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

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

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

It is a covariant derivative along $\gamma$, and it is often denoted by
:$\nabla_ =\frac.$

With respect to an arbitrary coordinate system $(x^)$, and with $(\Gamma^_)$ being the components of the connection (i.e., covariant derivative $\nabla_:=\nabla_$) relative to this coordinate system, defined by
:$\nabla_\frac= \Gamma^_\frac,$

for the acceleration vector field $a^:=(\nabla_)^$ one gets:
:$a^=v^\nabla_v^ =\frac+ \Gamma^_v^v^= \frac+ \Gamma^_\frac\frac,$

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

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 $\xi^\nabla_\xi^$.

See also

*Acceleration
*Covariant derivative

Notes

References

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Differential geometry
Manifolds

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