D O C U M E N T 4 5 9 K A L U Z A’ S T H E O R Y, PA R T 1 4 5 3 , (4) [9] where m stands for the numbers 1, 2, 3, 4. Thus, for a given R5, one hypersurface can be arbitrarily chosen that satisfies the sharpened cylinder condi- tion ((2a) and (3)). From (4) one obtains . (5)[10] If we substitute here by 1, and introduce the R4 tensor , (6) then we obtain, instead of (5), . (5a) If a Hamiltonian invariant in , which is formed out of the “metric coefficients” , the “electric potentials,” as well as the derivatives of these quantities, is also supposed to be invariant with respect to the -transfor- mations—which is necessary for Kaluza’s interpretation of the formal connection between gravitation and electricity—then this invariant may only contain the ’s in the combination . (7) This follows from the second of equations (5a). Hence, Kaluza’s idea offers a deeper understanding of the fact that besides the symmetric metric tensor ( ) only the antisymmetric tensor ( ) of the electro- magnetic field (which is derivable from a potential) plays a role.1) [11] 1) If one does not operate with the sharpened cylinder condition, our result changes in the follow- ing way. In this case, a scalar ( ) enters the Hamiltonian function (in four dimensions) in addition to a symmetric and an antisymmetric tensor.[12] xm xm = x0 x0 ψ( x1, x2, x3, x4) += [p. 25] x0 const. = γmn γmn ∂xm ∂ψ γ0n ∂xn ∂ψ γ0m ∂xm∂xn ∂ψ ∂ψ γ00 + + + = γ0n γ0n ∂xn ∂ψ γ00 += γ00 γ00 = γ00 gmn γmn γ0mγ0n –= gmn gmn = γ0m γ0m ∂xm ∂ψ += R4( x1, x2, x3, x4) gmn γ0m x0 γ0m φmn ∂xn ∂γ0m ∂xm ∂γ0n –= gmn φmn γ00
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