Lorentz transformation

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In physics, the Lorentz transformation is a set of equations that converts back and forth between two different observers' measurements of space and time. 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. 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.

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 and spatial rotations is known as the Poincare group.

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 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.

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[edit] Lorentz transformation for frames in standard configuration

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.

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.

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

t = \gamma (t' - \frac{v x'}{c^{2}})\ ,
x = \gamma (x' - v t')\ ,
y = y' \ ,
z = z' \ ,

where \gamma = 1/\sqrt{1 - v^2/c^2} is called the Lorentz factor. This Lorentz transformation is called a "boost" in the x-direction and is often expressed in matrix form as

\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}\ .

where \beta = \frac{v}{c}. The Lorentz transformations may be cast into a more useful form by introducing a parameter φ called the rapidity (an instance of hyperbolic angle) through the equation:

e^{\phi} = \gamma \left( 1 + \frac{v}{c} \right) = \frac{c + v}{\sqrt{c^2 - v^2}}

The Lorentz transformations in standard configuration are then:

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

[edit] 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 t\\\vec{r} \end{bmatrix},

where I is the identity matrix.

[edit] Details

In a given coordinate system (xμ), if two events A and B are separated by

(\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

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,

\eta_{\mu\nu} = \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,

s^2= \eta_{\mu\nu} 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_{\mu\nu} 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

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_{\mu\nu}{\Lambda^\mu}_\alpha{\Lambda^\nu}_\beta = \eta_{\alpha\beta}\ .

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.

Taking the determinant of \eta_{\mu\nu}{\Lambda^\mu}_\alpha{\Lambda^\nu}_\beta = \eta_{\alpha\beta} gives us

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

Lorentz transformations with \det ({\Lambda^\mu}_\nu)=+1 are called proper Lorentz transformations and consist of spatial rotations and boosts. Those with \det({\Lambda^\mu}_\nu)=-1 are called improper Lorentz transformations and consist of (discrete) space and time reflections.

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.

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

[edit] 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.

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.

[edit] 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.

[edit] History

The transformations were first discovered and published by Joseph Larmor in 1897. 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 (but not in the modern form) in 1899 and 1904. Although physicists such as Lorentz, Larmor, and Voigt had been discussing ideas such as these for nearly 20 years by the time Einstein published his theory of relativity, their interpretation of them was couched in the concepts of classical physics. Even though the null result of the Michelson-Morley experiment had been published, there was a great deal of confusion about its statistical reliability, and also its interpretation. 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 synchronised 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".

[edit] See also

[edit] External links

[edit] References

  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.
  • 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.
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