# Special Relativity: Dynamics

## Contents

#### Introduction to Relativistic Dynamics

In the first SparkNote on kinematics and special relativity we examined how objects are observed when they are in motion. We did not pay any attention to how they came to be in motion, how they may have stayed in motion and how objects could interact in spacetime. All these concepts fall under the concepts of dynamics, which examines what happens to mass, momentum, energy, force and acceleration in Special Relativity. As we shall see Einstein's theory has remarkable consequences for these concepts too.

In the first section we will explore the concepts of relativistic energy and relativistic momentum. The quantities are so-called because the equations by which they are expressed bear some relation to the equations for Newtonian energy and momentum. However, the most important thing to remember is that 'energy' and 'momentum' are just labels which we attached to quantities which happen to be conserved in the interactions between particles we observe. It is this conservation, which can be verified only experimentally, that makes energy and momentum such important concepts. The second section will introduce the concept of a 4-vector. These are just like regular vectors except they have four components. 4-vectors and the concepts associated with them can be used to greatly simplify much of Special Relativity; indeed all of Special Relativity can be expressed in 4-vector equations. The final section will consider relativistic force and acceleration.

The treatment of energy and momentum given here will differ from many presentations in what is term 'mass.' Some textbooks refer to the mass of a particle at rest (a motionless particle) as its rest mass and the mass of a moving particle as a 'relativistic mass' ( m rel = γm ). Although this notation leads to a formula for momentum , which is familiar, it can ultimately only be confusing. Here we will only refer to one type of mass, the one which other authors call the 'rest mass.' This is the same mass that appears in all the Newtonian formulas (eg. ) and it is the same mass than one would find if one actually weighed the particle when it is at rest. This single concept of mass is frame-independent (it is the same in all frames) and avoids the confusion between whether we are talking about rest mass or relativistic mass.