Acceleration (differential geometry)

This article has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these messages) This article is missing information about space, time, and their modelling with manifolds. It mixes derivatives along a curve with covariant derivatives in a spacetime manifold. Please expand the article to include this information. Further details may exist on the talk page. (January 2025) This article needs attention from an expert in mathematics. The specific problem is: Confusion of covariant derivatives along a curve and covariant derivatives in a spacetime manifold. Missing relation of a curve parameter to the specific notion of time. WikiProject Mathematics may be able to help recruit an expert. (January 2025) This article may be confusing or unclear to readers. Please help clarify the article. There might be a discussion about this on the talk page. (January 2025) (Learn how and when to remove this message) This article contains content that may be misleading to readers. Please help improve it by clarifying such content. Relevant discussion may be found on the talk page. (January 2025) (Learn how and when to remove this message)

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]

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

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

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

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

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

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

where ${\displaystyle x^{\mu }(\tau ):=\gamma ^{\mu }(\tau )}$ 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.

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

• Acceleration
• Covariant derivative

• 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. Vol. 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. Vol. The Lecture Notes in Physics. Volume 897. Heidelberg: Springer. doi:10.1007/978-3-319-15036-9. ISBN 978-3-319-15035-2.

This differential geometry-related article is a stub. You can help Wikipedia by expanding it.

• v
• t
• e