Fundamentals of Time and Relativity

Relativistic Force

A unique (frame independent) representation of the force accelerating a particle in any inertial frame is obtained by equating it formally with the force defined according to Newton’s second law in the inertial frame in which the particle is momentarily at rest.
Conservation of (rest) mass is assumed; this limits the usability of the concept of force to energies below the mass threshold.

We now seek a geometric generalization of Newton’s law of motion to describe accelerated motion in spacetime. The Newtonian relation between force and change of momentum is a natural starting point:

8.6

F : = \frac{d}{d τ} p = m \frac{d u}{d τ}

This equation of motion agrees with Newton’s second law in the frame where the particle is momentarily at rest.

The work done on the particle along the path $d s = u d τ = v d t$ by the force $F$ so defined is:

8.7

F \cdot d s = m \frac{d u}{d τ} \cdot u d τ = \frac{1}{2} m \frac{d u^{2}}{d τ} d τ

On account of (8.3) this gives the equation

8.8

F \cdot u = \frac{1}{2} m \frac{d u^{2}}{d τ} = m u^{0} \frac{d u^{0}}{d τ}

Let us now introduce the four-momentum $p = p^{μ} (τ) e_{μ}$ , with $p^{0} = γ m, p = m u$ . Then from (8.8) we have

8.9

\frac{F \cdot u}{u^{0}} = F \cdot v = \frac{d p^{0}}{d τ}

The equation of motion for a single particle can now be written in the Minkowski four-vector form

7.10

F = \frac{d p}{d τ} = m \frac{d u}{d τ} F \cdot u = 0

Here $u = u^{μ} (τ) e_{μ}$ is the four-velocity and $F = F^{μ} (τ) e_{μ}$ the relativistic four-force with temporal component $F^{0} = F \cdot u / u^{0} = F \cdot v$ and spatial component $F$ .

The force in the relativistic equation of motion (8.10) is (rest) mass preserving by the orthogonality condition $F \cdot u = 0$ . In high-energy collisions forces do not satisfy this condition.
The Lorentz Force of electromagnetism depends on the velocity of the charged particle on which it acts and is the prime example of a relativistic mass-preserving force.