Identity
Nov12-11, 12:10 AM
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
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