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 $$\gamma \colon\R \to M$$ 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:
 * $$a^{\mu}=v^{\rho}\nabla_{\rho}v^{\mu} =\frac{dv^{\mu}}{d\tau}+ \Gamma^{\mu}{}_{\nu\lambda}v^{\nu}v^{\lambda}= \frac{d^2x^{\mu}}{d\tau^2}+ \Gamma^{\mu}{}_{\nu\lambda}\frac{dx^{\nu}}{d\tau}\frac{dx^{\lambda}}{d\tau},$$

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 $$\xi^{b}\nabla_{b}\xi^{a}$$.