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 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

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.
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