by raattgift | 2025-06-12 08:33:29+00:00
No, there's no change in dimensionality.

The swapping of timelike and radial dimensions are a "game" frequently played with families of coordinates, including Schwarzschild coordinates. One can apply any system of coordinates on a physical system without changing the behaviour of the physical system: coordinates are unphysical. Think of navigating around in a neighbourhood: you can talk about going forward a few blocks then turning left, after which you go forward two more blocks; or for the same journey, going "city north" a few blocks then going "city west" two blocks. Here assuming that (initially) "forward" is in the "city north" direction (and "city north" is not necessarily exactly magnetic north nor a section of a meridian of longitude). After the left turn, "forward" is "city west". There's an analogue to the discussion's (ab)use of Schwarzschild coordinates.

In Schwarzschild spacetime, without applying any system of coordinates, just floating in free-fall far from a black hole extremizes your travel in the timelike dimension. (You can do this at home: you stay put at some point on Earth (whether you use GPS latitude/longitude/altitude or some other system of coordinates) but your wristwatch keeps ticking). Inside the black hole horizon, just floating in free-fall extremizes your travel in the direction of the singularity. Far from the black hole, accelerating as strongly as you can in any direction takes travel from the timelike dimension and puts it into one or more spatial dimensions. In particular, you have the freedom to increase the spacetime interval between you and the singularity. Within the horizon, however strongly you accelerate the spacetime interval between you and the singularity shrinks. This behaviour seems to invite the use a different set of coordinates applied to a patch of space around an observer far from the black hole and a patch of space around an observer inside the horizon. It's some human cognition thing, and in the early 20th century it took decades to discover systems of coordinates that work for observers far from the black hole, at the horizon, and inside the horizon. And even today, most people don't seem to try to enhance physical intuitions by swapping among arbitrary systems of coordinates (including no coordinates) on a single physical system like a black hole and a pair of observers (one inside the horizon and one far outside the black hole).

The Schwarzschild black hole interior is still locally Lorentz-invariant everywhere (because the whole Schwarzschild spacetime is a Lorentzian manifold).

The various local interactions of the Standard Model will keep working inside a black hole. In a really tiny patch around every point, everything behaves as if its in Minkowski space (flat 4-d (3 spatial + 1 time) spacetime).

(That's one of the problems of quantum field theory on curved spacetime: the "focusing-pressure" [for experts: this is encoded in the Weyl curvature tensor; my "scare quotes" take a view of this in a Raychaudhuri equation way] gets so high that the unknown ultraviolet behaviour of the Standard Model (a quantum field theory) becomes relevant. The Weyl behaviour in Schwarzschild is that quasispherical objects are strongly prolated with the long axis aligned radially: a soccer ball or basketball starts looking like an American or Canadian or Rugby football ball. The radial stretching "spaghettifies" by ripping apart weaker bonds (like intermolecular ones, and molecular ones, and ionizing atoms), but the tangential squashing ("focusing") must eventually generate more nuclear interactions, probably up to quantum chromodynamics (QCD) energies possibly before the radial stretching starts generating hadronization.

How this works in the Standard Model is just unknown. However simpler "test" quantum field theories (fewer, or even no, interactions; and often no colour-confinement-like processes) raise really difficult questions.

Finally, back to the Standard Model as local theory: how does any allegedly quantum nonlocality behaviour work? Local here in the sence that states can be distinguished by local measurements alone. Related questions: can you entangle particles deep inside a black hole? If an entangled pair fall in together, how does the entanglement evolve? Or obsessing black hole information people, what if you throw in only one half of an entangled pair and locally measure the outside half? Nobody has great answers for these sorts of questions at present, and there's no near-term hope of testing any proposals in laboratories or via astrophysical observation.