General Relativity: An Introduction for Physicists · Problem 5.2

Special relativity revisited — Hobson, Efstathiou & Lasenby

Physics · lorentz-transformation-covariant-components

Problem

Show that, under a Lorentz transformation, the covariant components of a vector transform as

v'_\mu = \Lambda_\mu{}^\nu v_\nu

. Hence show explicitly in component form that, for two 4-vectors

\boldsymbol{v}

and

\boldsymbol{w}

, the scalar product

\boldsymbol{v} \cdot \boldsymbol{w}

is invariant under a Lorentz transformation.