# The Lorentz Transformation

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

## A pleasing geometrical symmetry before we lay on the physics

The horizontal axis is the x axis, and the vertical axis is the t axis. For example, the point (2, 0) is two units to the right of the origin.

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.

## The physical interpretation

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.

## Le Rouge et le Noir

Now imagine a stationary second observer ("Mister Red", adorned with prime signs) whose origin is one unit to the right of the origin of the first observer (the stay-at-home "Mister Black"). Mister Red will believe that the x coördinate of any event is one less than what Mister Black thinks it is. In other words,
```x' = x - 1
```

## Mister Red begins to move

If Mister Red were moving to the right at velocity v, we would have
```x' = x - vt
```
and of course, before Lorentz and Einstein, the two observers agreed on the time at which each event took place:
```t' = t
```
The last two equations are known as the Galilei transformation.

## Mister Red begins to move at relativistic speeds

But nowadays we live in the age of the Lorenz transformation. The real formulæ for Mister Red's x' and t' are
```x' = (x - vt) / sqrt(1 - v * v / (c * c))
```
```t' = (t - vx / (c * c)) / 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).

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.