Elements of Relativistic Dynamics This is an extremely important part of Special Relativity Theory. There is so much highly interesting material that it would be sufficient to fill an entire quarter-long course. Unfortunately, we don’t have that much time, and therefore we will only “run fast” through this item. In class, we will skip most deri- vations and we will accept the equations without proof, focusing rather on the “physical picture”(you are encouraged, though, to study Chapter 2.7, where the formulae are derived, by yourself). The classical (Newtionian) momentum, as you remember, is: In the SRT the equation takes a new form: Which simply means that the classical momentum is the low-speed limit of the relativistic momentum.

Elements of Relativistic Dynamics (2) The classical (Newtonian) kinetic energy is given by the well-known formula: The relativistic kinetic energy has a form that is quite different: At the low-speed limit, i.e., for v<<c, this equation should reduce to the classical expression, right? Does it really? NO, INCORRECT! This is wrong reasoning. In order to handle the low-speed Limit correctly, one has to expand the first term in the right-hand sum into A Taylor series, and only then to check the behavior for small speed values Please refer to the hand-written notes, pp

The relativistic kinetic energy can be written as: where is called the “relativistic total energy” andis called the “rest energy”. The latter formula, as you surely know, is one of the most famous and MOST IMPORTANT equations in all physics! One EXTREMELY important conclusion from the Special Relativity Theory is (again, we will accept it without proof!) is that the total relativistic energy is conserved! In other words – it sets the equivalence between the mass and energy through the term. It means that: Mass can be converted into energy (e.g., kinetic) – and, conversely, energy (e.g., kinetic) can be converted into mass!

However, some people say that Einstein had not derived the E=mc 2 formula the way we did, but in a way shown in this cartoon:

Finally, another convenient relation between the total relativistic energy, momentum, and mass has the form: But this is not the end of relativistic dynamics. With the relativistic kinematics, time dilation, length contraction we are essentially done. From now on, you will deal with them only in the homework and in tests, but we will not use any of those equation in class any more. However, with relativistic dynamics it’s a different story: we will need to use the relativistic energy and momentum in several topics that we will discuss in the next seven weeks.