Differential geometry (which includes general relativity) often introduces the length differential, expressed as ds(adsbygoogle = window.adsbygoogle || []).push({}); ^{2}=g_{ab}dx^{a}dx^{b}, to introduce the covariant form of the metric tensor g_{ab}. However, this formulation scales ds^{2}incorrectly. The appearance of an index as a superscript, as in dx^{a}, indicates that under a dilation, dx^{a}scales as a contravariant tensor. That is, if we double the length of our measuring rod, then the value of dx^{a}falls in half. For a covariant tensor, such as a gradient, the index appears as a subscript, as in dx_{a}. If we double the length of our measuring rod, then the value of the of dx_{a}doubles. Rotations and other lorentz transformations do NOT affect covariant and contravarient vectors differently because the projection of one unit vector on another is reflexive. The common representation of a lorentz transformation as [tex]\gamma[/tex]^{a}_{b}displays invariance of a lorentz transformation under dilation. If ds^{2}=g_{ab}dx^{a}dx^{b}, then ds^{2}is invarient under dilation. Then ds does NOT describe the length of a differential displacement, which should scale contravariantly, like dx^{a}, because in another coordinate system, ds=dx'^{a}. Writing ds^{2}=dx^{a}dx^{a}for Cartesian coordinates gives ds proper scaling for a differential displacement. But, to describe a Minkowski space, the sign on dr^{2}must differ from that on dt^{2}. So one of them must have an explicit factor of i. This sign appears in the metric if we adopt David Hestenes' formulation g_{ab}=[tex]\gamma[/tex]_{a}[tex]\gamma[/tex]_{b}in terms of the basis vectors [tex]\gamma[/tex]^{a}. This formulation leaves the metric in a familiar form. If the negative sign attaches to dt^{2}, then [tex]\gamma[/tex]_{a}[tex]\gamma[/tex]^{a}=3-1=2 instead of 4. Then if [tex]\gamma[/tex]_{a}[tex]\gamma[/tex]_{b}=g_{(ab)}+g_{[ab]}operates on [tex]\gamma[/tex]^{b}, then 2[tex]\gamma[/tex]_{a}=[tex]\gamma[/tex]_{a}+g_{[ab]}[tex]\gamma[/tex]^{b}, or [tex]\gamma[/tex]_{a}=g_{[ab]}[tex]\gamma[/tex]^{b}, so that the antisymmetric part g_{[ab]}transforms tensors without dilation beyond that in the basis vectors, like the symmetric part g_{(ab)}. This symmetry might say something about why we observe variation in only one time dimension rather than the expected three. If [tex]\gamma[/tex]_{a}[tex]\gamma[/tex]^{a}=1-3=-2, then -[tex]\gamma[/tex]_{a}=g_{(ab)}[tex]\gamma[/tex]^{b}and -[tex]\gamma[/tex]_{a}=g_{[ab]}[tex]\gamma[/tex]^{b}, to preserve the absence of further dilation.

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# Scaling Paradox.

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