View source for PlanetPhysics/Simple Derivation of the Lorentz Transformation

\subsection{Simple Derivation of the Lorentz Transformation
(Supplementary to Section 11)}
From [[../SpecialTheoryOfRelativity/|Relativity: The Special and General Theory]] by [[../AlbertEinstein/|Albert Einstein]]
For the relative orientation of the co-ordinate [[../GenericityInOpenSystems/|systems]] indicated in
Fig. 2, the x-axes of both systems pernumently coincide. In the
present case we can divide the problem into parts by considering first
only events which are localised on the <math>x</math>-axis. Any such event is
represented with respect to the co-ordinate system <math>K</math> by the abscissa <math>x</math>
and the time <math>t</math>, and with respect to the system <math>K^1</math> by the abscissa <math>x'</math>
and the time <math>t'</math>. We require to find <math>x'</math> and <math>t'</math> when <math>x</math> and <math>t</math> are given.

A light-signal, which is proceeding along the positive axis of <math>x</math>, is
transmitted according to the equation

<math>x = ct</math>
or
<blockquote><math>
(1)
x - ct = 0
</math></blockquote>

Since the same light-signal has to be transmitted relative to <math>K^1</math> with
the [[../Velocity/|velocity]] <math>c</math>, the propagation relative to the system <math>K^1</math> will be
represented by the analogous [[../Formula/|formula]]
<blockquote><math>
(2)
x' - ct' = 0
</math></blockquote>

Those [[../SR/|space-time]] points (events) which satisfy (24) must also satisfy
(25). Obviously this will be the case when the [[../Bijective/|relation]]
<blockquote><math>
(3)
(x' - ct') = \lambda (x - ct)
</math></blockquote>

\noindent is fulfilled in general, where <math>\lambda</math> indicates a constant; for, according
to (26), the disappearance of <math>(x - ct)</math> involves the disappearance of
<math>(x' - ct')</math>.

If we apply quite similar considerations to light rays which are being
transmitted along the negative x-axis, we obtain the condition

<blockquote><math>
(4)
(x' + ct') = \mu (x + ct)
</math></blockquote>

By adding (or subtracting) equations (26) and (27), and introducing for
convenience the constants <math>a</math> and <math>b</math> in place of the constants <math>\lambda</math> and <math>\mu</math>,
where

<math>a = \frac{\lambda+\mu}{2}</math>

\noindent and

<math>a = \frac{\lambda-\mu}{2}</math>

\noindent we obtain the equations

<blockquote><math>
(5)
\left. \begin{matrix}{rcl} x' &=& ax-bct \\ ct' &=& act-bx \end{matrix} \right\}
</math></blockquote>

We should thus have the solution of our problem, if the constants <math>a</math>
and <math>b</math> were known. These result from the following discussion.

For the origin of <math>K^1</math> we have permanently <math>x' = 0</math>, and hence according
to the first of the equations (28)

<math>x = \frac{bc}{a}t</math>

If we call <math>v</math> the velocity with which the origin of <math>K^1</math> is moving
relative to <math>K</math>, we then have

<blockquote><math>
(6)
v=\frac{bc}{a}
</math></blockquote>

The same value <math>v</math> can be obtained from equations (28), if we calculate
the velocity of another point of <math>K^1</math> relative to <math>K</math>, or the velocity
(directed towards the negative <math>x</math>-axis) of a point of <math>K</math> with respect to
<math>K'</math>. In short, we can designate <math>v</math> as the relative velocity of the two
systems.

Furthermore, the principle of relativity teaches us that, as judged
from <math>K</math>, the length of a unit measuring-rod which is at rest with
reference to <math>K^1</math> must be exactly the same as the length, as judged from
<math>K'</math>, of a unit measuring-rod which is at rest relative to <math>K</math>. In order
to see how the points of the <math>x</math>-axis appear as viewed from <math>K</math>, we only
require to take a "snapshot" of <math>K^1</math> from <math>K</math>; this means that we have
to insert a particular value of <math>t</math> (time of <math>K</math>), {\it e.g.} <math>t = 0</math>. For this
value of <math>t</math> we then obtain from the first of the equations (5)

<math>x' = ax</math>

Two points of the <math>x'</math>-axis which are separated by the distance <math>\Delta x' = I</math>
when measured in the <math>K^1</math> system are thus separated in our instantaneous
photograph by the distance

<blockquote><math>
(7)
\Delta x = \frac{I}{a}
</math></blockquote>

\noindent But if the snapshot be taken from <math>K'(t' = 0)</math>, and if we eliminate <math>t</math>
from the equations (28), taking into account the expression (29), we
obtain

<math>x' = a \left( I - \frac{v^2}{c^2} \right) x</math>

\noindent From this we conclude that two points on the <math>x</math>-axis separated by the
distance <math>I</math> (relative to <math>K</math>) will be represented on our snapshot by the
distance

<math>\Delta x' = a \left( I - \frac{v^2}{c^2} \right) \quad . \quad . \quad . \quad \mbox{(7a)}</math>

But from what has been said, the two snapshots must be identical;
hence <math>\Delta x</math> in (7) must be equal to <math>\Delta x'</math> in (7a), so that we obtain

<math>a = \frac{I}{I-\frac{v^2}{c^2}} \quad . \quad . \quad . \quad \mbox{(7b)} </math>

The equations (29) and (7b) determine the constants <math>a</math> and <math>b</math>. By
inserting the values of these constants in (28), we obtain the first
and the fourth of the equations given in [[../IsomorphicObjectsUnderAnIsomorphism/|section]] 11.

<blockquote><math>
(8)
\left. \begin{matrix}{rcl}
x' &=& \frac{x-vt}{\sqrt{I-\frac{v^2}{c^2}}} \\
~ \\
t' &=& \frac{t-\frac{v}{c^2}x}{\sqrt{I-\frac{v^2}{c^2}}} \end{matrix} \right\}
</math></blockquote>

Thus we have obtained [[../LorentzTransformation/|The Lorentz transformation]] for events on the
<math>x</math>-axis. It satisfies the condition

<math>x'^2 - c^2t'^2 = x^2 - c^2t^2 \quad . \quad . \quad . \quad \mbox{(8a)} </math>

The extension of this result, to include events which take place
outside the <math>x</math>-axis, is obtained by retaining equations (31) and
supplementing them by the relations

<blockquote><math>
(9)
\left. \begin{matrix}{rcl} y' &=& y \\ z' &=& z \end{matrix} \right\}
</math></blockquote>

In this way we satisfy the postulate of the constancy of the velocity
of light in vacuo for rays of light of arbitrary direction, both for
the system <math>K</math> and for the system <math>K'</math>. This may be shown in the following
manner.

We suppose a light-signal sent out from the origin of <math>K</math> at the time </math>t
= 0<math>. It will be propagated according to the equation

<math>r = \sqrt{x^2+y^2+z^2} = ct</math>

\noindent or, if we [[../PiecewiseLinear/|square]] this equation, according to the equation

<blockquote><math>
(10)
x^2 + y^2 + z^2 = c^2t^2 = 0
</math></blockquote>

It is required by the law of propagation of light, in conjunction with
the postulate of relativity, that the [[../FluorescenceCrossCorrelationSpectroscopy/|transmission]] of the signal in
question should take place---as judged from <math>K^1</math>---in accordance with
the corresponding formula

<math>r' = ct'</math>

\noindent or,

<math>x'^2 + y'^2 + z'^2 - c^2t'^2 = 0  \quad . \quad . \quad . \quad \mbox{(10a)} </math>

In order that equation (10a) may be a consequence of equation (33), we
must have

<blockquote><math>
(11)
x'^2 + y'^2 + z'^2 - c^2t'^2 = \sigma (x^2 + y^2 + z^2 - c^2t^2)
</math></blockquote>

Since equation (8a) must hold for points on the <math>x</math>-axis, we thus have </math>\sigma
= I<math>. It is easily seen that the Lorentz transformation really
satisfies equation (34) for <math>\sigma = I</math>; for (34) is a consequence of (8a)
and (32), and hence also of (31) and (32). We have thus derived the
Lorentz transformation.

The Lorentz transformation represented by (31) and (32) still requires
to be generalised. Obviously it is immaterial whether the axes of <math>K^1</math>
be chosen so that they are spatially parallel to those of <math>K</math>. It is
also not essential that the velocity of translation of <math>K^1</math> with respect
to <math>K</math> should be in the direction of the <math>x</math>-axis. A simple consideration
shows that we are able to construct the Lorentz transformation in this
general sense from two kinds of transformations, {\it viz.} from [[../CosmologicalConstant/|Lorentz transformations]] in the special sense and from purely spatial
transformations. which corresponds to the replacement of the
rectangular co-ordinate system by a new system with its axes pointing
in other directions.

Mathematically, we can characterise the generalised Lorentz
transformation thus:

It expresses <math>x', y', x', t'</math>, in terms of linear homogeneous [[../Bijective/|functions]] of <math>x, y, x, t</math>, of such a kind that the relation

<math>x'^2 + y'^2 + z'^2 - c^2t'^2 = x^2 + y^2 + z^2 - c^2t^2  \quad . \quad . \quad . \quad \mbox{(11a)} </math>

\noindent is satisficd identically. That is to say: If we substitute their
expressions in <math>x, y, x, t</math>, in place of <math>x', y', x', t'</math>, on the
left-hand side, then the left-hand side of (11a) agrees with the
right-hand side.

===References===
This article is derived from the Einstein Reference Archive (marxists.org) 1999, 2002. [http://www.marxists.org/reference/archive/einstein/index.htm Einstein Reference Archive] which is under the FDL copyright.

{{CourseCat}}