See the Java source code for this applet and the complete Java class hierarchy.

Now drag the
*v*
slider
(for "velocity")
slowly to the right.
A red
coördinate
system will slowly diverge from the original one.
(The original system will turn black.)

Follow the point (2, 0) in the red system as it drifts upwards
and then to the upper right.
Note that when the slider reaches
velocity .745
(i.e, sqrt(5)/3), the red point (2, 0)
has drifted so far to the right that its black
*x*
coördinate
is now 3.
In fact, the red
*x*
coördinate of
*every*
point along the red
*x*
axis is now 2/3 of the point's black
*x*
coördinate.

Now look back at our original point whose black
coördinates are (2, 0).
Although it hasn't moved, its red
*x*
coördinate is now 3.
In fact, the black
*x*
coördinate of
*every*
point along the black
*x*
axis is now 2/3 of the point's red
*x*
coördinate.

The same symmetry occurs along the black and red
*t*
axes.
Keeping the velocity at .745 of the speed of light,
note that the point whose red
coördinates are (0, 2) now has a black
*t*
coördinate of 3.
In fact, the red
*t*
coördinate of
*every*
point along the red
*t*
axis is now 2/3 of the point's black
*x*
coördinate.
Meanwhile, our point whose black
coördinates are (0, 2).
now has a red
*t*
coördinate of 3.
In fact, the black
*t*
coördinate of
*every*
point along the black
*t*
axis is now 2/3 of the point's red
*t*
coördinate.

To achieve a foreshortening of 1/2 instead of 2/3, drag the velocity all the way to .866 (i.e., sqrt(3)/2). The second slider shows the amount of foreshortening.

Finally, observe that the black diagonal line is the graph of the function
*x* = *t*
in both the black and the red
coördinate
systems.

Imagine events taking place along a horizontal line.
We can use one number (*x*)
to represent the position of the event along the line,
and another number (*t*) to represent the time of the event.
Thus every event has a pair of coördinates (*x*, *t*).

For example,
the events along any horizontal black line will be simultaneous:
they will all have the same
*t*
coördinate.
And the events along any vertical black line will all be at the same place:
they will all have the same
*x*
coördinate.

The
*t*
coördinates are measured in seconds.
For convenience, the
*x*
coördinates are therefore measured in light seconds.
In other words, each unit along the
*x*
axis represents 186,284 miles.
The diagonal black line connects all the events that would take place
along a beam of light leaving the origin.
Note that one of these events has the
coördinates
(1, 1):
the speed of light is 186,284 miles per second.

x' =x- 1

and of course, before Lorentz and Einstein, the two observers agreed on the time at which each event took place:x' =x-vt

The last two equations are known as thet' =t

x' = (x-vt) / sqrt(1 -v*v/ (c*c))

A simple derivation of the Lorentz Transformation is in Appendix I of Relativity: the Special and the General Theory by Albert Einstein (Crown Publishers).t' = (t-vx/ (c*c)) / sqrt(1 -v*v/ (c*c))

From the point of view of Mister Red,
the events along any fairly horizontal red line will be simultaneous:
they will all have the same
*t'*
coördinate.
And the events along any fairly vertical red line
will all be at the same place:
they will all have the same
*x'*
coördinate.
Move Mister Red's velocity up to 0.745 of the speed of light.
(That's sqrt(5)/3).
Note that the event whose red
coördinates
are (2, 0)
now has an black
*x*
coördinate
of 3.
In other words,
from the point of view of Mister Red,
Mister Black's miles have shrunk to 2/3 of their normal length.

Meanwhile,
the event whose black
coördinates
are (2, 0)
now has a red
*x*
coördinate
of 3.
In other words,
from the point of view of Mister Black,
Mister Red's miles have shrunk to 2/3 normal length.

And as you can see,
the same relative shrinking happens to black time and red time.
Finally, observe that the event whose red
coördinates
are (1, 1) will always lie along the diagonal black line for any value of
*v*.
In other words, the two observers always agree that the speed of light
is 186,284 miles per second.