FACTOID # 154: Women make up more than 10% of the prison population in only six countries: Thailand, , Qatar, Paraguay, Costa Rica, and Singapore.
 
 Home   Encyclopedia   Statistics   Countries A-Z   Flags   Maps   Education   Forum   FAQ   About 
 
WHAT'S NEW
RECENT ARTICLES
More Recent Articles »
 

SEARCH ALL

FACTS & STATISTICS    Advanced view

Search encyclopedia, statistics and forums:

 

 

(* = Graphable)

 

 


Encyclopedia > Lorentz transformation

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. In classical physics (Galilean relativity), the only conversion believed necessary was x' = x − vt, describing how the origin of one observer's coordinate system slides through space with respect to the other's, at speed v and along the x-axis of each frame. According to special relativity, this is only a good approximation at speeds small compared to the speed of light, and in general the result is not just an offsetting of the x coordinates; lengths and times are distorted as well. Image File history File links No higher resolution available. ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... In general, the principle of relativity is the requirement that the laws of physics be the same for all observers. ... For a less technical and generally accessible introduction to the topic, see Introduction to special relativity. ...


If space is homogeneous, then the Lorentz transformation must be a linear transformation. 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. The Lorentz transformations describe only the transformations in which the event at x=0, t=0 is left fixed, so they can be considered as a rotation of Minkowski space. The more general set of transformations that also includes translations is known as the Poincaré group. Look up Homogeneous in Wiktionary, the free dictionary. ... In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... In special relativity and general relativity, time and three-dimensional space are treated together as a single four-dimensional pseudo-Riemannian manifold called spacetime. ... In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ... This article is about rotation as a movement of a physical body. ... In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ... In physics and mathematics, the Poincaré group is the group of isometries of Minkowski spacetime. ...


Henri Poincaré (1905) named the Lorentz transformations after the Dutch physicist and mathematician Hendrik Lorentz (1853-1928). They form the mathematical basis for Albert Einstein's theory of special relativity. The Lorentz transformations remove contradictions between the theories of electromagnetism and classical mechanics. They were derived by Joseph Larmor (1897) and Lorentz (1899, 1904). In 1905 Einstein derived them under the assumptions of Lorentz covariance and the constancy of the speed of light in any inertial reference frame. Jules TuPac Henri Poincaré (April 29, 1854 – July 17, 1912) (IPA: [][1]) was one of Frances greatest mathematicians and theoretical physicists, and a philosopher of science. ... Not to be confused with physician, a person who practices medicine. ... Leonhard Euler, considered one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is the field of mathematics. ... Hendrik Antoon Lorentz (July 18, 1853, Arnhem – February 4, 1928, Haarlem) was a Dutch physicist who shared the 1902 Nobel Prize in Physics with Pieter Zeeman for the discovery and elucidation of the Zeeman effect. ... 1853 was a common year starting on Saturday (see link for calendar). ... Year 1928 (MCMXXVIII) was a leap year starting on Sunday (link will display full calendar) of the Gregorian calendar. ... “Einstein” redirects here. ... For a less technical and generally accessible introduction to the topic, see Introduction to special relativity. ... Electromagnetism is the physics of the electromagnetic field: a field which exerts a force on particles that possess the property of electric charge, and is in turn affected by the presence and motion of those particles. ... Classical mechanics (commonly confused with Newtonian mechanics, which is a subfield thereof) is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. ... Sir Joseph Larmor (11 July 1857 – 19 May 1942), an Northern Irish physicist, mathematician and politician, researched electricity, dynamics, and thermodynamics. ... In physics, Lorentz covariance is a key property of spacetime that follows from the special theory of relativity, where it applies globally. ...

Contents

Lorentz transformation for frames in standard configuration

Diagram 1. Views of spacetime along the world line of a rapidly accelerating observer.Vertical direction indicates time. Horizontal indicates distance, the dashed line is the spacetime trajectory ("world line") of the observer. The lower quarter of the diagram shows the events visible to the observer. Upper quarter shows the light cone- those that will be able to see the observer. The small dots are arbitrary events in spacetime. The slope of the world line (deviation from being vertical) gives the relative velocity to the observer. Note how the view of spacetime changes when the observer accelerates.
Diagram 1. Views of spacetime along the world line of a rapidly accelerating observer.

Vertical direction indicates time. Horizontal indicates distance, the dashed line is the spacetime trajectory ("world line") of the observer. The lower quarter of the diagram shows the events visible to the observer. Upper quarter shows the light cone- those that will be able to see the observer. The small dots are arbitrary events in spacetime.

The slope of the world line (deviation from being vertical) gives the relative velocity to the observer. Note how the view of spacetime changes when the observer accelerates.

Assume there are two observers O and Q, each using their own Cartesian coordinate system to measure space and time intervals. O uses (t,x,y,z) and Q uses (t',x',y',z'). Assume further that the coordinate systems are oriented so that the x-axis and the x' -axis overlap, the y-axis is parallel to the y' -axis, as are the z-axis and 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. If all this holds, then the coordinate systems are said to be in standard configuration. A symmetric presentation between the forward Lorentz Transformation and the inverse Lorentz Transformation can be achieved if coordinate systems are in symmetric configuration. The symmetric form highlights that all physical laws should be of such a kind that they remain unchanged under a Lorentz transformation. Image File history File links Source of program used to generate image: //GPL #include <stdio. ... In physics, the world line of an object is the unique path of that object as it travels through 4-dimensional spacetime. ... For other uses of this term, see Spacetime (disambiguation). ... In physics, the world line of an object is the unique path of that object as it travels through 4-dimensional spacetime. ... In special relativity, a light cone is the pattern describing the temporal evolution of a flash of light in Minkowski spacetime. ... Fig. ... Assume there are two observers and , each using their own Cartesian coordinate system to measure space and time intervals. ... Assume there are two observers and , each using their own Cartesian coordinate system to measure space and time intervals. ...


The Lorentz transformation for frames in standard configuration can be shown to be:

begin{align}t' &= gamma left( t - frac{v x}{c^{2}} right) x' &= gamma left( x - v t right) y' &= y z' &= z end{align}

where gamma = { 1 over sqrt{1 - v^2/c^2} }, is called the Lorentz factor. It has been suggested that Lorentz term be merged into this article or section. ...


Matrix form

This Lorentz transformation is called a "boost" in the x-direction and is often expressed in matrix form as In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. ...

begin{bmatrix} c t' x' y' z' end{bmatrix} = begin{bmatrix} gamma&-beta gamma&0&0 -beta gamma&gamma&0&0 0&0&1&0 0&0&0&1 end{bmatrix} begin{bmatrix} c,t x y z end{bmatrix} .

or more generally for the x, y, and z-directions:

begin{bmatrix} c,t' x' y' z' end{bmatrix} = begin{bmatrix} gamma&-beta_x,gamma&-beta_y,gamma&-beta_z,gamma -beta_x,gamma&1+(gamma-1)frac{beta_{x}^{2}}{beta^{2}}&(gamma-1)frac{beta_{x}beta_{y}}{beta^{2}}&(gamma-1)frac{beta_{x}beta_{z}}{beta^{2}} -beta_y,gamma&(gamma-1)frac{beta_{y}beta_{x}}{beta^{2}}&1+(gamma-1)frac{beta_{y}^{2}}{beta^{2}}&(gamma-1)frac{beta_{y}beta_{z}}{beta^{2}} -beta_z,gamma&(gamma-1)frac{beta_{z}beta_{x}}{beta^{2}}&(gamma-1)frac{beta_{z}beta_{y}}{beta^{2}}&1+(gamma-1)frac{beta_{z}^{2}}{beta^{2}} end{bmatrix} begin{bmatrix} c,t x y z end{bmatrix} .

where beta = frac{v}{c}=frac{|vec{v}|}{c} and gamma = frac{1}{left( 1-beta^2 right)^frac{1}{2}}.


Rapidity

The Lorentz transformation can be cast into another useful form by introducing a parameter φ called the rapidity (an instance of hyperbolic angle) through the equation: A hyperbolic angle in standard position is the angle at (0,0) between the ray to (1,1) and the ray to (x,1/x) where x > 1. ...

e^{phi} = gamma(1+beta) = gamma left( 1 + frac{v}{c} right) = sqrt frac{1 + v/c}{1 - v/c}

Equivalently:

phi = ln left[gamma(1+beta)right] , -phi = ln left[gamma(1-beta)right] ,

Then the Lorentz transformation in standard configuration is:

c t-x = e^{- phi}(c t' - x') ,
c t+x = e^{phi}(c t' + x') ,
y = y' ,
z = z' .

Hyperbolic trigonometric expressions

It can also be shown that:

gamma = cosh(phi) = { e^{phi} + e^{-phi} over 2 }
beta = tanh(phi) = { e^{phi} - e^{-phi} over e^{phi} + e^{-phi} }

and therefore,

beta gamma = sinh(phi) = { e^{phi} - e^{-phi} over 2 }

Hyperbolic rotation of coordinates

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

begin{bmatrix} c t' x' y' z' end{bmatrix} = begin{bmatrix} cosh(phi) &-sinh(phi)&0&0 -sinh(phi) & cosh(phi) &0&0 0&0&1&0 0&0&0&1 end{bmatrix} begin{bmatrix} c t x y z end{bmatrix} .

Thus, the Lorentz transformation can be seen as a hyperbolic rotation of coordinates in Minkowski space, where the rapidity φ represents the hyperbolic angle of rotation. In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ...


General boosts

For a boost in an arbitrary direction with velocity vec{v}, it is convenient to decompose the spatial vector vec{r} into components perpendicular and parallel to the velocity vec{v}: vec{r}=vec{r}_perp+vec{r}_|. Then only the component vec{r}_| in the direction of vec{v} is 'warped' by the gamma factor:

t' = gamma left(t - frac{vec{r} cdot vec{v}}{c^{2}} right)
vec{r'} = vec{r}_perp + gamma (vec{r}_| - vec{v} t)

where now gamma equiv frac{1}{sqrt{1 - vec{v} cdot vec{v}/c^2}}. The second of these can be written as:

vec{r'} = vec{r} + left(frac{gamma -1}{v^2} (vec{r} cdot vec{v}) - gamma t right) vec{v}

These equations can be expressed in matrix form as

begin{bmatrix} c t' vec{r'} end{bmatrix} = begin{bmatrix} gamma&-frac{vec{v^T}}{c}gamma -frac{vec{v}}{c}gamma&I+frac{vec{v} cdot vec{v}^T}{v^2}(gamma-1) end{bmatrix} begin{bmatrix} c tvec{r} end{bmatrix} ,

where I is the identity matrix.


Spacetime interval

In a given coordinate system (xμ), if two events A and B are separated by For other uses of this term, see Spacetime (disambiguation). ...

(Delta t, Delta x, Delta y, Delta z) = (t_B-t_A, x_B-x_A, y_B-y_A, z_B-z_A) ,

the spacetime interval between them is given by For other uses of this term, see Spacetime (disambiguation). ...

s^2 = - c^2(Delta t)^2 + (Delta x)^2 + (Delta y)^2 + (Delta z)^2 .

This can be written in another form using the Minkowski metric. In this coordinate system, In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ...

eta_{munu} = begin{bmatrix} -1&0&0&0 0&1&0&0 0&0&1&0 0&0&0&1 end{bmatrix} .

Then, we can write

s^2 = begin{bmatrix}c Delta t & Delta x & Delta y & Delta z end{bmatrix} begin{bmatrix} -1&0&0&0 0&1&0&0 0&0&1&0 0&0&0&1 end{bmatrix} begin{bmatrix} c Delta t Delta x Delta y Delta z end{bmatrix}

or, using the Einstein summation convention, For other topics related to Einstein see Einstein (disambig) In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate equations or formulas. ...

s^2= eta_{munu} x^mu x^nu .

Now suppose that we make a coordinate transformation x^mu rightarrow x'^mu. Then, the interval in this coordinate system is given by

s'^2 = begin{bmatrix}c Delta t' & Delta x' & Delta y' & Delta z' end{bmatrix} begin{bmatrix} -1&0&0&0 0&1&0&0 0&0&1&0 0&0&0&1 end{bmatrix} begin{bmatrix} c Delta t' Delta x' Delta y' Delta z' end{bmatrix}

or

s'^2= eta_{munu} x'^mu x'^nu .

It is a result of special relativity that the interval is an invariant. That is, s^2 = s'^2 . It can be shown[1] that this requires the coordinate transformation to be of the form For a less technical and generally accessible introduction to the topic, see Introduction to special relativity. ... In physics, invariants are usually quantities conserved (unchanged) by the symmetries of the physical system. ...

x'^mu = x^nu {Lambda^mu}_nu + C^mu .

Here, C^mu is a constant vector and {Lambda^mu}_nu a constant matrix, where we require that

eta_{munu}{Lambda^mu}_alpha{Lambda^nu}_beta = eta_{alphabeta} .

Such a transformation is called a Poincaré transformation or an inhomogeneous Lorentz transformation.[2] The Ca represents a space-time translation. When C^a , = 0, the transformation is called an homogeneous Lorentz transformation, or simply a Lorentz transformation. In physics and mathematics, the Poincaré group is the group of isometries of Minkowski spacetime. ...


Taking the determinant of eta_{munu}{Lambda^mu}_alpha{Lambda^nu}_beta = eta_{alphabeta} gives us

det ({Lambda^a}_b) = pm 1 .

Lorentz transformations with det ({Lambda^mu}_nu)=+1 are called proper Lorentz transformations. They consist of spatial rotations and boosts and form a subgroup of the Lorentz group. Those with det({Lambda^mu}_nu)=-1 are called improper Lorentz transformations and consist of (discrete) space and time reflections combined with spatial rotations and boosts. They don't form a subgroup, as the product of any two improper Lorentz transformations will be a proper Lorentz transformation.


The composition of two Poincaré transformations is a Poincaré transformation and the set of all Poincaré transformations with the operation of composition forms a group called the Poincaré group. Under the Erlangen program, Minkowski space can be viewed as the geometry 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. In physics and mathematics, the Poincaré group is the group of isometries of Minkowski spacetime. ... An influential research programme and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen über neuere geometrische Forschungen. ... In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ... For other uses, see Geometry (disambiguation). ... The Lorentz group is the group of all Lorentz transformations of Minkowski spacetime. ...


A quantity invariant under Lorentz transformations is known as a Lorentz scalar. In physics a Lorentz scalar is a scalar which is invariant under a Lorentz transformation. ...


Special relativity

One of the most astounding predictions of special relativity was the idea that time is relative. In essence, each observer's frame of reference is associated with a unique clock, the result being that time passes at different rates for different observers. This was a direct prediction from the Lorentz transformations and is called time dilation. We can also clearly see from the Lorentz transformations that the concept of simultaneity varies between reference frames. Another startling result is length contraction. Time dilation is the phenomenon whereby an observer finds that anothers clock which is physically identical to their own is ticking at a slower rate as measured by their own clock. ... The relativity of simultaneity is the dependence of the notion of simultaneity on the observer. ... A frame of reference in physics is a set of axes which enable an observer to measure the aspect, position and motion of all points in a system relative to the reference frame. ... Length contraction, according to Albert Einsteins special theory of relativity, is the decrease in length experienced by people or objects traveling at a substantial fraction of the speed of light. ...


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 give rise to a magnetic field. These two fields are unified in the concept of the electromagnetic field. In physics, the electromagnetic force is the force that the electromagnetic field exerts on electrically charged particles. ... In physics, the space surrounding an electric charge or in the presence of a time-varying magnetic field has a property called an electric field. ... Magnetic field lines shown by iron filings Magnetostatics Electrodynamics Electrical Network Tensors in Relativity This box:      In physics, the magnetic field is a field that permeates space and which exerts a magnetic force on moving electric charges and magnetic dipoles. ... The electromagnetic field is a physical field that is produced by electrically charged objects and which affects the behaviour of charged objects in the vicinity of the field. ...


The correspondence principle

For relative speeds much less than the speed of light, the Lorentz transformations reduce to the Galilean transformation in accordance with the correspondence principle. The correspondence limit is usually stated mathematically as v rightarrow 0, so it is usually said that classical physics is a physics of "instant action on a distance" c rightarrow infty. 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. ... In physics, the correspondence principle is a principle, first invoked by Niels Bohr in 1923, which states that the behavior of quantum mechanical systems reduce to classical physics in the limit of large quantum numbers. ...


History

See also History of Lorentz transformations.

The transformations were first discovered and published by Joseph Larmor in 1897. In 1905, Henri Poincaré[3][4] named them after the Dutch physicist and mathematician Hendrik Antoon Lorentz (1853-1928) who had published a first order version of these transformations in 1895[5] and the final version in 1899 and 1904. The Lorentz transformations relate the space-time coordinates, (which specify the position and time of an event) relative to a particular inertial frame of reference (the rest system), and the coordinates of the same event relative to another coordinate system moving in the positive x-direction at a constant speed... Sir Joseph Larmor (11 July 1857 – 19 May 1942), an Northern Irish physicist, mathematician and politician, researched electricity, dynamics, and thermodynamics. ... Jules TuPac Henri Poincaré (April 29, 1854 – July 17, 1912) (IPA: [][1]) was one of Frances greatest mathematicians and theoretical physicists, and a philosopher of science. ... Not to be confused with physician, a person who practices medicine. ... Leonhard Euler, considered one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is the field of mathematics. ... Hendrik Antoon Lorentz (July 18, 1853, Arnhem – February 4, 1928, Haarlem) was a Dutch physicist who shared the 1902 Nobel Prize in Physics with Pieter Zeeman for the discovery and elucidation of the Zeeman effect. ... 1853 was a common year starting on Saturday (see link for calendar). ... Year 1928 (MCMXXVIII) was a leap year starting on Sunday (link will display full calendar) of the Gregorian calendar. ...


Actually many physicists, including FitzGerald, Larmor, Lorentz and Woldemar Voigt, had been discussing the physics behind these equations since 1887.[6][7] Larmor and Lorentz, who believed the luminiferous aether hypothesis, were seeking the transformations under which Maxwell's equations were invariant when transformed from the ether to a moving frame. In early 1889, Heaviside 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. This 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.[8] Their explanation was widely accepted as correct before 1905.[9] Larmor gets credit for discovering the basic equations in 1897 and for being first in understanding the crucial time dilation property inherent in his equations.[10] Woldemar Voigt (September 2, 1850 - December 13, 1919) was a German physicist. ... The luminiferous aether: it was hypothesised that the Earth moves through a medium of aether that carries light In the late 19th century luminiferous aether (light-bearing aether) was the term used to describe a medium for the propagation of light. ... The Michelson-Morley experiment, one of the most important and famous experiments in the history of physics, was performed in 1887 by Albert Michelson and Edward Morley at what is now Case Western Reserve University, and is considered by some to be the first strong evidence against the theory of...


Larmor's (1897) and Lorentz's (1899, 1904) final equations are algebraically equivalent to those published and interpreted as a theory of relativity by Albert Einstein (1905) but it was the French mathematician Henri Poincaré who first recognized that the Lorentz transformations have the properties of a mathematical group.[11] Both Larmor and Lorentz discovered that the transformation preserved Maxwell's equations. Paul Langevin (1911) said of the transformation In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... For thermodynamic relations, see Maxwell relations. ... Paul Langevin (January 23, 1872 â€“ December 19, 1946) was a prominent French physicist who developed Langevin dynamics and the Langevin equation. ...

"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".

Derivation

The usual treatment (e.g. Einstein's original work) is based on the invariance of the speed of light. However, this must not necessarily be the starting point: indeed (as is exposed, for example, in the second volume of the Course in Theoretical Physics 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 the vacuum. It is interesting to know that the need for locality in physical theories was already seen by Newton (see Koestler's "The Sleepwalkers"), who considered "philosophically absurd" the notion of an action at a distance and believed that gravity must be transmitted by an agent (interstellar aether) which obeys certain physical laws.


In an 1964 paper,[12] Erik Christopher Zeeman showed that a, in a mathematical sense, weaker condition, the causality preserving property, is enough to assure that the coordinate transformations be the Lorentz-transformations. Sir Erik Christopher Zeeman (born February 4, 1925), is a mathematician known for work in geometric topology and singularity theory. ... In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ...


From group postulates

Group Postulate Derivation

Following is a classical derivation based on group postulates and isotropy of the space.


Let us consider two inertial frames, K and K', the latter moving with velocity vec{v} 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

begin{bmatrix} t' z' end{bmatrix} = begin{bmatrix} gamma & delta beta & alpha end{bmatrix} begin{bmatrix} t z end{bmatrix},

where α,β,γ, and δ are some yet unknown functions of the relative velocity v.


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

begin{bmatrix} t' 0 end{bmatrix} = begin{bmatrix} gamma & delta beta & alpha end{bmatrix} begin{bmatrix} t vt end{bmatrix},

from which we get

beta=-valpha ,.

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

begin{bmatrix} t' -vt' end{bmatrix} = begin{bmatrix} gamma & delta beta & alpha end{bmatrix} begin{bmatrix} t 0 end{bmatrix},

from which we get

beta=-vgamma ,.

Combining these two gives α = γ and the transformation matrix has simplified a bit,

begin{bmatrix} t' z' end{bmatrix} = begin{bmatrix} gamma & delta -vgamma & gamma end{bmatrix} begin{bmatrix} t z end{bmatrix},

Now let us consider the inverse transformation. On one hand the inverse transformation is done simply by the inverse matrix,

begin{bmatrix} t z end{bmatrix} = frac{1}{gamma^2+vdeltagamma} begin{bmatrix} gamma & -delta vgamma & gamma end{bmatrix} begin{bmatrix} t' z' end{bmatrix}.

On the other hand the inverse transformation is the one where v is substituted by − v,

begin{bmatrix} t z end{bmatrix} = begin{bmatrix} gamma(-v) & delta(-v) vgamma(-v) & gamma(-v) end{bmatrix} begin{bmatrix} t' z' end{bmatrix},

Now the function γ can not depend upon the direction of v 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 v. Thus, γ( − v) = γ(v) and comapring the two matrices, we get

gamma^2+vdeltagamma=1. ,

At last 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, in particular the diagonal elements should be equal. Calculating the product of two transformation matrices, one with v the other with v' and comparing the diagonal elements gives

frac{vgamma(v)}{delta(v)}=frac{v'gamma(v')}{delta(v')}

Since this holds for any arbitrary v and v' this combination of function must be a universal constant, one and the same for all inertial frames. Let's define this constant as frac{vgamma(v)}{delta(v)}=-c^2 where c has a dimension of velocity (we have not yet assumed, that c2 > 0). Using the equation from the inverse transformation we finally get gamma=1/sqrt{1-v^2/c^2} and the transformation matrix is given by

begin{bmatrix} t' z' end{bmatrix} = frac{1}{sqrt{1-v^2/c^2}} begin{bmatrix} 1 & -v/c^2 -v & 1 end{bmatrix} begin{bmatrix} t z end{bmatrix}.

Apparently c2 cannot be negative because otherwise there would be a transformation which transforms time into spatial coordinate and vice versa. This is no good (at least in special relativity) since time can only run in the positive direction while coordinates in both. If then c2 > 0 it is apparently the highest achievable velocity. Theoretically it can be either infinitely large, which gives Galilean transformation and Euclidean world with absolute time, or it can be finite, which gives Lorentz transformation and Minkowski world of special relativity. The experiment tells us that it is finite, c = 299792458m/s.

See also

Wikibooks
Wikibooks has a book on the topic of
special relativity

Image File history File links Wikibooks-logo-en. ... Wikibooks logo Wikibooks, previously called Wikimedia Free Textbook Project and Wikimedia-Textbooks, is a wiki for the creation of books. ... The electromagnetic field is a physical field that is produced by electrically charged objects and which affects the behaviour of charged objects in the vicinity of the field. ... 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. ... Wikisource has original text related to this article: Relativity: The Special and General Theory A principle of relativity is a criterion for judging physical theories, stating that they are inadequate if they do not prescribe the exact same laws of physics in certain similar situations. ... In mathematics, the split-complex numbers are an extension of the real numbers defined analogously to the complex numbers. ... Invariance mechanics, in its simplest form, is the rewriting of the laws of quantum field theory in terms of invariant quantities only. ... // If a ship is moving relative to the shore at velocity , and a fly is moving with velocity as measured on the ship, calculating the velocity of the fly as measured on the shore is what is meant by the addition of the velocities and . ...

External links

Footnotes

  1. ^ Steven Weinberg (1972). Gravitation and Cosmology. Wiley. : Section 2.1
  2. ^ Steven Weinberg (1995). The Quantum Theory of Fields, Volume 1. Cambridge University Press. 
  3. ^ Jacques Fric, Henri Poincaré: A Decisive Contribution to Special Relativity
  4. ^ A. A. Logunov, Henri Poincaré and Relativity Theory
  5. ^ History of Special Relativity
  6. ^ J. J. O'Connor and E. F. Robertson, A History of Special Relativity
  7. ^ Supurna Sinha, Poincaré and the Special Theory of Relativity
  8. ^ Harvey R. Brown, Michelson, FitzGerald and Lorentz: the Origins of Relativity Revisited
  9. ^ Tony Rothman, Lost in Einstein's Shadow
  10. ^ Macrossan, Michael N. (1986). "A Note on Relativity Before Einstein". Brit. Journal Philos. Science 37: 232-34. 
  11. ^ Shaul Katzir, Poincaré’s Relativistic Physics: Its Origins and Nature
  12. ^ Zeeman, E. C. (April 1964). "Causality Implies the Lorentz Group". Journal of Mathematical Physics 5 (4): 490-493. 

Steven Weinberg (born May 3, 1933) is an American physicist. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... Sir Erik Christopher Zeeman (born February 4, 1925), is a mathematician known for work in geometric topology and singularity theory. ...

References

  • Giulini, Domenico. Algebraic and geometric structures of Special Relativity. arXiv eprint server. Retrieved on February 19, 2005.
  • Ernst, A. and Hsu, J.-P. (2001) “First proposal of the universal speed of light by Voigt 1887”, Chinese Journal of Physics, 39(3), 211-230.
  • Langevin, P. (1911) "L'évolution de l'espace et du temps", Scientia, X, 31-54
  • Larmor, J. (1897) "Dynamical Theory of the Electric and Luminiferous Medium" Philosophical Transactions of the Royal Society, 190, 205-300.
  • Larmor, J. (1900) Aether and Matter, Cambridge University Press
  • Lorentz, H. A. (1899) "Simplified theory of electrical and optical phenomena in moving systems", Proc. Acad. Science Amsterdam, I, 427-43.
  • Lorentz, H. A. (1904) "Electromagnetic phenomena in a system moving with any velocity less than that of light", Proc. Acad. Science Amsterdam, IV, 669-78.
  • Lorentz, H. A. (1913) The theory of electrons (book)
  • Poincaré, H. (1905) "Sur la dynamique de l'électron", Comptes Rendues, 140, 1504-08.
  • Voigt, W. (1887) "Über das Doppler'sche princip" Nachrichten von der Königlicher Gesellschaft den Wissenschaft zu Göttingen, 2, 41-51.
  • Thornton, S., Marion, J., (2004) Classical Dynamics of Particles and Systems Fifth Edition, Thomson Learning, 546-579.

  Results from FactBites:
 
Lorentz group - Wikipedia, the free encyclopedia (3542 words)
Lorentz transformations are examples of linear transformations; general isometries of Minkowski spacetime are affine transformations.
In the article on Möbius transformations, it is explained how this classification arises by considering the fixed points of Möbius transformations in their action on the Riemann sphere, which corresponds here to null eigenspaces of restricted Lorentz transformations in their action on Minkowski spacetime.
The Möbius transformations are precisely the conformal transformations of the Riemann sphere (or celestial sphere).
  More results at FactBites »


 

COMMENTARY     


Share your thoughts, questions and commentary here
Your name
Your comments
Please enter the 5-letter protection code

Want to know more?
Search encyclopedia, statistics and forums:
 


Lesson Plans | Student Area | Student FAQ | Reviews | Press Releases |  Feeds | Contact
The Wikipedia article included on this page is licensed under the GFDL.
Images may be subject to relevant owners' copyright.
All other elements are (c) copyright NationMaster.com 2003-5. All Rights Reserved.
Usage implies agreement with terms.