# What does it mean the condition “flatness at the center”?

I am reading this paper Formation and evaporation of non-singular black holes by S.A. Hayward.

It deals with the possible metric of a non singular black-hole. The autor consider the metric $$ds^2=-F(r)dt^2+\dfrac{1}{F(r)} dr^2+r^2d\omega^2.$$ In order to find the non singular metric, the autor imposes two conditions:

1) $$F(r)\rightarrow 1-\dfrac{2M}{r}$$ for $$r\rightarrow \infty$$

2) a flatness conditon: $$F(r)\rightarrow 1-\dfrac{r^2}{l^2}$$ for $$r\rightarrow0$$

What does the second condition mean?

Mathematically, it just means that the metric is forced (by this construction) to go to Minkowski space at the origin in a smooth way since $$F(0)=1$$.
• Would it be the same using $F(r)\rightarrow 1-\dfrac{r}{l}$? – mattiav27 Nov 14 '19 at 14:40
• Also note that the $r^2/l^2$ term is there to provide for a possible non-zero energy density near the origin (which works out to be $3/(8 \pi l^2)$ assuming Einstein's equation holds.) – Michael Seifert Nov 14 '19 at 14:41
• @MichaelSeifert so $F(r)\rightarrow 1-\dfrac{r}{l}$ is not an option? – mattiav27 Nov 14 '19 at 14:42
• @mattiav27: I'm pretty sure that in that case the curvature tensor would not be smooth at the origin. For example, if you take a metric $$ds^2 = \left(1 - \frac{\sqrt{x^2 + y^2 + z^2}}{l}\right)(dx^2 + dy^2 + dz^2)$$ (which is what you're proposing), then its first derivatives are not continuous at the origin because of the square root, which means that the second derivatives (and hence the curvature) are not well-defined there. – Michael Seifert Nov 14 '19 at 14:44