Chapter 2 - The Smooth, Expanding Universe
Exercise 4
Find how the energy of a massive, nonrelativistic particle changes as the universe expands. Recall that in the massless case we used the fact that gμνPμPν=0. In this case, it is equal not to zero, but to −m2.
Solution
We consider the zeroth component of the geodesic equation to obtain d2xμdλ2=−Γμαβdxαdλdxβdλ
Recall that we define λ via the energy-momentum four vector: Pα=dxαdλ Thus we obtain ddλ=dx0dλddx0=Eddt This enables us to rewrite the zeroth component of the geodesic equation: EdEdt=−Γ0ijPiPj =−δija2PiPj Another important fact that we are employing is that for a massive particle we have gμνPμPν=−E2+δija2PiPj=−m2 and so we can, again, rewrite the geodesic equation to read EdEdt=−˙aa(E2−m2) EE2−m2dEdt=−1adadt where we can cancel the dt terms and integrate making use of a substitution u=m2+E2 to obtain 12ln(m2+E2)=−ln(a)+c m2+E2∝a−2 and finally E∝1a.
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