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 $M$ with a given connection $\Gamma$. Let $\gamma \colon \mathbb{R} \to M$ be a curve in $M$ with tangent vector, i.e. velocity, ${\displaystyle {\dot {\gamma }}(\tau )}$, 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

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

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 ${\displaystyle \nabla _{\mu }:=\nabla _{\partial /x^{\mu }}}$) relative to this coordinate system, defined by

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

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 $\gamma$, 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 $M$ must be given.

See also

• Acceleration
• Covariant derivative

Notes

1. Friedman, M. (1983). Foundations of Space-Time Theories. Princeton: Princeton University Press. p. 38. ISBN 0-691-07239-6.
2. Benn, I.M.; Tucker, R.W. (1987). An Introduction to Spinors and Geometry with Applications in Physics. Bristol and New York: Adam Hilger. p. 203. ISBN 0-85274-169-3.

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. Volume 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.