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Concept
Four-acceleration
Summary
In the theory of relativity, four-acceleration is a four-vector (vector in four-dimensional spacetime) that is analogous to classical acceleration (a three-dimensional vector, see three-acceleration in special relativity). Four-acceleration has applications in areas such as the annihilation of antiprotons, resonance of strange particles and radiation of an accelerated charge.
In inertial coordinates in special relativity, four-acceleration is defined as the rate of change in four-velocity with respect to the particle's proper time along its worldline. We can say:
where
with the three-acceleration and the three-velocity, and
and
is the Lorentz factor for the speed (with ). A dot above a variable indicates a derivative with respect to the coordinate time in a given reference frame, not the proper time (in other terms, ).
In an instantaneously co-moving inertial reference frame , and , i.e. in such a reference frame
Geometrically, four-acceleration is a curvature vector of a worldline.
Therefore, the magnitude of the four-acceleration (which is an invariant scalar) is equal to the proper acceleration that a moving particle "feels" moving along a worldline. A worldline having constant four-acceleration is a Minkowski-circle i.e. hyperbola (see hyperbolic motion)
The scalar product of a particle's four-velocity and its four-acceleration is always 0.
Even at relativistic speeds four-acceleration is related to the four-force: where m is the invariant mass of a particle.
When the four-force is zero, only gravitation affects the trajectory of a particle, and the four-vector equivalent of Newton's second law above reduces to the geodesic equation. The four-acceleration of a particle executing geodesic motion is zero. This corresponds to gravity not being a force. Four-acceleration is different from what we understand by acceleration as defined in Newtonian physics, where gravity is treated as a force.
Official source
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