Deriving the 4-momentum of a free particle moving in curved spacetime

[jcap]

Consider a free particle with rest mass $m$ moving along a geodesic in some curved spacetime with metric $g_{\mu\nu}$:
$$S=-m\int d\tau=-m\int\Big(\frac{d\tau}{d\lambda}\Big)d\lambda=\int L\ d\lambda$$
$$L=-m\frac{d\tau}{d\lambda}=-m\Big(-g_{\mu\nu}\frac{dx^\mu}{d\lambda}\frac{dx^\nu}{d\lambda}\Big)^{1/2}$$
The canonical 4-momentum $P_\alpha$ can be derived from the Lagrangian $L$ using the following calculation:
\begin{eqnarray*}
P_\alpha &=& \frac{\partial L}{\partial(dx^\alpha/d\lambda)} \\
&=& \frac{m}{2}\frac{d\lambda}{d\tau}\Big(g_{\alpha\nu}\frac{dx^\nu}{d\lambda}+g_{\mu\alpha}\frac{dx^\mu}{d\lambda}\Big) \\
&=& m\ g_{\alpha\nu}\frac{dx^\nu}{d\tau} \\
&=& m\ \frac{dx_\alpha}{d\tau}
\end{eqnarray*}
where we have used the fact that the metric $g_{\mu\nu}$ is symmetric.

Thus, expressed in contravariant form, we have derived an expression for the 4-momentum $P^\alpha$ given by
$$P^\alpha=m\ \frac{dx^\alpha}{d\tau}$$
using a completely general metric $g_{\mu\nu}$.

Is it correct to interpret the components of $P^\alpha$ in the following manner:

$P^0$ is the energy of the particle,

$P^i$ is the 3-momentum of the particle in the $\partial_i$ direction?

 

[Orodruin]

Consider a free particle with rest mass $m$ moving along a geodesic in some curved spacetime with metric $g_{\mu\nu}$:
$$S=-m\int d\tau=-m\int\Big(\frac{d\tau}{d\lambda}\Big)d\lambda=\int L\ d\lambda$$
$$L=-m\frac{d\tau}{d\lambda}=-m\Big(-g_{\mu\nu}\frac{dx^\mu}{d\lambda}\frac{dx^\nu}{d\lambda}\Big)^{1/2}$$
The canonical 4-momentum $P_\alpha$ can be derived from the Lagrangian $L$ using the following calculation:
\begin{eqnarray*}
P_\alpha &=& \frac{\partial L}{\partial(dx^\alpha/d\lambda)} \\
&=& \frac{m}{2}\frac{d\lambda}{d\tau}\Big(g_{\alpha\nu}\frac{dx^\nu}{d\lambda}+g_{\mu\alpha}\frac{dx^\mu}{d\lambda}\Big) \\
&=& m\ g_{\alpha\nu}\frac{dx^\nu}{d\tau} \\
&=& m\ \frac{dx_\alpha}{d\tau}
\end{eqnarray*}
where we have used the fact that the metric $g_{\mu\nu}$ is symmetric.

Warning! The quantity $x_\alpha$ does not make any sense in GR. $x^\alpha$ are just coordinates, they are not the components of a vector and you cannot raise and lower the index. You also cannot move the $g_{\alpha\nu}$ inside the derivative at will. It depends on the coordinates and therefore on $\tau$ along the world line.

Is it correct to interpret the components of $P^\alpha$ in the following manner:

$P^0$ is the energy of the particle,

$P^i$ is the 3-momentum of the particle in the $\partial_i$ direction?

No. The split into momentum and energy requires the frame of some observer. You are not guaranteed that your coordinates define such a frame.

 

[jcap]

Thanks for the reply