Relativistic Dynamics

We have been introduced in Introduction to Special Relativity to the Lorentz transformation, which describes how spacetime coordinates change between two inertial frames moving at a constant relative velocity. On this page, we attempt a relativistic treatment of Newtonian dynamics, which is the study of how forces cause objects to move.

Mechanics in Special Relativity

Energy and Momentum in Special Relativity

In classical mechanics, the kinetic energy of a particle is given by \(K = \frac{1}{2} m v^2\), where \(m\) is the mass of the particle and \(v\) is its velocity. The momentum of the particle is \(p = m v\).

In special relativity, we these definitions are modified to give the total energy and momentum of a particle as

\[E = \gamma m c^2 \, , \quad \mathbf{p} = \gamma m \mathbf{v} \, ,\]

where \(\gamma = 1/\sqrt{1 - v^2/c^2}\) is the Lorentz factor. Note that even if the particle is at rest, it has a nonzero energy \(E = m c^2\), which is called the rest energy. The kinetic energy of the particle is then \(K = E - m c^2 = (\gamma - 1) m c^2\). In the non-relativistic limit \(v \ll c\), we have \(\gamma \approx 1 + \frac{1}{2} v^2/c^2\), and the kinetic energy reduces to the classical expression.