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.
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Related concepts (16)
Four-velocity
In physics, in particular in special relativity and general relativity, a four-velocity is a four-vector in four-dimensional spacetime that represents the relativistic counterpart of velocity, which is a three-dimensional vector in space. Physical events correspond to mathematical points in time and space, the set of all of them together forming a mathematical model of physical four-dimensional spacetime. The history of an object traces a curve in spacetime, called its world line.
Four-force
In the special theory of relativity, four-force is a four-vector that replaces the classical force. The four-force is defined as the rate of change in the four-momentum of a particle with respect to the particle's proper time: For a particle of constant invariant mass , where is the four-velocity, so we can relate the four-force with the four-acceleration as in Newton's second law: Here and where , and are 3-space vectors describing the velocity, the momentum of the particle and the force acting on it respectively.
Four-gradient
In differential geometry, the four-gradient (or 4-gradient) is the four-vector analogue of the gradient from vector calculus. In special relativity and in quantum mechanics, the four-gradient is used to define the properties and relations between the various physical four-vectors and tensors. This article uses the (+ − − −) metric signature. SR and GR are abbreviations for special relativity and general relativity respectively. indicates the speed of light in vacuum. is the flat spacetime metric of SR.
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