By Kronig R.
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Additional resources for A Theorem on Space Quantization
I5... the form T = Tkl... , ti... kl... , tkl... , . . , tk... , tl... , where tkl... = Tkl... , (1) (2) (N −1) (1) (N ) 5j... tj... kl... = Tkl... , and which is projected onto the four-dimensional (2) physical space-time R4 . For instance, for ψ = ψ i gi + σn then i ∇j ψ = ψ;j gi + ψ i gi;j + σ;j n + σn;j = 1 i 1 ψ Fij n + σ;j n − σF·ji gi = 2 2 1 1 i − σF·ji gi + σ;j + Fij ψ i n . ψ;j 2 2 i = ψ;j gi + = The projection of this onto R4 manifests in i i ψ|j = ψ;j − 1 σF·ji . 2 The stroke | represents the ϑn -covariant derivative which takes up the notion of cylindricity.
Let’s denote by m0 , ρ and c, the pointmass, material density and speed of light in vacuum. Then the vanishing 32 I. p) leads to the equation of motion for a charged particle: e Dui ≡ ui;k uk = F i uk . Ds m0 c2 ·k However, this does not provide a real hint to the supposedly missing link between matter and electromagnetism. We hope that there’s no need to add an external matter term to the stress-energy tensor. p) as telling us that matter and electromagnetism are already incorporated, in other words, the electromagnetic field produces material density out of the electromagnetic current J.
First Gik ;j → G| j = G;j − 2 X F·j − 2 Y F·j where X i = Gi5 = Y i (due to the symmetry of the tensor Gik , G5i = Ri5 ), so 1 ik k i that Gik |k = G;k − 2 Y F·k . Now the five-dimensional curvature tensors (the Riemann and Ricci tensors) in ϑn are A A E A E A R·BCD = ΓA BD,C − ΓBC,D + ΓBD ΓEC − ΓBC ΓED , C E C E C RAB = ΓC AB,C − ΓAC,B + ΓAB ΓEC − ΓAC ΓEB . 11d). Hence Gik |k = G;k . 32b) can also be easily proven this way. As a brief digression, consider a six-dimensional manifold V6 =ϑn ⊗m where m is the second normal coordinate with respect to R4 .
A Theorem on Space Quantization by Kronig R.