Hyperbolic motion (relativity)

Hyperbolic motion can be visualized on a Minkowski diagram, where the motion of the accelerating particle is along the <math>X</math>-axis. Each hyperbola is defined by <math>x=\pm c^2/\alpha</math> and <math>\eta=\alpha\tau/c</math> (with <math>c=1, \alpha=1</math>) in equation (Template:EquationNote).

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.<ref>Template:Harvnb</ref>

HistoryEdit

Hermann Minkowski (1908) showed the relation between a point on a worldline and the magnitude of four-acceleration and a "curvature hyperbola" (Template:Langx).<ref>Template:Cite journal</ref> In the context of Born rigidity, Max Born (1909) subsequently coined the term "hyperbolic motion" (Template:Langx) 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" (Template:Langx).<ref name=born>Template:Cite journal</ref> Born's formulas were simplified and extended by Arnold Sommerfeld (1910).<ref name=sommerfeld /> For early reviews see the textbooks by Max von Laue (1911, 1921)<ref name=laue /> or Wolfgang Pauli (1921).<ref name=pauli /> See also Galeriu (2015)<ref name=Galeriu /> or Gourgoulhon (2013),<ref>Template:Cite book</ref> and Acceleration (special relativity)#History.

WorldlineEdit

The proper acceleration <math>\alpha</math> 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 <math>a=du/dT</math> by

<math>\alpha=\gamma^3 a=\frac{1}{\left(1-u^2/c^2\right)^{3/2}}\frac{du}{dT},</math>

where <math>u</math> is the instantaneous speed of the particle, <math>\gamma</math> the Lorentz factor, <math>c</math> is the speed of light, and <math>T</math> is the coordinate time. Solving for the equation of motion gives the desired formulas, which can be expressed in terms of coordinate time <math>T</math> as well as proper time <math>\tau</math>. For simplification, all initial values for time, location, and velocity can be set to 0, thus:<ref name=laue>Template:Cite book; First edition 1911, second expanded edition 1913, third expanded edition 1919.</ref><ref name=pauli>Template:Citation
In English: Template:Cite book</ref><ref>Template:Cite book</ref><ref>Template:Cite book</ref><ref>PhysicsFAQ (2016), "Relativistic rocket", see external links</ref>

Template:NumBlk}\\

& =c\tanh\left(\operatorname{arsinh}\frac{\alpha T}{c}\right)\\

X(T) & =\frac{c^{2}}{\alpha}\left(\sqrt{1+\left(\frac{\alpha T}{c}\right)^{2}}-1\right)\\

& =\frac{c^{2}}{\alpha}\left(\cosh\left(\operatorname{arsinh}\frac{\alpha T}{c}\right)-1\right)\\

c\tau(T) & =\frac{c^{2}}{\alpha}\ln\left(\sqrt{1+\left(\frac{\alpha T}{c}\right)^{2}}+\frac{\alpha T}{c}\right)\\

& =\frac{c^{2}}{\alpha}\operatorname{arsinh}\frac{\alpha T}{c}

\end{align}

& \begin{align}u(\tau) & =c\tanh\frac{\alpha\tau}{c}\\

\\ X(\tau) & =\frac{c^{2}}{\alpha}\left(\cosh\frac{\alpha\tau}{c}-1\right)\\ \\ cT(\tau) & =\frac{c^{2}}{\alpha}\sinh\frac{\alpha\tau}{c}\\ \\ \\ \end{align} \end{array}</math>|Template:EquationRef}}

This gives <math>\left(X+c^{2}/\alpha\right)^{2}-c^{2}T^{2}=c^{4}/\alpha^{2}</math>, which is a hyperbola in time T and the spatial location variable <math>X</math>. In this case, the accelerated object is located at <math>X=0</math> at time <math>T=0</math>. If instead there are initial values different from zero, the formulas for hyperbolic motion assume the form:<ref>Template:Cite book</ref><ref>Template:Cite journal</ref><ref name=fraundorf>Template:Cite arXiv</ref>

<math>{\scriptstyle \begin{array}{c|c}

\begin{align}u(T) & =\frac{u_{0}\gamma_{0}+\alpha T}{\sqrt{1+\left(\frac{u_{0}\gamma_{0}+\alpha T}{c}\right)^{2}}}\quad\\

& =c\tanh\left\{ \operatorname{arsinh}\left(\frac{u_{0}\gamma_{0}+\alpha T}{c}\right)\right\} \\

X(T) & =X_{0}+\frac{c^{2}}{\alpha}\left(\sqrt{1+\left(\frac{u_{0}\gamma_{0}+\alpha T}{c}\right)^{2}}-\gamma_{0}\right)\\

& =X_{0}+\frac{c^{2}}{\alpha}\left\{ \cosh\left[\operatorname{arsinh}\left(\frac{u_{0}\gamma_{0}+\alpha T}{c}\right)\right]-\gamma_{0}\right\} \\

c\tau(T) & =c\tau_{0}+\frac{c^{2}}{\alpha}\ln\left(\frac{\sqrt{c^{2}+\left(u_{0}\gamma_{0}+\alpha T\right){}^{2}}+u_{0}\gamma_{0}+\alpha T}{\left(c+u_{0}\right)\gamma_{0}}\right)\\

& =c\tau_{0}+\frac{c^{2}}{\alpha}\left\{ \operatorname{arsinh}\left(\frac{u_{0}\gamma_{0}+\alpha T}{c}\right)-\operatorname{artanh}\left(\frac{u_{0}}{c}\right)\right\}

\end{align}

& \begin{align}u(\tau) & =c\tanh\left\{ \operatorname{artanh}\left(\frac{u_{0}}{c}\right)+\frac{\alpha\tau}{c}\right\} \\

\\ X(\tau) & =X_{0}+\frac{c^{2}}{\alpha}\left\{ \cosh\left[\operatorname{artanh}\left(\frac{u_{0}}{c}\right)+\frac{\alpha\tau}{c}\right]-\gamma_{0}\right\} \\ \\ cT(\tau) & =cT_{0}+\frac{c^{2}}{\alpha}\left\{ \sinh\left[\operatorname{artanh}\left(\frac{u_{0}}{c}\right)+\frac{\alpha\tau}{c}\right]-\frac{u_{0}\gamma_{0}}{c}\right\} \end{align} \end{array}}</math>

RapidityEdit

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

<math>X=\frac{c^{2}}{\alpha}\left(\cosh\frac{\alpha\tau}{c}-1\right)</math>

can be subjected to a spatial shift of amount <math>c^2/\alpha</math>, thus

<math>X=\frac{c^{2}}{\alpha}\cosh\frac{\alpha\tau}{c}</math>,<ref>Pauli (1921), p. 628, used the notation <math>x^{4}=a\operatorname{ch}\frac{ct}{a}</math> where <math>a=\frac{c^{2}}{b}</math></ref>

by which the observer is at position <math>X=c^2/\alpha</math> at time <math>T=0</math>. Furthermore, by setting <math>x=c^2/\alpha</math> and introducing the rapidity <math>\eta=\operatorname{artanh}\frac{u}{c}=\frac{\alpha\tau}{c}</math>,<ref name=fraundorf /> the equations for hyperbolic motion reduce to<ref name=sommerfeld>Template:Cite journal</ref><ref name=sommerfeld2>Sommerfeld (1910), pp. 670-671 used the form <math>x=r\cos(\varphi)</math> and <math>l=r\sin(\varphi)</math> with the imaginary angle <math>i\psi</math> and imaginary time <math>l=ict</math>.</ref>

Template:NumBlk

with the hyperbola <math>X^{2}-c^{2}T^{2}=x^{2}</math>.

Charged particles in hyperbolic motionEdit

Born (1909),<ref name=born /> Sommerfeld (1910),<ref name=sommerfeld /> von Laue (1911),<ref name=laue /> Pauli (1921)<ref name=pauli /> also formulated the equations for the electromagnetic field of charged particles in hyperbolic motion.<ref name=Galeriu>Template:Cite journal</ref> This was extended by Hermann Bondi & Thomas Gold (1955)<ref name=Bondi>Template:Cite journal</ref> and Fulton & Rohrlich (1960)<ref name=Fulton>Template:Cite journal</ref><ref name=Rohrlich>Template:Cite journal</ref>

<math>\begin{align}E_{\rho'}'= & \frac{\left(8e/\alpha^{2}\right)\rho'z'}{\xi^{\prime3}}\\

E_{z'}'= & \frac{-\left(4e/\alpha^{2}\right)1/\alpha^{2}+t^{\prime2}+\rho^{\prime2}-z^{\prime2}}{\xi^{\prime3}}\\ E_{\varphi'}'= & H_{\varphi'}'=H_{z'}'=0\\ H_{\varphi'}'= & \frac{\left(8e/\alpha^{2}\right)\rho't'}{\xi^{\prime3}}\\ \xi'= & \sqrt{\left(1/\alpha^{2}+t^{\prime2}-\rho^{\prime2}-z^{\prime2}\right)^{2}+\left(2\rho'/\alpha\right)^{2}} \end{align}</math>

This is related to the controversially<ref name=Lyle>Template:Cite book</ref><ref name=Gron3>Template:Cite journal</ref> discussed question, whether charges in perpetual hyperbolic motion do radiate or not, and whether this is consistent with the equivalence principle – even though it is 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<ref name=Bondi /> and Fulton & Rohrlich<ref name=Fulton /><ref name=Rohrlich /> showed that radiation does indeed arise.

Proper reference frameEdit

File:Event-horizon-particle.svg

The light path through E marks the apparent event horizon of an observer P in hyperbolic motion.

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} In equation (Template:EquationNote) for hyperbolic motion, the expression <math>x</math> was constant, whereas the rapidity <math>\eta</math> was variable. However, as pointed out by Sommerfeld,<ref name=sommerfeld2 /> one can define <math>x</math> as a variable, while making <math>\eta</math> constant. This means, that the equations become transformations indicating the simultaneous rest shape of an accelerated body with hyperbolic coordinates <math>(x,y,z,\eta)</math> 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 possess an apparent event horizon, beyond which no signal can reach them.

Special conformal transformationEdit

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.<ref>Galeriu, Cǎlin (2019) "Electric charge in hyperbolic motion: the special conformal solution", European Journal of Physics 40(6) {{#invoke:doi|main}}</ref> 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).<ref name=kastrup /> It has the form

Using only one spatial dimension by <math>x^{\mu}=(t,x)</math>, and further simplifying by setting <math>x=0</math>, and using the acceleration <math>a^{\mu}=(0,-\alpha/2)</math>, it follows<ref name=Fulton2>Template:Cite journal</ref>

<math>T=\frac{t}{1-\frac{1}{4}\alpha{}^{2}t^{2}},\quad X=\frac{-\alpha t^{2}}{2\left(1-\frac{1}{4}\alpha{}^{2}t^{2}\right)}</math>

with the hyperbola <math>\left(X-1/\alpha\right)^{2}-T^{2}=1/\alpha^{2}</math>. It turns out that at <math>t=\pm(x+2/\alpha)</math> the time becomes singular, to which Fulton & Rohrlich & Witten<ref name=Fulton2 /> remark that one has to stay away from this limit, while Kastrup<ref name=kastrup>Template:Cite journal</ref> (who is very critical of the acceleration interpretation) remarks that this is one of the strange results of this interpretation.

ReferencesEdit

Template:Reflist

External linksEdit

Physics FAQ: The Relativistic Rocket
Mathpages: Accelerated Travels, Does A Uniformly Accelerating Charge Radiate?