I'm reading the article https://arxiv.org/abs/hep-th/9705200 and part of it has left me very confused. In order to speak about their equation (1) the authors make the following statement: \begin{align*} K(t) &= \lim_{x' \to x} \int \dfrac{d^4 k}{(2\pi)^4} e^{i k \cdot (x - x')} \left( e^{-i k \cd...

If black holes are just regions of spacetime, where the slope of spacetime is infinite at it's center, how can black holes even move? When matter moves through spacetime, it bends the spacetime around it, but if black holes are just regions of spacetime, how can a region in spacetime bend other r...