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".^[1][2]

Formal definition

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

The acceleration vector of [math]\displaystyle{ \gamma }[/math] is defined by [math]\displaystyle{ \nabla_{\dot\gamma}{\dot\gamma} }[/math], where [math]\displaystyle{ \nabla }[/math] denotes the covariant derivative associated to [math]\displaystyle{ \Gamma }[/math].

It is a covariant derivative along [math]\displaystyle{ \gamma }[/math], and it is often denoted by

[math]\displaystyle{ \nabla_{\dot\gamma}{\dot\gamma} =\frac{\nabla\dot\gamma}{d\tau}. }[/math]

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

[math]\displaystyle{ \nabla_{\partial/\partial x^\mu}\frac{\partial}{\partial x^{\nu}}= \Gamma^{\lambda}{}_{\mu\nu}\frac{\partial}{\partial x^{\lambda}}, }[/math]

for the acceleration vector field [math]\displaystyle{ a^{\mu}:=(\nabla_{\dot\gamma}{\dot\gamma})^{\mu} }[/math] one gets:

[math]\displaystyle{ 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}, }[/math]

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

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

Using abstract index notation, the acceleration of a given curve with unit tangent vector [math]\displaystyle{ \xi^a }[/math] is given by [math]\displaystyle{ \xi^{b}\nabla_{b}\xi^{a} }[/math].^[3]

See also

• Acceleration
• Covariant derivative

Notes

References

• Friedman, M. (1983). Foundations of Space-Time Theories. Princeton: Princeton University Press. ISBN 0-691-07239-6. 
• Dillen, F. J. E.; Verstraelen, L.C.A. (2000). Handbook of Differential Geometry. 1. Amsterdam: North-Holland. ISBN 0-444-82240-2. 
• Pfister, Herbert; King, Markus (2015). Inertia and Gravitation. The Fundamental Nature and Structure of Space-Time. The Lecture Notes in Physics. Volume 897. Heidelberg: Springer. doi:10.1007/978-3-319-15036-9. ISBN 978-3-319-15035-2. 

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