Acceleration (differential geometry)
# 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".

## Formal definition

Consider a differentiable manifold $M$ with a given connection $\Gamma$. Let $\gamma \colon \mathbb {R} \to M$ be a curve in $M$ with tangent vector, i.e. velocity, ${\dot {\gamma }}(\tau )$, with parameter $\tau$.

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

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

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

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

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

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

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

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

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

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

