Four-force

In the special theory of relativity four-force is a four-vector that replaces the classical force; the four-force of the four-vector a is defined as the change in four-momentum over the particle's own time:

<math>F^a:= {dP^a \over d\tau}</math>.

Since <math>P^a = m_0U^a</math> where m₀ is the rest mass and U^a is the four-velocity, we can relate the four-force with the four-acceleration as like Newton's second law:

<math>F^a = m_0A^a = \left(\gamma\dot mc,\gamma\mathbf f\right)</math>.

Here, <math>m = m_0\gamma</math> is the relativistic mass and <math>\mathbf f=m_0\left(\dot\gamma\mathbf u+\gamma\mathbf{\dot u}\right)</math>.