Since Maxwell’s equations, written in terms of the classically measurable electromagnetic field tensor F [Eqs. (2.48)] involve only single gradients, it is reasonable to expect them to be lifted into curved spacetime without curvature-coupling additions. Assume this is true. It can be shown that: 

(i) If one writes the electromagnetic field tensor F in terms of a 4-vector potential A(vector) as

then half of the curved-spacetime Maxwell equations, F_αβ;γ + F_βγ;α + F_γα;β = 0 [the second of Eqs. (2.48)] are automatically satisfied

(ii) F is unchanged by gauge transformations in which a gradient is added to the vector potential,

(iii) By such a gauge transformation one can impose the Lorenz-gauge condition ∇(vector) · A(vector) = 0 on the vector potential.

Show that, when the charge-current 4-vector vanishes, J(vector) = 0, the other half of the Maxwell equations, F^αβ_;β = 0 [the first of Eqs. (2.48)] become, in Lorenz gauge and in curved spacetime, the wave equation with curvature coupling [Eq. (25.60)].

Eq. 25.60

Equation 2.48.

Transcribed Image Text:

Faß = AB;a - Aa;ß› (25.61)