# Trying to understand the weak gravitational field metric (3)

I've worked through Carroll's explanation of the Newtonian limit as far as $h_{00}=-2\phi$ (page 106 - Lecture Notes on General Relativity). As he's previously stated that $\left|h_{\mu\upsilon}\right|\ll1$ , I'm assuming $\phi$ should be tiny, and I'm therefore assuming there's a $\frac{1}{c^{2}}$ term missing from the value of $h_{00}$, which should read $h_{00}=\frac{-1}{c^{2}}2\phi$ (as it does in Foster and Nightingale). However, I can't see how the $\frac{1}{c^{2}}$ term arises in his argument. He starts off with the geodesic equation $\frac{d^{2}x^{\mu}}{d\lambda^{2}}+\Gamma_{jk}^{\mu}\frac{dx^{j}}{d\lambda}\frac{dx^{k}}{d\lambda}=0$ , moves on to the Christoffel symbol $\Gamma_{00}^{\mu}=-\frac{1}{2}g^{\mu\lambda}\left(\frac{\partial g_{00}}{\partial x^{\lambda}}\right)$ , then to another geodesic equation $\frac{d^{2}x^{\mu}}{d\tau^{2}}=\frac{1}{2}\eta^{\mu\lambda}\left(\frac{\partial h_{00}}{\partial x^{\lambda}}\right)\left(\frac{dt}{d\tau}\right)^{2}$ , then to the final geodesic equation $\frac{d^{2}x^{i}}{d\tau^{2}}=\frac{1}{2}\left(\frac{\partial h_{00}}{\partial x^{i}}\right)\left(\frac{dt}{d\tau}\right)^{2}$ . But how and where does the missing $\frac{1}{c^{2}}$ arise? I admit to a general confusion moving between equations using c=1 and equations using c=c.

Thank you.

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After thinking about this, can I try to answer my own question by saying that all I need to do is replace $dt$ with $d\left(ct\right)$ which becomes $cdt$. This will then eventually give the correct value of $h_{00}=\frac{-2\phi}{c^{2}}$ . –  Peter4075 Sep 15 '11 at 20:39
Factors of c are dimensional analysis. –  Ron Maimon Sep 16 '11 at 8:32