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]

## Contents

* 1 Formal definition
* 2 See also
* 3 Notes
* 4 References

## Formal definition

Consider a differentiable manifold M with a given connection Γ. Let γ:ℝ->M be a curve in M with tangent vector, i.e. velocity, γ˙(τ), with parameter τ.

The acceleration vector of γ is defined by ∇γ˙γ˙, where ∇ denotes the covariant derivative associated to Γ.

It is a covariant derivative along γ, and it is often denoted by

∇γ˙γ˙=∇γ˙dτ.

With respect to an arbitrary coordinate system (xμ), and with (Γλμν) being the components of the connection (i.e., covariant derivative ∇μ:=∇∂/∂xμ) relative to this coordinate system, defined by

∇∂/∂xμ∂∂xν=Γλμν∂∂xλ,

for the acceleration vector field aμ:=(∇γ˙γ˙)μ one gets:

aμ=vρ∇ρvμ=dvμdτ+Γμνλvνvλ=d2xμdτ2+Γμνλdxνdτdxλdτ,

where xμ(τ):=γμ(τ) is the local expression for the path γ, and vρ:=(γ˙)ρ.

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.

Using abstract index notation, the acceleration of a given curve with unit tangent vector ξa is given by ξb∇bξa.[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. 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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