First, I want to thank azalynn and beetlebomb for inspiring this in some ways
I made comments in the past here on this blog referring to the fourth dimension as a spatial one. There are several arguments against this though.
The single dimension time and the three dimensions of space are unique. While one can "go through" time (the verbiage suggests a kind of spacial movement), it is illusory to think of time and space as anything similar. On another note, the fourth dimension, the kalabi-yao spaces and the extra dimensions of string theory are not "spaces" in any way. The fourth dimension has a unique character and is something we already "experience". We experience the first dimension as time. We experience the second but I leave that mystery to you if you don't already know. I do know and it's been stated elsewhere. Hence, when discussing things like wormholes in "startrek", these things are really "NOT POSSIBLE". This is because wormholes are based on a metaphor that is not truly analogous.
The classic discourse on wormholes starts with a 2D to 3D analogy that is easier to understand in terms of a calculus style dimension. (we can't merely assign arbitrary things like color to these dimensions except to make theoretical models but such do not reflect "reality" like a model of a real life block of wood...iow, graphs and charts don't really exist exactly as they are in reality, only as a mental construct and abstraction).
Back to the 2D to 3D analogy. Consider an ant on a donut. If the ant is confined to the 2D surface of the donut, it must go around the circumference or the "long way"....but if the ant can ignore the surface of the donut and hop off, run on the table top and jump back on the other side (hence utilizing the 3rd dimension that is not "on the 2D" surface of the torus or "donut"), it has essentially "3D wormholed there". 2D is a subset of 3D space in terms of describing movement and so is 1D for that matter.
What happens next in this discourse is a major aberration of logic. If we can have an ant wormhole to the other side of a donut and race say another ant there who must travel the long way, why not extrapolate to 3D and 4D? ie: space and 4D wormholes? IOW, we are no different than ants on the surface of a donut but to get to the other side of the universe, we merely have to jump off the donut surface of space and race to the otherside using a kind of 4D shortcut. Unfortunately, such is really not physically possible given that there is no such thing as an "ether" or "fabric of space". Sure, there is the ZEF energy that some might consider a kind of fabric but it really isn't. It is merely an energy like all others that occupy space. Scientists can make maps of space-time mathematically on computers and play with them like hypercubes on 2D screens and the like and warp them into shapes but space and time are not real shapes and are not the same as space itself. Sure, you could theorize relativistic "curving" and such from a modelled point of view but it is a model no different than a bar chart. Bar charts are not reality, they model something. Bars in real life have only height width and depth in terms of "real" dimensions that we can treat as a "real" axis and behave as such. Sure, we can play with the axis of a model but such is not reality.
The thing to consider here though is that the donuts is a curved surface. It curves in 3D space. The 1D path of the ant though fills a 2D circle around the top of the donut though. This analogy gives rise to two theories that are highly questionable but becoming exceedingly accepted in physics today without question and we need to question it more. It is not satisfactory to apply liberally any math construct we want on reality. For example, the idea that the big bang expanded space itself, that gravity compresses space or "curves" space. The idea that space can have any number of dimensions. Can we prove this? We can't use a frame of reference really to do this. The best measure is to observe space curving. Do we see this with lensing? Black holes? Or are we seeing something else. Is light really refracted by gravity? Sure, it isn't really hard to think that space curving is possible since we can bend paper etc. Can we bend something of an "ether" that has been also proven not to exist?
Back to 4D not being space. If 4D is not space, we cannot wormhole. We simply cannot shortcut real space....no matter how much gravity is present. Space by design is not shortcuttable IOW. Another issue is this. All things in curved space must retain their shape and volume in real space. But because there is no curved space, we don't need to worry about the major contradictions there.