A simple derivation of Einstein's field equation, based on the extended relativity principle only, and the stunning consequences for Kaluza's theory

It is shown that it is the (extended) relativity principle alone from which all of General and Special Relativity, namely Einstein’s field equation, is derived. This is done by operations in which the principles of conservation of mass and momentum (whose observation is required by the relativity principle for any observer at rest), and the covariant divergence of tensors play crucial roles. Kaluza’s attempt of a unification of gravitation and electromagnetism by the introduction of a fourth spatial dimension (so that the accessible universe constitutes an ultra-thin brane) must then be re-considered. Instead of introducing five new tensor elements g40 , g41 , g42, g43, g44 (as Kaluza did), five new tensor elements T40 , T41 , T42, T43, T44 are introduced that are an expression of conservation of charge. In the face of the new method of deriving Einstein’s field equation, the adding of the principle of conservation of charge and hence of a fourth spatial dimension is a necessity. The new elements of T have two physical meanings each: Mass-/momentum-flux in the 4th spatial direction on the one hand, and charge-density/charge-flux in all four spatial directions on the other hand. But it turns out that their dimensions are identical in basic units. Thus no ambiguity exists. The result of the tensor-expansion is stunning (even though electric force cannot be “transformed away”): Maxwell’s equations can be extracted, and the introduction of evenly distributed electric charge in the interior of a non-spinning spherical mass affects the metric tensor gµ nu not only because of the energy of the electric field, but in an additional manner. The necessity of adding a fourth spatial dimension, that is, the switching from symmetrical 4 x 4 to 5 x 5 tensors, comes with testable consequences. These consequences are solutions both to the Trouton-Noble and the Ehrenfest paradox.