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A wormhole usually helps us to create a shortcut to a distant object through a higher dimensional space. Can the time factor in that higher dimensional space differ from our three-dimensional space? I mean that for example staying in a wormhole for a second (higher dimensional space) may cost a year in our three-dimensional space. Can such a time dilation happen in higher dimensions? If there could be more, please answer in detail.

It's an important factor in my story. Would it be too illogical if I added it?

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    $\begingroup$ In the classic Einstein-Rosen bridge concept I'm thinking of, wormholes do not pass through "a higher dimension." They instead use the properties of spacial warping to decrease the distance between two points. Now, I don't know a lot of theoretical physics to really answer your question, but I have read a lot of science fiction, and many popular authors I can think of used the concept of time dialtion inside wormholes. In Peter F. Hamilton's Commonwealth Wealth Saga, it's a crucial plot point to suspend a great portion of people inside of a wormhole for them to emerge later on. $\endgroup$ – B.fox Jan 6 '18 at 14:47
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As @B.fox explains in his comment, our understanding of wormholes does not include another dimension. When you pass through a wormhole, you are still in 3D linear time space.1 However, it is thought that gravity must still exist in a wormhole. Therefore, your question may still be answerable.

I suspect people think that wormholes pass through other dimensions thanks to movies like Interstellar, which correctly portray the opening of a wormhole as a sphere (3D), not a hole (2D, think Star Trek: Deep Space Nine, ), in space ... This might lead people to conclude that traveling through it must push you to a "fourth" or another dimension. This is not the case, insofar as we understand wormholes. An amazingly simple presentation of wormholes is found at Space.com

Think of a wormhole as a shortcut. Space is twisted around, like twisting a piece of paper, to bring two points closer together. Because the effects of gravity still exist in a wormhole, if that shortcut "passed near" a black hole, your passage through a wormhole would experience time dilation.2 If (much more likely) one of the mouths of the wormhole were near a black hole, then, too, you would experience dilation. No matter where the wormhole forms, if you pass through it with enough velocity, you will experience time dilation (this assumes the structure of the wormhole will allow for such speeds...).

As for whether or not time dilation can happen at higher dimensions... That depends on what you mean by a "high dimension" (and since we have yet to empirically prove they exist, it's pure speculation). One way of looking at a "higher dimension" is to add dimensions of existence, e.g., time. A building is 3D, but its existence is 4D due to the passage of time (it begins new and changes or decays during a measurable period of time). Because this perspective is no different than the lives we live now, time dilation is the act of modifying the 4th dimension, which isn't something you "pass-through" as much as it is something you "measure."

If what your talking about is a dimension of physical expression, such that a tesseract is a 3D shadow of a 4D cube having four lines drawn from the same vertex, each at 90° offset from the other, then we really don't know other than to conclude that such a cube must still have mass, and therefore gravity, and therefore could be affected by the gravity of a black hole, which means travel through that dimension would still experience time dilation due to proximity to a black hole or speed.

Conclusion

Yes, you can experience time dilation when traveling through a wormhole and yes, if you consider other dimensions to be dimensions of physical expression, then you can experience time dilation in those dimensions, too.

Finally, let me offer some advice. A good story has more to do with how you write it than it does the precision of the science you use as its foundation. A good example is Star Trek IV: The Voyage Home, where Checkov and Uhura collect photons through a closed bulkhead door. We, the moviegoers, don't really care that what they're doing makes no sense because we're enjoying the story, not the science. Even in Science Fiction, science is only the window dressing that lets you tell a good story. Don't get too hung up on whether theoretical science supports the story you want to tell. Focus on telling a good story.


1A lot of assumptions are occurring here. The most common example of demonstrating a wormhole is to fold a piece of paper, basically creating a 3D shortcut through a 2D existence. Maybe this means when space is folded we're taking a 4D shortcut through a 3D universe ... maybe not. I'm fond of a scene from Stargate SG1 where aliens are talking about a wormhole with Samantha Carter. "Oh," she exclaims," you mean like an Einstein-Rosen bridge!" "No," responds the alien, "it's not like that." Underscoring the simple reality that we barely understand what we're talking about and claiming wormholes do or do not take us through another dimension is jumping to a conclusion.

2The problem with this concept is that if space is only "folded" like folding a piece of paper, then it can't happen. If space, being 3D, is "twisted," then it might. It's a murky subject, and explaining it as if space were 2D is very much part of the problem.

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Time dilation is very possible "probably even most probable" in theory.

The Einstein-Rosen bridge concept was developed in 1935 to solve the singularities in space-time generated by a point mass as calculated by Karl Schwarzschield 1916. A black hole would arguably suffice as point mass to create a wormhole (Einstein-Rosen bridge). The important thing here is that not only space but also time converge to a singularity, so time will behave differently inside a black hole. Some mathematical solutions from string theory even suggest it can run backwards if I remember correctly.

Now the downside, since the wormhole is inside the black hole you will never be able to enter it before being ripped apart into your sub atomic particles. Even these particles will likely not come through due to the very high instability of the wormhole. Fuller and Wheeler 1962 calculated that the worm hole would not stay open long enough for a single photon to get through.

As conclusion currently worm holes only exist as mathematical construct that even if correct could not be used by space ships or possibly even single particles. That being said don't get hung up on the real physics in a science fiction but just invent the physics as part of your sci-fi

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