We have learned that the Lorentz transformation of a space-time coordinate is simplest and most reasonable
if the space coordinate and the time coordinate are in the same units.
This is not true in our SI system.
The unit of distance is one meter and the unit of time is one second.
**
for one second is
meters**.
This is one of the reasons it was hard for us to understand transformations in the
plane.
We would be better off with the foot and the nanosecond, but lets not do that.
(Consider how messy rotations would be is we measured
in miles and
in microns.)

The laws of Physics would be most easily understood if
but we will not make that simplification either.
So we will
**just use the coordinate
for time**.
We will also have to deal with this problem for many other quantities, like Energy for example.

Working problems
**without a universal time** is a complication to which we are not accustomed.

**Consider a muon (an unstable elementary particle) that is produced by cosmic rays (mainly protons) in the upper atmosphere,
with a velocity of
**.
The muon has a mean lifetime of about
microseconds (2000 feet in the units we don't use).
So ignoring the Lorentz transformation, the muon could typically travel a distance of
meters,
however, with the Lorentz transformation, we will find that it can travel much farther if its velocity is near the
speed of light.

Start with a frame with the muon is at rest at the origin (just as it is produced) at . In this frame, its rest frame, it decays at , at the origin. Now we can compute the lifetime in the frame of the earth, in which the muon is moving very fast. We can also compute how far it travels before it decays. In the muon's rest frame, the earth is coming toward it with a velocity of 0.9999 , meaning for the transformation.

transform to earth frame | |||

decay position | |||

decay time | |||

prodution position | |||

production time |

The lifetime viewed in the frame of the earth is , about 140 microseconds. This phenomenon is called

**Consider a stick of length
with one end at the origin
and the other at
meters** in the unprimed coordinate system.
It is at rest so the
positions of both ends is time independent in this frame.
What is its length in a coordinate system moving in the
direction?
How do we properly measure its length in a moving coordinate system?
This problem is a little harder to solve.
A reasonable definition of the length is
with the measurements made at
**same
**,
even though there is no simple way to make the measurement of both ends at the same time.
The way to calculate the length is a little conterintuitive because of the restriction that we have to use the same
at each end.
Lets do it for
,
and
.
**Transform back to the frame in which the stick is at rest** and has a length
.

Since is always greater than or equal to one, the stick is shorter in the moving frame. This is the phenomenon of

Let us also consider the transverse length.
Since we were transforming in the
plane, we
**assumed so far that
and
are unchanged by the boost**.
Consider a stick of length
along the
axis in the rest frame.
At any time
, its two ends are at
and
.
We must face the possibility that in the primed frame, the stick will not be parallel to the
axis.
At
, one end will still be at the origin since both
and
.
The other end will in general be at
.
As before, we do the transformation back into the unprimed frame.
The other end of the stick will be at.

Since we have shown that and the

It is clear from these examples that Special Relativity changes our understanding of many things, including length and time. While the length of vectors is invariant under rotations, it changes under boosts. Time intervals also change when the reference frame is boosted clearly showing that there will be no way to recover an absolute time. Even Euclidean geometry is affected.

**If 3 dimensional dot products and time intervals are not invariant under Lorentz boosts, what is?**
Lets take another look at our muon in the two frames and see if we can find anything that is invariant.
In the rest frame, we have

In the moving frame (of the earth), we have

transformation | |||

decay | |||

production | |||

This is the same as in the rest frame. At least in this case

Jim Branson 2012-10-21