showed the relation between a point on a worldline and the magnitude of four-acceleration and a "curvature hyperbola". In the context of Born rigidity, Max Born subsequently coined the term "hyperbolic motion" 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". Born's formulas were simplified and extended by Arnold Sommerfeld. For early reviews see the textbooks by Max von Laue or Wolfgang Pauli. See also Galeriu or Gourgoulhon, and Acceleration #History.
Worldline
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. In this case, the accelerated object is located at at time. If instead there are initial values different from zero, the formulas for hyperbolic motion assume the form:
Rapidity
The worldline for hyperbolic motion can be simplified in several ways. For instance, the expression can be subjected to a spatial shift of amount, thus by which the observer is at position at time. Furthermore, by setting and introducing the rapidity, the equations for hyperbolic motion reduce to with the hyperbola.
Born, Sommerfeld, von Laue, Pauli also formulated the equations for the electromagnetic field of charged particles in hyperbolic motion. This was extended by Hermann Bondi & Thomas Gold and Fulton & Rohrlich This is related to the controversially discussed question, whether charges in perpetual hyperbolic motion do radiate or not, and whether this is consistent with the equivalence principle – even though it's about an ideal situation, because perpetual hyperbolic motion is not possible. While early authors such as Born or Pauli argued that no radiation arises, later authors such as Bondi & Gold and Fulton & Rohrlich showed that radiation does indeed arise.
In equation for hyperbolic motion, the expression was constant, whereas the rapidity was variable. However, as pointed out by Sommerfeld, one can define as a variable, while making constant. This means, that the equations become transformations indicating the simultaneous rest shape of an accelerated body with hyperbolic coordinates as seen by a comoving observer By means of this transformation, the proper time becomes the time of the hyperbolically accelerated frame. These coordinates, which are commonly called Rindler coordinates, can be seen as a special case of Fermi coordinates or Proper coordinates, and are often used in connection with the Unruh effect. Using these coordinates, it turns out that observers in hyperbolic motion possesses an apparent event horizon, beyond which no signal can reach them.
A lesser known method for defining a reference frame in hyperbolic motion is the employment of the special conformal transformation, consisting of an inversion, a translation, and another inversion. It is commonly interpreted as a gauge transformation in Minkowski space, though some authors alternatively use it as an acceleration transformation. It has the form Using only one spatial dimension by, and further simplifying by setting , and using the acceleration , it follows with the hyperbola. It turns out that at the time becomes singular, to which Fulton & Rohrlich & Witten remark that one has to stay away from this limit, while Kastrup remarks that this is one of the strange results of this interpretation.
