By D. Ostvang
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Extra info for Non-Metric Approach to Space, Time and Gravitation [thesis]
We then include the local effects of the NKE into the family gt which we construct from the conformally flat metrics. That is, when the global effects of the NKE can be neglected the metric families g ¯t and gt can be represented by single metrics g ¯ and g since there is no explicit dependence on t in this case. Moreover, for the static, spherically symmetric case it turns out that g agrees with the Schwarzschild metric to sufficient order in the quantity rs /r, where rs is the Schwarzschild radius.
That is, one must know how it relates to the passive stress-energy tensor Tt in N . Now the relationship between Tt and Tt depends in principle on the general nature of the matter source. To illustrate this, consider Tt and Tt for a perfect fluid: Tt = (˜ ρm + c−2 p˜)¯ ut ⊗¯ ut + p˜g ¯t , Tt = (ˆ ρm + c−2 pˆ)ut ⊗ut + pˆgt , (27) where ρ˜m , p˜ are the active mass-energy density and the pressure in the local rest frame of √ √ ¯ t ≡ˆ ¯ t ≡ˆ the source. Here we have used the definitions ρm h ρm ht and p h p ht , where ρm 34 ¯ t and ht are defined and p are the passive counterparts to ρ˜m and p˜, respectively.
Furthermore we may expect that the abandoned dynamical degree of freedom will be put back implicitly via the NKE when constructing gt from g ¯t . It turns out that 25 this will work and that we get a consistent theory if we choose the basic 3-geometry to be S3 , as we have done in equation (6). Next an obvious question is which sort of affine structure is induced on N by the gt ). To family of metrics gt (and analogously which affine structure is induced on N by ¯ answer this question it is convenient to perceive gt as a single degenerate 5-dimensional ⋆ metric on M×R and construct a torsion-free connection ∇ (almost) compatible with this degenerate metric.
Non-Metric Approach to Space, Time and Gravitation [thesis] by D. Ostvang