Let's approach this mathematically: Xzibit at Aperture.

[Whirligig Girl]

Yo dawg, I put a portal in your portal so you can... well, what would happen? Let's try and figure this mathematically if possible.

Edited December 13, 2014 by GregroxMun

[Bill Phil]

What are you trying to accomplish?

[ZetaX]

Also, whatever you actually want to do: mathematics is a rigid and exact science. You will need to define precisely and accurately how portals supposedly work.

[Whirligig Girl]

Hmm. Well that'd be a problem. Let's ask a new question first then. How do (ignoring real scientific physics and how you'd actually accomplish this) portals work? They must work in the same way as they do in the game, boundaries aside.

[rtxoff]

Warping spacetime they do. Bringing close together two far away places they will.

[ZetaX]

To avoid problems of the kind "what is at the back of a portal, e.g. if I dig through the wall", we should probably assume that every portal is actually a portal on both sides. In other words, the backsides are linked, too; we simply can't see it in normal use.

Then we can define what they do: they change spacetime. So what does this mean exactly¿ For the following ignore everything in round brackets unless you want the formal mathematical definition; and even for that I will now ignore the time component because this makes it really ugly, but will come back to it at the end:

If we create portals, we actually choose two two-dimensional shapes of the same size and form (formaly: they are two 2-dim. smooth submanifolds which are closed subsets, come with a fixed smooth isometry and a fixed orientation).

Then we glue the points on the boundary together, so for each pair of points that "are the same" we now have a single point in spacetime. After that, we "fill in the insides" which is essentially how you would imagine it: a point close to portal A is now very close to the corresponding point at the other side of portal B, while the points that were once close to portal A's other side are now far away.

(

Formaly, we change the topology as follows. Let f be the isometry from portal A to B. Let the orientation give us a notion of "above" and "below", at least close to the portal. And let "eps-n" be a shortcut for epsilon-neighbourhood, i.e. balls of radius epsilon around a point. A basis of open sets, i.e. a system that tells us what is close and what is not, is now given as follows:

- For a point disjoint from the portals, just take all eps-n small enough to not intersect any portal.

- For a point x on the inside of portal A we let an eps-n of x be the union of {y| y in eps-n of x and on the positive side of A} and {z| z in eps-n of f(x) and on the negative side of B}. Note that this is only well-defined if epsilon is small enough.

- Similiar for points f(x) on portal B, with positive and negative interchanged.

- Points on the boundary of portal A get identified with the point f(x) on B's boundary, and we let an eps-n of x=f(x) be the union of the original eps-n of x and the original eps-n of f(x).

)

As you see, this is quite a mouthful. I don't think there is a much easier formal way, but the above should at least be what one usually means.

Now I want to briefly get back to the -time in spacetime:

We actually could have portals between different times. Actually, we necessarily get them as soon as we allow moving portals relative to each other or consider sources of gravity. By time dilation, one of them will then age slower than the other and the only way to consistently fix this will be to allow them to not go to the same time period. So just stop thinking of portals purely as between the same place, but let them be between two points in spacetime instead. We should probably assume some resolution of all the time-traveling paradoxes that could come with this, e.g. by having a deterministic paradox-free universe.

(Formaly, we would need to change the above construction to consider the portals' movement through spacetime and then would need to add some more cases on what happens with the topology at the moment the portal is formed or closed.)

tl;dr: Formalizing portals is lengthy, but gives you time-travel and therefore neat tricks: e.g. doing the infinite falling trick, but traveling backwards in time on portal-crossing; it ends when a portal "vanishes", i.e. when we travelled back far enough to reach a point where the next portal to go through doesn't exist yet.

Warping spacetime they do. Bringing close together two far away places they will.

Not exactly. This is not a continuous operation and will make some points further apart. We will this need to tweak the accepted laws of physics a bit, otherwise this won't be allowed at all.

Edited December 14, 2014 by ZetaX

[Three1415]

Also, just a note: Two-dimensional "portals," like the ones seen in the eponymous game, are impossible in a three-dimensional universe; any wormhole in our universe would appear to be a three-dimensional bubble-like structure, and entering from any angle would cause one to be transported to the other end, emerging with the same velocity (i.e, the same speed in the same direction).

[ZetaX]

The three dimensions are not a problem, see my post above for a way to do it. The actual laws of physics obviously are a problem, but that may as well hold for "normal" wormholes as you describe them (we don't know yet as far as I know).

Edited December 14, 2014 by ZetaX

[ZedNova]

Hi, Cave Johnson here. I just wanted to remind you all that how and why handheld portal devices work is a closely held secret, most of our trusted scientists don't even know how it works.

Edited December 14, 2014 by ZedNova

[lajoswinkler]

Also, whatever you actually want to do: mathematics is a rigid and exact science. You will need to define precisely and accurately how portals supposedly work.

No, math is not science. Math is math.

[ZetaX]

No, math is not science. Math is math.

It's indeed quite difficult to properly put it down, unless you just want to give it its own class. Mathematicians make predictions and try to check of they are true or false [or undecidable]. Yet it all happens in the mind, apart from that there is no direct relation to reality.

Thus it satisfies half of each, being some hybrid.

[Three1415]

After approximately two hours simply sitting here at my computer thinking about this, I have concluded that portals, which I define as "wormholes with surfaces topologically glued together" (or, alternately, "wormholes in n-dimensional space that consist of at least two n-1 dimensional balls whose surfaces are considered identical") are impossible in any n-dimensional space, not just for n=3.

First, considering the specific case of a portal composed of two 2-balls in three dimensional space, I have concluded that the structure would resemble a black hole whose event horizon had been "unfurled" from a 3-sphere into a 2-ball. For obvious reasons, this structure would likely be impossible to produce in this universe; it could simply not be generated, with any attempt yielding a "standard" black hole (more on this topic later in this post).

Generally, the reason such an object would have an appearance like that of an "unfurled" black hole is that neither light nor matter, once "in" the portal, would be able to get out again. This is more easily understood with light, and thus I shall use it as an example: As light in our three-dimensional world is transverse, with a magnetic component perpendicular to an electric component both perpendicular to the direction of motion, such a wave attempting to pass through a two-dimensional boundary region should get "stuck" and not be able to re-emerge, analogous to the behavior of a true event horizon. As matter also possesses wave-like characteristics, it, too, could get lodged in the two-ball as well. Likewise, this phenomenon would give rise to Hawking radiation emanating from the portal, and the description of this structure as a whole seems to demonstrate the holographic principle, wherein an object's surface (in this case the unfurled event horizon) should have characteristics indistinguishable from the actual object (in this case the black hole itself), which seems to emerge naturally from this hypothesis. As all of the above principles may be generalized to any number of dimensions, one can draw similar conclusions for these objects for any value of n.

I now consider the case of a mapped set of n-balls in n-dimensional space. This is equivalent to "rolling up" the object from the previous paragraph back into what we would think of as a "normal" (if that can really be applied to such an extreme structure) black hole: An n-ball with an n-sphere event horizon that swallows light and matter alike (due to gravitational forces this time, but this is essentially equivalent to the catastrophic tear in space-time that the portal displayed), more specifically a Schwarzschild (or perhaps the other types as well, depending on their parameters) black hole (to use an example from our universe). Like the mapped balls here, Schwarzchild black holes are considered to have two "ends," but the other side can never be reached; this is likely a generalization of the non-traversability of mapped balls in any number of dimensions.

Finally, I come to true wormholes, as in their classic formulation as Einstein-Rosen bridges. These are, like in the paragraph immediately preceding this one, n-balls in n-dimensional space, but they are connected by an n+1 dimensional "tunnel" traversing each mouth. Unlike the previously presented models of a wormhole discussed earlier, these do appear to be traversable, but must be stabilized with such strange things as exotic matter in order to not instantly collapse. Thus, my verdict on the subject of instantaneous transportation is:

[also serves as a tl;dr]

Portals (mapped n-1 dimensional balls in n-dimensional space): Impossible to create, and not traversable anyway. Analogous to a "standard" black hole.

Black holes (mapped n dimensional balls in n-dimensional space): Extant, but likely not traversable.

Wormholes (n dimensional balls in n-dimensional space, connected by an n+1 dimensional bridge): Possibly extant, probably traversable.

_________________________________________________________________________________________________________________________________________________________________________________

Hooray for walls of text! Also, keep in mind that much of this is speculative, if reasoned out, and by no means proven; gedankenexperiments can only take one so far, and not necessarily to the correct conclusion (although I am fairly confident in this analysis due to the emergence of real-world characteristics).

[ZetaX]

There is no "2-dimensional boundary region" as you describe. For example, the version I gave does not have one, the space where the portals are looks exactly like every other part of spacetime. The only exception is the one-dimensional boundary of the portals, but that's not the effect you describe.

First, considering the specific case of a portal composed of two 2-balls in three dimensional space, I have concluded that the structure would resemble a black hole whose event horizon had been "unfurled" from a 3-sphere into a 2-ball. For obvious reasons, this structure would likely be impossible to produce in this universe; it could simply not be generated, with any attempt yielding a "standard" black hole (more on this topic later in this post).

I am reading "I have concluded" or "for obvious reasons" and such way too often in your post. Please give an actual argument instead of repeatedly claming that.

Generally, the reason such an object would have an appearance like that of an "unfurled" black hole is that neither light nor matter, once "in" the portal, would be able to get out again. This is more easily understood with light, and thus I shall use it as an example: As light in our three-dimensional world is transverse, with a magnetic component perpendicular to an electric component both perpendicular to the direction of motion, such a wave attempting to pass through a two-dimensional boundary region should get "stuck" and not be able to re-emerge, analogous to the behavior of a true event horizon. As matter also possesses wave-like characteristics, it, too, could get lodged in the two-ball as well.

There is an error: you are assuming that everything that passes through a portal has to be at some point completely contained in the portal. The example I gave does not do that, for example. As another example, you could use the same faulty reasoning to argue that all universe is motionless and there is no light: consider any frame of reference and in that a single moment in time; both matter and photons are at some time in exactly that slice of spacetime; but as you said, photons and matter cannot exist in such a lower-dimensional thing alone.

Likewise, this phenomenon would give rise to Hawking radiation emanating from the portal, and the description of this structure as a whole seems to demonstrate the holographic principle, wherein an object's surface (in this case the unfurled event horizon) should have characteristics indistinguishable from the actual object (in this case the black hole itself), which seems to emerge naturally from this hypothesis.

While all of this being technically existing terms, this usage sounds a lot like technobabble. At least it is missing any arguments.

I now consider the case of a mapped set of n-balls in n-dimensional space.

You are aware that any soccer ball is an example of a 3-ball in 3-dimensional space¿ I simply don't get what you want to contradict here when there obviously exist such mappings.

Finally, I come to true wormholes, as in their classic formulation as Einstein-Rosen bridges. These are, like in the paragraph immediately preceding this one, n-balls in n-dimensional space, but they are connected by an n+1 dimensional "tunnel" traversing each mouth.

No, they are n-dimensional tunnels in n-dimensional spacetime. The (probably ill-defined) boundary would be (n-1)-dimensional.

Sorry of the above sounds harsh. But I have serious troubles to understand what you want to convey, the above is the result from trying to guess.

Edited December 14, 2014 by ZetaX