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

The 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]

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.
3. Malament, David B. (2012). Topics in the Foundations of General Relativity and Newtonian Gravitation Theory. Chicago: University of Chicago Press. ISBN 978-0-226-50245-8.

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. 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.