the equations for hyperbolic motion reduce to with the hyperbola . Charged particles in hyperbolic motion Born (1909), Sommerfeld (1910), von Laue (1911), Pauli (1921) also formulated the equations for the electromagnetic field of charged particles in hyperbolic motion. This was extended by Hermann Bondi & Thomas Gold (1955) and Fulton & Rohrlich (1960) 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 (1909) or Pauli (1921) argued that no radiation arises, later authors such as Bondi & Gold and Fulton & Rohrlich showed that radiation does indeed arise. Proper reference frame 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 (similar variants are called Kottler-Møller coordinates or Lass 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. Special conformal transformation 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 (see Kastrup for a critical historical survey). 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 (who is very critical of the acceleration interpretation) remarks that this is one of the strange results of this