A particle of rest mass

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

The Geometry of Gravity, Ch Tensors in flat spacetime: index notation and covariant field equations

Problem

A particle of rest mass MM, at rest in frame SS, decays into two photons travelling along +x+x and −x-x, each carrying half the rest energy: p1μ=(Mc/2, Mc/2, 0, 0)p_1^\mu = (Mc/2,\ Mc/2,\ 0,\ 0) and p2μ=(Mc/2, −Mc/2, 0, 0)p_2^\mu = (Mc/2,\ -Mc/2,\ 0,\ 0). In SS the parent's four-momentum is pμ=(Mc,0,0,0)p^\mu = (Mc, 0, 0, 0), and p1μ+p2μ=pμp_1^\mu + p_2^\mu = p^\mu as required by conservation. A second inertial frame S′S' moves at v=0.6cv=0.6c along xx relative to SS (γ=1.25\gamma = 1.25). ((a)) Apply the tensor transformation law for a type (1,0)(1,0) object (the Lorentz transformation) to p1μp_1^\mu, p2μp_2^\mu and pμp^\mu separately, to find their components in S′S', in units of McMc. ((b)) Verify that p1′μ+p2′μ=p′μp_1'^\mu + p_2'^\mu = p'^\mu still holds in S′S', and explain why checking conservation in SS alone was already enough to guarantee this, without redoing the calculation.

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