Acceleration (differential geometry)

Let be given a differentiable manifold $M$, considered as spacetime (not only space), with a connection $\backslash Gamma$. Let $\backslash gamma\; \backslash colon\backslash R\; \backslash to\; M$ be a curve in $M$ with tangent vector, i.e. (spacetime) velocity, $\{\backslash dot\backslash gamma\}(\backslash tau)$, with parameter $\backslash tau$.

The (spacetime) acceleration vector of $\backslash gamma$ is defined by $\backslash nabla\_\{\backslash dot\backslash gamma\}\{\backslash dot\backslash gamma\}$, where $\backslash nabla$ denotes the covariant derivative associated to $\backslash Gamma$.

It is a covariant derivative along $\backslash gamma$, and it is often denoted by

:$\backslash nabla\_\{\backslash dot\backslash gamma\}\{\backslash dot\backslash gamma\}\; =\backslash frac\{\backslash nabla\backslash dot\backslash gamma\}\{d\backslash tau\}.$

With respect to an arbitrary coordinate system $(x^\{\backslash mu\})$, and with $(\backslash Gamma^\{\backslash lambda\}\{\}\_\{\backslash mu\backslash nu\})$ being the components of the connection (i.e., covariant derivative $\backslash nabla\_\{\backslash mu\}:=\backslash nabla\_\{\backslash partial/\backslash partial\; x^\backslash mu\}$) relative to this coordinate system, defined by

:$\backslash nabla\_\{\backslash partial/\backslash partial\; x^\backslash mu\}\backslash frac\{\backslash partial\}\{\backslash partial\; x^\{\backslash nu\}\}=\; \backslash Gamma^\{\backslash lambda\}\{\}\_\{\backslash mu\backslash nu\}\backslash frac\{\backslash partial\}\{\backslash partial\; x^\{\backslash lambda\}\},$

for the acceleration vector field $a^\{\backslash mu\}:=(\backslash nabla\_\{\backslash dot\backslash gamma\}\{\backslash dot\backslash gamma\})^\{\backslash mu\}$ one gets:

:$a^\{\backslash mu\}=v^\{\backslash rho\}\backslash nabla\_\{\backslash rho\}v^\{\backslash mu\}\; =\backslash frac\{dv^\{\backslash mu\}\}\{d\backslash tau\}+\; \backslash Gamma^\{\backslash mu\}\{\}\_\{\backslash nu\backslash lambda\}v^\{\backslash nu\}v^\{\backslash lambda\}=\; \backslash frac\{d^2x^\{\backslash mu\}\}\{d\backslash tau^2\}+\; \backslash Gamma^\{\backslash mu\}\{\}\_\{\backslash nu\backslash lambda\}\backslash frac\{dx^\{\backslash nu\}\}\{d\backslash tau\}\backslash frac\{dx^\{\backslash lambda\}\}\{d\backslash tau\},$

where $x^\{\backslash mu\}(\backslash tau):=\; \backslash gamma^\{\backslash mu\}(\backslash tau)$ is the local expression for the path $\backslash gamma$, and $v^\{\backslash rho\}:=(\{\backslash dot\backslash gamma\})^\{\backslash rho\}$.

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 $\backslash xi^a$ is given by $\backslash xi^\{b\}\backslash nabla\_\{b\}\backslash xi^\{a\}$.{{cite book | last = Malament | first = David B. | title = Topics in the Foundations of General Relativity and Newtonian Gravitation Theory | publisher = University of Chicago Press | location = Chicago | year=2012 |author-link= David Malament|isbn= 978-0-226-50245-8}}

• Acceleration
• Covariant derivative

{{Reflist}}

• {{cite book | last = Friedman | first = M. | title = Foundations of Space-Time Theories | publisher = Princeton University Press | location = Princeton | year=1983 |author-link= Michael Friedman (philosopher)| isbn=0-691-07239-6}}
• {{cite book |last1= Dillen |first1= F. J. E.| last2 = Verstraelen | first2 = L.C.A. | title = Handbook of Differential Geometry | publisher = North-Holland | location = Amsterdam | year=2000 |volume=1 | isbn=0-444-82240-2}}
• {{cite book |last1= Pfister |first1= Herbert| last2 = King | first2 = Markus | title = Inertia and Gravitation. The Fundamental Nature and Structure of Space-Time | publisher = Springer | location = Heidelberg | year=2015 |volume=The Lecture Notes in Physics. Volume 897|doi =10.1007/978-3-319-15036-9| isbn=978-3-319-15035-2}}

Category:Differential geometry

Category:Manifolds

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