Metric Tensor Division: Is It Proper?

[redstone]

If you know that
$${{x}^{a}}{{g}_{ab}}={{x}_{b}}$$

is it proper to say that you also know
$${{g}_{ab}}=\frac{{{x}_{b}}}{{{x}^{a}}}$$

 

[PeterDonis]

If you know that
$${{x}^{a}}{{g}_{ab}}={{x}_{b}}$$

is it proper to say that you also know
$${{g}_{ab}}=\frac{{{x}_{b}}}{{{x}^{a}}}$$

No, because the expression ${{x}^{a}}{{g}_{ab}}={{x}_{b}}$ is not a single term, it's a sum of terms. Repeated indexes are summed over, so what your first equation really means is

$$\Sigma_a {{x}^{a}}{{g}_{ab}} = x^0 g_{0b} + x^1 g_{1b} + x^2 g_{2b} + x^3 g_{3b} = {{x}_{b}}$$

(I've assumed that we're working in a 4-dimensional manifold.) There's no way to transform that into your second equation.

 

[redstone]

Ah, yes, of course. Thank you.