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

\mathbf{F} = {d\mathbf{P} \over d\tau}.

For a particle of constant invariant mass m > 0, \mathbf{P} = m\mathbf{U} \, where \mathbf{U}=\gamma(c,\mathbf{u}) \, is the four-velocity, so we can relate the four-force with the four-acceleration as in Newton's second law:

\mathbf{F} = m\mathbf{A} = \left(\gamma {\mathbf{f}\cdot\mathbf{u} \over c},\gamma{\mathbf f}\right).


{\mathbf f}={d \over dt} \left(\gamma m {\mathbf u} \right)={d\mathbf{p} \over dt}


{\mathbf{f}\cdot\mathbf{u}}={d \over dt} \left(\gamma mc^2 \right)={dE \over dt}.

where \mathbf{u}, \mathbf{p} and \mathbf{f} are 3-vectors describing the velocity and the momentum of the particle and the force acting on it respectively.

In general relativity the relation between four-force, and four-acceleration remains the same, but the elements of the four-force are related to the elements of the four-momentum through a covariant derivative with respect to proper time.

F^\lambda := \frac{DP^\lambda }{d\tau} = \frac{dP^\lambda }{d\tau } + \Gamma^\lambda {}_{\mu \nu}U^\mu P^\nu


In special relativity, Lorentz 4-force (4-force acting to charged particle situated in electromagnetic field) can be expressed as:

F_\mu = qE_{\mu\nu}U^\nu, where E_{\mu\nu} - electromagnetic tensor, U^\nu - 4-velocity, q - electric charge.

See also[edit]


  • Rindler, Wolfgang (1991). Introduction to Special Relativity (2nd ed.). Oxford: Oxford University Press. ISBN 0-19-853953-3.