# Hyperbolic motion (relativity)
**Hyperbolic motion** is the rectilinear motion of a body whose *proper acceleration*, the acceleration measured in its own instantaneous rest frame, is constant. In a spacetime diagram its worldline is a hyperbola rather than the parabola of Newtonian mechanics, and the name comes from that curve[1](https://www.lehman.edu/faculty/anchordoqui/517_5.pdf) • [2](https://phys.libretexts.org/Bookshelves/Relativity/Special_Relativity_%28Crowell%29/03%3A_Kinematics/3.05%3A_The_Velocity_and_Acceleration_Vectors). The study of this motion goes back more than a century, starting with [Max Born](https://www.edgepedia.org/max-born)[3](https://www.scielo.org.mx/scielo.php?pid=S0035-001X2022000600007&script=sci_arttext), and it underlies the Rindler coordinate system, the Rindler horizon, and the analysis of accelerated observers in both special and general relativity.
| Key fact | Detail |
|---|---|
| Defining property | Constant proper acceleration α, the acceleration felt by an accelerometer carried along the worldline[2](https://phys.libretexts.org/Bookshelves/Relativity/Special_Relativity_%28Crowell%29/03%3A_Kinematics/3.05%3A_The_Velocity_and_Acceleration_Vectors) |
| Worldline equation | x² − c²t² = c⁴/α², a hyperbola in the (t, x) plane[1](https://www.lehman.edu/faculty/anchordoqui/517_5.pdf) |
| Parametric form | t = (1/α) sinh(ατ), x = (1/α) cosh(ατ), with τ the proper time[4](https://preskill.caltech.edu/ph236c/Preskill_QFT_curved_spacetime.pdf) |
| Velocity | v = tanh(ατ), approaching the speed of light asymptotically but never reaching it[4](https://preskill.caltech.edu/ph236c/Preskill_QFT_curved_spacetime.pdf) |
| Horizon | A light signal emitted far enough behind the particle never catches up; the asymptote of the hyperbola is a Rindler horizon[1](https://www.lehman.edu/faculty/anchordoqui/517_5.pdf) • [2](https://phys.libretexts.org/Bookshelves/Relativity/Special_Relativity_%28Crowell%29/03%3A_Kinematics/3.05%3A_The_Velocity_and_Acceleration_Vectors) |
| Rigid frame | A family of such worldlines with acceleration varying as 1/ρ forms a Born-rigid Rindler frame; equal proper acceleration for all members is incompatible with Born rigidity[5](https://vitenti-physics.github.io/courses/electrodynamics/accel.html) |
## The worldline
[Proper acceleration](https://www.edgepedia.org/proper-acceleration) is the acceleration measured in the momentarily comoving inertial frame, what a passenger feels as weight or an onboard accelerometer reads[2](https://phys.libretexts.org/Bookshelves/Relativity/Special_Relativity_%28Crowell%29/03%3A_Kinematics/3.05%3A_The_Velocity_and_Acceleration_Vectors) • [6](https://math.ucr.edu/home/baez/physics/Relativity/SR/BellSpaceships/spaceship_puzzle.html). It is a Lorentz-invariant quantity: all inertial observers who watch the accelerating object agree on its value[4](https://preskill.caltech.edu/ph236c/Preskill_QFT_curved_spacetime.pdf). The coordinate acceleration measured in a lab frame is different and falls with speed; for motion along the acceleration direction it is g = g′/γ³, where g′ is the proper acceleration and γ the [Lorentz factor](https://www.edgepedia.org/lorentz-factor)[6](https://math.ucr.edu/home/baez/physics/Relativity/SR/BellSpaceships/spaceship_puzzle.html).
**Deriving the hyperbola.** Let τ be the particle's proper time and dots denote derivatives with respect to τ. The four-velocity has unit magnitude, and the four-acceleration has magnitude α in the instantaneous rest frame. Solving these differential equations with the particle momentarily at rest at t = 0 gives[2](https://phys.libretexts.org/Bookshelves/Relativity/Special_Relativity_%28Crowell%29/03%3A_Kinematics/3.05%3A_The_Velocity_and_Acceleration_Vectors)
\[ t(\tau) = \frac{1}{\alpha} \sinh(\alpha\tau), \qquad x(\tau) = \frac{1}{\alpha} \cosh(\alpha\tau), \]
or, eliminating τ,
\[ x^2 - c^2 t^2 = \frac{c^4}{\alpha^2}. \]
This is the equation of a hyperbola whose asymptotes are lightlike lines[1](https://www.lehman.edu/faculty/anchordoqui/517_5.pdf). The corresponding Newtonian calculation gives the parabola x = ½αt², and the relativistic curve coincides with it only in the early stages of acceleration, when speeds are small[2](https://phys.libretexts.org/Bookshelves/Relativity/Special_Relativity_%28Crowell%29/03%3A_Kinematics/3.05%3A_The_Velocity_and_Acceleration_Vectors) • [7](https://sites.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/general_relativity_pathway/uniform_acceleration.html).
The instantaneous velocity is v = tanh(ατ)[4](https://preskill.caltech.edu/ph236c/Preskill_QFT_curved_spacetime.pdf). Because relativistic velocities add through hyperbolic angles (rapidities), a small proper-time interval dτ changes the velocity by dv = α dτ in the particle's own frame, which is exactly the statement that α is the proper acceleration[4](https://preskill.caltech.edu/ph236c/Preskill_QFT_curved_spacetime.pdf). As τ grows, v approaches the speed of light but never reaches it, and the coordinate acceleration in the lab frame decreases toward zero[2](https://phys.libretexts.org/Bookshelves/Relativity/Special_Relativity_%28Crowell%29/03%3A_Kinematics/3.05%3A_The_Velocity_and_Acceleration_Vectors). In the same limit x increases exponentially with proper time, so that in principle very large distances can be covered within a human lifetime of ship time[2](https://phys.libretexts.org/Bookshelves/Relativity/Special_Relativity_%28Crowell%29/03%3A_Kinematics/3.05%3A_The_Velocity_and_Acceleration_Vectors).
## The Rindler horizon
The lightlike asymptote of the hyperbola has a physical meaning. A photon emitted a distance behind the particle at the moment it is momentarily at rest travels along that asymptote and cannot catch up[1](https://www.lehman.edu/faculty/anchordoqui/517_5.pdf). More generally, provided the rocket never stops accelerating, the entire region of spacetime beyond the asymptote is forever hidden from its passengers[2](https://phys.libretexts.org/Bookshelves/Relativity/Special_Relativity_%28Crowell%29/03%3A_Kinematics/3.05%3A_The_Velocity_and_Acceleration_Vectors). An observer in hyperbolic motion therefore has an event horizon, the Rindler horizon, even in flat spacetime with no gravity at all[2](https://phys.libretexts.org/Bookshelves/Relativity/Special_Relativity_%28Crowell%29/03%3A_Kinematics/3.05%3A_The_Velocity_and_Acceleration_Vectors). Unlike Newtonian uniform acceleration, which can cover the whole spacetime, hyperbolic motion is confined to a wedge bounded by two lightlike curves[7](https://sites.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/general_relativity_pathway/uniform_acceleration.html).
## Rigid frames and Bell's spaceship paradox
A family of hyperbolic worldlines can be used to build an accelerated reference frame. In [Rindler coordinates](https://www.edgepedia.org/rindler-coordinates) the worldlines satisfy X² − T² = ρ², and the proper acceleration of each observer is α = 1/ρ: observers farther back accelerate harder[5](https://vitenti-physics.github.io/courses/electrodynamics/accel.html). This spatial variation is the [Born rigidity](https://www.edgepedia.org/born-rigidity) condition, first derived by Max Born, which ensures that all points of an accelerated body pass from one instantaneous rest frame to the next while preserving their mutual distances[5](https://vitenti-physics.github.io/courses/electrodynamics/accel.html) • [8](https://doi.org/10.1088/2399-6528/acdc83). The resulting Rindler frame is Born-rigid, stationary, and has a horizon[5](https://vitenti-physics.github.io/courses/electrodynamics/accel.html).
**Equal acceleration is not rigid.** If instead every point of an extended body has the same proper acceleration, the rear observer must acquire a slightly larger velocity than the front one for the separation to stay fixed in the instantaneous rest frame, so Born rigidity fails; the Rindler arrangement with an acceleration gradient is the rigid alternative[5](https://vitenti-physics.github.io/courses/electrodynamics/accel.html). This is the kinematic core of [Bell's spaceship paradox](https://www.edgepedia.org/bells-spaceship-paradox), devised by Edmond Dewan and Michael Beran and popularized by John Bell: two rockets with equal accelerations in the lab frame keep a constant lab separation, yet their proper separation grows roughly in proportion to γ, so a string connecting them eventually snaps[6](https://math.ucr.edu/home/baez/physics/Relativity/SR/BellSpaceships/spaceship_puzzle.html). If the accelerations are retuned so the two hyperbolae share an asymptote, the rockets form a Rindler frame, measure themselves at rest relative to each other, and agree on simultaneity; this construction is closely tied to the reasoning behind Einstein's equivalence principle[6](https://math.ucr.edu/home/baez/physics/Relativity/SR/BellSpaceships/spaceship_puzzle.html).
## Extensions and uses
Wolfgang Rindler generalized the differential equations of hyperbolic motion to curved spacetime in *Physical Review* (volume 119, page 2082), obtaining them by generalizing the differential-geometric properties of a rectangular hyperbola in Minkowski spacetime. Solving them for de Sitter spacetime, he found that a radially uniformly accelerated particle ultimately moves at constant relative velocity through the cosmological substratum, that a critical galaxy on its line of motion can never be overtaken, that a light signal emitted at or after a certain critical time never catches the particle, and that a particle with available acceleration α passing beyond a certain proper distance, the α horizon, can no longer return to its origin; he also examined applications to intergalactic rocketry[9](https://doi.org/10.1103/PhysRev.119.2082).
Hyperbolic motion also serves as the standard model of an accelerated observer in quantum field theory in flat spacetime. A detector moving with constant proper acceleration along the hyperbola z² − t² = 1/a² is the setup in which the response of an accelerated detector to the vacuum is analyzed, the calculation behind the [Unruh effect](https://www.edgepedia.org/unruh-effect)[4](https://preskill.caltech.edu/ph236c/Preskill_QFT_curved_spacetime.pdf). Recent work has extended the kinematics to higher proper-time derivatives, giving explicit formulas for all kinematic quantities up to the fourth derivative (the snap) in 1+1 and 1+3 dimensional Minkowski spacetime[3](https://www.scielo.org.mx/scielo.php?pid=S0035-001X2022000600007&script=sci_arttext).
## References
1. [Special relativity lecture notes, L. Anchordoqui, Lehman College (rectilinear motion with constant proper acceleration)](https://www.lehman.edu/faculty/anchordoqui/517_5.pdf)
2. [Special Relativity, B. Crowell, Physics LibreTexts, section 3.5: The Velocity and Acceleration Vectors](https://phys.libretexts.org/Bookshelves/Relativity/Special_Relativity_%28Crowell%29/03%3A_Kinematics/3.05%3A_The_Velocity_and_Acceleration_Vectors)
3. [Relativistic hyperbolic motion and its higher order kinematic quantities, Mexican Journal of Physics / Revista Mexicana de Física](https://www.scielo.org.mx/scielo.php?pid=S0035-001X2022000600007&script=sci_arttext)
4. [J. Preskill, lecture notes on quantum field theory in curved spacetime, Caltech](https://preskill.caltech.edu/ph236c/Preskill_QFT_curved_spacetime.pdf)
5. [Accelerated Observers and Born Rigidity, S. Vitenti, physics lecture notes](https://vitenti-physics.github.io/courses/electrodynamics/accel.html)
6. [Bell's Spaceship Paradox, Usenet Physics FAQ (math.ucr.edu)](https://math.ucr.edu/home/baez/physics/Relativity/SR/BellSpaceships/spaceship_puzzle.html)
7. [Uniform Acceleration in a Minkowski Spacetime, J. D. Norton, University of Pittsburgh](https://sites.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/general_relativity_pathway/uniform_acceleration.html)
8. [Bending of electric field lines and light-ray trajectories in a static gravitational field, IOPscience](https://doi.org/10.1088/2399-6528/acdc83)
9. [W. Rindler, Hyperbolic Motion in Curved Space Time, Physical Review 119, 2082 (1960)](https://doi.org/10.1103/PhysRev.119.2082)
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*Topic: Encyclopedia › Physical world and mathematics › Physics › Relativity and gravitation › Special relativity › Relativistic dynamics › Relativistic force and acceleration*
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