According to MTW,

$$F^{\alpha\beta}{}_{;\beta} = 4\pi J^\alpha$$

and we can infer that the four-current must be an ordinary vector field because the left side is tensorial.

But Wikipedia says that the four-current is a vector density and gives the Maxwell Lagrangian density as follows

$$\mathcal{L} = -\frac{1}{4\mu_0} F_{\alpha\beta} F^{\alpha\beta} \sqrt{-g} + A_\alpha J^\alpha$$

which makes it clear that the four-current defined in the way in which the article uses it must be a vector density since there is no $\sqrt{-g}$ factor in the second term. But a definition isn't given.

So which is it? Are there two different conventions in use for the four-current? Which is more common?

According to MTW,

$$F^{\alpha\beta}{}_{;\beta} = 4\pi J^\alpha$$

and we can infer that the four-current must be an ordinary vector field because the left side is tensorial.

But Wikipedia says that the four-current is a vector density and gives the Maxwell Lagrangian density as follows

$$\mathcal{L} = -\frac{1}{4\mu_0} F_{\alpha\beta} F^{\alpha\beta} \sqrt{-g} + A_\alpha J^\alpha$$

which makes it clear that the four-current defined in the way in which the article uses it must be a vector density since there is no $\sqrt{-g}$ factor in the second term. But a definition isn't given.

So which is it? Are there two different conventions in use for the four-current? Which is more common?