Lorentz Transformation

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	CATEGORIES ABOUT LORENTZ TRANSFORMATION
	equations
	minkowski spacetime
	special relativity
	fundamental physics concepts
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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: :$s^2\; \backslash ,=\; -\; c^2(t\_2\; -\; t\_1)^2\; +\; (x\_2\; -\; x\_1)^2\; +\; (y\_2\; -\; y\_1)^2\; +\; (z\_2\; -\; z\_1)^2$ and is an Invariant .: :$x^a\; x\_a\; \backslash ,\; =\; x\text{'}^a\; x\text{'}\_a$ i.e., $\backslash eta\_\{ab\}\text{'}x\text{'}^ax\text{'}^b\; \backslash ,\; =\; \backslash eta\_\{cd\}\; x^c\; x^d$ where $\backslash eta\_\{ab\}$ is the Minkowski Metric :$\backslash eta\_\{\backslash mu$ u}= \begin{bmatrix} -1&0&0&0\ 0&1&0&0\ 0&0&1&0\ 0&0&0&1 \end{bmatrix} and the Einstein Summation Convention is being used. From this relation follows the linearity of the coordinate transformation: :$x\text{'}^a\; \backslash ,\; =\; \backslash Lambda\; ^a\{\}\_b\; x^b\; +\; C^a$ where $C^a$ and $\backslash ,\; \backslash Lambda\; ^a\{\}\_b$ satisfy: :$\backslash Lambda^a\{\}\_b\; \backslash eta\; \_\{ac\}\; \backslash Lambda^c\; \{\}\_d\; \backslash ,=\; \backslash eta\; \_\{bd\}$ :$C^a\; \backslash eta\_\{ac\}\; \backslash Lambda\; ^c\{\}\_b\; \backslash ,\; =\; 0$ :$\backslash eta\_\{ab\}\; C^a\; C^b\; \backslash ,\; =\; 0$ Such a transformation is called a Poincaré Transformation . The $C^a$ represents a space-time translation; when $C^a\; \backslash ,\; =\; 0$, the transformation is a Lorentz transformation. Taking the determinant of the first equation gives $\backslash det\; (\backslash Lambda\; ^a\{\}\_b)\; \backslash ,\; =\; \backslash pm\; 1$ Lorentz transformations with $\backslash det\; (\backslash Lambda\; ^a\{\}\_b)\; \backslash ,\; =\; +\; 1$ are called proper Lorentz transformations and consist of spatial rotations and boosts. Those with $\backslash det\; (\backslash Lambda\; ^a\{\}\_b)\; \backslash ,\; =\; -\; 1$ 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 . 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, $\backslash ,\; (t\_1,\; x\_1,\; y\_1,\; z\_1)$ and $\backslash ,\; (t\_2,\; x\_2,\; y\_2,\; z\_2)$, 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: : : $x\_1\; =\; \backslash gamma\; (x\_2\; -\; v\; t\_2)\; \backslash ,$ : $y\_1\; =\; y\_2\; \backslash ,$ : $z\_1\; =\; z\_2\; \backslash ,$ :::where ::::$x\_1$ is the X Component (in $S\_1$) of the position of a point (or object) stationary relative to ''observer $S\_1$'', ::::$x\_2$ is the ''x component'' (in $S\_2$) of the position of the same point (or object) that is stationary relative to ''observer $S\_1$'' (moving with respect to $S\_2$), ::::$t\_1$ is the time since time zero (in $S\_1$) of ''observer $S\_1$'', ::::$t\_2$ is the time since time zero (in $S\_1$) of ''observer $S\_2$'' (as seen by ''observer $S\_1$''), ::::$y\_1$ and $y\_2$ are both the ''y component'' of the position of a point in both reference frames (same with $z\_1$ and $z\_2$), ::::      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 : $$ \begin{bmatrix} c t_1 \x_1 \y_1 \z_1 \end{bmatrix} = \begin{bmatrix} \gamma&-rac{v}{c} \gamma&0&0\ -rac{v}{c} \gamma&\gamma&0&0\ 0&0&1&0\ 0&0&0&1\ \end{bmatrix} \begin{bmatrix} c t_2\x_2\y_2\z_2 \end{bmatrix}. where the coordinate $t\_2$ is replaced by $ct\_2$ (and similarly for $t\_1$). The Lorentz transformations in SC may be cast into a more useful form by introducing a parameter $\backslash phi$ called the rapidity or '''hyperbolic parameter''' through the equation: : The Lorentz transformations in SC are then: :$c\; t\_1-x\_1\; \backslash ,\; =\; e^\{\backslash phi\}(c\; t\_2\; -\; x\_2)$ :$c\; t\_1+x\_1\; \backslash ,\; =\; e^\{-\; \backslash phi\}(c\; t\_2\; +\; x\_2)$ :$y\_1\; \backslash ,\; =\; y\_2$ :$z\_1\; \backslash ,\; =\; z\_2$ GENERAL BOOSTS

:

These equations can be expressed in matrix form as

: $$
\begin{bmatrix}
c t' \ ec{r'}
\end{bmatrix}
=
\begin{bmatrix}
\gamma&-rac{ec{v^T}}{c}\gamma\
-rac{ec{v}}{c}\gamma&I+rac{ec{v} \cdot ec{v}^T}{v^2}(\gamma-1)\
\end{bmatrix}
\begin{bmatrix}
c t\ec{r}
\end{bmatrix}
,
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 Frame s.

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 $v\; ightarrow\; 0$.


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
:$t\text{'}\; =\; t\; -\; vx/c^2,\; \backslash ;\backslash ;x\text{'}\; =\; x\; -\; vt,\; \backslash ;\backslash ;\; y\text{'}\; =\; \{y\}/\; \{\backslash gamma\},\backslash ;\backslash ;\; z\text{'}\; =\{z\}/\; \{\backslash gamma\}$
which incorporated Relativity Of Simultaneity ("local time") and Time Dilation . For Voigt, clocks ran slower by the factor $\backslash gamma\; ^2$ which is greater than the now accepted value of $\backslash gamma$ 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 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

• Carroll Group


• Electromagnetic Field


• Galilean Transformation


• Principle Of Relativity


• Special Relativity

EXTERNAL LINKS

• Nothing but Relativity There are many ways to derive the Lorentz transformation without invoking Einstein's constancy of light postulate. The path preferred in this paper restates a simple, established approach.


• 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.


• "A note on relativity before Einstein", ''Brit. Journal Philos. Science'', 37, 232-34 (1986) . A brief discussion of the work of Voigt, Larmor and Lorentz.


• Warp Special Relativity Simulator . A computer program demonstrating the Lorentz transformations on everyday objects.


• A history of special relativity , including a discussion of the independent derivations of the Lorentz transformations by Voigt and Larmor.

REFERENCES

• ¹


• 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 of 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.