Key Points from Chapter 3:
As a result of time dilation, momentum (as classically defined) is not necessarily conserved when two objects collide.
In order to restore the principle that is should always be conserved, it is necessary to redefine momentum:
This is sometimes (controversially) interpreted as an increase in mass when an object moves but this concept is not necessary to develop the rest of the theory.
When work is performed on an object to make it move, the object accrues an equal amount of kinetic energy. This allows us to derive an expression for the kinetic energy by quantifying the work done as:
The force is the rate of change of momentum, dp/dt.
The relativistic redefinition of momentum has the knock-on effect of leading (eventually) to a new, relativistic expression for kinetic energy, which is startlingly different from the classical definition:
This means that any particle with mass would have infinite kinetic energy, if it were travelling at the speed of light and this, in turn, means that infinite work would need to be done to accelerate it to this point. This explains why light speed is not attainable for objects with mass.
The kinetic energy expression can be interpreted as meaning that the kinetic energy is the difference between the total energy of an object and an intrinsic energy which the object possesses even when it is at rest. The energy of an object at rest is given by:
E = mc2
The total energy of a particle can be expressed by the energy-momentum relation:
E = √(m2c4 + p2c2)
This equation leads to the conclusion – confirmed by experiment – that a photon, even though it has no mass, has a momentum, given by the relationship:
p = h / λ
