I was trying to imagine what 3D objects would look like in 4D and I came across this question about visualizing the 4th dimension here: Visualizing the 4th dimension.

It was said that you could visualize 3D objects as projections/shadows of 4D objects. Almost all the questions I could see about this talk about 3D cubes being considered like an intersection of a Tesseract. This following question contains an answer with a nice visualization of the 3D intersections of a Tesseract as an animation: What would a tesseract actually look like?

Which lead me to thinking about what would a visualization of a black hole (in our 3D universe) look like as a 4D object? i.e. What kind of 4D object would have to exist to produce an intersection that would look like a black hole? In particular what would the singularity look like, also an infinitesimal point in 4D? I guess the event horizon could be considered as a (hollow?) sphere?

  • We already visualize black holes in 4D. 4 coordinates are needed to fully specify a black hole. Often, we begin by working with "inertial coordinates," i.e. coordinates that, far from the black hole, are just the usual $x,y,z,t$ coordinates on flat space-time. But sometimes these aren't the best coordinates for working with a black hole, so we define new coordinates in terms of the old one that give us a better picture of what's going on. In any case, we always need $4$ to fully describe the system. – Charles Hudgins Sep 08 '19 at 09:22
  • @CharlesHudgins Thanks for the reply. Perhaps I misunderstood what it means to be a 4D object. I was thinking about how a cube (3D) could be considered an intersection of a Tesseract (4D). So I was curious what kind of 4D object would produce an intersection (3D) that looked like a black hole. Like what be the equivalent of the Tesseract to the Cube. Or am I getting completely confused about the whole subject? Since you already mentioned that 4 coordinates are needed already to describe a black hole, does that mean 4 coordinates are always used to specify 3D objects like a cube? – CasualUser24 Sep 08 '19 at 10:46
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    @CasualUser24 "*what would the singularity look like, also an infinitesimal point in 4D?*" - A Schwarzschild singularity is not a point, but asymptotically an infinitely long spacelike line representing a moment of time when the 3-cylinder space of the inner Schwarzschild metric shrinks down to this line and time ends. A singularity is not an object in space, but a moment of time. It does not "exist", but "happens" everywhere at once when "everywhere" becomes "nowhere". See: https://math.stackexchange.com/questions/2929400/is-the-schwarzschild-singularity-stretched-in-space-as-a-straight-line – safesphere Sep 08 '19 at 12:51
  • @safesphere Thank you, that was really helpful. It seems I had quite a bit of misunderstanding about the subject and perhaps my question wasn't well formed. Although I have some math background I only have a layman's understanding around the subject. – CasualUser24 Sep 09 '19 at 15:27
  • @CharlesHudgins Describing a black hole using a 4D metric is not the same as visualizing. The Schwarzschild spacetime can be isometrically embedded in a manifold with the minimum of 6 dimensions. So a black hole can be fully visualized only n 6D. For example, a sphere is a 2D surface described using 2 coordinates, but it cannot be visualized in 2D, only in 3D. – safesphere Sep 09 '19 at 17:38

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