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] Let be given a differentiable manifold
, considered as spacetime (not only space), with a connection
γ :
with tangent vector, i.e. (spacetime) velocity,
The (spacetime) acceleration vector of
γ ˙
denotes the covariant derivative associated to
It is a covariant derivative along
γ
, and it is often denoted by With respect to an arbitrary coordinate system
μ ν
being the components of the connection (i.e., covariant derivative
) relative to this coordinate system, defined by for the acceleration vector field
γ ˙
is the local expression for the path
The concept of acceleration is a covariant derivative concept.
In other words, in order to define acceleration an additional structure on
Using abstract index notation, the acceleration of a given curve with unit tangent vector
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