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Is a Kruskal diagram a 2D flat space projection of Schwarzschild space-time diagram? If not, isn't it true that one could not draw one accurately on a paper?

BTW, I am not referring to Penrose Diagrams. What I mean is, if I try to draw a diagram with $R$ and $T$ axes, it won't fit on a 2D paper. Instead, I would need to bend it in a third dimension (for example, a Flams paraboloid is a Schwarzschild diagram with $R$ and $\theta$ and is a 3d structure looking like a well).

is my reasoning correct?

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  • $\begingroup$ I assume the diagram you are referring to is the Penrose diagram. And yes, it is a 2D representation in which the angular part is suppressed. $\endgroup$
    – Pipe
    Commented Oct 19, 2023 at 5:12
  • $\begingroup$ As I understand it, there are two separate "projections" (I & II are usually paired together as below, and III & IV are "duplicates", they are sometimes used to "illustrate a white hole", but I'm not sure that is 100% rigorous). $\endgroup$
    – m4r35n357
    Commented Oct 19, 2023 at 14:43
  • $\begingroup$ On further consideration, I am pretty sure the white hole argument is wrong. The light cones in regions III and IV must be aligned so as to prevent particles traversing from region IV to region I etc., in other words pointing downwards. That points the "arrow of time" towards the singularity in IV, making it merely a (flipped) "reflection" of the BH in region I, not a "white hole". $\endgroup$
    – m4r35n357
    Commented Oct 19, 2023 at 15:43

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Is Kruskal diagram a 2D flat space projection of Schwarzchild space-time diagram?

No, Kruskal–Szekeres diagram, such as this one, taken from Wikipedia: Kruskal–Szekeres diagram from Wikipedia

is a 2D Minkowski diagram of a conformal map of radial plane (i.e. $\theta=\mathrm{const}, \phi=\mathrm{const}$ section) of Schwarzchild spacetime onto (part of) $1+1$ Minkowski spacetime.

Fixing the angular variables, the radial plane metric of the Schwarzchild spacetime in Kruskal–Szekeres coordinates reads: $$ ds^2= \frac{32G^3M^3}{r}e^{-r/2GM}(-dT^2 + dX^2).$$ The part in parentheses is the 2D Minkowski metric in Cartesian coordinates $(T, X) $, while the $\frac{C} {r} e^{(…)} $ is a conformal factor. Conformal factor is not represented in the diagram, while lines of constant Schwarzchild coordinates $(t, r) $ are plotted in the $(T, X)$- plane.

If not, it couldn’t be drawn accurately on a paper.

Arguably, any 1+1 spacetime cannot be accurately represented on paper since paper is not a Lorentzian spacetime. But yes, because this is a conformal diagram it does not accurately represents distances and durations even for horizontal and vertical separations.

Nevertheless, such diagram allows one to reason about causal structure of the Schwarzchild spacetime since radial null curves here are the lines with $\pm 45°$ slant.

If you are interested in an isometric embedding of Schwarzchild radial plane, it could be realized as a surface in $1+2$ Minkowski spacetime such as in this figure:

Figure from Marolf's paper

taken from the paper:

The paper contains many more figures, and a follow-up paper helps visualize other spherical symmetric black hole solutions.

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  • $\begingroup$ Can you have a look at your "follow up" link - it seems to be broken. Cheers. $\endgroup$
    – m4r35n357
    Commented Oct 19, 2023 at 12:17
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    $\begingroup$ @m4r35n357 Fixed, thanks! $\endgroup$
    – A.V.S.
    Commented Oct 19, 2023 at 12:58

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