Four-force

$mathbf\{F\}\; =\; \{dmathbf\{P\}\; over\; dtau\}$.

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

$mathbf\{F\}\; =\; mmathbf\{A\}\; =\; left(gamma\; \{dgamma\; over\; dt\}\; mc,gammamathbf\; fright)$.

Here $mathbf\; f=mleft(\{dgamma\; over\; dt\}\; mathbf\; u+gamma\{d\; mathbf\{u\}\; over\; dt\}\; right)$.

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.

$F^lambda\; :=\; frac\{DP^lambda\; \}\{dtau\}\; =\; frac\{dP^lambda\; \}\{dtau\; \}\; +\; Gamma^lambda\; \{\}\_\{mu\; nu\}U^mu\; P^nu$

See also

• four-vector
• four-velocity
• four-acceleration
• four-momentum

References

• Rindler, Wolfgang (1991). Introduction to Special Relativity (2nd). Oxford: Oxford University Press. ISBN 0-853971-853951-5.