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By Allen. Arthur Ormiston

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86) The last manoeuvre is a sleight of hand! Although Φ may be the same function of x as the Newtonian potential, x itself is not a Cartesian coordinate for spacetime, except in an approximate sense. We now retrieve hµν from l = l 00 = l00 , 1 1 1 h00 = l00 − l = l00 = Φ, 2 2 8πκ G 32 2 From Minkowski Spacetime to General Relativity 1 1 1 hi j = li j + δi j l = δi j l00 = δi j Φ, 2 2 8πκ G h0i = l0i = 0 . 69) on p. 26, if we transform to another quasi-stationary coordinate system in the same class, hi j and h0i will take other values, but h00 will remain unchanged to this level of approximation.

With coordinates satisfying the conditions listed on p. , describing a free-falling object encountered by the observer, takes the form [11, p. 135]. 55) with suitable interpretation of the terms, where ai and Ω i j are determined by the connection coefficients according to ai := Γ00i , Γ0i0 = Γi00 = ai Ω i j := Γj0i = Γ0ij = −Γ0ij (i = 1, 2, 3) , (i, j = 1, 2, 3) . , Ω i j = 0, for i, j = 1, 2, 3, by suitable choice of coordinates (basically, using Fermi–Walker transport). 55) holds exactly only on the observer worldline, although it will hold approximately nearby.

And as we just noted when examining the worldlines of free particles, different SE observers can attribute any Newtonian potential they like to the spacetime neighbourhood, precisely because changing from one uniformly accelerating SE coordinate system to another does not keep one inside the same class of quasi-canonical coordinate systems. The linearised Ricci tensor and curvature scalar are 1 Rµν = η σ τ Rµσ ντ = − η σ τ hµν ,σ τ + hσ τ ,µν − hσ ν ,µτ − h µτ ,σ ν , 2 R = η µν Rµν = −h,µ µ + h µν ,µν .

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