Introduction to Electrodynamics · Problem 12.39

Electrodynamics and Relativity — David J. Griffiths

Physics · proper-acceleration

Problem

Define proper acceleration in the obvious way: αμ≡dημdτ=d2xμdτ2.(12.75)\alpha^\mu \equiv \frac{d\eta^\mu}{d\tau} = \frac{d^2 x^\mu}{d\tau^2}. \tag{12.75} (a) Find α0\alpha^0 and α\boldsymbol{\alpha} in terms of u\mathbf{u} and a\mathbf{a} (the ordinary acceleration). (b) Express αμαμ\alpha_\mu \alpha^\mu in terms of u\mathbf{u} and a\mathbf{a}. (c) Show that ημαμ=0\eta^\mu \alpha_\mu = 0. (d) Write the Minkowski version of Newton's second law, Eq. 12.68, in terms of αμ\alpha^\mu. Evaluate the invariant product KμημK^\mu \eta_\mu.
Intuitive explanation

Proper acceleration is the rate of change of four-velocity with respect to proper time—the acceleration felt by the moving object itself. The key insight is that the four-velocity ημ\eta^\mu has constant magnitude in spacetime (it's normalized), so its derivative must be orthogonal to it, just like how a vector moving on a circle stays perpendicular to the radius. This orthogonality and the mixing of time and space components under Lorentz transformations will structure all your answers.

Intuitive explanation

Proper acceleration is the rate of change of four-velocity with respect to proper time—the acceleration felt by the moving object itself. The key insight is that the four-velocity ημ\eta^\mu has constant magnitude in spacetime (it's normalized), so its derivative must be orthogonal to it, just like how a vector moving on a circle stays perpendicular to the radius. This orthogonality and the mixing of time and space components under Lorentz transformations will structure all your answers.

Related theory

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