)

In general, homogeneity is defined as the quality or state of being homogeneous . For instance, a uniform electric field would be compatible with homogeneity...

, then the Lorentz transformation must be a linear transformation

Linear transformation

In mathematics, a linear map, linear mapping, linear transformation, or linear operator is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. As a result, it always maps straight lines to straight lines or 0...

. Also, since relativity postulates that the speed of light is the same for all observers, it must preserve the spacetime interval between any two events in Minkowski space

Minkowski space

In physics and mathematics, Minkowski space or Minkowski spacetime is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated...

. The Lorentz transformation describes only the transformations in which the spacetime event at the origin is left fixed, so they can be considered as a hyperbolic rotation of Minkowski space

Minkowski space

In physics and mathematics, Minkowski space or Minkowski spacetime is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated...

. The more general set of transformations that also includes translations is known as the Poincaré group

Poincaré group

In physics and mathematics, the Poincaré group, named after Henri Poincaré, is the group of isometries of Minkowski spacetime.-Simple explanation:...

History

History of lorentz transformations

The Lorentz transformations relate the space-time coordinates, relative to a particular inertial frame of reference , 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...

Many physicists, including George FitzGerald

George FitzGerald

George Francis FitzGerald was an Irish professor of "natural and experimental philosophy" at Trinity College in Dublin, Ireland, during the last quarter of the 19th century....

, Joseph 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...

, Hendrik 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...

and Woldemar 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...

, had been discussing the physics behind these equations since 1887.
Larmor and Lorentz, who believed the luminiferous ether hypothesis, were seeking the transformation under which Maxwell's equations

Maxwell's equations

Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...

were invariant when transformed from the ether to a moving frame. Early in 1889, Oliver Heaviside

Oliver Heaviside

Oliver Heaviside was a self-taught English electrical engineer, mathematician, and physicist who adapted complex numbers to the study of electrical circuits, invented mathematical techniques to the solution of differential equations , reformulated Maxwell's field equations in terms of electric and...

had shown from Maxwell's equations that the electric field surrounding a spherical distribution of charge should cease to have spherical symmetry once the charge is in motion relative to the ether. FitzGerald then conjectured that Heaviside’s distortion result might be applied to a theory of intermolecular forces. Some months later, FitzGerald published his conjecture in Science to explain the baffling outcome of the 1887 ether-wind experiment of Michelson and Morley

Michelson-Morley experiment

The Michelson–Morley experiment was performed in 1887 by Albert Michelson and Edward Morley at what is now Case Western Reserve University in Cleveland, Ohio. Its results are generally considered to be the first strong evidence against the theory of a luminiferous ether and in favor of special...

. This idea was extended by Lorentz
and Larmor
over several years, and became known as the FitzGerald-Lorentz explanation of the Michelson-Morley null result, known early on through the writings of Lodge, Lorentz, Larmor, and FitzGerald.
Their explanation was widely known before 1905.
Larmor is also credited to have been the first to understand the crucial time dilation property inherent in his equations.

In 1905, Henri Poincaré

Henri Poincaré

Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...

was the first to recognize that the transformation has the properties of a mathematical group

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

,
and named it after Lorentz.
Later in the same year Einstein derived the Lorentz transformation under the assumptions of 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....

and the constancy of the speed of light in any inertial reference frame,
obtaining results that were algebraically equivalent to
Larmor's (1897) and Lorentz's (1899, 1904), but with a different interpretation.

Paul Langevin

Paul Langevin

Paul Langevin was a prominent French physicist who developed Langevin dynamics and the Langevin equation. He was one of the founders of the Comité de vigilance des intellectuels antifascistes, an antifascist organization created in the wake of the 6 February 1934 far right riots...

(1911) said of the transformation:

"It is the great merit of H. A. Lorentz to have seen that the fundamental equations of electromagnetism admit a group of transformations which enables them to have the same form when one passes from one frame of reference to another; this new transformation has the most profound implications for the transformations of space and time".

Lorentz transformation for frames in standard configuration

Consider two observers O and O' , each using their own Cartesian coordinate system

Cartesian coordinate system

A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length...

to measure space and time intervals. O uses (t, x, y, z) and O ' uses (t' , x' , y' , z' ). Assume further that the coordinate systems are oriented so that the x-axis and the x' -axis are collinear

Line (geometry)

The notion of line or straight line was introduced by the ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects...

, the y-axis is parallel to the y' -axis, and the z-axis parallel to the z' -axis. The relative velocity between the two observers is v along the common x-axis. Also assume that the origins of both coordinate systems are the same, that is, coincident times and positions.

If all these hold, then the coordinate systems are said to be in standard configuration. A symmetric presentation

Lorentz transformation under symmetric configuration

In physics, the Lorentz transformation converts between two different observers' measurements of space and time, where one observer is in constant motion with respect to the other....

between the forward Lorentz Transformation and the inverse Lorentz Transformation can be achieved if coordinate systems are in symmetric configuration

Lorentz transformation under symmetric configuration

In physics, the Lorentz transformation converts between two different observers' measurements of space and time, where one observer is in constant motion with respect to the other....

. The symmetric form highlights that all physical laws should remain unchanged under a Lorentz transformation.

Below the Lorentz transformations are called "boosts" in the stated directions.

Boost in the x-direction

These are the forms most usually introduced. The Lorentz transformation for frames in standard configuration can be shown to be (see for example and ):

where:

v is the relative velocity between frames in the x-direction,
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...

,
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...

(Greek
Greek alphabet
The Greek alphabet is the script that has been used to write the Greek language since at least 730 BC . The alphabet in its classical and modern form consists of 24 letters ordered in sequence from alpha to omega...

lowercase gamma
Gamma
Gamma is the third letter of the Greek alphabet. In the system of Greek numerals it has a value of 3. It was derived from the Phoenician letter Gimel . Letters that arose from Gamma include the Roman C and G and the Cyrillic letters Ge Г and Ghe Ґ.-Greek:In Ancient Greek, gamma represented a...

),
(Greek
Greek alphabet
The Greek alphabet is the script that has been used to write the Greek language since at least 730 BC . The alphabet in its classical and modern form consists of 24 letters ordered in sequence from alpha to omega...

lowercase beta
Beta
Beta is the second letter of the Greek alphabet. Beta or BETA may also refer to:-Biology:*Beta , a genus of flowering plants, mostly referred to as beets*Beta, a rank in a community of social animals...

), again for the x-direction.

The use of β and γ is standard throughout the literature. For the remainder of the article - they will be also used throughout unless otherwise stated. Since the above is a linear system of equations (more technically a linear transformation

Linear transformation

), they can be written in matrix

Matrix

- Science and mathematics :* Matrix , a mathematical object generally represented as an array of numbers** Matrix calculus, a notation for calculus operations on matrix spaces** Identity matrix...

form:

Boost in the y or z directions

The above collection of equations apply only for a boost in the x-direction. If the standard configuration used the y or z directions instead of x, the results would be similar.

For the y-direction:

summarized by

where v and so β are now in the y-direction. For the z-direction:

summarized by

where v and so β are now in the z-direction. These are easily obtained by Cyclic permutation

Cyclic permutation

A cyclic permutation or circular permutation is a permutation built from one or more sets of elements in cyclic order.The notion "cyclic permutation" is used in different, but related ways:- Definition 1 :right|mapping of permutation...

s of x, y, z. If we couldn't do this - it would imply the laws of physics would be different in each direction. This is not the case, by experimentation and observation.

The Lorentz or boost matrix is usually denoted by Λ (Greek

Greek alphabet

The Greek alphabet is the script that has been used to write the Greek language since at least 730 BC . The alphabet in its classical and modern form consists of 24 letters ordered in sequence from alpha to omega...

capital lambda

Lambda

Lambda is the 11th letter of the Greek alphabet. In the system of Greek numerals lambda has a value of 30. Lambda is related to the Phoenician letter Lamed . Letters in other alphabets that stemmed from lambda include the Roman L and the Cyrillic letter El...

). Above the transformations have been applied to the four-position

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...

The Lorentz transform for a boost in one of the above directions can be compactly written as a single matrix equation:

However, the transformation matrix is universal for all four-vector

Four-vector

s . If A is any four-vector, then:

Boost in any one direction by index permutation

The previous sets of equations can be summarized using index notation

Index notation

Index notation is used in mathematics and computer programming to specify the elements of matrices or the components of a vector. The formalism of how indices are used varies according to the discipline...

, rather than Cartesian coordinates :

where:

x₀ is the time coordinate,
x_i, x_j, x_k are spatial coordinates,
β_i and v_i are in the direction of relative motion,
The indices i, j, k each correspond a direction mutually perpendicular
Perpendicular
In geometry, two lines or planes are considered perpendicular to each other if they form congruent adjacent angles . The term may be used as a noun or adjective...

to the others, so x_i is mutually perpendicular to x_j and x_k, x_j mutually perpendicular to x_i and x_k etc., for all cyclic permutations of i, j, k.

Boost in any direction

More generally for a boost in any arbitrary direction at velocity v = (v_x, v_y, v_z), or equivalently β = (β_x, β_y, β_z),

where:

(cartesian notation) equivalently written (component notation),
equivalently written
equivalently written
applies for the resultant velocity v, not only one component.

Again the transformation can be written in the same form as before,

Although the matrix Λ is symmetric, it appears daunting and unwieldy. To make it easier to remember and use, we could simply write the matrix in terms of components.

The above transformation has the structure:

where the components are:

Note that this transformation is only the "boost," i.e., a transformation between two frames whose x, y, and z axis are parallel and whose spacetime origins coincide (see The "Standard configuration" Figure). The most general proper Lorentz transformation also contains a rotation of the three axes, because the composition of two boosts is not a pure boost but is a boost followed by a rotation. The rotation gives rise to Thomas precession

Thomas precession

In physics the Thomas precession, named after Llewellyn Thomas, is a special relativistic correction that applies to the spin of an elementary particle or the rotation of a macroscopic gyroscope and relates the angular velocity of the spin of a particle following a curvilinear orbit to the angular...

. The boost is given by a symmetric matrix, but the general Lorentz transformation matrix need not be symmetric.

Composition of two boosts

The composition of two Lorentz boosts B(u) and B(v) of velocities u and v is given by:

,

where

is the velocity-addition

Velocity-addition formula

In physics, a velocity-addition formula is an equation that relates the velocities of moving objects in different reference frames.- Galilean addition of velocities :...

, and Gyr[u,v] (capital G) is the rotation arising from the composition, gyr (lower case g) being the gyrovector space

Gyrovector space

A gyrovector space is a mathematical concept for studying hyperbolic geometry in analogy to the way vector spaces are used in Euclidean geometry. This vector-based approach has been developed by Abraham Albert Ungar from the late 1980s onwards...

abstraction of the gyroscopic Thomas precession, and B(v) is the 4x4 matrix that uses the components of v, i.e. v₁, v₂, v₃ in the entries of the matrix, or rather the components of v/c in the representation that is used above.

The composition of two Lorentz transformations L(u,U) and L(v,V) which include rotations U and V is given by:

If the 3x3 matrix form of the rotation applied to spatial coordinates is given by gyr[u,v], then the 4x4 matrix rotation applied to 4-coordinates is given by:

For a boost in an arbitrary direction with velocity

, it is convenient to decompose the spatial vector

into components perpendicular and parallel to the velocity

. Then only the component

in the direction of

is 'warped' by the gamma factor:

where now

. The second of these can be written as:

These equations can be expressed in matrix form as

where I is the identity matrix, v is velocity written as a column vector, v^T is its transpose (a row vector) and

is its versor.

Rapidity

The Lorentz transformation can be cast into another useful form by defining a parameter

called the rapidity

Rapidity

In relativity, rapidity is an alternative to speed as a framework for measuring motion. On parallel velocities rapidities are simply additive, unlike speeds at relativistic velocities. For low speeds, rapidity and speed are proportional, but for high speeds, rapidity takes a larger value. The...

(an instance of hyperbolic angle

Hyperbolic angle

In mathematics, a hyperbolic angle is a geometric figure that divides a hyperbola. The science of hyperbolic angle parallels the relation of an ordinary angle to a circle...

) such that

so that

Equivalently:

Then the Lorentz transformation in standard configuration is:

Hyperbolic trigonometric expressions

From the above expressions for e^φ and e^−φ

and therefore,

Hyperbolic rotation of coordinates

Substituting these expressions into the matrix form of the transformation, we have:

Thus, the Lorentz transformation can be seen as a hyperbolic rotation of coordinates in Minkowski space

Minkowski space

In physics and mathematics, Minkowski space or Minkowski spacetime is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated...

, where the parameter

represents the hyperbolic angle of rotation, often referred to as rapidity

Rapidity

. This transformation is sometimes illustrated with a 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...

Lorentz transformation of the electromagnetic field

The fact that the electromagnetic field shows relativistic effects becomes clear by carrying out a simple thought experiment:

Consider an observer measuring a charge at rest in a reference frame F. The observer will detect a static electric field. As the charge is stationary in this frame, there is no electric current, so the observer will not observe any magnetic field.

If another observer in frame F' moving at relative velocity v (to F and the charge), this observer will notice the electric field changing as a result of relative motion, in addition interpret the charge in motion travelling at relative velocity −v (to F') as an electric current having a magnetic field around it.

This shows that the Lorentz transformation also applies to electromagnetic field quantities when changing the frame of reference.

For the electric and magnetic field quantities, the following transformations apply:

These formulae can be summarized in the matrix:

In non-relativistic approximation, i. e. for speeds

, the relativistic factor

, so that there is no need to distinguish between the spatial and temporal coordinates in Maxwell's equations. This yields the following transformations:

Spacetime interval

In a given coordinate system (

), if two events

and

are separated by

the spacetime interval between them is given by

This can be written in another form using the Minkowski metric. In this coordinate system,

Then, we can write

or, using the Einstein summation convention,

Now suppose that we make a coordinate transformation

. Then, the interval in this coordinate system is given by

It is a result of special relativity

Special relativity

that the interval is an invariant

Invariant (physics)

In mathematics and theoretical physics, an invariant is a property of a system which remains unchanged under some transformation.-Examples:In the current era, the immobility of polaris under the diurnal motion of the celestial sphere is a classical illustration of physical invariance.Another...

. That is,

. It can be shown that this requires the coordinate transformation to be of the form

Here,

is a constant vector and

a constant matrix, where we require that

Such a transformation is called a Poincaré transformation

Poincaré group

In physics and mathematics, the Poincaré group, named after Henri Poincaré, is the group of isometries of Minkowski spacetime.-Simple explanation:...

or an inhomogeneous Lorentz transformation. The

represents a spacetime translation. When

, the transformation is called an homogeneous Lorentz transformation, or simply a Lorentz transformation.

Taking the determinant of

gives us

Lorentz transformations with

form a subgroup called proper Lorentz transformations which is the special orthogonal group

. Those with

are called improper Lorentz transformations which is not a subgroup, as the product of any two improper Lorentz transformations will be a proper Lorentz transformation. From the above definition of

it can be shown that

, so either

, called orthochronous and non-orthochronous respectively. An important subgroup of the proper Lorentz transformations are the proper orthochronous Lorentz transformations which consist purely of boosts and rotations. Any Lorentz transform can be written as a proper orthochronous, together with one or both of the two discrete transformations; space inversion (

) and time reversal (

), whose non-zero elements are:

The set of Poincaré transformations satisfies the properties of a group and is called the Poincaré group

Poincaré group

In physics and mathematics, the Poincaré group, named after Henri Poincaré, is the group of isometries of Minkowski spacetime.-Simple explanation:...

. Under the Erlangen program

Erlangen program

An influential research program and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen über neuere geometrische Forschungen...

, Minkowski space

Minkowski space

In physics and mathematics, Minkowski space or Minkowski spacetime is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated...

can be viewed as the geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

defined by the Poincaré group, which combines Lorentz transformations with translations. In a similar way, the set of all Lorentz transformations forms a group, called the 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...

.

A quantity invariant under Lorentz transformations is known as a Lorentz scalar

Lorentz scalar

In physics, a Lorentz scalar is a scalar which is invariant under a Lorentz transformation. A Lorentz scalar may be generated from multiplication of vectors or tensors...

Special relativity

One of the most astounding consequences of Einstein's clock-setting method is the idea that time is relative. In essence, each observer's frame of reference is associated with a unique set of clocks, the result being that time passes at different rates for different observers. This was a direct result of the Lorentz transformations and is called time dilation

Time dilation

. We can also clearly see from the Lorentz "local time" transformation that the concept of the relativity of simultaneity and of the relativity of length contraction are also consequences of that clock-setting hypothesis.

Lorentz transformations can also be used to prove that magnetic and electric fields are simply different aspects of the same force — the electromagnetic force. If we have one charge or a collection of charges which are all stationary with respect to each other, we can observe the system in a frame in which there is no motion of the charges. In this frame, there is only an "electric field". If we switch to a moving frame, the Lorentz transformation will predict that a "magnetic field" is present. This field was initially unified in Maxwell's concept of the "electromagnetic field".

The correspondence principle

For relative speeds much less than the speed of light, the Lorentz transformations reduce to the Galilean transformation

Galilean transformation

in accordance with the correspondence principle

Correspondence principle

In physics, the correspondence principle states that the behavior of systems described by the theory of quantum mechanics reproduces classical physics in the limit of large quantum numbers....

.

The correspondence limit is usually stated mathematically as: as

. In words: as velocity approaches 0, the speed of light (seems to) approach infinity. Hence, it is sometimes said that nonrelativistic physics is a physics of "instant action at a distance".

Derivation

The usual treatment (e.g., Einstein's original work) is based on the invariance of the speed of light. However, this is not necessarily the starting point: indeed (as is exposed, for example, in the second volume of the Course of Theoretical Physics

Course of Theoretical Physics

The Course of Theoretical Physics is a ten-volume series of books covering theoretical physics that was initiated by Lev Landau and written in collaboration with his student Evgeny Lifshitz starting in the late 1930s....

by Landau and Lifshitz), what is really at stake is the locality of interactions: one supposes that the influence that one particle, say, exerts on another can not be transmitted instantaneously. Hence, there exists a theoretical maximal speed of information transmission which must be invariant, and it turns out that this speed coincides with the speed of light in vacuum. The need for locality in physical theories was already noted by Newton (see Koestler's The Sleepwalkers

The Sleepwalkers

The Sleepwalkers: A History of Man's Changing Vision of the Universe is a 1959 book by Arthur Koestler, and one of the main accounts of the history of cosmology and astronomy in the Western World, beginning in ancient Mesopotamia and ending with Isaac Newton. The book challenges the habitual idea...

), who considered the notion of an action at a distance "philosophically absurd" and believed that gravity must be transmitted by an agent (such as an interstellar aether) which obeys certain physical laws.

Michelson and Morley in 1887 designed an experiment, employing an interferometer and a half-silvered mirror, that was accurate enough to detect aether flow. The mirror system reflected the light back into the interferometer. If there were an aether drift, it would produce a phase shift and a change in the interference that would be detected. However, no phase shift was ever found. The negative outcome of the Michelson-Morley experiment left the concept of aether (or its drift) undermined. There was consequent perplexity as to why light evidently behaves like a wave, without any detectable medium through which wave activity might propagate.

In a 1964 paper, Erik Christopher Zeeman

Erik Christopher Zeeman

Sir Erik Christopher Zeeman FRS , is a Japanese-born British mathematician known for his work in geometric topology and singularity theory....

showed that the causality preserving property, a condition that is weaker in a mathematical sense than the invariance of the speed of light, is enough to assure that the coordinate transformations are the Lorentz transformations.

From group postulates

Following is a classical derivation (see, e.g., http://arxiv.org/abs/gr-qc/0107091 and references therein) based on group postulates and isotropy of the space.

Coordinate transformations as a group

The coordinate transformations between inertial frames form a group

Group (mathematics)

(called the proper 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...

) with the group operation being the composition of transformations (performing one transformation after another). Indeed the four group axioms are satisfied:

Closure: the composition of two transformations is a transformation: consider a composition of transformations from the inertial frame to inertial frame , (denoted as ), and then from to inertial frame , , there exists a transformation, , directly from an inertial frame to inertial frame .
Associativity: the result of and is the same, .
Identity element: there is an identity element, a transformation .
Inverse element: for any transformation there exists an inverse transformation .

Transformation matrices consistent with group axioms

Let us consider two inertial frames, K and K', the latter moving with velocity

with respect to the former. By rotations and shifts we can choose the z and z' axes along the relative velocity vector and also that the events (t=0,z=0) and (t'=0,z'=0) coincide. Since the velocity boost is along the z (and z') axes nothing happens to the perpendicular coordinates and we can just omit them for brevity. Now since the transformation we are looking after connects two inertial frames, it has to transform a linear motion in (t,z) into a linear motion in (t',z') coordinates. Therefore it must be a linear transformation. The general form of a linear transformation is

where

and

are some yet unknown functions of the relative velocity

.

Let us now consider the motion of the origin of the frame K'. In the K' frame it has coordinates (t',z'=0), while in the K frame it has coordinates (t,z=vt). These two points are connected by our transformation

from which we get

.
Analogously, considering the motion of the origin of the frame K, we get

from which we get

.
Combining these two gives

and the transformation matrix has simplified a bit,

Now let us consider the group postulate inverse element. There are two ways we can go from the

coordinate system to the

coordinate system. The first is to apply the inverse of the transform matrix to the

coordinates:

The second is, considering that the

coordinate system is moving at a velocity

relative to the

coordinate system, the

coordinate system must be moving at a velocity

relative to the

coordinate system. Replacing

with

in the transformation matrix gives:

Now the function

can not depend upon the direction of

because it is apparently the factor which defines the relativistic contraction and time dilation. These two (in an isotropic world of ours) cannot depend upon the direction of

. Thus,

and comparing the two matrices, we get

According to the closure group postulate a composition of two coordinate transformations is also a coordinate transformation, thus the product of two of our matrices should also be a matrix of the same form. Transforming

and from

gives the following transformation matrix to go from

In the original transform matrix, the main diagonal elements are both equal to

, hence, for the combined transform matrix above to be of the same form as the original transform matrix, the main diagonal elements must also be equal. Equating these elements and rearranging gives:

The denominator will be nonzero for nonzero v as

is always nonzero, as

. If v=0 we have the identity matrix which coincides with putting v=0 in the matrix we get at the end of this derivation for the other values of v, making the final matrix valid for all nonnegative v.

For the nonzero v, this combination of function must be a universal constant, one and the same for all inertial frames. Let's define this constant as

where

has the dimension of

. Solving

we finally get

and thus the transformation matrix, consistent with the group axioms, is given by

were positive, then there would be transformations (with

>> 1) which transform time into a spatial coordinate and vice versa. We exclude this on physical grounds, because time can only run in the positive direction. Thus two types of transformation matrices are consistent with group postulates: i) with the universal constant

=0 and ii) with

<0.

Galilean transformations

then we get the Galilean-Newtonian kinematics with the Galilean transformation,

where time is absolute,

, and the relative velocity

of two inertial frames is not limited.

Lorentz transformations

is negative, then we set

which becomes the invariant speed

Invariant speed

The invariant speed or observer invariant speed is the speed an object or particle must be traveling at for its speed to have the same measure in all reference frames. The invariance of the speed of light is a consequence of the postulates of special relativity, and the terms speed of light and...

, 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...

in vacuum. This yields

and thus we get special relativity with Lorentz transformation

where the speed of light is a finite universal constant determining the highest possible relative velocity between inertial frames.

If

the Galilean transformation is a good approximation to the Lorentz transformation.

Only experiment can answer the question which of the two possibilities,

=0 or

<0, is realised in our world. The experiments measuring the speed of light, first performed by a Danish physicist Ole Rømer, show that it is finite, and the Michelson–Morley experiment showed that it is an absolute speed, and thus that

<0.

From physical principles

The problem is usually restricted to two dimensions by using a velocity along the x axis such that the y and z coordinates do not intervene. It is similar to that of Einstein.
As in the Galilean transformation, the Lorentz transformation is linear since the relative velocity of the reference frames is constant as a vector; otherwise, inertial forces

Fictitious force

A fictitious force, also called a pseudo force, d'Alembert force or inertial force, is an apparent force that acts on all masses in a non-inertial frame of reference, such as a rotating reference frame....

would appear. They are called inertial or Galilean reference frames. According to relativity no Galilean reference frame is privileged. Another condition is that the speed of light must be independent of the reference frame, in practice of the velocity of the light source.

Galilean reference frames

In classical kinematics, the total displacement x in the R frame is the sum of the relative displacement x′ in frame R' and of the distance between the two origins x-x. If v is the relative velocity of R' relative to R, the transformation is: x = x′ + vt, or x′ = x − vt. This relationship is linear for a constant v, that is when R and R' are Galilean frames of reference.

In Einstein's relativity, the main difference with Galilean relativity is that space is a function of time and vice-versa: t ≠ t′. The most general linear relationship is obtained with four constant coefficients, A, B, γ, and b:

The Lorentz transformation becomes the Galilean transformation when γ = B = 1 , b = -v.

An object at rest in the R' frame at position x′=0 will move with constant velocity v in the R frame. Hence the transformation must yield x′=0 if x=vt. Therefore, b=-γ v and it may written as:

Principle of relativity

According to the principle of relativity, there is no privileged Galilean frame of reference.
Therefore, the inverse transformation for the position from frame R′ to frame R must be

with the same value of γ (which must therefore be an even function of v).

Speed of light independent of the velocity of the source

If the speed of light must be independent of the reference frame, the transformation must ensure that x = ct if x′ = ct′. In other words, the light emitted at t=t′=0 moves at velocity c in both frames. Replacing x and x′ in the preceding equations, one has:

Multiplying these two, one finds

From which

called the "Lorentz factor".

Transformation of time

The factors A and B in the transformation for time can now be obtained. Substituting the derived expression for x′

in the inverse transformation equation

gives

Solving for t′, this results in

and identification with the general transformation

results in

and thus finally in

Extentions

A recent optical experiment reported an interesting result
. Three authors performed a modified 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...

to test the velocity-addition formula

Velocity-addition formula

In physics, a velocity-addition formula is an equation that relates the velocities of moving objects in different reference frames.- Galilean addition of velocities :...

of light speed in air flow between still water. The visible shift of interference fringes implies that parameter c in the rule and Lorentz transformation should be replaced by c/n (n=1.33 is refractive index of water in this experiment). It will reduce to that of old Fizeau's experiments and Lorentz transformation in case of n=1(vacuum) or n~1 (air).

External links

Derivation of the Lorentz transformations. This web page contains a more detailed derivation of the Lorentz transformation with special emphasis on group properties.
The Paradox of Special Relativity. This webpage poses a problem, the solution of which is the Lorentz transformation, which is presented graphically in its next page.
Relativity - a chapter from an online textbook
Special Relativity: The Lorentz Transformation, The Velocity Addition Law on Project PHYSNET
Warp Special Relativity Simulator. A computer program demonstrating the Lorentz transformations on everyday objects.
Animation clip visualizing the Lorentz transformation.
Lorentz Frames Animated from John de Pillis. Online Flash animations of Galilean and Lorentz frames, various paradoxes, EM wave phenomena, etc.

The source of this article is wikipedia, the free encyclopedia. The text of this article is licensed under the GFDL.

History

Lorentz transformation for frames in standard configuration

Boost in the x-direction

Boost in the y or z directions

Boost in any one direction by index permutation

Boost in any direction

Composition of two boosts

Rapidity

Hyperbolic trigonometric expressions

Hyperbolic rotation of coordinates

Lorentz transformation of the electromagnetic field

Spacetime interval

Special relativity

The correspondence principle

Derivation

From group postulates

Coordinate transformations as a group

Transformation matrices consistent with group axioms

Galilean transformations

Lorentz transformations

From physical principles

Galilean reference frames

Principle of relativity

Speed of light independent of the velocity of the source

Transformation of time

Extentions

See also

External links