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

\nabla _{\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^{2}x^{\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.

Notes

↑ Friedman, M. (1983). Foundations of Space-Time Theories. Princeton: Princeton University Press. p. 38. ISBN 0-691-07239-6.
↑ 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.

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Acceleration (differential geometry)

Formal definition

See also

Notes

References