Note: You may obtain a PDF version of Lab Note 2, with parts 2 and 3 combined, at Lab Note 2, with parts 2 and 3.
Also Note: This Lab Note picks up where Lab Note 2, Part 2, left off, following section 7 thereof. Equation numbers here, reference this earlier Lab Note.
8. The Electrodynamic Potential as the Axial Component of the Gravitational Potential
Working from the relationship which generalizes (5.4) to five dimensions, and recognizing that the field strength tensor is related to the four-vector potential according to , let us now examine the relationship between and the metric tensor . This is important for several reasons, one of which is that these are both fields and so should be compatible in some manner at the same differential order, and not the least of which is that the vector potential is necessary to establish the QED Lagrangian, and to thereby treat electromagnetism quantum-mechanically. (See, e.g., Witten, E., Duality, Spacetime and Quantum Mechanics, Physics Today, May 1997, pg. 28.)
Starting with , expanding the Christoffel connections , making use of which as shown in (6.5) is equivalent to , and using the symmetry of the metric tensor, we may write:
It is helpful to lower the indexes in field strength tensor and connect this to the covariant potentials , generalized into 5-dimensions as , using , as such:
The relationship expresses clearly, the antisymmetry of in terms of the remaining connection terms involving the gravitational potential. Of particular interest, is that we may deduce from (8.2), the proportionality
(If one forms from (8.3) and then renames indexes and uses , one arrives back at (8.2).) Further, we well know that , i.e., that the covariant derivatives of the potentials cancel out so as to become ordinary derivatives when specifying , i.e., that is invariant under the transformation . Additionally, the Maxwell components (7.10) of the Einstein equation, are also invariant under , because (7.10) also employs only the field strength . Therefore, let is transform in the above, then perform an ordinary integration and index renaming, to write:
In the four spacetime dimensions, this means that the axial portion of the metric tensor is proportional to the vector potential, , and that the field strength tensor and the gravitational field equations are invariant under the transformation used to arrive at (8.4). We choose to set , and can thereby employ the integrated relationship (8.4) in lieu of the differential equation (8.3), with no impact at all on the electromagnetic field strength or the gravitational field equations, which are invariant with respect to this choice.
9. Unification of the Gravitational and QED Lagrangians
The Lagrangian density for a gravitational field in vacuo is , where g is the metric tensor determinant and is the Ricci tensor. Let us now examine a Lagrangian based upon the 5-dimensional Ricci scalar, which we specify by: