In the Kaluza-Klein theory the scalar field was added as a fifth dimension. Was there a specific reason to do that? Could it also be inserted between the three dimensions and time? So, can the scalar field be considered independent of the time curvature? Would a swap between the two change anything in the formulas and in the interpretation?
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2scalar field was added as a fifth dimension I think you are mischaracterizing K-K theory. In any case, a field is not a dimension, and there is no concept of “insertion between dimensions”. – G. Smith Jan 22 '20 at 01:24
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I am no expert on K-K but my understanding is that you extend GR to 5 dimensions and find that the additional components of the metric tensor act like an electromagnetic four-potential and a scalar. These new fields are not dimensions. They are reinterpretations of parts of the higher-dimensional metric field. – G. Smith Jan 22 '20 at 05:32