*Note*: An older *Mathematica*-formatted version may be found
here.

The Lorentz transformation
equations follow simply from
*Minkowski*'s space-time version of *Pythagoras*' theorem,
the metric equation. *Pythagoras*' theorem is
for right triangles just "the hypoteneuse squared
equals the sum of sides squared", or in other words
** ds^{2} = dx^{2} + dy^{2}**.
This also works in three dimensions. To get the metric
equation, one simply adds in a "side" of length

To move to Lorentz transforms,
we adapt here the strategy employed by Professor Thomas A. Moore
in "Book R" of the excellent McGraw-Hill
introductory text "Six ideas that shaped physics". The basic idea
is simple. For a *time-like interval* in the frame of a moving
observer (illustrated below by points **O** and **P**), one can define
*coordinate-velocity* v as *d*x/*d*t relative to the
"home frame", β as v/c, and γ as 1/Sqrt[1-β^{2}].
Solving the metric equation above for *d*t in terms of
time elapsed in the traveling frame (eliminating *d*x), one gets that
**c dt = γ c dτ**. Solving the metric
equation instead for

Equations for time-like intervals in
the frame of a moving observer are quite important. For example,
they prove that γ = *d*t/*d*τ is
"speed-of-map-time" from the perspective of the traveler, and
that γv = *d*x/*d*τ is
the map-distance traveled per unit traveler-time
or *proper-velocity* w. These two quantities, with
the first multiplied by c, make up the *velocity four-vector*.
It is purely time-like in the traveler frame, but takes on space-like
components (proper-velocity) if the traveler frame is moving.
Multiply the four-velocity by rest mass m, and these quantities
describe our traveler's *energy-momentum* as well.
Since infinity is a reasonable upper-bound on proper-velocity and
γ, coordinate-velocity v never gets larger than c.

A similar strategy can be applied to a
drawing for
*spacelike intervals* between events simultaneous to a traveling
observer (e.g. events lying on the traveler's x'-axis).
From the metric equation, one can show that
** dx = γ dσ**, and

These four relationships for
time-like and space-like intervals elapsed between events
can be applied to events **O** and **Q** in the drawing below.
It is then simple to calculate the rest-frame space and time
intervals (blue) in terms of the space and time intervals (green)
experienced by the traveling observer. As you can see from the
drawing, in each case a sum of two terms is needed.
The resulting "inverse Lorentz transform" equations are
**c dt = γ (c dt' + β dx')**, and

Given the above two equations in terms
of two unknowns (*d*x' and *d*t'),
one can solve for them to get the "forward Lorentz
transform" equations. In effect, the sign of β
is reversed, giving:
**c dt' = γ (c dt - β dx)** and

Note that the time-elapsed between these two
events (i.e. Δt') is shortest to constant-speed observers,
here with β' = 3/5 and γ' = 5/4, for whom both events happen
at the same place. Further study of
these diagrams shows that *accelerated observers present at both
events* experience even shorter elapsed times between **O** and **Q**!
An extreme example might be a photon of light, emitted by
event **O**, bouncing off a mirror, and returning to be absorbed by
event **Q**. Since γ is infinite for photons, time on their
clocks doesn't pass at all! Ironically, therefore, a
principle
of extremal aging is also hidden in the metric equation, whereby
*the straight path between two events takes the most* (not
the least) *time* on the part of the traveler. It turns out
that this principle also works in curved space-time, and has
connections to quantum mechanics'
explore-all-paths rule
as well.