The Spatial Components of the Metric Tensor

For the spatial components of the metric tensor, we have the tensor transformation expression

$$g_{kl}={\partial X^{\mu} \over \partial x^k} {\partial X^{\nu} \over \partial x^l}G_{\mu \nu}.$$ (43)

$$g_{kl}=-{V^k V^l \over c^2} + \delta_{ij} \left(1-{2 \gamma \Phi ... ...c^2}\right) {\partial X^i \over \partial x^k} {\partial X^j \over \partial x^l}$$ (44)

Using the transformation coefficients calculated in an earlier section, expanding and keeping only linear terms, almost all terms cancel out and the result is: (it takes a page of calculation)

$$g_{ij} =\delta_{ij} +O(x^i x^j).$$ (45)