3 ms·

I haven't read the paper, but I suspect the whole concept of a 4, 5, 6, or 7-dimensional black hole is nonphysical, but the math works out if you just close you

I haven't read the paper, but I suspect the whole concept of a 4, 5, 6, or 7-dimensional black hole is nonphysical, but the math works out if you just close your eyes and assume such a thing can exist. The reason I suspect n=7 is as high as it goes is because the volume to surface area ratio of an n-dimensional sphere is r/n, which means that the higher n you have, an n-sphere of radius r has a much larger boundary for the same volume. Conversely, that means that for a given surface area, you have less volume, so, for a constant mass density, you're enclosing less "stuff" inside.

akomtu 3y ago
What prevents the formation of many sub-universes with different space-time configurations? We happen to live a large 3+1 one, but others may exist. Speculating further, some black holes could be bridges between e.g. a 3+1 world and a 4+3 world. The observation about 7-dimensional spaces would mean that above that level there are no such boundaries between world, they all become one universe.
- tsimionescu 3y ago
  I believe that when people talk about N-dimensional spaces they typically talk about spaces with a Euclidean metric unless they explicitly say otherwise. The properties of a sphere in an N-dimensional Euclidean space are not all the same as those in an N-dimensional Minkowski-like spacetime (either 6+1 or 4+3 or any combination).
codethief 3y ago
> The reason I suspect n=7 is as high as it goes is because the volume to surface area ratio of an n-dimensional sphere is r/n, which means that the higher n you have, an n-sphere of radius r has a much larger boundary for the same volume. No, I haven't read the paper in detail either, but the reason is very likely the same for why Schoen & Yau (mentioned in the article) could initially only prove their Positive-Mass Theorem up to spatial dimension n=7: In n=8 spatial dimensions minimal hypersurfaces can suddenly develop singularities[0] and you can no longer treat them as manifolds but need tools from geometric measure theory (so-called minimal currents) to describe them. This makes proofs relying on minimal-surface theory in dimensions n > 7 much, much harder (though not necessarily impossible). [0]: The singularities, when zooming in, essentially look like the tip of the 7-dimensional Simons' cone, https://encyclopediaofmath.org/wiki/Simons_cone https://encyclopediaofmath.org/wiki/Simons_cone , making the latter the prototypical example. In fact, the existence of Simons' cone is the whole reason why (i.e. essential to the proof that) stable minimal hypersurfaces in n > 7 can have singularities but in lower dimensions they cannot and are instead smooth: There are no singular stable minimal hypercones in lower dimensions.

akomtu 3y ago

What prevents the formation of many sub-universes with different space-time configurations? We happen to live a large 3+1 one, but others may exist. Speculating further, some black holes could be bridges between e.g. a 3+1 world and a 4+3 world. The observation about 7-dimensional spaces would mean that above that level there are no such boundaries between world, they all become one universe.

tsimionescu 3y ago
I believe that when people talk about N-dimensional spaces they typically talk about spaces with a Euclidean metric unless they explicitly say otherwise. The properties of a sphere in an N-dimensional Euclidean space are not all the same as those in an N-dimensional Minkowski-like spacetime (either 6+1 or 4+3 or any combination).

codethief 3y ago

> The reason I suspect n=7 is as high as it goes is because the volume to surface area ratio of an n-dimensional sphere is r/n, which means that the higher n you have, an n-sphere of radius r has a much larger boundary for the same volume. No, I haven't read the paper in detail either, but the reason is very likely the same for why Schoen & Yau (mentioned in the article) could initially only prove their Positive-Mass Theorem up to spatial dimension n=7: In n=8 spatial dimensions minimal hypersurfaces can suddenly develop singularities[0] and you can no longer treat them as manifolds but need tools from geometric measure theory (so-called minimal currents) to describe them. This makes proofs relying on minimal-surface theory in dimensions n > 7 much, much harder (though not necessarily impossible). [0]: The singularities, when zooming in, essentially look like the tip of the 7-dimensional Simons' cone, https://encyclopediaofmath.org/wiki/Simons_cone https://encyclopediaofmath.org/wiki/Simons_cone , making the latter the prototypical example. In fact, the existence of Simons' cone is the whole reason why (i.e. essential to the proof that) stable minimal hypersurfaces in n > 7 can have singularities but in lower dimensions they cannot and are instead smooth: There are no singular stable minimal hypercones in lower dimensions.