I am aware of the approximation generally used for low speeds to calculate the Lorentz factor, that being, $$\gamma \approx 1 + \frac{1}{2} \left(\frac{v}{c} \right)^2$$ But I need the exact opposite thing -- is there any suitable approximation for when v is extremely close to c?

If we take two vectors and want to contract them with the metric tensor to find some frame invariant quantity: $$A^a B^b g_{ab}=\vec A\cdot \vec B$$ is there a convention on where the metric tensor should be placed in this equation to emphasise that we are contracting one tensor or the other? B...