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Tuesday, July 15, 2014

Dodelson - Problem 2.4

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(E2m2)
EE2m2dEdt=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+E2a2
and finally E1a
.

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