General Relativity: An Introduction for Physicists · Problem 1.15

The spacetime of special relativity — Hobson, Efstathiou & Lasenby

Physics · uniform-acceleration-proper-time

Problem

A spaceship travels at a variable speed u(t)u(t) in some inertial frame SS. An observer on the spaceship measures its acceleration to be f(τ)f(\tau), where τ\tau is the proper time. If at τ=0\tau = 0 the spaceship has a speed u0u_0 in SS show that u(τ)−u01−u(τ)u0/c2=ctanh⁡ψ(τ),\frac{u(\tau) - u_0}{1 - u(\tau)u_0/c^2} = c\tanh\psi(\tau), where cψ(τ)=∫0τf(τ′) dτ′c\psi(\tau) = \int_0^\tau f(\tau')\,d\tau'. Show that the velocity of the spaceship can never reach cc.

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