# The zeroth component of the four-momentum

Why is it that the Energy of a system is identified with $p_0$ and not $p^0$?

This is especially concerning to me in non-Minkowski metrics, such as the Schwarzschild metric, where the difference between $p^0$ and $p_0$ can be quite dramatic.

In the schwarzschild metric, we have:

$$p_0 = -\tilde{E}\,m$$

where $\tilde{E}$ is energy per unit mass, and for a photon, $p_0=-E$

But that means that for a massive particle,

$$p^0 = m\left(1-\frac{2M}{r}\right)^{-1}\tilde{E}$$

and for a photon,

$$p^0 = \left(1-\frac{2M}{r}\right)^{-1}E$$

Clearly $p_0$ has been more closely related with energy than its contravariant form