Acceleration (differential geometry) explained
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
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
\nabla\mu
| :=\nabla | |
| \partial/\partialx\mu |
) relative to this coordinate system, defined by
| \nabla | |
| \partial/\partialx\mu |
=\Gammaλ{}\mu\nu
,
for the acceleration vector field
one gets:
a\mu=v\rho\nabla\rhov\mu=
+\Gamma\mu{}\nuλv\nuvλ=
+\Gamma\mu{}\nuλ
,
where
x\mu(\tau):=\gamma\mu(\tau)
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.
is given by
.
[3] See also
References
- Book: Friedman, M. . Foundations of Space-Time Theories . Princeton University Press . Princeton . 1983 . Michael Friedman (philosopher). 0-691-07239-6.
- Book: Dillen . F. J. E.. Verstraelen . L.C.A. . Handbook of Differential Geometry . North-Holland . Amsterdam . 2000 . 1 . 0-444-82240-2.
- Book: Pfister . Herbert. King . Markus . Inertia and Gravitation. The Fundamental Nature and Structure of Space-Time . Springer . Heidelberg . 2015 . The Lecture Notes in Physics. Volume 897. 10.1007/978-3-319-15036-9. 978-3-319-15035-2.
Notes and References
- Book: Friedman, M. . Foundations of Space-Time Theories . Princeton University Press . Princeton . 1983 . 38 . 0-691-07239-6.
- Book: Benn . I.M.. Tucker . R.W. . An Introduction to Spinors and Geometry with Applications in Physics . Adam Hilger . Bristol and New York . 1987 . 203 . 0-85274-169-3.
- Book: Malament, David B. . Topics in the Foundations of General Relativity and Newtonian Gravitation Theory . University of Chicago Press . Chicago . 2012 . David Malament. 978-0-226-50245-8.
