360
6 Special Theory of Relativity
This shows that small emission angles of photons in the lab system are
favored.
6.102 In Fig. 6.16, AB and AD represent the momentum vectors of the two photons
in the LS. BC is drawn parallel to AD so that ABC forms the momentum
triangle, that is
AB + BC = AC
Fig. 6.16 Locus of the tip of
the momentum vector of
γ -rays from π
0 decay is an
ellipse
From energy conservation
h ν 1 + h ν 2 = γ mc
2
/2(1 + β cos θ
∗ ) + (γ mc
2
/2)(1 + β cos(π − θ
∗ ))
= γ mc
2
= const
where we have used the fact that the angles of emission of the two photons
in the rest frame of π
0 , are supplementary.
Since momentum is given by p = h/c, it follows that AB + BC = constant,
which means that the locus of the tip of the momentum vector is an ellipse.
E = mc
2
/2γ (1 − β cos θ)
(14)
Compare this with the standard equation for ellipse
r = a(1 − ε
2 )/(1 − ε cos θ)
We find ε = β
a(1 − β
2 ) = mc
2
/2 γ
Or a = γ mc
2
/2
The larger the velocity of π
0 , the greater will be the eccentricity, ε.