Einstein postulated that the
**speed of light is the same in any inertial frame of reference**.
It is not possible to meet this condition if the transformation from one inertial reference frame to another
is done with a universal time, that is,
.
Let us study
**a transformation from one inertial reference frame to another that is moving with
a constant velocity
** in the
direction.
Such a transformation is usually referred to as
**a boost**.

Newton's motion with constant velocity
transforming to
requires that the
**transformation be linear**,
like a rotation.
We therefore
**try a linear transformation in which both the position and the time transform**.
Since this linear transformation will mix
and
, it is reasonable to try to
**transform quantities that have the same units**
^{2},
so we will try a dimensionless transformation of
and
.

We will
**work in just two dimensions**,
and
, like a rotation in the
plane.
(For the boost along the
direction,
and
are not changed.)

By the

the given transformation | |||

definition of the boost | |||

So we can

The
**inverse of the transformation must be the same as a transformation with
** as the velocity of moving frame.
and
are scale factors that depend on the velocity of the transformation.
Since there is no difference between the
and
directions in physics,
we must use the
**same
and
in the inverse transformation** which has the same magnitude of velocity.

the transformation | |||

transformation as matrix eq. | |||

transformation matrix as function of | |||

inverse is | |||

except depends on | |||

rhs is just inverse of 2X2 | |||

upper left | |||

lower right | |||

must be | |||

plug | |||

upper left | |||

lower left | |||

the transformation now |

Now,
**consider a pulse of light** moving in the
direction emitted at
and
in one inertial frame.
Since the origins of the two systems coincide at
, this light is emitted in the primed frame with
, and
.
At a later time, the position of the light pulse will be at
.
By Einstein's postulate that the speed of light is independent of inertial frame,
(and by the Michelson-Morley measurement).
**The pulse of light should be at
** in the primed frame.
Our transformation must give this result, so lets try it.
Transforming the later position of the light pulse, we get.

plug in | |||

plug in | |||

from the 2 eq. |

The condition that the

We have shown that the
**most general transformation** to a frame moving with a velocity
,
that is
**consistent with Newton's laws and the isotropy of space**,
and that satisfies the condition that the
**inverse of the transformation is a transformation with velocity
**,
is given by:

It took
**surprisingly little physics input** to derive the Lorentz transformation for the space-time coordinates.

Jim Branson 2012-10-21