Four-acceleration

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In special relativity, four-acceleration is a four-vector and is defined as the change in four-velocity over the particle's proper time:

<math>A^a:=\frac{dU^a}{d\tau}=\left(\gamma\dot\gamma c,\gamma\mathbf a\right)</math>

where

<math>\mathbf a=\dot\gamma\mathbf u+\gamma\mathbf{\dot u}</math>

and <math>\gamma</math> is the Lorentz factor. It should be noted that a dot above a variable indicates a derivative with respect to the time in a given reference frame, not the proper time <math>\tau</math>.

The scalar product of a four-velocity and the corresponding four-acceleration is always 0.

As in classical mechanics (at low velocity) the four acceleration is related to the four-force by Newton's second law such that

<math>F^a = m_0A^a</math>

where m0 is the rest mass of a particle.

See also: four-vector, four-velocity, four-momentum, four-force.

References