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Kaluza–Klein theory
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== Kaluza's hypothesis for the matter stress–energy tensor == Kaluza proposed<ref name=kal/> a five-dimensional matter [[stress–energy tensor|stress tensor]] <math>\widetilde{T}_M^{ab}</math> of the form : <math>\widetilde{T}_M^{ab} = \rho \frac{dx^a}{ds} \frac{dx^b}{ds},</math> where <math>\rho</math> is a density, and the length element <math>ds</math> is as defined above. Then the spacetime component gives a typical [[Dust solution|"dust"]] stress–energy tensor: : <math> \widetilde{T}_M^{\mu\nu} = \rho \frac{dx^\mu}{ds} \frac{dx^\nu}{ds}. </math> The mixed component provides a 4-current source for the Maxwell equations: : <math> \widetilde{T}_M^{5\mu} = \rho \frac{dx^\mu}{ds} \frac{dx^5}{ds} = \rho U^\mu \frac{q}{kmc}. </math> Just as the five-dimensional metric comprises the four-dimensional metric framed by the electromagnetic vector potential, the five-dimensional stress–energy tensor comprises the four-dimensional stress–energy tensor framed by the vector 4-current.
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