To motivate the Lorentz transformation, recall the **Galilean
transformation** between moving coordinate systems:

(17.1) | |||

(17.2) | |||

(17.3) | |||

(17.4) |

(where is fixed and is moving in the 1-direction at speed ).

Then

(17.5) |

or

But

(17.6) |

and so

(17.7) | |||

(17.8) | |||

(17.9) | |||

(17.10) |

Thus if

(17.11) |

then

(17.12) |

(!) so that the wave equation, and hence Maxwell's equations which lead directly to the wave equation in free space,

The simplest linear transformation of coordinates is that preserves the form
of the wave equation is easy to determine. It is one that keeps the speed of
the (light) wave equal in both the
and the
frames. Geometrically, if
a flash of light is emitted from the (coincident) origins at time
, it will appear to expand like a sphere out from *both* coordinate
origins, each in its own frame:

(17.13) |

and

(17.14) |

are

(17.15) |

where, describes a possible change of scale between the frames. If we insist that the coordinate transformation be homogeneous and symmetric between the frames

(17.16) |

Let us define

(17.17) | |||

(17.18) | |||

(17.19) | |||

(17.20) | |||

Minkowski metric |
(17.21) |

Then we need a **linear transformation** of the coordinates that mixes **x** and (ct) in the direction of
in such a way that the length

(17.22) |

is conserved

(17.23) | |||

(17.24) | |||

(17.25) | |||

(17.26) |

where at ,

(17.27) |

Then

(17.28) |

leads to

(17.29) |

or

(17.30) |

so

(17.31) |

where we choose the sign

(17.32) |

as we all know and love.

Now, let me remind you that when ,

(17.33) |

to lowest surviving order in . As we shall see, this is why ``kinetic energy'' in non-relativistic systems (being defined as the total energy minus the potential energy and the rest mass energy) is the usual .

The inverse transformation (from
to
) is also of some interest.

(17.34) | |||

(17.35) | |||

(17.36) | |||

(17.37) |

which is perfectly symmetric, with . It appears that which frame is at ``rest'' and which is moving is mathematically, at least, a matter of perspective.

Finally, if we let

(17.38) |

(in an arbitrary direction) then we have but to use dot products to align the vector transformation equations with this direction: x_0' & = & &gamma#gamma;(x_0 - &beta#beta; ·

**Solution:**
Note that
**x**_&par#parallel;= (&beta#beta; ·**x**)&beta#beta;&beta#beta;^2
Lorentz transform it according to this rule and one gets (by inspection)
**x**_&par#parallel;' = &gamma#gamma;(**x**_&par#parallel;- &beta#beta; x_0)
as one should. The
transform is obvious. Finally, the other two
(
) components do not get a contribution from
. That is,
**x**' = (**x**_&perp#perp;+ **x**_&par#parallel;) + &gamma#gamma;**x**_&par#parallel;
- **x**_&par#parallel;- &gamma#gamma;&beta#beta; x_0
(reconstructing the result above directly) QED.

This is not the most general or convenient way to write the final transform. This is because and are both related functions; it should not be necessary to use two parameters that are not independent. Also, the limiting structure of the transformation is not at all apparent without considering the functional forms in detail.

It is easy to see from the definition above that
(&gamma#gamma;^2 - &gamma#gamma;^2 &beta#beta;^2) = 1.
The range of
is determined by the requirement that the transformation
be non-singular and its symmetry:
0 &le#le;&beta#beta;< 1 so that 1 &le#le;&gamma#gamma;< &infin#infty;.
If we think about functions that ``naturally'' parameterize these ranges, they
are:
^2 &xi#xi;- ^2 &xi#xi;= 1
where
&beta#beta;& = & &xi#xi; = e^-&xi#xi; - e^&xi#xi;e^-&xi#xi; +
e^&xi#xi; &isin#in; [0,1)

&gamma#gamma;& = & &xi#xi; = 12 (e^-&xi#xi; + e^&xi#xi;)
&isin#in; [1,&infin#infty;)

&gamma#gamma;&beta#beta;& = & &xi#xi; = 12 (e^-&xi#xi; -
e^&xi#xi;) &isin#in; [0,&infin#infty;) .

The parameter
is called the **boost parameter** or **rapidity**.
You will see this used frequently in the description of relativistic problems.
You will also hear about ``boosting'' between frames, which essentially means
performing a Lorentz transformation (a ``boost'') to the new frame. This will
become clearer later when we generalize these results further. To give you
something to meditate upon, consider in your minds the formal similarity
between a ``boost'' and a ``rotation'' between the
and
coordinates where the rotation is through an imaginary angle
.
Hmmmm.

To elucidate this remark a wee tad more, note that in this parameterization,
x_0' & = & x_0 &xi#xi;- x_&par#parallel;&xi#xi;

x_&par#parallel;' & = & -x_0 &xi#xi;+ x_&par#parallel;&xi#xi;

**x**_&perp#perp;' & = & **x**_&perp#perp;.
What is the
transformation matrix (in four dimensions)
for this result? Does it look like a ``hyperbolic rotation''^{17.2} or what?

We have just determined that the (vector) coordinate system transforms a
certain way. What, then, of vector fields, or any other vector quantity? How
do general vectors transform under a boost? This depends on the nature of the
vectors. Many vectors, if not most, transform like the underlying coordinate
description of the vectors. This includes the ones of greatest interest in
physics. To make this obvious, we will have to generalize a vector quantity
to **four dimensions**.