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We know that a black hole behaves like a whirlpool or a tornado or any of the other rotating phenomena we experience on Earth.But the thing is, all these phenomena, except the black hole, are 2-Dimensional (plus 1 dimension of time) rotating vortices, they move objects from 2D planes to other 2D planes instantly, but through a 3D medium. My question is this, since a black hole is a rotating sphere(3D for that matter, plus 1D of time), does it mean that its endpoint in the vortex is 5D?

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The short answer is no, with some caveats to the effect of sort of, depending on how loose an analogy you want to make.

Sound propagation in a fluid is limited by the speed of sound, which can be used to define a "sound cone" structure analogous to the causal light cone structure in spacetime. This is a described by an acoustric metric, which could have an acoustic horizon when the speed of the fluid exceeds the speed of sound, and even an analogue of Hawking radiation.

A general acoustic metric for a perfect fluid has the form $$g_{\mu\nu}=\alpha^2\begin{bmatrix}-(c^2-v^2)&-\vec{v}\\-\vec{v}&\mathbf{1}\end{bmatrix}\text{,}$$ where $\alpha$ is a conformal factor. The Schwarzschild black hole can be put in this form, as in the Gullstrand–Painlevé chart is not just spatially conformally flat, but exactly Euclidean at every instant of time. One can imagine that a Schwarzschild black hole is like a drain sucking space down to the singularity at the local escape velocity.

The rotating Kerr black hole cannot be put in this form, although its equatorial slice can be. See Visser and Winfurtner (2005) for details, as well for a discussion as to why interpreting the metric as corresponding to a physical fluid is problematic even in the Schwarzschild case due to the the conformal factor.

However, Hamilton and Lisle (2008) found that the Doran chart of Kerr spacetime can be interpreted as a six-dimensional "Lorentz river" characterized not only by a velocity but also a twist. This really quite different from the acoustic fluid analogy, as the 'river' does not spiral inwards, but rather has an intrinsic twist that rotates infalling objects. Still, it is interesting in its own right.

River model of Kerr black hole, by A. Hamilton

Image by Andrew Hamilton. (Twist not shown.)


References:

  1. Visser, M., Weinfurtner, S. "Vortex analogue for the equatorial geometry of the Kerr black hole", Class. Quant. Grav. 22:2493-2510 (2005) [arXiv:gr-qc/0409014]
  2. Hamilton, A. J. S., Lisle, J. P., "The river model of black holes", Am. J. Phys. 76:519-532 (2008) [arXiv:gr-qc/0411060]
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