D O C U M E N T 4 8 0 K A L U Z A’ S T H E O R Y: PA R T 2 4 7 5 §2. The Geodesic Line in R5 We introduce the following notation: . (8)[4] Then we know that in covariant relations only the and the antisymmetric derivatives of the may occur.[5] The matrix of the is expressed in terms of the and the thus: . (9) From this it follows that assumes the form or . (10) Now let us assume that τ is an arbitrary parameter in R5 and that . (10a) Then geodesic lines are characterized in the familiar way by the equation , (11) that is, by the equations . (11a) For α = 0 one obtains . If one now chooses τ so that W = const., then this equation yields . (12) Because of (10a), it is the case that, on the geodesic line . (13) gmn γmn γ0mγ0n –= φm γ0m = gmn φm γμv gmn φm g11 φ1φ1 + g12 φ1φ2 + . . φ1 g21 φ2φ1 + g22 φ2φ2 + . . φ2 . . . . . . . . . φ1 φ2 . . 1 dσ2 gmn φmφn)dxmdxn + ( 2φmdxmdx0 dx02 + + dσ2 gmndxmdxn dx0 φmdxm)2 + ( + = W2 gmn-------- dxm dτ --------- dxn dτ dx0 dτ -------- φm--------¹ dxm· dτ - + © § 2 + = δ® ³Wdτ¾ ¯ ¿ ½ 0= ∂xα ∂W dτ© d · ∂xα¹ ∂W § · – 0 = dτ d 1 2W ------- - 2( x0 · φmxm) · + ⋅ 0= x0 · φmxm · + A = gmn dxm dτ -------- -dxn dτ ------- - W2 A2 – const. = =