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# Einstein's special relativity

You can imagine these vectors like arrows, pointing in a certain direction. Since our world has three dimensions (left-right, forward-backward, up-down) a vector has three vector components, describing the speed in each of the three fundamental directions.
In physics you normally denote them with x, y, and z, and the overall vector with a small arrow over its symbol:

Since in many physical processes, the so called ``momentum'' (product of mass and velocity) is conserved, it is normally written as:

As you can see, multiplying a vector with itself results in the ``pythagoras equation'', giving us the length of the vector squared. It is very natural to assume, that this length stays the same, even when we rotate the vector, or (even more so) we redefine where the basic directions x, y, and z are pointing to, like when we turn our head, and left-right becomes forward-backward... This last change is called a transformation of the reference frame.
In our little example of driving in a car on the freeway, this transformation connects the vector of your car seen by observer on a bridge behind you and the police car in front of you:

Both obervers see the same length of the vector although the components of v are different. For the observer on the bridge you are moving away, but for the police man you are coming closer.
This means, all physical processes calculated in each reference frame, will give the same results, if they use vector lengths in their mathematical formulation, like the kinetic energy does.

has been very successful.

Albert Einstein found in his theory of Special Relativity that in order to avoid contradictions in the formulation of physic laws one has to combine time with the three coordinates of space to a new object, called 4-vector (``four vector'').
Physicist distinguish those vectors from normal vectors by using a greek overscript symbolizing the four components 0 to 3.

The new scalar product with these 4-vectors looks like this:

Note the - sign in the last equation. This product value stays constant under all legal transformation (the ``Lorentz transformations'').

The first equation is obtained by using the ``new'' scalar product for 4-vectors.
The second equation is an approximation for very slow particles and shows that the normal equation for the kinematic energy is recovered, but there is an additional term that connects mass with energy.
Since the length of the 4-vector is again constant under all allowed transformation, this mass is called the ``invariant mass''.

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CLEO WEB PAGES
Updated: 28. November 1995
Author: Andreas H. Wolf (ahw@mps.ohio-state.edu)