Clearly the symmetry transformation in the
(
**a boost**) is not identical to
that in the
plane (
**a rotation**)

because there is some difference in the geometry,
but they are closely related.
Lets try to put in the hyperbolic functions by
**setting
** as the off diagonal terms in the matrix would indicate.

So we see that

and the matrix becomes something very
**similar to the rotation**,
where the
**rapidity
** plays a role similar to an angle.
Like an angle when two subsequent rotations are made in the same plane,
**the rapidity just adds**
if two boosts along the same direction are made.

This can be
**easily demonstrated** by multiplying the two matrices and using the identities for hyperbolic sine and cosine.

This gives us our simplest calculation of the
**velocity addition formula**.

This never becomes bigger than one and therefore
**no velocity can exceed the speed of light**.
The velocity addition formula can also be derived by considering the derivative of the position vector with
respect to
**proper time**, which is time in the rest frame.
This derivative is a 4-vector while
is not a 4-vector.

**Subsections**
Jim Branson
2012-10-21