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Warning, pop science coming.. please correct what I’m getting wrong. Einstein’s equations of relativity showed the potential for existence of wormholes that can connect different points in space time. I understand the mechanisms for their practical implementation are nothing near feasible. However, based on the equations of gravitational “tunneling”, I can traverse back and forth between times and locations. Wouldn’t this require a higher dimension than 4d space time?

That is, we’re moving from a point that we would think of as the present to another point we would think of as the present. If this were feasible, Would These “presents” need to be on a traversable continuum?

To my lay brain, This seems as though there are points along a higher dimension where what we would consider the future is currently present, and what we consider the past is also present. That the world we see is determined and laid out as slices in a higher dimension that would be traversed with a wormhole, and that we normally traverse in a single direction.

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  • $\begingroup$ what you are describing is not "determinism" as used in physics. en.wikipedia.org/wiki/Determinism $\endgroup$ – anna v Oct 14 '20 at 4:48
  • $\begingroup$ Thanks, even more evidence for my laymanism, if anyone needed it. I’ll Change the title. $\endgroup$ – Michael James Oct 14 '20 at 4:49
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Wormholes in GR do not require higher dimensions. It easier to imagine curved spacetime as being embedded in higher dimensions, but the usual mathematical description of curved spaces does not require that.

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    $\begingroup$ Does a curved space by itself not already imply existing in a higher dimension than the space itself? $\endgroup$ – Flater Oct 14 '20 at 11:28
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    $\begingroup$ @Flater: It is completely possible to describe a curved surface mathematically without any reference to how it is "embedded" in a higher-dimensional space. In other words, there is no need to refer to any higher-dimensional space to describe the physics. And from that point, Sir Isaac said it best: "We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances." $\endgroup$ – Michael Seifert Oct 14 '20 at 13:08
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    $\begingroup$ @Flater For a very simple example, the standard hyperbolic metric on the upper half plane is typically defined without any reference to an embedding, although such an embedding exists. $\endgroup$ – Denis Nardin Oct 14 '20 at 13:40
  • $\begingroup$ @DenisNardin Well, any surface can have some density function on it so as to have a "shape" - but if this function resolves clearly as an object in a higher dimension I'd say the implication is there. A sphere surface is two dimensional, but the properties of such a surface implies the existence of a higher dimensioned object. Such resolution can, of course, be arbitrary or random but it does beg the question. $\endgroup$ – Stian Yttervik Oct 14 '20 at 13:53
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    $\begingroup$ @StianYttervik The Nash embedding theorem is true, but not obviously so. and in particular I don't even know how many dimensions you need to embed the hyperbolic plane. Another example is that the universal cover of any (pseudo)Riemmanian manifold has a canonical (pseudo)Riemmanian structure, but it doesn't come with any preferred embedding. I am quite unsure of what you mean with "implies the existence of a higher dimensional object"... For the record, when I think about manifolds I rarely think of them as embedded somewhere (but I am a mathematician :)) $\endgroup$ – Denis Nardin Oct 14 '20 at 13:57
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Sadly I do not really understand everything you said. But I can comment on this

wormholes that can connect different points in space time

The thing is, that all you really need to know is exactly which points are connected or "next to each other". You do not need any higher dimensional space for this.

Take for example 6 points called P1, P2, ..., P6. I will use notation A<->B to say, that A and B are connected.

To represent line, the information required is that P1<->P2, P2<->P3, ...,P5<->P6

To represent circle you have P1<->P2, P2<->P3, ...,P5<->P6 and P1<->P6, which connects the end points together.

On this "space" you can form a "wormhole" by connecting P2 to P4.

The thing is, that these connections require no knowledge of some higher dimensional space. All the information is encoded using the points of the space you have.

If you wish to read more about the topic, the mathematical structure that encodes this information is called topology.

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  • $\begingroup$ But your circle is, from the perspective of someone in that space, a one-dimensional space Movement is one-directional, and whether this space is curved or not doesn't really change that. In order for that new path (wormhole) to make sense, you need it to not lie on the original path (because then it wouldn't be new), which requires a 2D space. I think your example mostly works because you chose to represent a linear 1D space in a 2D space from the get go, thus not needing to upgrade it later down the line. Make your example a straight line and it won't work. $\endgroup$ – Flater Oct 14 '20 at 10:52
  • $\begingroup$ @Flater I did not mention any dimensions. I just have a set of 6 elements and said which ones are neighbors. This elements might be numbers (1,2,3,4,5,6) for example. There is also no curvature on this set defined yet. The difference between line and a curve is that end points are connected, while on the line they are not. You can define the distance along the path on this set (topology) by the number of elements you need to go through from one element to the other. Thus on the line, there is only one path from 1 to 6 and it goes through 5 elements, thus the distance is 5. (cont.) $\endgroup$ – Umaxo Oct 14 '20 at 13:09
  • $\begingroup$ @Flater (cont.) but on a circle, you can go directly from 1 to 6, thus the distance is 1. The distance from 2 to 4 on a line is 2, but if you connect them, you can go directly and it reduces to 1. I know this is all very abstract, but the point is, that I do not need to assume any higher set in which my original set is contained to make this structure. For an ant living on the line this is the same. He has knowledge about how many paths there are from 2 to 4 without needing any knowledge about two dimensional space in which his 1D universe might be embedded $\endgroup$ – Umaxo Oct 14 '20 at 13:13
  • $\begingroup$ @Flater In GR we have no knowledge of additional dimensions, we live entirely in our 4D spacetime. Thus the theory talks only about distances, angles, paths and so on. It does not talk about higher dimensional spaces and embeddings, it only talks about geometry induced on the 4D spacetime. Take for example paper rolled into a cylinder. From 3D view, it is curved. But from 2D view, some ant living on it would never notice this. For it, the rolled paper is flat. All the lines, distances, angles obey (based on its measurements) euclidean geometry. $\endgroup$ – Umaxo Oct 14 '20 at 13:20
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Agree to Rd Basha. Embedding spaces are only necessary for the mathematical constructions. They don't necesserily have physical reality.

Like the mathematics of a 2-sphere is easier if it's embedded into a 3-dimensional Euclidean space. But the 2-sphere happily exists without a third physical dimension.

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    $\begingroup$ An embedding space is not necessary for the mathematical construction, i.e. the definition of a differential manifold. However, it can be used to construct a manifold. Moreover, each manifold can be embedded in a higher dimensional space, see for example the Nash embedding theorem for the Riemannian manifold version. $\endgroup$ – Johnny Longsom Oct 14 '20 at 10:15
  • $\begingroup$ Yeah, thanks. So we agree about embedding spaces not necessarily have physical reality. $\endgroup$ – RobertSzili Oct 14 '20 at 10:27
  • $\begingroup$ Depending on your definitions & philosophical leanings, you might not consider any of the spaces (or spacetime, or the Universe) to necessarily have physical reality. $\endgroup$ – D. Halsey Oct 14 '20 at 18:00
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I guess so. At least according to the illustrations/analogies of folding paper. However there is nothing in Einstein's equations that require an existence of a higher dimension unlike in string theory. But if wormholes are proven to exist, then yes this could prove the possibility of higher dimensions since there is no other way for wormholes to work.

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