4 5 2 D O C U M E N T 4 5 9 K A L U Z A S T H E O R Y , P A R T 1 It will be possible to choose the coordinate system in such a way that only the 0-component of differs from zero and that everywhere has the same value. Then (2) reduces to1) . (2a) Furthermore, we shall assume that the five-dimensional continuum is such that the infinitesimal displacement vector has the same magnitude everywhere, i.e. that is independent of all variables (“sharpened cylinder condition”).[4] Given tho coordinate system we chose this implies that and hence also , is constant. Therefore, without restricting generality, we may set For the sake of simplicity, let us use the positive sign in the following the negative sign leads to the same result. Using the “adapted” coordinate system, we can set[5] , (3) where the sum with respect to the indices m and n runs from 1 to 4. One easily sees that dτ2 and are invariants with respect to the transformations of . Kaluza interpreted them as metric and electric invariants, respec- tively, in the four dimensional continuum (R4), which is possible as a consequence of (2a). Thus, he gave a formal unification of the two fundamental invariants by the introducing a five-dimensional, cylindrical continuum (R5). However, this does not constrain the laws of nature any more than the usual method of the general theory of relativity, which introduces and as indepen- dent invariants.[6] Kaluza attains such a constraint by only allowing equations that are covariant with respect to arbitrary point transformations in R5 and that depend only on the ’s. Kaluza obtains field equations for gravitation and electricity that are correct in the first approximation by setting the once-contracted Riemann cur- vature tensor of R5 to zero. We will not introduce this further hypothesis here but instead limit ourselves to deriving a consequence from the existence of the metric in R5 and from the sharpened cylinder condition.[7] Apart from an arbitrary point transformation of in R4, the “adapted coordinate system” allows for the transformation (“ -transformation”), 1) The detour via equation (2) was only chosen in order to make the invariant character of the “cylinder condition” explicit.[8] [p. 24] ξβ ξ0 ∂x0 ∂γμν 0= γμνξμξν γ00ξ0ξ0, γ00 γ00 ±1. = dσ2 dτ2 2dφdx0 dx02 + + = dτ2 γmndxmdxn = γ0mdxm = x1, x2, x3, x4 γμν x1, x2, x3, x4 x0
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