(a) By conservation of four-momentum: ( (m,0,0,0) = (E_\gamma, E_\gamma,0,0) + (E_\gamma, -E_\gamma,0,0) ) in natural units ( c=1 ). This gives ( 2E_\gamma = m ), so ( E_\gamma = m/2 ). Restoring ( c ): ( E_\gamma = \frac{m c^2}{2} ).

The are arranged by difficulty and topic, each with a complete, self-contained solution that explains not only the mathematics but also the physical reasoning. No steps are skipped.

(b) In the lab frame, boost the photon four-momenta. For a photon emitted at angle ( \theta'=0 ) in the rest frame, the lab energy is ( E = \gamma E' (1+\beta) ). The second photon (emitted at ( \theta'=\pi ) in rest frame) has lab energy ( E = \gamma E' (1-\beta) ). Their directions are not opposite in the lab frame. Using the aberration formula, the lab angle between the two photons is found to be ( 2 \arctan\left(\frac{1}{\gamma\beta}\right) ) (for ( \beta = v/c )). Full derivation shows that for ( v\to c ), the angle approaches ( 0 ) (both photons forward), consistent with beaming.

Special relativity (150 problems) builds fluency with Lorentz transformations, four-vectors, and relativistic dynamics. General relativity (150 problems) starts from the equivalence principle and walks through curved spacetime, geodesics, Einstein’s equations, and key applications.

(The complete solution spans half a page with all intermediate algebra and a spacetime diagram.) “Relativity is often taught as a collection of astonishing results — time slows down, space contracts, black holes trap light. Yet without solving problems, these insights remain abstract. This book bridges the gap between conceptual understanding and technical mastery.