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Use the negative curvature convention defined in curvature-sign convention in Killing derivative identities, and $C_{ab}=C^c{}_{acb}$. Covariant derivatives of a vector obey $[{\nabla }_b,{\nabla }^a]A^b=-C^a{}_b{A}^b$. Hence the flat-space equivalent expressions $\square A^a-{\partial }_b{\partial }^a{A}^b$ and $\square A^a-{\partial }^a{\partial }_b{A}^b$ acquire a difference $+C^a{}_b{A}^b$ under the comma-to-semicolon prescription. Covariantizing $F_{ab}=2{\nabla }_{[a}{A}_{b]}$ and its Maxwell equations gives $\square A^a-{\nabla }^a({\nabla }_b{A}^b)+C^a{}_b{A}^b={\mu }_0{J}^a$. This form respects gauge invariance. Dropping the Ricci term is generally not gauge invariant. The two expressions coincide in a Ricci-flat vacuum spacetime; vanishing matter alone with nonzero cosmological constant does not imply Ricci flatness.

The strong equivalence principle asserts universality of free fall, including gravitational self-energy, and local independence of nongravitational and gravitational experiments from a freely falling laboratory's location and velocity. In a sufficiently small local inertial frame, uniform gravitational acceleration is removed and local physics takes its special-relativistic form. The Riemann curvature tensor remains measurable through tidal effects; a finite laboratory is not made exactly flat. Freely falling test bodies follow geodesics, clocks measure proper time, and minimally coupled light follows null geodesics in geometrical optics. Local nongravitational laws retain their special-relativistic form even when different laboratories observe gravitational redshift.

The comma-to-semicolon prescription replaces flat partial derivatives by the Levi-Civita connection, and the Minkowski metric by the curved metric tensor. It implements this local correspondence for minimally coupled matter. However, an identity obtained by commuting partial derivatives need not survive unchanged when they become covariant derivatives.

Raising the indices of the field-strength definition in flat spacetime gives $F^{ab}={\partial }^a{A}^b-{\partial }^b{A}^a$. Substitution into the sourced Maxwell equations yieldsso the first potential equation isPartial derivatives commute, giving the second formNo Lorenz gauge assumption was used.

Apply the comma-to-semicolon prescription before commuting derivatives. The two candidate left-hand sides becomeUse the cover curvature $C$ from Question 1, with $C_{ab}=C^c{}_{acb}$. The vector commutator is $[{\nabla }_c,{\nabla }_d]A^a=-C^a{}_{bcd}{A}^b$. HenceThis is the derivative-order ambiguity in covariant Maxwell potential equations. At a point in Riemann normal coordinates, the connection itself vanishes but its derivatives need not; the strong equivalence principle does not dispose of this curvature term by a coordinate choice.

Covariantizing the antisymmetric field strength and its Maxwell equations directly gives ${\mathcal{P}}_1^a={\mu }_0{J}^a$, orThis formulation has gauge invariance under $A_a\mapsto A_a+{\nabla }_a\chi$, since $F_{ab}$ is unchanged. In contrast, the uncorrected ${\mathcal{P}}_2$ acquires $-C^a{}_b{\nabla }^b\chi$ under that transformation. Thus preserving the original electromagnetic gauge symmetry singles out the directly covariantized Maxwell equations, rather than treating the two potential equations as interchangeable in matter. The equivalence principle alone is not a unique recipe for every possible higher-derivative or curvature coupling.

In a Ricci-flat vacuum spacetime, $C_{ab}=0$, so ${\mathcal{P}}_1={\mathcal{P}}_2$ and this particular ambiguity disappears even though the full Riemann curvature tensor may be nonzero. This uses vacuum with zero cosmological constant; an Einstein vacuum with nonzero cosmological constant is not Ricci flat.

For the terrestrial estimate, the relevant effect is Ricci curvature inside matter, not merely the tidal field outside the Earth. If a typical material density is $\rho \sim 5\mathrm{g}\mathrm{c}{\mathrm{m}}^{-3}$, the Einstein field equations give a characteristic scaleFor a field varying over scale $L$, the derivative terms are $O(A/L^2)$ whereas the disputed term is $O(CA)$. Their relative size is $O(C{L}^2)$: about $10^{-22}$ for a metre-scale experiment. Even substituting the Earth's radius gives onlyA curvature length comparable to $C^{-1/2}\sim 10^{13}\mathrm{cm}$ would be needed for an order-one effect. In principle non-vacuum measurements could distinguish different curvature couplings, but an ordinary terrestrial laboratory has an exceedingly small signal, overwhelmed by its ordinary electromagnetic response to matter. This is a practical scale estimate, not a theorem that an arbitrarily precise experiment is impossible.