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

In the schwarzschild metric, we have:

[tex]p_0 = -\tilde{E}\,m[/tex]

where [itex]\tilde{E}[/itex] is energy per unit mass, and for a photon, [itex]p_0=-E[/itex]

But that means that for a massive particle,

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

and for a photon,

[tex]p^0 = \left(1-\frac{2M}{r}\right)^{-1}E[/tex]

Clearly [itex]p_0[/itex] has been more closely related with energy than its contravariant form