We started with the
**position vector** in Minkowski space.

An important dot product is that of the difference between two spacetime points.
The dot product above gives the ``distance''
in Minkowski space from the origin.
The
**difference between spacetime points** for a single particle is an important case.
We use the dot product of this difference with itself.

The time difference in the particles rest frame
is called the
**proper time** and
is demonstrated to be a scalar quantity in the above equation.
We define the
**velocity 4-vector** with the equation.

We define the
**momentum 4-vector** with.

We have shown that in the non-relativistic limit, the 4-momentum is consistent with.
We accept this as being the
**components of the momentum 4-vector**
giving us the
**dot product of the momentum 4-vector with itself**.
The
**dot product of the momentum 4-vector and the position 4-vector**

is related to the phase of waves.
For example in
**quantum mechanics**, a free particle with a definite momentum is represented by the plane wave.

We define the
**Force 4-vector**.

Jim Branson
2012-10-21