General Relativity: An Introduction for Physicists · Problem 5.15

Special relativity revisited — Hobson, Efstathiou & Lasenby

Physics · pure-4-force-3-force-relation

Problem

For a pure 4-force

\boldsymbol{f}

acting on a particle of rest mass

m_0

, show that the corresponding 3-force

\vec{f}

satisfies

\vec{f} = \gamma_u m_0 \vec{a} + \frac{\vec{f}\cdot\vec{u}}{c^2}\vec{u}.

Hence show that

\vec{a}

is only parallel to

\vec{f}

when

\vec{f}

is either parallel or orthogonal to

\vec{u}

. Show further that, in these two cases, one has

\vec{f} = \gamma_u^3 m_0 \vec{a}

and

\vec{f} = \gamma_u m_0 \vec{a}

respectively.