The chapter derives only for a four-force that leaves the particle's rest mass unchanged

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Special relativity: spacetime and kinematics · Relativistic dynamics and fields in flat spacetime

The Geometry of Gravity, Ch Minkowski spacetime and relativistic dynamics, completed

Problem

The chapter derives fμuμ=0f^\mu u_\mu=0 only for a four-force that leaves the particle's rest mass mm unchanged. A rocket burning fuel does not: in its own instantaneous rest frame, its rest mass decreases, dm/dτ<0dm/d\tau<0 (an idealised, continuous burn), while some four-force fμ=dpμ/dτf^\mu=dp^\mu/d\tau acts on it. ((a)) Using pμ=muμp^\mu=mu^\mu with m=m(τ)m=m(\tau), show that fμ=dmdτ uμ+m aμ.f^\mu=\frac{dm}{d\tau}\,u^\mu+m\,a^\mu. ((b)) Contract this with uμu_\mu to show that fμuμ=−c2 dm/dτf^\mu u_\mu=-c^2\,dm/d\tau, and say which of the two terms in (a) changes the rest mass and which changes only the direction of uμu^\mu. ((c)) Confirm that the result of (b) reduces to the chapter's fμuμ=0f^\mu u_\mu=0 when dm/dτ=0dm/d\tau=0. ((d)) For a burn rate dm/dτ=−2.0 g s−1dm/d\tau=-2.0\ \mathrm{g\,s^{-1}}, evaluate fμuμf^\mu u_\mu in SI units and state its sign, and find the time component f0f^0 in the rocket's instantaneous rest frame. ((e)) The rocket starts from rest with rest mass m0m_0 and moves along +x+x. At every instant, in its instantaneous rest frame, the spatial four-force points along +x+x and has size c ∣dm/dτ∣c\,|dm/d\tau|, as it would if the exhaust were pure light. Find the rapidity φ\varphi of the rocket, relative to its starting frame, when its rest mass has fallen to mm, and its speed, as a fraction of cc, when half the starting rest mass remains.

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