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Lorentz transformation

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A Lorentz transformation (LT) is a linear transformation that preserves the spacetime interval between any two events in Minkowski space, while leaving the origin fixed (=rotation of Minkowski space). The transformation describes how space and time coordinates are related as measured by observers in different inertial reference frames. Poincaré (1905) named them 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 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.

Contents

Lorentz boost in 2 dimensions

With respect to the form [xy]
[\begin
e^\phi & \\& e^\end] In these coordinates, it is a squeeze mapping.

With respect to the form [x^2-y^2]

[\begin\gamma&-\beta \gamma\\-\beta \gamma&\gamma\end]

Lorentz transformation for frames in standard configuration

Given two observers [S_1] and [S_2], each using a Cartesian coordinate system to measure space and time intervals, [\, (t_1, x_1, y_1, z_1)] and [\, (t_2, x_2, y_2, z_2)], assume that the coordinate systems are oriented so that the relative velocity of [S_1] and [S_2] is v along the common [x_1]-[x_2] axis with parallel (but not common) [y_2] and [y_1] axes (same for the [z_2] and [z_1] axes). Also, assume that their origins meet at the common time [t_1]=[t_2]=0. Then the frames are said to be in standard configuration (SC). The Lorentz transformation for frames in SC can be shown to be:

[t_1 = \gamma (t_2 - \frac})]
[x_1 = \gamma (x_2 - v t_2) \,]
[y_1 = y_2 \,]
[z_1 = z_2 \,]
::where
:::[x_1] is the x component in [S_1] of the position of an event,
:::[x_2] is the x component in [S_2] of the position of the same event,
:::[t_1] is the time since time zero in [S_1] of the event,
:::[t_2] is the time since time zero in [S_2] of the event,
:::[y_1] and [y_2] are both the y component of the position of the event in both reference frames (same with [z_1] and [z_2]),
:::[\gamma \equiv \frac} = \frac} ]      is called the Lorentz factor,
:::[c] is the speed of light in a vacuum, and
:::[v] is the relative velocity between the two observers.
This Lorentz transformation is called a boost in the x-direction and is often expressed in matrix form as

[\beginc t_1 \\x_1 \\y_1 \\z_1\end=\begin\gamma&-\beta \gamma&0&0\\-\beta \gamma&\gamma&0&0\\0&0&1&0\\0&0&0&1\\\end\beginc t_2\\x_2\\y_2\\z_2\end.]
where,
the coordinate [t_1] is replaced by [ct_1] (and similarly for [t_2]), and
[ \beta \gamma ] is [ \frac} ].
The Lorentz transformations in SC may be cast into a more useful form by introducing a parameter [\phi] called the rapidity or hyperbolic parameter through the equation:

[e^ \equiv \gamma \left( 1 + \frac \right) = \frac}]
The Lorentz transformations in SC are then:

[c t_1-x_1 \, = e^(c t_2 - x_2)]
[c t_1+x_1 \, = e^(c t_2 + x_2)]
[y_1 \, = y_2]
[z_1 \, = z_2]

Details

In a given coordinate system [(x^a)], the spacetime interval between two events [A] and [B] with coordinates [(t_1, x_1, y_1, z_1)] and ([t_2, x_2, y_2, z_2)] respectively is given by:

[s^2 \,= - c^2(t_2 - t_1)^2 + (x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2 ]
and is an invariant.:

[x^a x_a \, = x'^a x'_a]
i.e.,

[\eta_'x'^ax'^b \, = \eta_ x^c x^d]

where [\eta_] is the Minkowski metric

[\eta_=\begin-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end]
and the Einstein summation convention is being used. From this relation follows the linearity of the coordinate transformation:

[x'^a \, = \Lambda ^a_b x^b + C^a]
where [C^a] and [\, \Lambda ^a_b] satisfy:

[\Lambda^a_b \eta _ \Lambda^c _d \,= \eta _]
[C^a \eta_ \Lambda ^c_b \, = 0]
[ \eta_ C^a C^b \, = 0]
Such a transformation is called a Poincaré transformation. The [C^a] represents a space-time translation; when [C^a \, = 0], the transformation is a Lorentz transformation.

Taking the determinant of the first equation gives

[\det (\Lambda ^a_b) \, = \pm 1]

Lorentz transformations with [\det (\Lambda ^a_b) \, = + 1] are called proper Lorentz transformations and consist of spatial rotations and boosts. Those with [\det (\Lambda ^a_b) \, = - 1] are called improper Lorentz transformations and consist of (discrete) space and time reflections.

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

General boosts

For a boost in an arbitrary direction with velocity [\vec], it is convenient to decompose the spatial vector [\vec] into components perpendicular and parallel to the velocity [\vec]: [\vec=\vec_\perp+\vec_\|]. Then only the component [\vec_\|] in the direction of [\vec] is 'warped' by the gamma factor:

[t' = \gamma \left(t - \frac \cdot \vec}} \right)]
[\vec = \vec_\perp + \gamma (\vec_\| - \vec t)]
where now [\gamma \equiv \frac \cdot \vec/c^2}}]. The second of these can be written as:

[\vec = \vec + \left(\frac (\vec \cdot \vec) - \gamma t \right) \vec]
These equations can be expressed in matrix form as

[\beginc t' \\ \vec\end=\begin\gamma&-\frac}\gamma\\-\frac}\gamma&I+\frac \cdot \vec^T}(\gamma-1)\\\end\beginc t\\\vec\end],
where [I] is the identity matrix.

Lorentz and Poincaré groups

The composition of two Lorentz transformations is a Lorentz transformation and the set of all Lorentz transformations with the operation of composition forms a group called the Lorentz group.

Under the Erlangen program, Minkowski space can be viewed as the geometry defined by the Poincaré group, which combines Lorentz transformations with translations.

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. Time dilation was also used to prove that simultaneity varies between reference frames.

Lorentz transformations can also be used to prove that magnetic and electric fields are the same force. The reason they are interpreted differently is due to Lorentz transformations done on the charges moving relative to the observer. Changing reference frames shows that an electric field will look like both a magnetic and electric field to a moving observer. There also exist frames in which a magnetic and electric field can be observed as a single magnetic, or electric 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/c \rightarrow 0].

History

The transformations were first discovered and published by Joseph Larmor in 1897, although Woldemar Voigt had published a slightly different version of them in 1887, for which he showed that Maxwell's equations were invariant. In 1905, Henri Poincaré named them after the Dutch physicist and mathematician Hendrik Antoon Lorentz (1853-1928) who had published a first order version of these transformations in the 1890s and the final version in 1899 and 1904. The development of these transformations was encouraged by the null result of the Michelson-Morley experiment.

The Lorentz transformations were published in 1897 and 1900 by Joseph Larmor and by Hendrik Lorentz in 1899 and 1904. Voigt (1887) had published a form of the equations

[ t' = t - vx/c^2, \;\;x' = x - vt, \;\; y' = / ,\;\; z' =/ ]
which incorporated relativity of simultaneity ("local time") and time dilation. For Voigt, clocks ran slower by the factor [\gamma ^2] which is greater than the now accepted value of [\gamma] predicted by Larmor (1897). Note that Voigt equations have a length expansion in the transverse direction. Voigt derived these transformations as those which would make the speed of light the same in all reference frames. Lorentz believed Voigt's transformations were equivalent to his (apparently not seeing the significance of the different time dilation) and wrote
"In a paper which to my regret has escaped my notice all these years, Voigt has applied a transformation equivalent to [the Lorentz treansformation]. The idea of the transformations used above might therefore have been borrowed form Voigt and the proof that it does not alter the form of the equations for the free ether is contained in his paper" (Lorentz 1913)
In a similar vein, 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.

Henri Poincaré in 1900 attributed the invention of local time to Lorentz and showed how Lorentz's first version of it (which applies to invariant clock rates) arose when clocks were sychronised by exchanging light signals which were assumed to travel at the same speed against and with the motion of the reference frame (see relativity of simultaneity).

Larmor's (1897) and Lorentz's (1899, 1904) final equations were not in the modern notation and form, but were algebraically equivalent to those published (1905) by Henri Poincaré, the French mathematician, who revised the form to make the four equations into the coherent, self-consistent whole we know today. Both Larmor and Lorentz discovered that the transformation preserved Maxwell's equations. Paul Langevin (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".

See also

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References

 

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