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The covariant component $-P_0$ is the conserved total energy (kinetic, potential and rest energy) in stationary spacetimes and if $x^0=t$. The transformation from the spatial components of the 4-momentum into the local 3-momentum is best done over the local linear velocity $v$ (we use the +--- signature and natural units):

$$P_{\alpha}= \left( \sum_{\beta=1}^{4} \ g_{\alpha \beta} \ \dot{x}^{\beta} \right) - q \ A_{\alpha} = \frac{v_{\alpha}}{\sqrt{1-||v||^2}} \sqrt{-g_{\alpha \alpha}} - (1-||v||^2) \ q \ A_{\alpha}$$

If you regard only geodesics for uncharged particles the electromagnetic vector potential $A$ and the particle charge $q$ can be set to $0$.

The covariant component $-P_0$ is the conserved total energy (kinetic, potential and rest energy) in stationary spacetimes and if $x^0=t$. The transformation from the spatial components of the 4-momentum into the local 3-momentum is best done over the local linear velocity $v$ (we use the +--- signature and natural units):

$$P_{\alpha}= \left( \sum_{\beta=1}^{4} \ g_{\alpha \beta} \ \dot{x}^{\beta} \right) - q \ A_{\alpha} = \frac{v_{\alpha}}{\sqrt{1-||v||^2}} \sqrt{-g_{\alpha \alpha}} - (1-||v||^2) \ q \ A_{\alpha}$$

The contravariant momentum is simply $P^{\alpha}=\dot{x}^{\alpha}$. If you regard only geodesics for uncharged particles the electromagnetic vector potential $A$ and the particle charge $q$ can be set to $0$.

The covariant component $-P_0$ is the conserved total energy (kinetic, potential and rest energy) in stationary spacetimes and if $x^0=t$. The transformation from the spatial components of the 4-momentum into the local 3-momentum is best done over the local linear velocity $v$ (the signature used here is +---):

$$P_{\alpha}= \left( \sum_{\beta=1}^{4} \ g_{\alpha \beta} \ \dot{x}^{\beta} \right) - q \ A_{\alpha} = \frac{v_{\alpha}}{\sqrt{1-||v||^2}} \sqrt{-g_{\alpha \alpha}} - (1-||v||^2) \ q \ A_{\alpha}$$

If you regard only geodesics for uncharged particles the electromagnetic vector potential $A$ and the particle charge $q$ can be set to $0$.

The covariant component $-P_0$ is the conserved total energy (kinetic, potential and rest energy) in stationary spacetimes and if $x^0=t$. The transformation from the spatial components of the 4-momentum into the local 3-momentum is best done over the local linear velocity $v$ (we use the +--- signature and natural units):

$$P_{\alpha}= \left( \sum_{\beta=1}^{4} \ g_{\alpha \beta} \ \dot{x}^{\beta} \right) - q \ A_{\alpha} = \frac{v_{\alpha}}{\sqrt{1-||v||^2}} \sqrt{-g_{\alpha \alpha}} - (1-||v||^2) \ q \ A_{\alpha}$$

If you regard only geodesics for uncharged particles the electromagnetic vector potential $A$ and the particle charge $q$ can be set to $0$.

The covariant component $P_0$ is the conserved total energy (kinetic, potential and rest energy) in stationary spacetimes and if $x^0=t$. The transformation from the spatial components of the 4-momentum into the local 3-momentum is best done over the local velocity $v$:

$$P_{\alpha}= \left( \sum_{\beta=1}^{4} \ g_{\alpha \beta} \ \dot{x}^{\beta} \right) - q \ A_{\alpha} = \frac{v_{\alpha}}{\sqrt{1-||v||^2}} \sqrt{-g_{\alpha \alpha}} - (1-||v||^2) \ q \ A_{\alpha}$$

If you regard only geodesics for uncharged particles the electromagnetic vector potential $A$ and the particle charge $q$ can be set to $0$.

The covariant component $-P_0$ is the conserved total energy (kinetic, potential and rest energy) in stationary spacetimes and if $x^0=t$. The transformation from the spatial components of the 4-momentum into the local 3-momentum is best done over the local linear velocity $v$ (the signature used here is +---):

$$P_{\alpha}= \left( \sum_{\beta=1}^{4} \ g_{\alpha \beta} \ \dot{x}^{\beta} \right) - q \ A_{\alpha} = \frac{v_{\alpha}}{\sqrt{1-||v||^2}} \sqrt{-g_{\alpha \alpha}} - (1-||v||^2) \ q \ A_{\alpha}$$

If you regard only geodesics for uncharged particles the electromagnetic vector potential $A$ and the particle charge $q$ can be set to $0$.