# Smoothness of the metric when going from the interior to the exterior Schwarzschild metric

Following (1) and (2), the Schwarzschild metric can be written as: $$ds^2 = a(r)\cdot c^2 dt^2 - b(r)\cdot dr^2 - r^2\cdot(d\theta^2 + sin^2(\theta)\cdot d\varphi^2)$$ where: $$a(r) = \begin{cases} \frac14\left(3\sqrt{1-\frac{r_s}{r_g}} - \sqrt{1-\frac{r^2 r_s}{r_g^3}}\right)^2 \qquad \text{if } 0\leq r \leq r_g\\ 1-\frac{r_s}{r}\qquad \text{if } r_g\leq r \end{cases}$$ $$b(r) = \begin{cases} \left(1-\frac{r^2 r_s}{r_g^3}\right)^{-1} \qquad \text{if } 0\leq r \leq r_g\\ \left(1-\frac{r_s}{r}\right)^{-1} \qquad \text{if } r_g\leq r \end{cases}$$ Here :

• $$r_s=2GM/c^2$$ is the Schwarzschild radius of a spherical body of mass $$M$$
• $$r_g$$ is the value of $$r$$ at the surface of the spherical body

The part $$r\in [0,r_g]$$ is the interior Schwarzschild solution while the part $$r\geq r_g$$ is the exterior Schwarzschild solution.

Assume $$r_s < r_g$$.

Now, I graph $$a(r)$$ and $$b(r)$$. I see that $$a(r)$$ and $$b(r)$$ are continuous functions. I see that $$a(r)$$ looks fairly smooth. However, at the transition $$r=r_g$$, the function $$b(r)$$ does not look smooth. Is it normal? Why would $$b(r)$$ be not differentiable at the transition from $$r to $$r>r_g$$?

Here is a screenshot for $$r_s=1$$ and $$r_g=2$$.

The distribution of mass has a discontinuity at $$r=r_g$$ so it entirely reasonable that the metric has a feature that reflects this.
The main point is that while the $$a(r)$$ function does not have a continuous derivative (slope), it is continuous, so there is no discontinuity in the metric itself. That is, the metric does not have to be smooth, only continuous.
• Yes, I was thinking about the non-continuous distribution of mass. However, in classical mechanics the potential energy is differentiable when going from inside $r^2/2 + const.$ to the outside $-1/r$. So I was thinking maybe this should be reflected in the above metric. But yes, if the mass is not continuous, so is $T_{\mu\nu}$, so is $G_{\mu\nu}$, so should some first derivatives of $g_{\mu\nu}$. – Noé AC Jun 2 '19 at 20:21