# Relativistic force

In special relativity, force is redefined so that Newton's second law remains consistent with the Lorentz invariance of physical law. The central object is the **four-force**, a four-vector defined as the rate of change of a particle's four-momentum with respect to the particle's proper time (time measured by the moving particle's own clock), which replaces the classical three-vector force.<sup>[@0](https://phys.libretexts.org/Bookshelves/University_Physics/Mechanics_and_Relativity_(Idema)/15%3A_Relativistic_Forces_and_Waves/15.01%3A_The_Force_Four-Vector)</sup> For a particle of constant invariant mass, the four-force equals the mass multiplied by the four-acceleration, the direct analog of F = ma.<sup>[3](https://scipp-legacy.pbsci.ucsc.edu/~haber/webpage/avector.pdf)</sup> In the limit of speeds far below light speed (v/c ≪ 1), the spatial components of the four-force law reduce to Newton's second law.<sup>[1](https://interactivetextbooks.tudelft.nl/classical-mechanics-and-special-relativity/_sources/content/ChXX1_RelDynColl.md)</sup>

| Key fact | Detail |
|---|---|
| Definition | Four-force is the derivative of four-momentum with respect to proper time<sup>[@0](https://phys.libretexts.org/Bookshelves/University_Physics/Mechanics_and_Relativity_(Idema)/15%3A_Relativistic_Forces_and_Waves/15.01%3A_The_Force_Four-Vector)</sup> |
| Constant-mass case | Four-force equals invariant mass times four-acceleration<sup>[3](https://scipp-legacy.pbsci.ucsc.edu/~haber/webpage/avector.pdf)</sup> |
| Newtonian limit | Spatial components reduce to Newton's second law when v/c ≪ 1<sup>[1](https://interactivetextbooks.tudelft.nl/classical-mechanics-and-special-relativity/_sources/content/ChXX1_RelDynColl.md)</sup> |
| Three-force transformation | Component parallel to the relative velocity is unchanged under a Lorentz boost; the perpendicular component scales as 1/γ<sup>[2](https://phys.libretexts.org/Bookshelves/Relativity/Special_Relativity_(Crowell)/04%3A_Dynamics/4.05%3A__Force)</sup> |
| Speed limit | A particle under sustained constant acceleration never reaches the speed of light in a finite time<sup>[3](https://scipp-legacy.pbsci.ucsc.edu/~haber/webpage/avector.pdf)</sup> |

## Three-force and four-force

The ordinary three-vector relation \(F = dp/dt\) between force and momentum remains valid in special relativity when momentum and energy are understood relativistically, with power defined as the time rate of change of relativistic energy.<sup>[3](https://scipp-legacy.pbsci.ucsc.edu/~haber/webpage/avector.pdf)</sup> However, a force defined with respect to coordinate time is not itself a four-vector. If force is required to be a four-vector, it must be differentiated with respect to proper time, giving the four-force as the derivative of four-momentum.<sup>[1](https://interactivetextbooks.tudelft.nl/classical-mechanics-and-special-relativity/_sources/content/ChXX1_RelDynColl.md)</sup> The force measured by an observer comoving with the particle, \(F_{o} = dp/dt\), coincides with the four-force only in that comoving frame.<sup>[2](https://phys.libretexts.org/Bookshelves/Relativity/Special_Relativity_(Crowell)/04%3A_Dynamics/4.05%3A__Force)</sup>

The time component of the four-force is F^0 = (γ/c)(dE/dt), where E is the particle's total relativistic energy, t is coordinate time, and γ is its [Lorentz factor](https://www.edgepedia.org/lorentz-factor) in that frame; this relation does not require heat exchange to vanish.<sup>[4](https://www.damtp.cam.ac.uk/user/tong/relativity/eight.pdf)</sup>

## Transformation of force between frames

Force components transform between inertial frames differently depending on their orientation relative to the motion. Under a Lorentz boost from the frame in which the particle is momentarily at rest, the force component parallel to the relative velocity is unchanged, while the perpendicular component scales as \(F_{\mathrm{o'},\perp} = F_{\mathrm{o},\perp}/\gamma\), where γ is the Lorentz factor.<sup>[2](https://phys.libretexts.org/Bookshelves/Relativity/Special_Relativity_(Crowell)/04%3A_Dynamics/4.05%3A__Force)</sup> Equivalently, if the correct expression for force is known in the frame where the particle is momentarily at rest, the relativistic force in another frame moving at constant velocity follows from a [Lorentz transformation](https://www.edgepedia.org/lorentz-transformation).<sup>[@0](https://phys.libretexts.org/Bookshelves/University_Physics/Mechanics_and_Relativity_(Idema)/15%3A_Relativistic_Forces_and_Waves/15.01%3A_The_Force_Four-Vector)</sup>

## Limits on speed under sustained force

Because relativistic momentum grows without bound as speed approaches the speed of light, a sustained force does not produce unbounded speed. A particle moving with constant acceleration never reaches the speed of light in a finite amount of time.<sup>[3](https://scipp-legacy.pbsci.ucsc.edu/~haber/webpage/avector.pdf)</sup> The applied work increasingly goes into raising the particle's energy rather than its speed, which is why the four-force, not the three-force, is the frame-independent measure of the interaction.

## Four-force in general relativity and electromagnetism

In general relativity, the relation between four-force and four-acceleration is unchanged, but the four-force is related to the four-momentum through a covariant derivative with respect to proper time.

## References

1. [Relativistic dynamics and collisions - TU Delft interactive textbook](https://interactivetextbooks.tudelft.nl/classical-mechanics-and-special-relativity/_sources/content/ChXX1_RelDynColl.md)
2. [Force - Physics LibreTexts (Special Relativity, Crowell)](https://phys.libretexts.org/Bookshelves/Relativity/Special_Relativity_(Crowell)/04%3A_Dynamics/4.05%3A__Force)
3. [Acceleration and Force in Special Relativity - UC Santa Cruz (Haber)](https://scipp-legacy.pbsci.ucsc.edu/~haber/webpage/avector.pdf)
4. [damtp.cam.ac.uk](https://www.damtp.cam.ac.uk/user/tong/relativity/eight.pdf)

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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: Sep 30, 2026 · Edited: Sep 30, 2026; Oct 15, 2026; Oct 25, 2026 · Last review: Sep 30, 2026*

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