# Four-force
The **four-force** (also called the Minkowski force) is the Lorentz-covariant generalization of force in relativity: the derivative of a particle's four-momentum with respect to its own proper time τ, written \( K^{\mu} = dp^{\mu}/d\tau \).[1](https://notes.amittai.studio/relativity/relativistic-dynamics/four-momentum-force-and-accelerated-motion) Because proper time is a Lorentz scalar, the derivative with respect to it turns any four-vector into another four-vector, so \( K^{\mu} \) transforms the same way as the four-momentum \( p^{\mu} \) does.[2](https://www.youtube.com/watch?v=37jTbJssmwY)
| Key fact | Detail |
|---|---|
| Definition | \( K^{\mu} = dp^{\mu}/d\tau \), the proper-time derivative of the four-momentum; also called the Minkowski force[1](https://notes.amittai.studio/relativity/relativistic-dynamics/four-momentum-force-and-accelerated-motion) |
| Components | \( K^{\mu} = \left( \dfrac{\mathbf{f} \cdot \mathbf{v}/c}{\sqrt{1 - v^2/c^2}}, \dfrac{\mathbf{f}}{\sqrt{1 - v^2/c^2}} \right) \), with \( \mathbf{f} = d\mathbf{p}/dt \) the ordinary three-force[3](https://www.feynmanlectures.caltech.edu/II_26.html) |
| Orthogonality | For constant rest mass, \( K \cdot U = 0 \): the four-force is Minkowski-orthogonal to the four-velocity[1](https://notes.amittai.studio/relativity/relativistic-dynamics/four-momentum-force-and-accelerated-motion) |
| Relation to acceleration | \( K^{\mu} = m A^{\mu} \) with \( A^{\mu} = dU^{\mu}/d\tau \); the invariant length satisfies \( A \cdot A = \alpha^2 \), where \( \alpha \) is the proper acceleration an onboard accelerometer reads[1](https://notes.amittai.studio/relativity/relativistic-dynamics/four-momentum-force-and-accelerated-motion) |
| Electromagnetic case | The Lorentz force takes the covariant form \( f_{\mu} = q\, u_{\nu} F_{\mu\nu} \), with \( F_{\mu\nu} \) the electromagnetic field tensor[3](https://www.feynmanlectures.caltech.edu/II_26.html) |
| Direction dependence | A given three-force produces \( \gamma_u^2 \) times less acceleration along the motion than across it: \( f_{\parallel} = \gamma_u^3 m a_{\parallel} \), \( f_{\perp} = \gamma_u m a_{\perp} \)[1](https://notes.amittai.studio/relativity/relativistic-dynamics/four-momentum-force-and-accelerated-motion) |
## Why proper time
A naive four-component object built from the ordinary force, such as \( (\mathbf{f} \cdot \mathbf{v}/c, F_x, F_y, F_z) \), is not a four-vector, because the coordinate-time derivative \( d/dt \) requires the choice of a particular frame for measuring \( t \).[3](https://www.feynmanlectures.caltech.edu/II_26.html) The fix is the same one that turns velocity into the four-velocity: multiply by \( 1/\sqrt{1 - v^2/c^2} \), which is equivalent to differentiating with respect to the invariant interval \( s \), the proper time along the particle's path.[3](https://www.feynmanlectures.caltech.edu/II_26.html) The resulting equation of motion is \( dp_{\mu}/ds = f_{\mu} \), or \( f_{\mu} = m_0\, d^2 x_{\mu}/ds^2 \), a form reminiscent of \( F = ma \) but containing relativistic mechanics, which differs from Newton's law at high velocities.[3](https://www.feynmanlectures.caltech.edu/II_26.html)
**Components.** The space part of the four-force is the ordinary three-force multiplied by the [Lorentz factor](https://www.edgepedia.org/lorentz-factor) \( \gamma_u \), since \( d/d\tau = \gamma_u\, d/dt \); the time part is the rate at which the force does work, \( K^0 = (\gamma_u/c)\, dE/dt = (\gamma_u/c)\, \mathbf{f} \cdot \mathbf{u} \).[1](https://notes.amittai.studio/relativity/relativistic-dynamics/four-momentum-force-and-accelerated-motion) Dividing the four-force by \( \gamma \) recovers the classical force; this check reproduces the \( x \)- and \( y \)-components of the classical [Lorentz force](https://www.edgepedia.org/lorentz-force) from the covariant expression.[4](https://courses.physics.illinois.edu/phys436/sp2011/p436web/p436web/special_rel4-notes.pdf)
## Orthogonality and proper acceleration
For a particle of constant rest mass, the invariant \( p \cdot p = -m^2 c^2 \) is fixed along the worldline. Differentiating with respect to proper time gives \( 2 p \cdot K = 0 \), and since \( p = m U \), the four-force satisfies \( K \cdot U = 0 \).[1](https://notes.amittai.studio/relativity/relativistic-dynamics/four-momentum-force-and-accelerated-motion) In covariant terms, a force that merely redirects a particle while leaving its rest mass unaltered can never possess a component in the direction of that particle's four-velocity.[1](https://notes.amittai.studio/relativity/relativistic-dynamics/four-momentum-force-and-accelerated-motion) The same relation follows directly by differentiating the constant norm \( U \cdot U = -c^2 \), which makes the four-acceleration \( A^{\mu} = dU^{\mu}/d\tau \) always orthogonal to the four-velocity; in the particle's own instantaneous rest frame the four-acceleration therefore has only spatial components.[5](https://galileoandeinstein.phys.virginia.edu/Elec_Mag/2022_Lectures/EM_62_SR_Formalism.html)
In that instantaneous rest frame \( U^{\mu} = (c, 0) \), so orthogonality forces the time component of \( A^{\mu} \) to vanish and the four-acceleration is purely spatial, \( A^{\mu} = (0, a_{\mathrm{IRF}}) \). Its invariant length is the square of the proper acceleration \( \alpha \), the quantity an accelerometer carried by the particle reads: \( A \cdot A = \alpha^2 \).[1](https://notes.amittai.studio/relativity/relativistic-dynamics/four-momentum-force-and-accelerated-motion) In Minkowski geometry this orthogonality tilts the velocity-acceleration pair away from a Euclidean right angle when viewed from another frame.[1](https://notes.amittai.studio/relativity/relativistic-dynamics/four-momentum-force-and-accelerated-motion)
## The Lorentz force in covariant form
The classical Lorentz force \( \mathbf{F} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B}) \) is already compatible with relativity, and it can be written as a four-vector law. The electric and magnetic fields assemble into the antisymmetric field tensor \( F_{\mu\nu} \), and the four-force on a charge is \( f_{\mu} = q\, u_{\nu} F_{\mu\nu} \).[3](https://www.feynmanlectures.caltech.edu/II_26.html) The time component of the electromagnetic four-force is \( f_t = q\, \mathbf{v} \cdot \mathbf{E}/(c\sqrt{1 - v^2/c^2}) \), the rate of doing work by the electric field; the magnetic field does no work.[3](https://www.feynmanlectures.caltech.edu/II_26.html)
## What the four-force explains about acceleration
The ordinary three-force is not generally parallel to the acceleration. Differentiating \( \mathbf{p} = \gamma_u m \mathbf{u} \) gives \( \mathbf{f} = \gamma_u m \mathbf{a} + \gamma_u^3 m (\mathbf{u} \cdot \mathbf{a})\, \mathbf{u}/c^2 \), so the two are parallel only when the acceleration is purely along or purely across the velocity.[1](https://notes.amittai.studio/relativity/relativistic-dynamics/four-momentum-force-and-accelerated-motion) The two responses are \( f_{\parallel} = \gamma_u^3 m a_{\parallel} \) and \( f_{\perp} = \gamma_u m a_{\perp} \); these coefficients were once termed the longitudinal and transverse mass, though physicists now interpret the situation as a single rest mass \( m \), with the directional variation attributed to the kinematic factors.[1](https://notes.amittai.studio/relativity/relativistic-dynamics/four-momentum-force-and-accelerated-motion)
This asymmetry has practical consequences for accelerators. As $u \to c$, a linear accelerator must work ever harder to raise a particle's speed because of the longitudinal stiffness $\gamma_u^3 m$: pushing the particle forward yields less and less acceleration, by a factor of $\gamma_u^{-3}$, whereas deflecting it sideways magnetically costs far less effort, so storage rings use magnets to guide their energetic beams.[1](https://notes.amittai.studio/relativity/relativistic-dynamics/four-momentum-force-and-accelerated-motion) Rest mass is invariant; what grows without bound is the momentum, so a finite force yields a vanishing speed gain, and the speed limit is kinematic rather than a mass increase.[6](https://www.physics2u.com/vault/relativistic-force-and-acceleration.html)
## References
1. [Four-Momentum, Four-Force, and Accelerated Motion, study. (Amittai)](https://notes.amittai.studio/relativity/relativistic-dynamics/four-momentum-force-and-accelerated-motion)
2. [Force in Special Relativity | Four-Force, Pretty Much Physics (YouTube)](https://www.youtube.com/watch?v=37jTbJssmwY)
3. [The Feynman Lectures on Physics, Vol. II Ch. 26: Lorentz Transformations of the Fields](https://www.feynmanlectures.caltech.edu/II_26.html)
4. [Special Relativity lecture notes, Phys 436, University of Illinois](https://courses.physics.illinois.edu/phys436/sp2011/p436web/p436web/special_rel4-notes.pdf)
5. [SR Formalism, Galileo and Einstein (University of Virginia)](https://galileoandeinstein.phys.virginia.edu/Elec_Mag/2022_Lectures/EM_62_SR_Formalism.html)
6. [Relativistic Force and Four-Acceleration, physics2u](https://www.physics2u.com/vault/relativistic-force-and-acceleration.html)
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*Topic: Encyclopedia › Physical world and mathematics › Physics › Classical physics › Electromagnetism › Electric and magnetic fields › Maxwell's equations and field formulations › Covariant formulation of electromagnetism › Relativistic dynamics of charged particles*
*Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Sep 30, 2026; Oct 15, 2026 · Last review: Oct 15, 2026*
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