deant
Joined: 12 Jul 2010, 07:44 Posts: 154

Exercise [19.06]
I'm not exactly sure what this exercise is asking for, but I'll take a stab at it.
It can be observed that F, and J were constructed componentwise on page 443 from the components of the electric and magnetic fields, and charge flux and density, respectively. This construction depended on the use of standard Minkowski coordinates.
However, F and J themselves are coordinateindependent, tensor quantities, expressible as "geometrical" properties of the underlying spacetime manifold. Similarly the Maxwell equations expressed using these tensors don't depend on any particular coordinate system: They represent relations between these geometrical properties (including derivatives) at each spacetime point, and so remain perfectly welldefined even on manifolds with curvature.
The only question is how to relate F and J in curved spacetime back to the components they were constructed from on p. 443. In a general curvilinear coordinate system, these components will get all mixed up.
However, note two things:
(i) The initial component construction of these tensors did not depend on a particular choice of reference frame  it works in any reference frame parameterized by flat Minkowsi coordinates.
(ii) The tangent space at any point in curved spacetime is a flat Minkowski space that approximates some small region of the actual spacetime around that point (i.e. curved spacetimes are locally flat at small enough scales). So any coordinate system can be approximated in some small region around a given point by using standard, flat, orthogonal Minkowski coordinates. In these coordinates, the components of the electric and magnetic fields, flux, etc, can be extracted from F and J as on p. 443; the values will be exact at the given point, and an approximation in the region around it.
