Dear Marina,

Kaluza's idea is great. To get the electromagnetic field from the geometry of the fifth dimension of a curved 5-dimensional spacetime, some symmetry conditions have to be imposed to the geometry. The main problem is, what takes care of these conditions to be satisfied everywhere? If the fifth dimension is like the others, it does not necessarily have these symmetries. Also, from experimental viewpoint, there is no way to see the fifth dimension, unless those symmetry conditions are broken. Yet, the very existence of the electromagnetic field can be viewed as proof of the fifth dimension, subject to those conditions. Now, the thing is that, by imposing these conditions, what we obtain is a special case of space, named technically principal bundle, with gauge group (the group of symmetries) given by the rotations in plane. The group of these rotations is, from geometric viewpoint, a circle, so the fifth dimension is compactified, but it can't be probed anyway, not because it is small, but because of the gauge invariance. So, the gauge theory viewpoint, which is the most accepted, is in fact the Kaluza-Klein viewpoint. At least from the viewpoint of geometry. But most physicists see the phase parameter (the fifth coordinate) to be just a parameter, and don't load it with a geometric interpretation. This may be a smart move, because there are other ways to introduce gauge dimensions, than simply adding dimensions. For example, the role of the extra fifth dimension can be played by the phase of the wavefunction of the electron, whatever this is. So, we keep what is good, but try to assume as little as possible.

Best regards,

Cristi

Dear Cristi,

thank you for your reply! You speak of the 5th dimension, while I speak of only 4, all of them spatial though. In this regard, I have a question about the GR:

In GR, the spacetime is said to be 4-dimensional, 3 are spatial and 1 is time. GR is also said to be geometrical. But, in simple examples where geometry is commonly employed, like tracing a trajectory of a cannonball, we always align the time dimension with one of the spatial dimensions, in the direction of the movement.

Imagine a simple graph: a thick horizontal line at the bottom, a cannon 'standing' on this line on the left and a dash-line of the parabola traced by the cannonball in the 'sky'. On this simple 2D X,Y graph, Z spatial dimension is missing; however it is implied. Time dimension is also present: it is simply added, superimposed, with the X spatial dimension. Here the spacetime is 2D (idealized parabola on a plane). Both dimensions are spatial. The time dimension is superimposed on the pre-existing 2D space. And please note that the time dimension is always aligned in the direction of the movement -- imagine it was going perpendicularly, or even in the opposite direction lol -- but why not, if it is simply a dimension just like space?

Now, the Einstein's 4D field equations. Why can't I say that they too are about the 4-dimensional space (as in 4 spatial dimensions) with time being superimposed on this 4-space to show the 'movement'? Of course, the movement here is more complex than in the cannonball example, but it is still a movement in space. Doesn't it show the dynamic curving of a 3D surface of a 4D "object", just like artists commonly illustrate it in popular science magazines?

In other words, please give a good argument that in the GR field equations, the time dimension is not superimposed on 4D space, just like it is in the cannonball example.

Thank you very much for your feedback :)

Best regards,

-Marina

Dear Marina,

Kaluza-Klein theory has 4 space dimensions plus one time dimension. If I understand well, you want to replace time by a space dimension. But the analogy of cannonball really is in 1 space plus 1 time, not just in 1 space. So, if you want to use this analogy and add a fourth space dimension, it seems to me that the fifth is kept, and is time. Unless I don't understand what you mean.

Sometimes is useful to make time to look like space. There is a trick of this type, named Wick rotation, used to solve some problems in quantum field theory and quantum mechanics. Also you may want to look for imaginary time, and Hartle-Hawking state.

But I am sure you don't have this in mind either.

Maybe you will find what you want in this essay, I think you can contact the author on his this year's essay page. He may have some ideas related to what you said, but I don't know how far did he advance with them.

Frankly, I don't think I understand what you mean, and what problems do you think this can solve, to beat the way relativity is done. To know how to continue this discussion, may I ask you how well do you know general relativity? I would ask you to pick one of these:

(1) "I know about special and general relativity for pop-science books or documentaries, with no equations at all. I want to discuss using analogies with school level mechanics"

(2) "I understand special relativity including the equations, and also electrodynamics. I know general relativity only at pop-sci level."

(3) "I understand both special and general relativity including with equations, and differential geometry as it is used in general relativity."

(4) "(3), but in addition, I am familiar at the mathematical level with Kaluza-Klein and gauge theories."

Then, if your choice is greater than (1) you may try to translate your arguments from (1) to the level of your choice. Maybe you can compare your ideas with those from GR, and show what would be the advantages of your theory. You asked me for a good argument for GR against your proposal, but I don't see what are the arguments supporting your proposal. GR has plenty of arguments supporting it.

I want to confess that I don't find very productive the discussions about physics which remain at level (1). This may help understanding, build an intuition, but it is just a first step. More efficient arguments can be made at the more advanced levels.

Best regards,

Cristi

Dear Cristi,

thank you for your reply :) To answer your questionnaire, I am definitely above the first level, without the technicalities of course, which probably would place me at level 2 (-? I was not quizzed in school on this).

But here is what I meant with my question. I am not looking for ways to "to beat the way relativity is done". I am only looking for the evidence that the universe is actually 4-dimensional (as in 4 spatial dimensions) and this requires not invention of something new but a slightly different way of looking at the already familiar.

If you consider the historical context in which relativity and Minkowski spacetime appeared, the 4th spatial dimension was the hottest topic in both popular culture and science at the time (more so in the popular culture, a 'little' fact largely forgotten today). The relevant science was vigorously developed during the XIX century, and in the last quarter of it, the ideas spilled from the universities into the popular culture and captured the people's imagination.

What followed was that already by the turn of the XX century, the popular culture took over the subject of the 4th dimension to such an extent that it became unseemly for a serious scientist to even mention it.

Just to give you an idea of the atmosphere that surrounded the topic at the time, spiritualists were the first to "colonize" the 4th dimension. It became the "afterlife" where spirits roamed and from where they "communicated" with the living -- them being in the 4th dimension "explained" why they could not be seen directly. But this was just the tip of the iceberg. Several popular books were published, of which Edwin Abbott's Flatland is the most notable and survived to this day. The 4th dimension was mentioned in philosophy and literature, employed in art and even music. The marketplace was abuzz with most wild speculations. Basically, at the time, everything unexplained or simply misunderstood was promptly dispatched into the 4th dimension -- and this was supposed to "explain" it all.

This was the cultural context in which relativity and Minkowski spacetime appeared. To speak of the 4th spatial dimension, other than in the abstract areas pertaining exclusively to geometry and topology, was equivalent to debasing science to the level of the marketplace -- and physics is not an abstract science. Physics is about reality. To speak of the reality of the 4th spatial dimension, at the time, was akin to giving validity to all the craziness that went on. The 4th spatial dimension became almost an unspeakable topic in physics simply due to this cultural pressure.

Now, knowing this, I wonder how physics would develop without this cultural context. My hunch is that Minkowski went out of his way to veil the 4-dimensional setting by stressing that the 4th dimension was time -- after which it became the tradition and the way of looking at things. And so with my question I wanted to point out that, when we speak of movement, in all our models, time is always simply aligned with one of the preexisting spatial dimensions.

In any rate, thank you for allowing me to pick your brilliant brains. I found that I learn the quickest from a few words of a specialist like you than from studying hundreds of pages of books. But I understand your reluctance to participate in this one-way exchange. I only wanted to consider the reality of the extra spatial dimensions -- now that the cultural context is vastly different lol.

Again, thanks a lot :)

Best regards,

-Marina

Dear Marina,

"This was the cultural context in which relativity and Minkowski spacetime appeared."

I would not credit too much the "market" eager to find where to place the spirits, and avid of magical tricks involving the fourth dimension, for the following reasons.

In 1905, Einstein did not mention time as a fourth dimension. He only realized that the Lorentz-Fitzgeranld and Poincare transformations are a good replacement of Galilei's transformations, able to accommodate electrodynamics.

In 1905, Poincare realized that Lorentz transformations are in fact rotations in a four dimensional space, where one of the dimensions was imaginary. Rotations were already known in any number of dimensions. They are the linear transformations which preserve a quadratic form. It was known that not all quadratic forms can be diagonalized to have only 1 on the diagonal, some of them had some -1. This was the case for Lonrentz's transformations.

In 1907, Minkowski developed this idea more, and took it more literally than Poincare.

In my opinion, it was not the cultural context you mention which triggered this, but rather both were possible by the mathematical developments of the XIX century. Spaces with more dimensions were known to mathematicians. Quaternions, which live in 4 dimensions, were also known. An important role was also played by the non-Eucldiean geometries and Riemannian geometry.

I think that the main paradigm which made time in special relativity to be interpreted as a fourth dimension was Klein's Erlangen program. The idea that symmetry groups are at the core of the newly appeared geometries was so simple, and efficient, and the work of Lorentz, Einstein, Poincare, and Minkowski, made it to be applicable to special relativity too. This is why time was accepted as the fourth dimension. Einstein's idea of relativity of simultaneity becomes so natural when we realize that the different observers are oriented along different directions in a four dimensional spacetime. Lorentz transformations are rotations in the fourth dimension, so by changing the velocity, we rotate out of our initial space, in time. Of course, because the quadratic form which is preserved during the Lorentz transformations, is not positive definite, there are directions which cannot be rotated one into the other. Space cannot be rotated to become time and conversely. Space and time directions are separated by the light cone. This prevents us for accelerating until we can travel in our past. So, we may say that we cannot actually test that time is the fourth dimension, more than a parameter, by traveling in time as we travel in space. But except this, which is predicted by the theory itself, the evidence is abundant. Energy and momentum become married in four dimensions, electric and magnetic potentials too, and many other physical quantities are unified like this. By seeing that Poincare invariance is so universal, and that indeed lengths and durations depend on the observer, we realize that all possible evidence is there. Moreover, just by combining Poincare invariance with quantum mechanics, maintaining the spirit of the Erlangen program, we obtain the particles, classified by the spin. The evidence is overwhelming, and it can't be just a package of special relativity, to be sold to a marked intoxicated with ideas about fourth dimension. This evidence tastes better when you try to understand it mathematically. Then, you see that all parts are connected, and not a collection of disparate ideas that can be modified or replaced with other ideas so easily.

Best regards,

Cristi

Dear Cristi,

thank you for your reply :) I'm afraid you completely misunderstood what I was saying about how the marketplace influenced the development of physics in the early XX century. You said "In my opinion, it was not the cultural context ... which triggered this, but rather both were possible by the mathematical developments of the XIX century." Exactly!

My point was that, at the time, the whole society was so intoxicated with the idea of the 4th dimension that already by the turn of the XX century, the mere mention of it provoked in most people a knee-jerk-like reaction of the type 'please! not again!' The 4th dimension had become synonymous with the most wild, unbridled speculation. That's what I meant by the 'cultural context'.

Imagine there was no internet, no TV and not even radio. What most people did for entertainment was talking, speculating about various tings; and the 4th dimension was the main 'entertainment' for decades. A new generation, including those who later on defined the developments in physics, grew up hearing or reading about it almost daily, in various contexts, often one crazier than next.

And so my point was that young physicists in the first quarter of the XX century were negatively conditioned against the 4th dimension. I believe this is why the idea was not pursued as much as it should have been; and why, despite the overwhelming mathematical evidence, the mainstream physics --100 years later!-- still has not accepted its reality (and by extension, society as a whole).

I researched this info couple of years ago, trying to understand why this happened; and my conclusion was that in the beginning of the XX century on, it became almost impossible to present novel ideas that involved the 4th spatial dimension. The cultural setting was such that a mere mention of 4D was enough to make most people wince! At the time, it took courage to seriously speak of the 4th D -- no one wanted to be dismissed outright by mere association with the topic that by then had become synonymous with wild speculations. That's how, I believe, the marketplace influenced the development of physics in the XX century.

As for the history of the 4th dimension in mathematics, maybe you remember the last-year essay by Renate Quehenberger . I thought it contained one of the best historical reviews of the 4th dimension in science 'in 9 pages or less'.

Again, thank you for your feedback, I'm glad we agree that the mathematical evidence for the 4th spatial dimension is overwhelming. Now that the cultural setting is vastly different and it's once again cool to talk about the 4th dimension, let's hope it gets accepted by the mainstream soon :)

Again, thank you for your feedback!

Best regards,

-Marina

  • [deleted]

  • Post #260

PS

re mathematical evidence, I wanted to go further than what you said above and so well-reviewed by Renate in her 2012 essay. I wanted to make a case that our beloved general relativity, the jewel in the crown of physics, is in fact set in 4 spatial dimensions. It's a case of one way of looking at things vs another, which is conventional -- but it could have been the other way around. I am reading again Minkowski 1908 talk Space and Time - pages XV on.

..oops! I found this PDF looking for the best original Minkowski but only now, when I checked that the link I gave you worked, I looked at the preface and it's just what I've been looking for!

Thank you again, Cristi! I knew that talking to you would be most productive :)

-Marina

Dear Marina,

Thank you for the clarification. I think you have a very good sense of how the public's expectations were at that time. You say "I'm glad we agree that the mathematical evidence for the 4th spatial dimension is overwhelming". So far, the evidence pointed to a fourth dimension which is time, and the square of a timelike vector has opposite sign than the square of a spacelike vector. In this respect, there is a difference between space and time. So, the arguments I mentioned support the idea of time as the fourth dimension. This point is widely accepted by the mainstream, although they may not all acknowledge it. For example, the mainstream demands theories to be Lorentz invariant, hence they accept the fourth dimension. But many of them don't like the consequence, that space and time form a continuum, a block world, and this is why I said "they accept, but they may not acknowledge". On the other hand, I don't know of arguments supporting the idea of a fourth space dimension, other than those coming from the Kaluza-Klein theory and variations. In "variations" I include superstring theory, and here, again, there is a curious phenomenon. The mainstream is divided. There are so many working in superstring theory, and so many papers, and so many supporters, and there are in the same time many skeptics. But my feeling is that most accept with more ease the extra dimensions in superstring theory, but doubt the possibility that time itself is a dimension as well. Accepting extra dimensions as in string theory requires to accept something which is not directly seen, but seems to explain a lot of what we see. On the other hand, accepting time as a fourth dimension gives us the impression that future is already there, and it is fixed, and we know that we can choose freely. I explained why the two don't necessarily contradict one another in page 5 of my essay, and more in refs. [17-19].

Best regards,

Cristi

8 days later
2 months later

Hi Cristi,

Congrats for the Prize.

You Jennifer Nielsen and Douglas Singleton, Elias Vagenas, & Tao Zhu are the only positive news on the ridiculous and shameful "results" of this Essay Contest.

Cheers,

Ch.

    • [deleted]

    • Post #264

    Dear Christian,

    Thank you for the congratulations. It was a pleasure to meet you here. I liked your essay and the discussions we had on this forum.

    Best regards,

    Cristi

    Write a Reply...