# What's the distance metric for a compactified dimension?

• B
• Paige_Turner
In summary, the people who answered the question before it was closed said that compactified dimensions are not topology, that you can't ask the question until you learn QFT and M-theory, and that you can't even ask the question until you learn Euclidean 2D + 1 extra, better.
Paige_Turner
TL;DR Summary
They're dimensions, so they DO have a metric equation, right?
I asked that question on another forum here. The 2 answers I got before the question was closed by an angry mod said:
• You wouldn't understand the answer.
• You're too lazy to look up the answer in [a GR textbook that I don't own].
• Compactified dimensions aren't topology. Go ask got somewhere else.
• You can't even ask [that simple question] until you learn QFT and M-theory.
Maybe I asked it wrong because I'm autistic, but that's not the way Feynman would have answered it differently. First of all, he would've answered the question--starting with something like "Well, the metric for compactified manifolds works a little differently, see, and…"

Now I'm asking again: They're dimensions, so they have a metric equation, right?

Note that when I came here, i said that I'd be thrown out and not know why. Clearly it's started the way it always does: mysterious anger.

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Delta2, Motore and weirdoguy
Paige_Turner said:
Now I'm asking again: They're dimensions, so they have a metric equation, right?
What is your answer to the same question about the Euclidean space? It has dimensions, so what is their metric equation?

In more than four dimensions, the metric is written the same as in four dimensions, just with correspondingly more components.

edit: I just noticed you're asking specifically about a compactified dimension... Well, once you have a compact space, the distance between points becomes path-dependent - the distance you travel to get from one point to the other, can depend on which way you travel along the dimension!

You could define "the distance" as the minimum, the length of the shortest path. But since this is physics, it's probably time to learn Riemann's approach to geometry, in which the metric tensor encodes a deformation of the Pythagoras (or Minkowski) distance formula that varies from point to point, and the length of a path is the integral of the infinitesimal line element along the path.

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weirdoguy
Use just one extra dimension, which is the original idea of Kaluza (and Klein). Being compact, it is a circle. Or, violating a little the rules, use a compact but open line (a segment). Think euclidean 2D + 1 extra, better. You paste a segment in each point of a plane.

Well maybe @PeterDonis is trying a bit harder than he should to keep some quality standards for PF. He leaves absolutely no room for personal theories, speculations, crack pottery e.t.c.

Ok but I got the feeling that if we were living at the end of the 19th century, and there was internet and PF back then, and Einstein was coming to PF to discuss his ideas on special relativity (a few years BEFORE he publish his ground breaking paper "On the electrodynamics of moving bodies"), Peterdonis would have closed the thread as personal speculations.

I guess it is the price we pay for keeping high quality standards here in PF, some good ideas might be rejected as well along with the majority of ideas which are indeed crack pottery though.

Delta2 said:
and Einstein was coming to PF to discuss his ideas on special relativity (...) Peterdonis would have closed the thread as personal speculations.

Of course he would, and I see nothing wrong about it since PF is not for that. Besides, I think that first and foremost Einstein would have discussed his ideas with his collegues scientists, not on some random forum And that's what real scientists do.

weirdoguy said:
Of course he would, and I see nothing wrong about it since PF is not for that. Besides, I think that first and foremost Einstein would have discussed his ideas with his collegues scientists, not on some random forum And that's what real scientists do.
Oh come on, physics forums is not some random forum, it has quality and some real scientists as well (like you for example and of course ME e hehe, seriously speaking people like chestermiller, peterdonis, vanhees and many more are real scientists i think) .

I am afraid you will have to rephrase some of your comments :D.

Delta2 said:
Well maybe @PeterDonis is trying a bit harder than he should to keep some quality standards for PF. He leaves absolutely no room for personal theories, speculations, crack pottery e.t.c.

Ok but I got the feeling that if we were living at the end of the 19th century, and there was internet and PF back then, and Einstein was coming to PF to discuss his ideas on special relativity (a few years BEFORE he publish his ground breaking paper "On the electrodynamics of moving bodies"), Peterdonis would have closed the thread as personal speculations.

I guess it is the price we pay for keeping high quality standards here in PF, some good ideas might be rejected as well along with the majority of ideas which are indeed crack pottery though.
Why are you complaining about these things here?!

martinbn said:
Why are you complaining about these things here?!
well,sorry I understand its kind of off topic.

Paige_Turner said:
I asked that question on another forum here. The 2 answers I got before the question was closed by an angry mod said:
Look at the thread, fresh wasn't angry, or close to it.
Paige_Turner said:
Didn't see anyone say you wouldn't understand the question. Didn't see anyone tell you to ask about a Riemann sphere instead. He said he recommends you look INTO it, because that is a great place to start when talking about compactification of dimensions! It is one of the first examples you learn about when talking about compact surfaces! fresh isn't a physicist, he is a mathematician.
Paige_Turner said:
You're too lazy to look up the answer in [a GR textbook that I don't own]
"There's a pretty decent explanation in "Gravity" by James Hartle (the only undergraduate-level general relativity textbook I known of) including an example of a metric tensor for a manifold with a compactified dimension."

How you interpreted his statement, and what was actually written is completely different. Nothing about what he said implies you are lazy, he just cited a relevant textbook. You want to know the correct reply to this? "Hey, I don't have access to that textbook, would you be able to quote some relevant information?"

Paige_Turner said:
Compactified dimensions aren't topology. Go ask got somewhere else.
This is where your ignorance is shining. He said IF you want to talk about M-theory, come here. If you look at your opening post, you said "Does energy flow cyclically between pairs of dimensions?". Clearly, this has nothing to do with pure mathematics. His follow up IS correct, which was "It is something else to ask about the topology of compactifications. For questions about them, I recommend choosing some easier examples like the Riemann sphere." Now, if you wanted to take his advice to heart, you would have seen (even on the wiki!) that a Riemannian sphere IS A TYPE OF compact surface. It is a great place to start (as stated above)!
Paige_Turner said:
You can't even ask [that simple question] until you learn QFT and M-theory.
You're the only one saying this question is "simple". However, that wasn't what you were told, you were told: "However, it is an extremely technical subject, that cannot be dealt with on a "B" level thread. You will need to learn QFT and string theories first, and there is no shortcut." What about this says you can't ask this question? It just says you can't ask it with the "B" qualifier. It's a technical subject that assumes knowledge of QFT.

Paige_Turner said:
Maybe I asked it wrong because I'm autistic
No, you didn't ask it "wrong". What you're doing "wrong" is your interpretation of the responses you've received. As stated above, I detect no anger in any of the responses in your previous thread. If you want to see anger on physicsforums, go read some discussions between martinbn and demysterfier on bohmian mechanics! Fresh, and nuga were pretty normal from my perspective (and I'm sure theirs!).

Paige_Turner said:
that's not the way Feynman would have answered it differently. First of all, he would've answered the question--starting with something like "Well, the metric for compactified manifolds works a little differently, see, and…"

I doubt Feynman would be able to succinctly answer a question about compactified metrics because it isn't a physics question. Why you think he would say they work "differently" is bizarre as well because, it's just a metric. It either satisfies a set of axioms, or it doesn't.

Paige_Turner said:
Now I'm asking again: They're dimensions, so they have a metric equation, right?
If they satisfy the axioms of a metric, then yes. If they don't, then no. This isn't a "yes" or "no" question despite how badly you want it to be. You have to actually know how to check conditions of axioms in order to determine if a space qualifies to have additional structures. And since you refuse to (or can't) give us relevant examples of what you mean by "dimensions" or "compactified dimension", this is as far as you will be able to go.

Paige_Turner said:
Note that when I came here, i said that I'd be thrown out and not know why. Clearly it's started the way it always does: mysterious anger.
Once again, no one is angry. You may even interpret my replies above to be angry, but they are not. I would advise you not take thread closings, or people questioning you as "anger".

arivero, Delta2 and berkeman
Paige_Turner said:
Summary:: They're dimensions, so they DO have a metric equation, right?

I asked that question on another forum here. The 2 answers I got before the question was closed by an angry mod said:

berkeman said:
berkeman said:
Oh lordy. . . it's a downright miracle. . . . .

.

berkeman
So, can we focus in the question now?

I still think the best way to answer it is to consider a 2D surface when you add an extra dimension which is compact but has a boundary. This is, a segment of a line.

So now the 2D + 1 d surface is just a 3D volume, easy to visualize. Is this a good startpoint? I would like to hear feedback from OP and interested parties.

Assuming it is, let's us now to consider the obvious way to add an extra dimension compact and without boundary. Just add periodic conditions to the previous one.

The corresponding visualisation is a "lasagna", where all the layers are the same layer. The OP wants to know the minimum distance. This visualisation is powerful enough select all the points at distance k of a given point: draw a sphere, and the number of layers you must cross is the number of wraps in the extra dimension.

## 1. What is a compactified dimension?

A compactified dimension is a dimension that has been "rolled up" or made small and compact, typically to explain the behavior of particles at very small scales in physics. This concept is often used in theories such as string theory and Kaluza-Klein theory.

## 2. Why is a distance metric needed for a compactified dimension?

A distance metric is needed for a compactified dimension in order to quantify the distance between points in that dimension. This is important for understanding the behavior and properties of particles in that dimension.

## 3. What is the difference between a distance metric for a compactified dimension and a regular distance metric?

A distance metric for a compactified dimension is different from a regular distance metric in that it takes into account the compactification of the dimension. This means that the distance between points may not be linear, as it would be in a regular distance metric.

## 4. How is a distance metric for a compactified dimension calculated?

A distance metric for a compactified dimension is typically calculated using mathematical equations and concepts from differential geometry. This involves considering the curvature and topology of the compactified dimension.

## 5. Can a distance metric for a compactified dimension be visualized?

Yes, a distance metric for a compactified dimension can be visualized using mathematical tools such as graphs and diagrams. However, since compactified dimensions are often at very small scales, it may be difficult to visualize them in a physical sense.

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