If the embedding is smooth but not required to be isometric, an arbitrary spacetime should need at least eight dimensions [1]. This may be fine if the "seven dimensions" in the story refers to seven spatial ones (and some temporal ones unmentioned), except that we might have trouble embedding multiple universes into it in a physically reasonable manner.
However, things immediately get a lot more complicated if you insist that the higher-dimensional space has its own metric structure that is not independent from out spacetime. For globally isometric embeddings (i.e., the embedding preserving geometrical structure everywhere and not just topological structure). For general 3+1-dimensional spacetimes, the minimal bounds for a globally isometric embeddings are not known, but the best ones proven to work are ca. 90-dimensional (even in the globally hyperbolic case).
[1] Whitney's theorem holds this for four-dimensional Riemannian manifolds. I haven't checked the details for the pseudo-Riemannian (Lorentzian) case, but at least in the globally hyperbolic case, we can foliate the spacetime with spacelike hypersurfaces, define a universal cosmological time (possibly subject to some additional assumptions), and Wick-rotate it to get a Riemannian manifold. Not being globally hyperbolic can only make the situation worse, hence "at least".
Surlethe wrote:Does light need to behave the same way in the extra dimensions?
If we postulate that it's pseudo-Riemannian, yes. But if it is also flat (Minkowski), then continuing the attempt to embed spacetimes into it, we'd find that there solutions of the Einstein field equations, even vacuum solutions, that are not embeddable in flat Minkowski spacetime E^{1,n} for any n. For example, if a spacetime contains closed timelike curves (e.g., Gödel's rotating universe), then it cannot be imbedded in E^{1,n} because in the latter no timelike curve can meet itself again. The general embedding results alluded to above make use of either two or three timelike dimensions (but only one timelike dimension is needed for globally hyperbolic spacetimes, which never have CTCs).
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Starglider wrote:I came up with the notion of the spatial geometry between the 3d 'bubbles' of individual planes being chaotic, such that there is no direct correspondence between a point in hell and a point in our universe. A portal (or even strong enough signal) would locally flatten this geometry and create a 1:1 correlation over a small area of space in each plane.
Here's an alternative rationalization. Instead of trying to force globally isometric embeddings, which is something that is doomed to failure anyway due to an insufficient dimensionality, we can have postulate that the embedding is merely smooth and only isometric in the vicinity of the portal, and that this is a requirement for portal operation. This has certain immediate advantages:
1. It explains why objects in our universe do not normally experience the higher-dimensional space: in most places, the metrical structures of the universe and the higher space are mostly independent, and only the topological structure is embedded. Portals happen in regions of our universe in which the metric exactly matches the induced metric from the higher space, i.e., in regions where the embedding is locally isometric.
2. As per above, portals might even happen naturally by chance, when the metric structure of our universe just happens to match the higher space. Possibly, there is some additional "coupling" requirements, but overall, a higher probability of natural formation allows better consistency with the attitude of the demons not to actively investigate such things. (But see complication in #3.)
3. Since under that assumption, the only geometrical distortion required of the of the universe to locally match the background higher-dimensional space, energy requirements for a portal may vary depending on location in the higher space, but in some places they might happen to be very low. On the other hand, it may still be very hard. Even neglecting whatever cost to actively "couple" their geometries might be (I wouldn't know), there are two physically distinguishable kinds of curvature, Ricci and Weyl; the former represents gravitational sources, and the latter everything else. Manipulating the former is in principle much easier than the latter, and it may happen that the portal required no energy-momentum whatsoever (no Ricci curvature) but does require a gravitational wave emitted by wiggling binary stars (lots of Weyl curvature), or some other such practically impossible requirement.
The next bit might require a bit more explanation. The Einstein field equations state that the stress-energy-momentum tensor corresponds to the (trace-reversed) Ricci curvature tensor. Since the latter is a symmetric bilinear form, it has n+C(n,2) = C(n+1,2) independent components. In four dimensions, that's 10 independent components, and in five, 15. The full curvature depends on the second partial derivatives of the metric components, of which there are C(n+1,2)²-nC(n+2,3) = n²(n²-1)/12 [I can provide the counting argument if anyone's interested, but the left-hand form should be a substantial hint]. That's 20 in four dimensions and 50 in five. Observation: (4-D curvature components) 20 < 50-15 (5-D Weyl curvature components). That doesn't actually prove anything rigorously, but it does motivate the result of a certain embedding theorem:
4. Any four-dimensional spacetime can be locally isometrically embedded in a five-dimensional Ricci-flat spacetime. This has a potentially profound consequence for the 'elapsed time' discussion previously, for the simple reason that it four-dimensional stress-energy is converted into higher-dimensional "gravitational radiation" (Weyl curvature), and so would experiences no proper time whatsoever in transit at the speed of light. (And with certain exotic conditions in the higher-dimensional space, such as CTCs or even more than one temporal dimension, the coordinate time lag for a round-trip may be arbitrarily short, no matter what the distance between source and destination is.)
5. Additionally, it may somewhat explain why simply dumping more energy at the portal doesn't collapse it: once the conditions of the portal are established, it just keeps dumping more gravitational radiation into the higher-dimensional space (there may be a hard limit on that, but it's plausible that we have no device of sufficient energy and power that saturates it). On the other hand, the high requirements of arbitrarily-curved global (read: "for large regions") isometric embeddings (ca. 90 dimensions) mean that certain arrangements of energy-momentum in the region of the portal may cause it to no longer be isometrically embeddable, thus collapsing the portal.
Darth Wong wrote:If mages can "flatten" the geometry of Hell to make portals, this would imply that the spacetime of Hell is extraordinarily malleable, and that its shape is not correlated to mass/energy as it is in our universe. Which, in turn, makes you wonder how gravity works in Hell.
That may vary. See #3 right above.
Darth Wong wrote:Perhaps it's best not to inquire too deeply :)
That way might lie wisdom, but this way lies amusement...
I can't help it. I am a sick, sick man.