homework and exercises - Energy of a circular orbit in the Schwarzschild metric?

I am looking at circular orbits in the Schwarzschild metric and the energy associated with them. In places I look the energy is given as:

$$\newcommand{\p}[2]{\frac{\partial #1}{\partial #2}} \newcommand{\f}[2]{\frac{ #1}{ #2}} \newcommand{\l}[0]{\left(} \newcommand{\r}[0]{\right)} \newcommand{\mean}[1]{\langle #1 \rangle}\newcommand{\e}[0]{\varepsilon} \newcommand{\ket}[1]{\left|#1\right>} E=\l 1-\f{r_s}{r}\r\f{dt}{d\tau}=\f{r-2GM}{r(1-3GM)}$$ examples of where this is done is Moore, 2013 pg 126 and Baumgarte & Shapiro, 2010 pg10. But is the energy not simply given by the zeroth component of the 4-momentum i.e.: $$E=\f{dt}{d\tau}=\f{r-2GM}{r(1-3GM)(1-\f{r_s}{r})}$$ where deos the extra factor of $1-\f{r_s}{r}$ come from and which is the correct expression?