Editing Four-force (section)

== In special relativity ==
The four-force is defined as the rate of change in the [[four-momentum]] of a particle with respect to the particle's [[proper time]]. Hence,:

<math display="block">\mathbf{F} = {\mathrm{d}\mathbf{P} \over \mathrm{d}\tau}.</math>

For a particle of constant [[invariant mass]] <math>m > 0</math>, the four-momentum is given by the relation <math>\mathbf{P} = m\mathbf{U}</math>, where <math>\mathbf{U}=\gamma(c,\mathbf{u})</math> is the [[four-velocity]]. In analogy to [[Newton's second law]],  we can also relate the four-force to the [[four-acceleration]],  <math>\mathbf{A}</math>, by equation:

<math display="block">\mathbf{F} = m\mathbf{A} = \left(\gamma {\mathbf{f}\cdot\mathbf{u} \over c},\gamma{\mathbf f}\right).</math>

Here

<math display="block">{\mathbf f}={\mathrm{d} \over \mathrm{d}t} \left(\gamma m {\mathbf u} \right)={\mathrm{d}\mathbf{p} \over \mathrm{d}t}</math>

and

<math display="block">{\mathbf{f}\cdot\mathbf{u}}={\mathrm{d} \over \mathrm{d}t} \left(\gamma mc^2 \right)={\mathrm{d}E \over \mathrm{d}t} .</math>

where <math>\mathbf{u}</math>, <math>\mathbf{p}</math> and <math>\mathbf{f}</math> are [[3-space]] vectors describing the velocity, the momentum of the particle and the force acting on it respectively; and <math>E</math> is the total energy of the particle.