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" (German: Krümmungshyperbel). In the context of Born rigidity, Max Born (1909) subsequently coined the term "hyperbolic motion" (German: Hyperbelbewegung) 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" (German: hyperbolisch beschleunigtes Bezugsystem). Born's formulas were simplified and extended by Arnold Sommerfeld (1910). For early reviews see the textbooks by Max von Laue (1911, 1921) or Wolfgang Pauli (1921). See also Galeriu (2015) or Gourgoulhon (2013), and Acceleration (special relativity)#History.
Worldline The proper acceleration α {\displaystyle \alpha } 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 a = d u / d T {\displaystyle a=du/dT} by
α = γ 3 a = 1 ( 1 − u 2 / c 2 ) 3 / 2 d u d T , {\displaystyle \alpha =\gamma ^{3}a={\frac {1}{\left(1-u^{2}/c^{2}\right)^{3/2}}}{\frac {du}{dT}},}
where u {\displaystyle u} is the instantaneous speed of the particle, γ {\displaystyle \gamma } the Lorentz factor, c {\displaystyle c} is the speed of light, and T {\displaystyle T} is the coordinate time. Solving for the equation of motion gives the desired formulas, which can be expressed in terms of coordinate time T {\displaystyle T} as well as proper time τ {\displaystyle \tau } . For simplification, all initial values for time, location, and velocity can be set to 0, thus:
This gives ( X + c 2 / α ) 2 − c 2 T 2 = c 4 / α 2 {\displaystyle \left(X+c^{2}/\alpha \right)^{2}-c^{2}T^{2}=c^{4}/\alpha ^{2}} , which is a hyperbola in time T and the spatial location variable X {\displaystyle X} . In this case, the accelerated object is located at X = 0 {\displaystyle X=0} at time T = 0 {\displaystyle T=0} . If instead there are initial values different from zero, the formulas for hyperbolic motion assume the form:
u ( T ) = u 0 γ 0 + α T 1 + ( u 0 γ 0 + α T c ) 2 = c tanh { arsinh ( u 0 γ 0 + α T c ) } X ( T ) = X 0 + c 2 α ( 1 + ( u 0 γ 0 + α T c ) 2 − γ 0 ) = X 0 + c 2 α { cosh [ arsinh ( u 0 γ 0 + α T c ) ] − γ 0 } c τ ( T ) = c τ 0 + c 2 α ln ( c 2 + ( u 0 γ 0 + α T )
2 + u 0 γ 0 + α T ( c + u 0 ) γ 0 ) = c τ 0 + c 2 α { arsinh ( u 0 γ 0 + α T c ) − artanh ( u 0 c ) } u ( τ ) = c tanh { artanh ( u 0 c ) + α τ c } X ( τ ) = X 0 + c 2 α { cosh [ artanh ( u 0 c ) + α τ c ] − γ 0 } c T ( τ ) = c T 0 + c 2 α { sinh [ artanh ( u 0 c ) + α τ c ] − u 0 γ 0 c } {\displaystyle {\scriptstyle {\begin{array}{c|c}{\begin{aligned}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{aligned}}&{\begin{aligned}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{aligned}}\end{array}}}}
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
X = c 2 α ( cosh α τ c − 1 ) {\displaystyle X={\frac {c^{2}}{\alpha }}\left(\cosh {\frac {\alpha \tau }{c}}-1\right)}
can be subjected to a spatial shift of amount c 2 / α {\displaystyle c^{2}/\alpha } , thus
X = c 2 α cosh α τ c {\displaystyle X={\frac {c^{2}}{\alpha }}\cosh {\frac {\alpha \tau }{c}}} , by which the observer is at position X = c 2 / α {\displaystyle X=c^{2}/\alpha } at time T = 0 {\displaystyle T=0} . Furthermore, by setting x = c 2 / α {\displaystyle x=c^{2}/\alpha } and introducing the rapidity η = artanh u c = α τ c {\displaystyle \eta =\operatorname {artanh} {\frac {u}{c}}={\frac {\alpha \tau }{c}}} , the equations for hyperbolic motion reduce to
with the hyperbola X 2 − c 2 T 2 = x 2 {\displaystyle X^{2}-c^{2}T^{2}=x^{2}} .
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)
E ρ ′ ′ = ( 8 e / α 2 ) ρ ′ z ′ ξ ′ 3 E z ′ ′ = − ( 4 e / α 2 ) 1 / α 2 + t ′ 2 + ρ ′ 2 − z ′ 2 ξ ′ 3 E φ ′ ′ = H φ ′ ′ = H z ′ ′ = 0 H φ ′ ′ = ( 8 e / α 2 ) ρ ′ t ′ ξ ′ 3