mburns's Journal: General Relativity Made Simple

Each observer can do a specific simplification of the metric and Einstein tensors. For that observer there is usually just one orientation of coordinates to make those tensors diagonal or orthogonal. Than the red and blue partitions that I have written about before are orthogonal and make a completely accurate representation of the metric.

The remarkable result is that this orthogonality is Cartesian in appearance, not hyperbolic since that property is just needed to translate to other observers, which in turn can do their own orthogonalization. Further, the hyperbolic rotation property of spacetime is not needed to geometrically construct geodesics from these visualized red and blue partitions. It is as though Cartesian geometry is sufficient for this purpose. The red and blue partitions are dimensionally accurate when used to correctly construct a geodesic - there is no use of an incorrect tensor rank then, so the correction factor of cosine squared for geodesic curves oblique to the red or blue partition holds for timelike and mixed curves as well as spacelike.

If the field is changed with the addition or subtraction of a source, then the orthogonalization is generally spoiled. A readjustment is possible.

Hyperbolic geometry remains crucial for the conservation of the centers or sources of the red and blue partitions. By the Bianchi identity, the centers of the components of the field, the centers of the red and blue partitions before they are superimposed and orthogonalized, are conserved in the direction that is hyperbolically orthogonal to the surface of those partitions

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Michael J. Burns