12C (Dynamics and Relativity)

E_{1}, E_{2}

and their momentum angle be

θ

. Conservation of four-momentum and the null photon momenta give

M^{2} c^{2} = (p_{1} + p_{2})^{2} = 2 p_{1} \cdot p_{2} = \frac{2 E _{1} E _{2}}{c ^{2}} (1 - cos θ) .

Using

1 - cos θ = 2 sin^{2} (θ /2)

sin \frac{θ}{2} = \frac{M c ^{2}}{2 E _{1} E _{2}} .

Thus the opening angle depends only on the parent rest energy and the two photon energies.

b

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

p

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}} .

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

If it decays after proper time

τ

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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