PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Newtonian gravitation as a field theory · The equivalence principle and gravity as geometry
The Geometry of Gravity, Ch The tidal field: what free fall cannot remove
Problem
Working only in Newtonian gravity, a researcher proposes replacing the two-index tidal field Eij by a package of four numbers: the three components of the gravitational field g=−∇Φ together with the single number ∇⋅g.
((a)) Test the proposal against each condition in the specification for a geometric replacement of Newtonian gravity: that the object have two indices and be symmetric, that it not be removable by a change to free-fall coordinates, that it reduce to Eij/c2 for weak, slow fields, and that its trace be fixed by the mass density in that limit. State plainly which condition, or conditions, the proposal fails.
((b)) Using the change to freely falling coordinates x′=x−21gt2 centred on a point where the field is g (taken as uniform over the small region considered), show explicitly what happens to g and to ∇⋅g under this change, and explain why the four-number package still cannot carry all the information that Eij carries.