Hyperbolic motion (relativity)

Hyperbolic motion is the motion of an object with constant proper acceleration in special relativity. It is called hyperbolic motion because the equation describing the path of the object through spacetime is a hyperbola, as can be seen when graphed on a Minkowski diagram whose coordinates represent a suitable inertial (non-accelerated) frame. This motion has several interesting features, among them that it is possible to outrun a photon if given a sufficient head start, as may be concluded from the diagram. Hermann Minkowski (1908) showed the relation between a point on a worldline and the magnitude of four-acceleration and a "curvature hyperbola" (Krümmungshyperbel). In the context of Born rigidity, Max Born (1909) subsequently coined the term "hyperbolic motion" (Hyperbelbewegung) for the case of constant magnitude of four-acceleration, then provided a detailed description for charged particles in hyperbolic motion, and introduced the corresponding "hyperbolically accelerated reference system" (hyperbolisch beschleunigtes Bezugsystem). Born's formulas were simplified and extended by Arnold Sommerfeld (1910). For early reviews see the textbooks by Max von Laue (1911, 1921) or Wolfgang Pauli (1921). See also Galeriu (2015) or Gourgoulhon (2013), and Acceleration (special relativity)#History. The proper acceleration of a particle is defined as the acceleration that a particle "feels" as it accelerates from one inertial reference frame to another. If the proper acceleration is directed parallel to the line of motion, it is related to the ordinary three-acceleration in special relativity by where is the instantaneous speed of the particle, the Lorentz factor, is the speed of light, and is the coordinate time. Solving for the equation of motion gives the desired formulas, which can be expressed in terms of coordinate time as well as proper time . For simplification, all initial values for time, location, and velocity can be set to 0, thus: This gives , which is a hyperbola in time T and the spatial location variable .

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Accelerations in special relativity (SR) follow, as in Newtonian Mechanics, by differentiation of velocity with respect to time. Because of the Lorentz transformation and time dilation, the concepts of time and distance become more complex, which also leads to more complex definitions of "acceleration". SR as the theory of flat Minkowski spacetime remains valid in the presence of accelerations, because general relativity (GR) is only required when there is curvature of spacetime caused by the energy–momentum tensor (which is mainly determined by mass).

A proper reference frame in the theory of relativity is a particular form of accelerated reference frame, that is, a reference frame in which an accelerated observer can be considered as being at rest. It can describe phenomena in curved spacetime, as well as in "flat" Minkowski spacetime in which the spacetime curvature caused by the energy–momentum tensor can be disregarded.

Rindler coordinates are a coordinate system used in the context of special relativity to describe the hyperbolic acceleration of a uniformly accelerating reference frame in flat spacetime. In relativistic physics the coordinates of a hyperbolically accelerated reference frame constitute an important and useful coordinate chart representing part of flat Minkowski spacetime. In special relativity, a uniformly accelerating particle undergoes hyperbolic motion, for which a uniformly accelerating frame of reference in which it is at rest can be chosen as its proper reference frame.

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