Superscript vs. subscript indices in Euler-Lagrange equation in relativistic field theories In the literature of field theories in flat spacetime, both forms of Euler Lagrange equation are used. 
e.g Consider a real scalar field $\phi$ 
$$\partial_\mu\frac{\partial L}{\partial \phi,_\mu}=\frac{\partial L}{\partial \phi}\qquad\text{where}\qquad\phi,_\mu = \frac{\partial \phi}{\partial x^\mu}\tag{a}$$
$$\partial^\mu\frac{\partial L}{\partial \phi,^\mu}=\frac{\partial L}{\partial \phi}\qquad\text{where}\qquad\phi,^\mu = \frac{\partial \phi}{\partial x_\mu}\tag{b}$$
Since $\partial^\mu = \pm \partial_{\mu}$, the two forms are obviously equivalent to each other. However, sometimes I am quite uncomfortable with form b. Originally, the spacetime volume element was defined to be $dtdxdydz = \prod_{\mu}dx^\mu$. Form b of the Euler equation implies that we are thinking the Lagrangian and the field $\phi$ as a function of $x_\mu$ in the action integral and in the course of the variational calculation. I would like to know whether it is possible to regard the spacetime volume as $\prod_{\mu}dx_\mu$ so that everything is mathematically consistent?
(I think it is somewhat unnatural to consider the action $ S = \int L $ as an integration with respect to $x_{\mu}$. I do understand that this question may not bear any physical significance but I wish to know the answer for the sake of mathematical clarity.)
 A: *

*If $x^{\mu}$ with superscript denotes local spacetime coordinates, then we usually$^1$ define the subscript version as $$x_{\mu}  ~:=~ g_{\mu\nu}x^{\nu},$$ where $g_{\mu\nu}$ is the (0,2) spacetime metric tensor field.

*Within SR, we only allow affine coordinate transformations, so that $x^{\mu}$ transforms as (components of) a (1,0) tensor field and $x_{\mu}$ transforms as (components of) a (0,1) tensor field, and you can use both notations.

*Within GR, we allow general coordinate transformations, and none of $x^{\mu}$ and $x_{\mu}$ then transform as (components of) tensor fields. However $\partial/\partial x^{\mu}$ transforms as (components of) a (0,1) tensor field, while $\partial/\partial x_{\mu}$ doesn't transform as (components of) a (1,0) tensor field. Similarly, $\mathrm{d}x^{\mu}$ transforms as (components of) a (1,0) tensor field, while $\mathrm{d}x_{\mu}$ doesn't transform as (components of) a (0,1) tensor field. Hence the superscript $x^{\mu}$ is preferred in order to maintain covariance.
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$^1$ This is of course ultimately a question of convention.
