Acceleration (differential geometry)

Summary

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 edit

Consider a differentiable manifold   with a given connection  . Let   be a curve in   with tangent vector, i.e. velocity,  , with parameter  .

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

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

 

With respect to an arbitrary coordinate system  , and with   being the components of the connection (i.e., covariant derivative  ) relative to this coordinate system, defined by

 

for the acceleration vector field   one gets:

 

where   is the local expression for the path  , and  .

The concept of acceleration is a covariant derivative concept. In other words, in order to define acceleration an additional structure on   must be given.

Using abstract index notation, the acceleration of a given curve with unit tangent vector   is given by  .[3]

See also edit

Notes edit

  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 edit

  • 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.