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I am reading the book Gravitational waves vol 1 by Maggiore. On page 190 he writes down an equation for the magnitude of the velocity:

$$\frac{d|\vec{u}|}{dt}=-\frac{\dot{a}}{a}|\vec{u}|\tag{$*$}$$

where $|\vec{u}|^2= g_{ij} u^i u^j$. He is using the metric:

$$ds^2=-c^2 dt^2+a^2(t)\left[\frac{dr^2}{1-kk^2}+r^2d\theta^2+r^2\sin^2\theta d\phi^2\right]$$

I want to derive eq. $(*)$.

I start from the geodesic equation:

$$\frac{du^\mu}{dt}+\Gamma^\mu_{~\alpha\beta}u^\alpha u^\beta=0.$$

Then I evaluate this for $\mu=0$ and look up the Christoffel symbols which do not vanish. I end up with:

$$\frac{du^0}{dt}=-\frac{\dot{a}}{a}a^2\left(\frac{(u^1)^2}{1-kr^2}+r^2 (u^2)^2+r^2\sin^2\theta (u^3)^2\right)=-\frac{\dot{a}}{a} g_{ij}u^iu^j=-\frac{\dot{a}}{a}|\vec{u}|^2\tag{$**$}$$

Now I am using

$$0=g_{\mu\nu}u^\mu u^\nu=-(u^0)^2+g_{ij}u^i u^j\Rightarrow u^0=|\vec{u}|$$

The eq. $(**)$ is:

$$\frac{d|\vec{u}|}{dt}=-\frac{\dot{a}}{a}|\vec{u}|^2$$

This is almost eq. $(*)$ however I get $|\vec{u}|^2$ on the RHS and I don't see why I should get $|\vec{u}|$?

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The issue is subtle!

You're using an incorrect expression for the geodesic equation: the derivative of the velocity should be written with respect to the proper time $\tau$, not to (the non-covariant) coordinate time $t$.

In fact, $u^0 = \mathrm{d} t / \mathrm{d} \tau$, so you precisely get a $u^0 = \left|\vec{u}\right|$ factor which cancels the square!

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