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Hyperbolic Motion Relativity
Hyperbolic motion can be visualized on a Minkowski diagram, where the motion of the accelerating particle is along the -axis. Each hyperbola is defined by and (with ) in equation (2).
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.
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:
The worldline for hyperbolic motion (which from now on will be written as a function of proper time) can be simplified in several ways. For instance, the expression
can be subjected to a spatial shift of amount , thus
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
The light path through E marks the apparent event horizon of an observer P in hyperbolic motion.
In equation (2) 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 interpretation.
^ abcvon Laue, M. (1921). Die Relativitätstheorie, Band 1 (fourth edition of "Das Relativitätsprinzip" ed.). Vieweg. pp. 89-90, 155-166.; First edition 1911, second expanded edition 1913, third expanded edition 1919.
^ abSommerfeld (1910), pp. 670-671 used the form and with the imaginary angle and imaginary time .
^ abBondi, H., & Gold, T. (1955). "The field of a uniformly accelerated charge, with special reference to the problem of gravitational acceleration". Proceedings of the Royal Society of London. 229 (1178): 416-424. Bibcode:1955RSPSA.229..416B. doi:10.1098/rspa.1955.0098.CS1 maint: multiple names: authors list (link)
^ abFulton, T., Rohrlich, F., & Witten, L. (1962). "Physical consequences of a co-ordinate transformation to a uniformly accelerating frame". Il Nuovo Cimento. 26 (4): 652-671. Bibcode:1962NCim...26..652F. doi:10.1007/BF02781794.CS1 maint: multiple names: authors list (link)
Leigh Page & Norman I. Adams (Mar 1936). "A New Relativity. Paper II. Transformation of the Electromagnetic Field Between Accelerated Systems and the Force Equation". Physical Review. 49 (6): 466-469. Bibcode:1936PhRv...49..466P. doi:10.1103/PhysRev.49.466.