Acceleration (differential geometry)

In mathematics and physics, acceleration is the rate of change of velocity of a curve with respect to a given linear connection. This operation provides us with a measure of the rate and direction of the "bend".¹²

Let be given a differentiable manifold ${\displaystyle M}$, considered as spacetime (not only space), with a connection ${\displaystyle \Gamma }$. Let ${\displaystyle \gamma \colon \mathbb {R} \to M}$ be a curve in ${\displaystyle M}$ with tangent vector, i.e. (spacetime) velocity, ${\displaystyle {\dot {\gamma }}(\tau )}$, with parameter ${\displaystyle \tau }$.

The (spacetime) acceleration vector of ${\displaystyle \gamma }$ is defined by ${\displaystyle \nabla _{\dot {\gamma }}{\dot {\gamma }}}$, where ${\displaystyle \nabla }$ denotes the covariant derivative associated to ${\displaystyle \Gamma }$.

It is a covariant derivative along ${\displaystyle \gamma }$, and it is often denoted by

${\displaystyle \nabla _{\dot {\gamma }}{\dot {\gamma }}={\frac {\nabla {\dot {\gamma }}}{d\tau }}.}$

With respect to an arbitrary coordinate system ${\displaystyle (x^{\mu })}$, and with ${\displaystyle (\Gamma ^{\lambda }{}_{\mu \nu })}$ being the components of the connection (i.e., covariant derivative ${\displaystyle \nabla _{\mu }:=\nabla _{\partial /\partial x^{\mu }}}$) relative to this coordinate system, defined by

${\displaystyle \nabla _{\partial /\partial x^{\mu }}{\frac {\partial }{\partial x^{\nu }}}=\Gamma ^{\lambda }{}_{\mu \nu }{\frac {\partial }{\partial x^{\lambda }}},}$

for the acceleration vector field ${\displaystyle a^{\mu }:=(\nabla _{\dot {\gamma }}{\dot {\gamma }})^{\mu }}$ one gets:

${\displaystyle a^{\mu }=v^{\rho }\nabla _{\rho }v^{\mu }={\frac {dv^{\mu }}{d\tau }}+\Gamma ^{\mu }{}_{\nu \lambda }v^{\nu }v^{\lambda }={\frac {d^{2}x^{\mu }}{d\tau ^{2}}}+\Gamma ^{\mu }{}_{\nu \lambda }{\frac {dx^{\nu }}{d\tau }}{\frac {dx^{\lambda }}{d\tau }},}$

where ${\displaystyle x^{\mu }(\tau ):=\gamma ^{\mu }(\tau )}$ is the local expression for the path ${\displaystyle \gamma }$, and ${\displaystyle v^{\rho }:=({\dot {\gamma }})^{\rho }}$.

The concept of acceleration is a covariant derivative concept. In other words, in order to define acceleration an additional structure on ${\displaystyle M}$ must be given.

Using abstract index notation, the acceleration of a given curve with unit tangent vector ${\displaystyle \xi ^{a}}$ is given by ${\displaystyle \xi ^{b}\nabla _{b}\xi ^{a}}$.³