Confusion surrounding Noether's Theorem

The continuity equation for electric charge can be written $$frac{partial rho}{partial t} + nabla cdot vec J = 0$$

where $rho$ is the electric charge density and $mathbf J$ is the electric current density. In integral form, this expresses the idea that if the amount of charge in some arbitrary volume changes, it is because some charge passed through the boundary; in other words, charge does not simply appear or disappear.

If we define the current 4-vector as $mathbf J = (crho, vec J)$ and the position coordinates $x^mu = (ct,vec r)$, this can be expressed as

$$frac{partial }{partial x^mu} J^mu = 0$$

where the Einstein summation convention is used. In particular, note that $J^0$ is identified with the charge (density) and the spatial components $J^a$, $a=1,2,3$ are identified with the current (density). In some volume $V$, the total charge is given by $Q = int_V mathrm d^3 x J^0$.

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From Noether's theorem, we find that spacetime translation invariance yields $frac{partial}{partial x^mu} T^{munu} = 0$. That is, translation invariance in the $x^0$ direction (i.e. a time translation) yields the continuity equation $frac{partial}{partial x^mu} T^{mu 0}= 0$, with total "charge" given by $int mathrm d^3x T^{00}$. Similarly, translation invariance in the $x^3$ direction yields $frac{partial}{partial x^mu} T^{mu 3} = 0$ which has total charge $int mathrm d^3x T^{03}$.

Noether's theorem takes a continuous symmetry as an input and spits out a continuity equation of the form $partial_mu J^mu = 0$, where $J^0$ is the conserved Noether charge density (and $int mathrm d^3 x J^0$ is the total charge). In this framework, the definition of energy is the Noether charge corresponding to time translations, while momentum is the Noether charge corresponding to spatial translations.