Editing Four-acceleration (section)

== Four-acceleration in non-inertial coordinates ==

In non-inertial coordinates, which include accelerated coordinates in special relativity and all coordinates in [[general relativity]], the acceleration four-vector is related to the [[four-velocity]] through an [[absolute derivative]] with respect to proper time.

<math display="block">A^\lambda := \frac{DU^\lambda }{d\tau} = \frac{dU^\lambda }{d\tau } + \Gamma^\lambda {}_{\mu \nu}U^\mu U^\nu </math>

In inertial coordinates the [[Christoffel symbols]] <math>\Gamma^\lambda {}_{\mu \nu}</math> are all zero, so this formula is compatible with the formula given earlier.

In special relativity the coordinates are those of a rectilinear inertial frame, so the [[Christoffel symbols]] term vanishes, but sometimes when authors use curved coordinates in order to describe an accelerated frame, the frame of reference isn't inertial, they will still describe the physics as special relativistic because the metric is just a frame transformation of the [[Minkowski space]] metric. In that case this is the expression that must be used because the [[Christoffel symbols]] are no longer all zero.