# Proper acceleration

**Proper acceleration** is the physical acceleration experienced by an object, measured in its own instantaneous rest frame; it is what an accelerometer carried by the object reads. It is a Lorentz-invariant scalar in special relativity, defined as the magnitude of the four-acceleration, the derivative of the four-velocity with respect to the object's proper time.

| Key fact | Detail |
|---|---|
| Definition | Magnitude of the four-acceleration \( A^{\mu} = dU^{\mu}/d\tau \), the proper-time derivative of the four-velocity<sup>[1](https://farside.ph.utexas.edu/teaching/em/lectures/node115.html)</sup> |
| Invariant norm | In the particle's rest frame \( \vec{A} \cdot \vec{A} = |\mathbf{a}_0|^2 \), so the norm equals the square of the proper acceleration<sup>[2](https://home.ifa.hawaii.edu/users/kaiser/lectures/elements.pdf)</sup> |
| Orthogonality | The four-acceleration is always orthogonal to the four-velocity, \( U_{\mu} A^{\mu} = 0 \)<sup>[1](https://farside.ph.utexas.edu/teaching/em/lectures/node115.html)</sup> |
| Measurement | An accelerometer measures proper acceleration: one at rest on Earth's surface reads gravity, one in free fall reads zero<sup>[3](https://courses.physics.illinois.edu/phys371/sp2023/documents/week7.pdf)</sup> |
| Constant proper acceleration | Produces hyperbolic motion, with \( v = c \tanh(a_0 \tau / c) \) and a hyperbolic worldline in spacetime<sup>[4](https://www-thphys.physics.ox.ac.uk/people/AndreiStarinets/TUTORIALS/GR-HT-2021/solutions_general_relativity_B5_2021_set_1.pdf)</sup> |
| Relation to coordinate acceleration | For straight-line motion at constant proper acceleration \( a_0 \), \( \gamma^3 a = a_0 \)<sup>[5](https://users.physics.ox.ac.uk/~Steane/teaching/rel_A.pdf)</sup> |

## Definition through four-vectors

In special relativity a particle's worldline is parametrized by its proper time \( \tau \), the time read by a clock moving with the particle. The four-velocity is \( U^{\mu} = dx^{\mu}/d\tau \), and the four-acceleration is its derivative,<sup>[1](https://farside.ph.utexas.edu/teaching/em/lectures/node115.html)</sup>

\[ A^{\mu} = \frac{d^2 x^{\mu}}{d\tau^2} = \frac{dU^{\mu}}{d\tau}. \]

Because the four-velocity has constant magnitude, \( U^{\mu} U_{\mu} = -c^2 \), differentiating gives \( U_{\mu} A^{\mu} = 0 \): the four-acceleration is always orthogonal to the four-velocity. In the particle's rest frame, where \( U^{\mu} = (c, 0, 0, 0) \), the four-acceleration has only spatial components, \( A^{\mu} = (0, \mathbf{a}) \), and its norm is simply the square of the ordinary acceleration measured there.<sup>[1](https://farside.ph.utexas.edu/teaching/em/lectures/node115.html)</sup><sup> • </sup><sup>[6](https://galileoandeinstein.phys.virginia.edu/Elec_Mag/2022_Lectures/EM_62_SR_Formalism.html)</sup> That rest-frame magnitude is the proper acceleration.

The four-acceleration can be written in terms of the three-acceleration \( \mathbf{a} = d\mathbf{u}/dt \) measured in some inertial frame, and the components involve powers of the [Lorentz factor](https://www.edgepedia.org/lorentz-factor) \( \gamma \): terms scale with \( \gamma^2 \) and \( \gamma^4 \), which makes the four-acceleration an inconvenient object for direct computation even though its geometric meaning, the curvature of the worldline, is clean.<sup>[7](https://phys.libretexts.org/Bookshelves/University_Physics/Mechanics_and_Relativity_%28Idema%29/15%3A_Relativistic_Forces_and_Waves/15.02%3A_The_Four-Acceleration)</sup>

## What an accelerometer measures

An accelerometer is a tool designed to measure proper acceleration, the acceleration of a body in its own instantaneous rest frame; it does not measure coordinate acceleration in a fixed coordinate system. This is why an accelerometer at rest on the surface of the Earth reads an acceleration due to Earth's gravity, while accelerometers in free fall read zero.<sup>[3](https://courses.physics.illinois.edu/phys371/sp2023/documents/week7.pdf)</sup> In the terms of general relativity, an object falling solely under gravity follows a geodesic, a straight worldline in curved spacetime, and feels no acceleration at all; the accelerometer instead diagnoses forced, non-geodesic motion.<sup>[8](https://blackhole.org/physical-foundations/accelerated-observers-reference-frames/)</sup>

Consumer devices work on the same principle. A [MEMS accelerometer](https://www.edgepedia.org/mems-accelerometer) is a microscopic mass on a spring connected to a capacitor, where spring compression changes the electrical output; a phone lying on a table reads a magnitude of about 9.81 m/s² along the axis perpendicular to the table, and during a jump the graph shows the push-off, a free-fall interval where the raw reading falls to approximately 0 m/s² rather than the \( -9.8 \ \mathrm{m/s^2} \) of the body's acceleration, and the landing.<sup>[3](https://courses.physics.illinois.edu/phys371/sp2023/documents/week7.pdf)</sup><sup> • </sup><sup>[9](https://www.nasa.gov/wp-content/uploads/2015/09/aero_introductory_physics.pdf?emrc=dcb103)</sup><sup> • </sup><sup>[14](https://developer.android.com/develop/sensors-and-location/sensors/sensors_motion?authuser=56)</sup><sup> • </sup><sup>[15](https://developer.android.com/develop/sensors-and-location/sensors/sensors_motion?trk=article-ssr-frontend-pulse-little-text-block)</sup>

## Hyperbolic motion

Rectilinear motion with constant proper acceleration is called hyperbolic motion. Integrating the equations of motion with proper acceleration \( a_0 \), starting from rest, gives the coordinate velocity and position<sup>[4](https://www-thphys.physics.ox.ac.uk/people/AndreiStarinets/TUTORIALS/GR-HT-2021/solutions_general_relativity_B5_2021_set_1.pdf)</sup><sup> • </sup><sup>[7](https://phys.libretexts.org/Bookshelves/University_Physics/Mechanics_and_Relativity_%28Idema%29/15%3A_Relativistic_Forces_and_Waves/15.02%3A_The_Four-Acceleration)</sup>

\[ v(t) = \frac{a_0 t}{\sqrt{1 + a_0^2 t^2 / c^2}}, \qquad x(t) = \frac{c^2}{a_0} \left( \sqrt{1 + \frac{a_0^2 t^2}{c^2}} - 1 \right), \]

and, in terms of the proper time \( \tau \),

\[ v(\tau) = c \tanh \left( \frac{a_0 \tau}{c} \right), \qquad x = \frac{c^2}{a_0} \left( \cosh \frac{a_0 \tau}{c} - 1 \right), \qquad ct = \frac{c^2}{a_0} \sinh \frac{a_0 \tau}{c}. \]

The trajectory in the \( ct \)-\( x \) plane is a hyperbola, \( x^2 - c^2 t^2 = c^4 / a_0^2 \), whose branches are asymptotic to the light cone \( x = \pm ct \); this is the origin of the name.<sup>[4](https://www-thphys.physics.ox.ac.uk/people/AndreiStarinets/TUTORIALS/GR-HT-2021/solutions_general_relativity_B5_2021_set_1.pdf)</sup> The rapidity increases linearly with proper time, \( a_0 \tau / c \), while the coordinate velocity approaches \( c \) but never reaches it.<sup>[10](https://www.damtp.cam.ac.uk/user/tong/gr/grhtml/S1.html)</sup> For small \( t \) the formulas reduce to the Newtonian results \( v = a_0 t \) and \( x = \tfrac{1}{2} a_0 t^2 \).<sup>[7](https://phys.libretexts.org/Bookshelves/University_Physics/Mechanics_and_Relativity_%28Idema%29/15%3A_Relativistic_Forces_and_Waves/15.02%3A_The_Four-Acceleration)</sup>

**Coordinate acceleration falls off.** The acceleration measured in a fixed inertial frame is not constant even when the proper acceleration is. For straight-line motion at constant proper acceleration \( a_0 \), the two are related by \( \gamma^3 a = a_0 \), so the coordinate acceleration \( a \) decreases as \( \gamma \) grows.<sup>[5](https://users.physics.ox.ac.uk/~Steane/teaching/rel_A.pdf)</sup> A light signal emitted from far enough behind the accelerating particle, when the particle is momentarily at rest, cannot catch up with it: the light ray's graph is the asymptote of the particle's hyperbola.<sup>[11](https://www.lehman.edu/faculty/anchordoqui/517_5.pdf)</sup>

## Accelerated frames and horizons

The coordinates built on a uniformly accelerated observer's worldline, \( t = (1/\kappa) \sinh(\kappa \tau) \), \( x = (1/\kappa) \cosh(\kappa \tau) \) with \( \kappa \) the magnitude of the four-acceleration, cover only one quadrant of Minkowski spacetime, \( x > |t| \). The coordinate system becomes singular at \( x_1 = -1/\kappa \), where the metric coefficient \( g_{\hat{0}\hat{0}} \) vanishes; this boundary acts as a horizon for the accelerated observer.<sup>[12](https://pup-assets.imgix.net/onix/images/9780691207391/9780691207391.pdf?fm=pdf)</sup>

W. Rindler, a mathematician at Cornell University, extended the equations of uniform acceleration to curved spacetime in a 1960 *Physical Review* paper by generalizing the differential-geometric properties of the rectangular hyperbola in Minkowski spacetime. Solving them in de Sitter spacetime, relevant to the steady-state cosmology then under discussion, he found that a radially uniformly accelerated particle ultimately moves at constant relative velocity through the cosmological substratum, that it will never overtake a critical galaxy on its line of motion, and that a particle with a given available acceleration which passes beyond a certain proper distance, the horizon, can no longer return to its place of origin.<sup>[13](https://doi.org/10.1103/PhysRev.119.2082)</sup>

## Relativistic rocketry

Constant proper acceleration is the natural regime for a crewed relativistic rocket, because the passengers feel a steady, tolerable thrust rather than a changing one. Rindler's 1960 paper examined possible applications to intergalactic rocketry on this basis.<sup>[13](https://doi.org/10.1103/PhysRev.119.2082)</sup>

## References

1. [4-velocity and 4-acceleration, R. Fitzpatrick, UT Austin electromagnetism lectures](https://farside.ph.utexas.edu/teaching/em/lectures/node115.html)
2. [Elements of Astrophysics, N. Kaiser, University of Hawaii lecture notes](https://home.ifa.hawaii.edu/users/kaiser/lectures/elements.pdf)
3. [Back to basics ~ accelerometer, Phys 371, University of Illinois](https://courses.physics.illinois.edu/phys371/sp2023/documents/week7.pdf)
4. [Solutions, General Relativity B5 2021 Set 1, Oxford Theoretical Physics](https://www-thphys.physics.ox.ac.uk/people/AndreiStarinets/TUTORIALS/GR-HT-2021/solutions_general_relativity_B5_2021_set_1.pdf)
5. [Relativity lecture notes, A. Steane, Oxford University](https://users.physics.ox.ac.uk/~Steane/teaching/rel_A.pdf)
6. [SR Formalism, University of Virginia](https://galileoandeinstein.phys.virginia.edu/Elec_Mag/2022_Lectures/EM_62_SR_Formalism.html)
7. [15.2: The Four-Acceleration, T. Idema, Mechanics and Relativity, TU Delft via Physics LibreTexts](https://phys.libretexts.org/Bookshelves/University_Physics/Mechanics_and_Relativity_%28Idema%29/15%3A_Relativistic_Forces_and_Waves/15.02%3A_The_Four-Acceleration)
8. [Accelerated Observers and Reference Frames, blackhole.org](https://blackhole.org/physical-foundations/accelerated-observers-reference-frames/)
9. [WITH YOU WHEN YOU FLY: Aeronautical Physics, NASA](https://www.nasa.gov/wp-content/uploads/2015/09/aero_introductory_physics.pdf?emrc=dcb103)
10. [Geodesics in Spacetime, D. Tong, General Relativity, University of Cambridge](https://www.damtp.cam.ac.uk/user/tong/gr/grhtml/S1.html)
11. [Special Relativity lecture notes, L. Anchordoqui, Lehman College CUNY](https://www.lehman.edu/faculty/anchordoqui/517_5.pdf)
12. [Relativity and Cosmology, Thorne, Blandford, and collaborators, Princeton University Press](https://pup-assets.imgix.net/onix/images/9780691207391/9780691207391.pdf?fm=pdf)
13. [W. Rindler (1960). Hyperbolic Motion in Curved Space Time. Physical Review 119, 2082.](https://doi.org/10.1103/PhysRev.119.2082)
14. [Motion sensors, Sensors and location, Android Developers](https://developer.android.com/develop/sensors-and-location/sensors/sensors_motion?authuser=56)
15. [Motion sensors, Sensors and location, Android Developers](https://developer.android.com/develop/sensors-and-location/sensors/sensors_motion?trk=article-ssr-frontend-pulse-little-text-block)

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*Topic: Encyclopedia › Physical world and mathematics › Physics › Relativity and gravitation › Special relativity › Relativistic dynamics › Relativistic force and acceleration*

*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

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