Energy in general relativity - $p^t$ or $p_t$? I am slightly confused about what we call energy in general relativity. Minser, Thorne and Wheeler (pg657) use it as:
$$E=p^t$$
whilst this answer to one of my recent questions and other sources define it as
$$E=p_t$$
In most cases what we call 'energy' doesn't really matter. But for some it does, and in particular for the case of photons, in which according to Hobson, Efstathiou and Lasenby (pg12) has a four-momentum defined as:
$$p^\mu=(E/c,\vec p)$$
and hence this would suggest that $\nu\propto E\propto p^t$ but they then latter (pg222) say that:
$$E=p_\mu u^\mu$$
which indicates that $\nu\propto E\propto p_t$. My question is therefore, which of the equations in this question are correct, which are opinion based and which are wrong and why?
 A: If $\xi$ is a Killing vector of a certain metric $g_{\mu\nu}$, then the following quantity
$$
Q_\xi =  \xi^\mu p_\mu ~, \qquad p^\nu =  \frac{d x^\nu}{d\lambda}~, \quad p_\mu = g_{\mu\nu} p^\nu~.  
$$
is constant along all geodesics w.r.t. $g$. In particular, metrics that are globally hyperbolic have $\xi = - \partial_t$ has a Killing vector for some "time" coordinate $t$. The sign chosen here is a matter of convention. The conserved quantity corresponding to this Killing vector is what we call the energy. This is
$$
E = - p_t
$$
Thus, we note that the quantity that is conserved is $p_t$ not $p^t$. The former should be called the energy since it is the thing that is conserved. 
Of course, when in Minkowski space, the difference between $p_t$ and $p^t$ is a sign so they are both conserved. In this case, it is irrelevant which quantity you define as energy. Here, it is simply a matter of convention.
A: Energy and momentum organize themselves into a 4-vector $p_{\mu}$ in general relativity. Now, if you want to ask what a time-like observer measures as energy you need to first define the velocity vector of the observer as u^{\mu} such that $g_{\mu\nu}u^{\mu}u^{\nu}=1$. Then the energy the observer will measure at any point along their trajectory is $p_{\mu}u^{\mu}=p^{\mu}u_{\mu}$. In flat Minkowski space for an observer at rest $u^{\mu}={1,0,0,0}$, so they measured $E = p_{0}=-p^{0}$.
