Page 1 of Finding Particles Page 2

Einstein's special relativity

What are vectors?
Vectors are mathematical objects that can be used to describe real physical properties of the world around us. For instance, when you drive in car on the freeway it is not only important that you stay under the speed limit but also that you drive in the correct direction.
The property speed and direction can be put into math by using a velocity vector. This vector has a ``length'' that represents the speed, and a direction.

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:

Speed, momentum and kinetic energy
In physics all objects have a really serious speed limit, the speed of light! Nothing can move faster than light (in vacuum), called ``c''. That's a mind-boggling large speed of ca. 300,000 km per second, but still it's the absolut limit. Therefore all moving particles with high speed, like particles in HEP, have their velocity measured as ratio over the speed of light. For this vector we use the greek symbol ``beta'':

It is a well-known fact in physics that speed means the object has energy. That is the reason, why an abrupt stop of a moving car with a tree causes damage to both. The amount of ``movement energy'', called kinetic energy, is a very important number. In non-relativistic mechanics they are connected by:

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

The mathematical operation of muliplying two vectors, resulting in a single number (also called ``scalar'') is called ``scalar product''. For our vectors it is defined 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.

Vectors in special relativity
This fundamental approach of physics

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'').

Combining momentum and energy
Besides all those widely known consequences of this theory, like slower ticking clocks in moving rockets and shortened lengths of speeding cars, the momentum vector also has to become a vector with four components. Einstein discovered that this ``zero-th'' component has to be the total energy of the moving particle, and the length of the vector equals the mass of the object. This results in the most famous equation of this theory:

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''.

We in the HEP field use mass units that show this equality of mass and energy. We also ``redefine'' the world by using ``natural units'', in which ``c'' (the speed of light) equals 1. The mass of an object is measured in eV. 1 eV is the energy an object with the charge of the electron receives or looses as kinetic energy, when it passes the voltage difference of 1 V. This unit was chosen, since all particle accelerators work with high voltage differences to accelerate charged particles.
1 eV is equal to 1.8 10**-36 kg...

Page 1 of Finding Particles Page 2
Updated: 28. November 1995
Author: Andreas H. Wolf (