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Lines of constant $r$ and $t$ are often shown on conformal (Penrose) diagrams of Minkowski space. A google search readily gives many examples such as:

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Is anyone aware of a graphic that displays lines of constant $r$ and $t$ for the Penrose diagram depicting the collapse of matte r into a black hole, namely the following:

enter image description here

Bonus points for the case of the evaporating black hole, where I suspect the lines will be highly distorted due to the shift in the origin:

enter image description here

A precise depiction based on explicit coordinate transformations would be ideal, but even a sketch of the qualitative behaviour of the lines is useful!

Thanks!

pill
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  • Are you interested in all of the lines or only those outside of the collapsing star? The difference is that the outside of the star is always a Schwarzschild solution, so it is sort of straightforward to compute. The inner solution, however, will very likely depend on details of the collapse, and hence I don't believe there is a unique answer. – Níckolas Alves Nov 28 '22 at 02:49
  • I am interested in all the lines. As you say, the interior metric obviously depends on the details of the collapse, so I was hoping for some some examples (which have been provided by AVS) – pill Nov 29 '22 at 03:34

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Note, that for dynamical spacetimes such as black hole collapse models the shape of Penrose diagram is not unique depending on which part of spacetime is put at its center. This is illustrated by multiple variants of Penrose diagram for Oppenheimer–Snyder collapse presented in Andrew Hamilton's book General Relativity, Black Holes, and Cosmology (Fig. 7.20)

O-S collapse diagrams

Here, thin purple lines are curves of constant time, while thin violet lines are curves of constant $r$, while the outer boundary of collapsing dust cloud is thick red line.

A precise depiction based on explicit coordinate transformations would be ideal …

Algorithms for explicit calculation of Penrose diagrams are discussed in this paper:

This paper contains multiple examples of Penrose diagrams, including those for collapse of thin shell of “null dust” into a black hole:

Null shell collapse

Thin lines are curves of constant $r$, while the authors do not bother with curves of constant time, possibly because for dynamical spacetimes definition of time coordinate is inherently ambiguous, whereas radial coordinate is defined unambiguously for spherically symmetric spacetimes.

Python code utilized by this paper is available on GitHub but (currently) there seems to be almost no documentation.

Bonus points for the case of the evaporating black hole …

The authors of the above mentioned paper (with another researcher) also calculated multiple Penrose diagrams for various models of black hole evaporation:

The simplest such model (single null shell collapses into a black hole and subsequently evaporates in a single burst of Hawking radiation) corresponds to these diagrams:

BH collapse and evaporation

Thin lines are lines of constant $r$, green for $r<r_s$, pink for $r>r_s$. Two diagram variants correspond to black hole with central singularity (left) and without a singularity (right).

A.V.S.
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  • Great, thank you for drawing my attention to these papers. That is exactly what I was looking for. – pill Nov 29 '22 at 03:35
  • Great answer on Penrose diagrams +1. I wonder if you could please help with my perhaps much simpler question on a Penrose diagram of a white hole: https://physics.stackexchange.com/questions/740527/ - Classical GR (nothing quantum), not the maximally extended solution, just a simple time reversal of the original Schwarzschild / Droste solution. Thank you! – safesphere Dec 11 '22 at 17:26
  • @safesphere: The problem is that the causal structure of a white hole is very dependent on assumptions on how quantum gravity should work to allow such an entity. Lots of nice Penrose diagrams with white holes could be found in Martin-Dussaud&Rovelli. – A.V.S. Dec 13 '22 at 07:34
  • Thanks so much for the link, very helpful! I understand that quantum gravity is critical here in general to paint a complete picture, however I wonder if my specific and rather simple question can be answered without it. My question takes an astrophysical object described by the Schwarzschild/Droste solution (not maximally extended) attractive to matter today and reverses time (only inside, because outside the solution is time symmetric. Nothing changes outside, the object keeps being attractive to matter and we ignore what happens to it at the horizon. Can we draw a Penrose diagram for this? – safesphere Dec 13 '22 at 14:10
  • @safesphere: … Can we draw a Penrose diagram for this? Sure, see e.g. fig. 2. here (While that paper is about charged solutions that figure is not, look at the works cited there for original reference). – A.V.S. Dec 13 '22 at 14:57