# 4-acceleration

4-acceleration (four-acceleration) is a four-vector which in special relativity is the proper time derivative of 4-velocity.

 ${\displaystyle A^{\mu }={\frac {dU^{\mu }}{d\tau }}}$

When curvalinear coordinates are used such as in general relativity, the acceleration 4-vector is the covariant derivative of 4-velocity with respect to proper time (see operator of proper-time-derivative)

 ${\displaystyle A^{\lambda }={\frac {DU^{\mu }}{d\tau }}={\frac {d^{2}x^{\lambda }}{d\tau ^{2}}}+\Gamma _{\mu \nu }^{\lambda }{\frac {dx^{\mu }}{d\tau }}{\frac {dx^{\nu }}{d\tau }}}$

It relates to the force Four-force by Newton's second law

 ${\displaystyle F^{\lambda }=mA^{\lambda }}$

where the mass ${\displaystyle m}$ is an invariant.

And in the Absence of a 4-force this

 ${\displaystyle A^{\lambda }=0}$

gives the geodesic equation

 ${\displaystyle {\frac {d^{2}x^{\lambda }}{d\tau ^{2}}}+\Gamma _{\mu \nu }^{\lambda }{\frac {dx^{\mu }}{d\tau }}{\frac {dx^{\nu }}{d\tau }}=0}$