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What does the Penrose diagram of a super extremal black hole (naked singularity) looks like?

I have seen the ones for regular charged/rotating black holes, but I can't find a good clear one for super extremal black holes.

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  • $\begingroup$ researchgate.net/publication/344277256/figure/fig4/… $\endgroup$
    – safesphere
    Jun 2, 2022 at 4:09
  • $\begingroup$ Notice that the diagram is a 2D slice. To imagine it in 3D, you need to rotate it around the vertical axis, but keep in mind that no geodesics cross the vertical line of the singularity, but end on it. $\endgroup$
    – safesphere
    Jun 2, 2022 at 4:13

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A good reference on how to draw Penrose diagrams is Walker's Block Diagrams and the Extension of Timelike Two‐Surfaces, J. Math. Phys. 11, 2280 (1970). It describes how to obtain block diagrams for metrics simple enough that one can write them in the form $$\mathrm{d}s^2 = - F(r) \mathrm{d}t^2 + \frac{\mathrm{d}r^2}{F(r)},$$ while ignoring two dimensions (which is allowed due to some symmetry in the spacetime, such as spherical symmetry, or by choosing some particular slice of the spacetime). By generalizing the standard Kruskal extension construction that is done for Schwarzschild spacetime, one can treat way more general spacetimes by simply checking the zeros of $F(r)$ (assuming also that is is "well-behaved"). The trick is that this metric has a Killing vector field given by $\frac{\partial}{\partial t}$, and this field becomes null precisely at the points in which $F(r)$ vanishes. Hence, these are the null surfaces that "divide the blocks" in Penrose diagrams. By checking each of the zeroes, one can build each block and afterwards glue them together.

For a standard Schwarzschild black hole, one has $$F(r) = 1 - \frac{2M}{r}.$$ This function has a zero at $r = 2M$ and a singularity at $r = 0$. This leads to a block corresponding to the region with $2M < r < +\infty$ and a different block for the region $0 < r < 2M$. Gluing them together in the most general way compatible with the time-orientation of each block, one obtains the Penrose diagram for Schwarzschild.

In a super-extremal case, we have the same function, but now the coordinate singularity $r = 2M$ happens for $r < 0$. Hence, we get a block for $0 < r < +\infty$. There's no longer a second block, because the coordinates $2M < r < 0$ can't really exist past the physical singularity at $r$ (it's been some time since I read Walker's paper, so I don't recall whether he gives much detail into this issue, but I find it likely for this to be hidden in there somewhere). Hence, the only block we get is this one:

Penrose diagram for a super-extremal Schwarzschild black hole. There is a vertical zigzag line, met at its extrema by two lines at 45 degrees, which meet at the right, forming a 90 degree angle. Above the upper 45 degree line, one sees \mathscr{I}^+. Similarly, \mathscr{I}^- is written beneath the lower line. i^0 is written next to the place they meet.

While I never worked out the cases for charged and/or spinning black holes, I trust the same process will work and I'm quite confident that it will yield the same diagram for a charged black hole (as Tim Rias mentioned in the comments, a Kerr black hole allows for the observer to pass through $r = 0$, so it will be more complicated), since a null surface before the singularity would be an event horizon. Walker's paper develops the diagrams for the sub-extremal charged and spinning black holes, in case it interest you.

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    $\begingroup$ The Kerr one will be a bit more complicated because of the possibility of passing through the center of the ring singularity and entering the $r<0$ region. $\endgroup$
    – TimRias
    Jun 2, 2022 at 14:54
  • $\begingroup$ @TimRias fair enough! Thanks for pointing it out, I've updated the last paragraph accordingly =) $\endgroup$ Jun 2, 2022 at 17:05
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    $\begingroup$ Perhaps it is worth mentioning that this diagram is the same as for the flat Minkowski spacetime where the naked timelike singularity is like an object that exists in time just like any other object. So the wavy vertical line simply is the worldline of this object (naked timelike singularity) in the middle of the asymptotically flat spacetime. Here is a good discussion with a diagram (Figure $54$): damtp.cam.ac.uk/user/tong/gr/six.pdf - “The final result is exactly the same as Minkowski space, with one difference: there is a curvature singularity at $r = 0$“ $\endgroup$
    – safesphere
    Jun 4, 2022 at 22:22
  • $\begingroup$ It also may be worth mentioning that timelike naked singularities cannot form, because: (1) in the Schwarzschild case, a negative mass does not exist; (2) in the charged case, the electric repulsion exceeds the gravitational attraction; (3) in the rotation case, the centrifugal force exceeds the gravitational force. However, spacelike naked singularities can be physical, for example, the Big Bang (whose Penrose diagram is completely different). $\endgroup$
    – safesphere
    Jun 4, 2022 at 22:32
  • $\begingroup$ @safesphere While it is conjectured that naked singularities cannot arise from a gravitational collapse, as far as I know there is no proof of this statement within General Relativity. This is known as the Weak Cosmic Censorship Conjecture. $\endgroup$ Jun 8, 2022 at 17:41

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