Spacetime metrics extended to negative radius

Question

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.

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 $r$.

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

and

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 — Yukterez, Aug 01 '23 at 16:02
@Yukterez here gives details on negative space, but not on when or how can be constructed — john, Aug 01 '23 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°. — Yukterez, Aug 01 '23 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 01 '23 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 — Yukterez, Aug 01 '23 at 19:27
@Yukterez extremely interesting lecture, thank you very much. — john, Aug 01 '23 at 20:31

Pipe · Answer 1 · 2023-08-03T01:43:34.450

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.

1 https://jila.colorado.edu/~ajsh/insidebh/penrose.html

Spacetime metrics extended to negative radius

1 Answers1