Hyperbolic motion (relativity)

[[Image:HyperbolicMotion.svg|thumb|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 ().]]

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. It can be visualized when graphed on a Minkowski diagram, whose position coordinate represents a suitable inertial frame. This type of 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.

History

Hermann Minkowski (1908) 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 (1909) 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 (1910). and Acceleration (special relativity)#History.

Worldline

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:

}\\

& =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>|

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:

:<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>

Rapidity

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>,

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>,

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

Charged particles in hyperbolic motion

Born (1909), This was extended by Hermann Bondi & Thomas Gold (1955) and Fulton & Rohrlich (1960)

:<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 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 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).

:<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 (who is very critical of the acceleration interpretation) remarks that this is one of the strange results of this interpretation.

References

External links

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

History

Worldline

Rapidity

Charged particles in hyperbolic motion

References

Further reading

External links