subscript

The passage around this formula

Collect the pieces. Three charts, the Newtonian centroid $\mathbf R_N$ , the observer-dependent stress-energy centroid $X^\mu(u,\Sigma)$ , and the canonical Newton–Wigner operator $\mathbf X_{\mathrm{NW}}$ , form a genuine, well-behaved sub-atlas. Define, for any two of them evaluated on the same hypersurface $\Sigma$ , $\delta_{ij}(u,\Sigma) = \frac{\big|X_i^\mu(u,\Sigma) - X_j^\mu(u,\Sigma)\big|_\Sigma}{\rho_M}$. the proper spatial separation within $\Sigma$ between the two charts’ answers, normalized by the spin-to-mass length fixed in the first section. $\delta_{ij}$ is dimensionless by construction, symmetric, zero exactly when the two charts coincide, and it has now been checked, not merely asserted, in both directions this kind of construction is required to pass: it…

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