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Special Relativity

Lorentz Transformations

Let two inertial co-ordinate systems (ICR's),
*S* and ,
be in standard configuration. This means that
is moving with constant velocity
relative to *S* in the *x*-direction as in the following figure

and that the spatial origins, *O* and ,
and the three spatial axes,
and ,
co-incide at time .

We shall convert units of time (eg. seconds) to a length (eg. meters)
by multiplying time by the speed of light *c* - thus our measure of
time will be *ct* which has units of length (this assumes implicitly
that *c* is the same in both reference frames - one of the postulates
of relativity). We wish to determine Cartesian co-ordinates in , ,
as functions of Cartesian co-ordinates in *S*, (*ct*,*x*,*y*,*z*),
using reasonable assumptions. In other words
will be a function of *ct*,*x*,*y* and *z*, i.e. ,
etc. It will sometimes be convenient to adopt an index notation where the
four co-ordinates (*ct*,*x*,*y*,*z*)
are labelled by an index *a* taking on four possible values 0,1,2,3
with *x*^{0}=*ct*,
*x*^{1}=*x*, *x*^{2}=*y* and *x*^{3}=*z*
so that (*ct*,*x*,*y*,*z*)=(*x ^{a}*).
Similarly for primed co-ordinates an index notation is sometimes useful, ,
where the prime is placed on the index so that the index itself can be
used to distinguish between the two co-ordinates systems. The change from (

The derivation of the explicit form of the co-ordinate transformation
proceeds in four steps:

**Step 1): The transformations are linear**

Consider a clock moving with constant velocity, showing a time .
The path of the clock in *S* can be described by four functions.
Since the clock is moving with constant velocity in *S*equal increments
of
must correspond to equal increments of the co-ordinates (*x ^{a}*)
labeling the position of the clock in

= | (1) | ||

= | (2) |

Thus
and can
only be true if ,
in other words the transformations must be *linear* in *x ^{a}*.
In mathematical symbols this means

= | (3) |

where
are constants and
are sixteen functions, independent of *x ^{a}* but possibly
depending on

These conditions can be summarised in the single formula

which can be thought of as a matrix formula with
a
matrix and and
*x ^{a}* column vectors,

The matrix with components
can be inverted to give
*x ^{a}* in terms of ,

where *L*^{-1}(*v*) is the inverse matrix to *L*(*v*),
i.e.
with
the identity matrix. Since the (*x ^{a}*) co-ordinate system
is moving in the

We shall now determine the sixteen functions .**Step
2): **

At time
the two planes co-incide
for all *y* and *z*, as in the following figure

(4) | |||

(5) | |||

(6) |

At time
the two planes
co-incide. Since the relative motion is in the *x*-direction and there
is no rotation (by assumption), the planes co-incide
as in the following figure

thus

(7) | |||

(8) | |||

(9) |

We can apply the same argument with *S* and interchanged,
which requires that we replace *L*(*v*) by *L*^{-1}(*v*)=*L*(-*v*),
to deduce that .
Hence .

Now if we reflect ,
without changing the other co-ordinates in *S*, it should be clear
that *z* and do
not change (since we have just proven that is
independent of *x*). But changing the sign of *x*changes the
sign of *v*, since the relative motion is in the
*x*-direction.
Hence ,
thus ,
so .
The sign can be determined by the trivial observation that *v*=0 should
give the identity transformation, thus .

A similar argument applied to the two planes allows us to conclude that

and .

In summary, we have now that the transformation matrix must be of the form

Up until now we have only really used the postulates of relativity
to streamline the notation. Now it will be used to full effect. First suppose
a flash of light is emitted from the origin *O*=(0,0,0) of *S*
at *t*=0 (and so also from the origin
of
at ).
The flash expands with the speed of light,
*c* which is the *same*
in both reference frames, as a spherical shell whose radius at time *t*
is given by *x*^{2}+*y*^{2}+*z*^{2}=*c*^{2}*t*^{2}
in *S* and by
in .
Now we already know that
and
so

Also

so

Demanding that this hold true for all *t* and any *x*
with |*x*|<*ct* gives three conditions

1 | (10) | ||

1 | (11) | ||

0 | (12) |

on four unkowns. We can express these as four functions of a single
parameter by using the identity for
any real
to write

(13) | |||

(14) |

where
is a function of *v* which is yet to be determined (the minus sign
is for later convenience). We have now arrived at the following form for
the transformation matrix:

The spatial origin
of
is determined by .
In *S* the point
moves with speed *v* in the *x*-direction, i.e. it has *x*
co-ordinate *x*=*vt*. Thus

so
can be written as an inverse hyper-trigonometric function

Note that the properties of the
function now imply that -*c*<*v*<*c* (see the figure
below).
is called the *rapidity* of the transformation.

Using
we have

It is conventional to define

and then the transformation matrix can be written as

Thus we have finally arrived at the following form for the transformation

= | (15) | ||

= | (16) | ||

= | y |
(17) | |

= | z. |
(18) |

These are called **Lorentz Transformations** or sometimes Lorentz
Boosts, to distinguish them from rotations - the name ``boost'' is unfortunate
as there is no acceleration involved.

In matrix notation the Lorentz Transformations can be represented as

The rapidity has the useful property that it is *additive*
under successive transformations in the same direction with
and .
This is most easily established using matrix multiplication to show that

(use the hyperbolic trigonometric identities

= | (19) | ||

= | (20) |

). Thus these two transformations are equivalent to a single transformation
with rapidity .