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".

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

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

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

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.