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## The Nordstrom four dimensional theories

Although I have been attempting to understand electromagnetism in four dimensions for a while, I only recently learned that the first attempt at a four dimensional universe came from Gunnar Nordstrom from 1912-3 which Einstein considered to be a serious competitor to general relativity.

Nordstrom imagined a four dimensional spacetime endowed with an extra fifth dimension.  Both electromagnetism and gravity should be part of a Maxwell theory in five dimensions.

$L = 1/4 F^{ab}F_{ab} + J_a A^a$

where the Latin indices have five values.  The electromagnetic field potential includes the gravitational potential in the extra dimension.  For the Nordrom model, there is a separation of variable so that Maxwell equations for the spactime 3+1 dimensions separates from the extra dimension.  More details appear in the paper with which I have disagreements regarding experimental evidence for higher dimensions and expansion of the universe. RavndalScalarGravitationExtraDim

The Nordstrom approach could be applied to an S4 universe where one has additional possibilities, for example describing gravity in terms of an embedding of a three dimensional hypersurface and using the extra dimension to insert the scalar potential of electromagnetism.