Lets examine a typical GR metric:
$$ds^2=g_{00}dt^2-g_{11}dx^2-g_{22}dy^2-g_{33}dz^2$$
The "d" going with ds has its correct meaning when the path is specified with respect to a one dimensional manifold (remembering that ds is the proper time interval which will depend on path).
The physical distance (spatial) between two points along the x-axis between the points A and B is given by: $\int\sqrt{g_{11}}dx$ from A to B and not by $\int dx$ in curved space.
Infinitesimal separation between points on the x axis are given by $g_{11}dx$ and not by $dx$.
Now in Maxwell's equations in the covariant form we have quantities like delta-x,delta-y etc which are meaningful only in the Euclidean (rather in the flat space-time Lorentzian) context.But Maxwell's equations in the covariant form refer to curved space time (with respect to strongly curved spacetime also). Are these quantities ($\partial x$,$\partial y $ etc.) expected to retain their physical significance in curved space-time except that they remind us of an Euclidean background?
Better we could write (locally):
$$ds^2=dT^2-dX^2-dY^2-dZ^2$$
Where,
$$\begin{align}dT&=\sqrt{g_{00}}dt \\ dX &= \sqrt{g_{11}}dx \\ dY &= \sqrt{g_{22}}dy \\ dZ &= \sqrt{g_{33}}dz\end{align}$$
(The "d" going with T,X,Yand Z is as justified as the d going with s.)
Locally we have,
$$ds^2=dT^2-dX^2-dY^2-dZ^2$$
Therefore locally we have the same form of Maxwell's equations-- Maxwell's equations in the traditional form!
Though the form of Maxwell’s equations (traditional form being referred to here) remains unchanged locally, the values of the individual variables may change, preserving the traditional form of Maxwell’s equations in the local inertial frames.
We may consider a pair of local labels $x$ and $x+dx$. The physical distance between them along the x-axis is $g_{11}dx$. If the metric changes, say due to the advance of a heavy mass or a high density mass distribution, the physical intervals $dX$,$dY$ etc will change. To preserve the form of the equation the values of $E$,$B$,$j$ etc should also change.
So gravity can change the magnitudes of $E$,$B$ etc. (and of course their directions). If one thinks in the cosmological direction the curvature of space-time was very strong in the remote past and gradually it weakened casting a heavy influence on the values of the electric and the magnetic fields.
Query: How are they taking care of this in the LHC experiments in tracing the past?
[Incidentally the quantities x,y,z etc are simply labels in the curved spacetime context. dx should correspond to some "Euclidean memory"]
