Past exam of the mathematics course of the University of Cambridge / 2020 / ia / Paper 2 / 12C / b / Solution

In the parent's rest frame, momentum conservation makes the massive daughter and photon momenta equal and opposite, of magnitude

p

. The relativistic two-body decay formulas give

p = \frac{M ^{2} - m ^{2}}{2 M} c, E_{R} = \frac{M ^{2} + m ^{2}}{2 M} c^{2} .

Since

v = p c^{2} / E_{R}

and

r = m / M

,

v = c f (r), f (r) = \frac{1 - r ^{2}}{1 + r ^{2}} .

The daughter's Lorentz factor is

γ = \frac{E _{R}}{m c ^{2}} = \frac{1 + r ^{2}}{2 r} .

If it decays after proper time

τ

, time dilation gives

t = γ τ

, while uniform motion gives

x = v γ τ

. Therefore the event coordinates in the parent frame are

(c t, x) = (\frac{1 + r ^{2}}{2 r} c τ, \frac{1 - r ^{2}}{2 r} c τ) .

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