# Spacetime metrics extended to negative radius

In the Kerr metric $$ds^2=\left(1-\frac{2Mr}{\rho^2}\right)dt^2+\frac{4Mar\sin^2\theta}{\rho^2}dtd\varphi-\frac{\rho^2}{\Delta}dr^2-\rho^2d\theta^2-\left(r^2+a^2+\frac{2Ma^2r\sin^2\theta}{\rho^2}\right ) \sin^2\theta d\phi^2$$ with $$\rho^2=r^2+a^2\cos^2\theta,\quad \Delta=r^2-2Mr+a^2$$ the coordinate $$r$$ can be extended to negative values.
This because a variable which behaves like a radius can be built as $$R=r\sqrt{1+a^2\sin^2\theta /r^2}$$ and only asymptotically far from the source we have that $$R\sim r$$.
Everywhere else the coordinate $$r$$ is just a coordinate, and therefore can take also negative values.

1. There is a physical meaning to this? Or is just a mathematical consequence of the form of the metric?

The extension to negative value of $$r$$ is possible also on other spacetimes, such as Simpson-Visser, which looks a lot like Schwarzschild metric, but also in this case we can expand to $$r=0$$ and negative $$r$$, since the metric is regular for any $$r$$.

1. Why can't we then expand Schwarzschild to negative $$r$$, but without $$r=0$$? (just from a mathematical point of view)

and

1. Given an arbitrary metric, when can we expand it to negative $$r$$?
• It can be done, see here and here or here, it's called the negative space Aug 1, 2023 at 16:02
• @Yukterez here gives details on negative space, but not on when or how can be constructed
– john
Aug 1, 2023 at 17:38
• In the Eddington Finkelstein style Kerr Schild coordinates you can plot trajectories through the ring singularity into the negative r region. You can also transform to Null coordinates and use the u,v-axes tilted by 45°. Aug 1, 2023 at 17:49
• @Yukterez ok, but my question is more about the mathematical aspect of how we see that the negative space exists
– john
Aug 1, 2023 at 18:03
• I know, but since I can't answer your whole question I must restrict myself to these comments. Roy Black himself although does not believe in it, see here Aug 1, 2023 at 19:27

In general, what you describe is done, even in the case of black holes. Barring infinity identifications, the region with a negative radius is in some cases interpreted as another universe. In the case of black holes in the following diagram taken from 1, the region denoted as a "parallel universe" is the one with a negative radius. The interesting aspect is that the radial point at which the two universes meet is at $$r = 2M$$, since beyond this radius the radial and time coordinate switches their roles, so that $$r = 0$$ is at the top of the diagram. Given a general metric, nothing a priori prohibits extending the coordinates, even to negative radii, barring degeneracies or singularities. It obviously depends on the physical meaning you want to assign to the metric.