4
$\begingroup$

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 $r$?
$\endgroup$
6
  • $\begingroup$ It can be done, see here and here or here, it's called the negative space $\endgroup$
    – Yukterez
    Aug 1, 2023 at 16:02
  • $\begingroup$ @Yukterez here gives details on negative space, but not on when or how can be constructed $\endgroup$
    – john
    Aug 1, 2023 at 17:38
  • $\begingroup$ 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°. $\endgroup$
    – Yukterez
    Aug 1, 2023 at 17:49
  • $\begingroup$ @Yukterez ok, but my question is more about the mathematical aspect of how we see that the negative space exists $\endgroup$
    – john
    Aug 1, 2023 at 18:03
  • 1
    $\begingroup$ 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 $\endgroup$
    – Yukterez
    Aug 1, 2023 at 19:27

1 Answer 1

0
$\begingroup$

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

Penrose Diagram

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.