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# 9: Relativistic Kinematics

• • Contributed by Niels Walet
• Professor (Physics) at University of Manchester

One of the features of particle physics is the importance of special relativity. This occurs at a very fundamental level, since particle physics is all about creating and annihilating particles. This can only occur if we can convert mass to energy and vice-versa. Thus Einstein’s idea of the equivalence between mass and energy plays an extremely fundamental rôle in this field of physics. In order for this to be possible we typically need processes that occur at velocities near the light velocity $$c$$, so that the kinematics (i.e., the description of momenta and energy) of these processes requires relativity. In this chapter we shall succinctly introduce the few necessary concepts – I hope that for most of you this is a review, but this chapter is intended to be self-contained and contains everything I shall need in relativistic kinematics.

• 9.1: Lorentz Transformations of Energy and Momentum
• 9.2: Invariant Mass
One of the key numbers we can extract from mass and momentum is the invariant mass, a number independent of the Lorentz frame we are in.
• 9.3: Transformations between CM and lab frame
Even though the use of the invariant mass simplifies calculations considerably, it clearly does not provide all necessary information. It does suggest however, that a natural frame to analyse reactions is the center of mass (CM) frame. Often we shall analyze a process in this frame, and use a Lorentz transformation to get information about processes in the laboratory frame.
• 9.4: Elastic-inelastic
We shall often be interested in cases where we transfer both energy and momentum from one particle to another, i.e., we have inelastic collisions where particles change their character – e.g., their rest-mass.
• 9.E: Relativistic kinematics (Exercises)

Thumbnail: A sketch of a collision between two particles.