Can Distances Be Defined Without a Coordinate System?

[Antonio Lao]

Is the use of a frame of reference for the sole purpose of defining a distance?
Can distances be defined without a coordinate system?

 

[Antonio Lao]

Is the one and only reference frame (ether frame) another word for absolute rest? And if there are many reference frames can each defines a relative rest?

 

[sol2]

I think of the Quark to Quark measure and the metric. Consider the energy value

Hyperspace, by Michio Kaku Page 84 and 85,

"To see higher dimensions simplify the laws of nature, we recall that any object has length, width and depth. Since we have the freedom to rotate an object by 90 degrees, we can turn its length into width, and its width into depth. By a simple rotation, we can interchange any of the three spatial dimensions. Now if time is the fourth dimension then it is possible to make "rotations" that convert space into time, and vice versa. These four-dimensional "rotations" are precisely the distortions of space and time demanded by special relativity. In other words, space and time have mixed in a essential way, governed by relativity. The meaning of time as being the fourth dimension is that time and space can rotate into each other in a mathematical precise way. From now on, they must be treated as two aspects of the same quantity: space-time. Thus adding a higher dimension helped to unify the laws of nature."

http://www.superstringtheory.com/forum/extraboard/messages12/493.html

 

[PFanalog57]

Is the use of a frame of reference for the sole purpose of defining a distance?
Can distances be defined without a coordinate system?

A tensor equation that is valid for one coordinate system is good for other coordinate systems. Coordinates are used for convenience, it seems.

Worldlines seem much more "realistic" than imaginary paths and tangent vectors. Coordinate independence, is very interesting.

Richard Feynman's sum over histories-path integral, gives a particle's worldline, from point A to point B. The action principle applies, it appears that the universe has laws to maximize efficiency.

 

[arildno]

Is the use of a frame of reference for the sole purpose of defining a distance?
Can distances be defined without a coordinate system?

Isn't that what Ernst Mach was up to, using solely the relative distances between the particles as parameters?

 

[wisp]

Is the one and only reference frame (ether frame) another word for absolute rest? And if there are many reference frames can each defines a relative rest?

It is not necessary to have an absolute frame of reference to determine distance measurements, because with the ether model distances are invariant regardless of whether frames of reference are used.
It is also possible for the ether in different parts of the universe to be in relative motion, and so, locally it is possible to have several “local” absolute frames of reference. If our motion relative to the ether is zero, then we can say we are at rest in an absolute (local) frame of reference.
The term absolute is used in relation to the motion of the ether.
There can exist one theoretical absolute frame to which everything is references, but this has no affect on how the laws of physics will work at a local level.

 

[Antonio Lao]

sol2,

From now on, they must be treated as two aspects of the same quantity: space-time. Thus adding a higher dimension helped to unify the laws of nature."

Space is basically three dimensional. Can subtracting dimension (say from 3 to 2, 2 to 1, 1 to 0) also helped more to unify the laws of nature?

 

[Antonio Lao]

Isn't that what Ernst Mach was up to, using solely the relative distances between the particles as parameters?

I think what Mach was up to is more to do with absolute acceleration. In a universe devoid of matter, Mach argued that the distinction between spinning and not spinning is not possible, that is no motion or acceleration if there is no frame for comparison.

 

[Antonio Lao]

Russell,

Worldlines seem much more "realistic" than imaginary paths and tangent vectors. Coordinate independence, is very interesting.

Can you tell the dimensionality of worldlines or distances (curve lines)? Is this a stupid question?

Worldlines are embedded in 4-dim spacetime but they are still lines hence 1-dimensional (maybe a higher kind of one-dimensionality?).

If n is the level of dimensionality then the total dimension of spacetime is given by 3n+1. 4-dim is where n=1, 1st level. 7-dim is where n=2 (2nd level), 10-dim is where n=3 (3rd level).

 

[sol2]

sol2,



Space is basically three dimensional. Can subtracting dimension (say from 3 to 2, 2 to 1, 1 to 0) also helped more to unify the laws of nature?

The standard model in its formation has to follow a defined pathway? When you get to that point, what is revealled of that energy, but the "potential=a probabilistic deterination at Planck length?)" for this point to be expressed in the dimenisons you are asking? Rotation?


The effect of adding an extra compact dimension is more subtle than that. It causes the effective gravitational constant to change by a factor of the volume 2pR of the compact dimension. If R is very small, then gravity is going to be stronger in the lower dimensional compactified theory than in the full higher-dimensional theory.

http://superstringtheory.com/experm/exper5a.html

There is a much simplier generalization that I am looking for. You have to move accordingly from euclidian coordinates, to understand gaussian curvatures as well understanding you have unified the dynamcis into GR. But it goes past this with Kaluza and Klien.

The lesson here fro me was to understand this progression from the work of the Fifth Postulate, and with the help Sacchheri, Gauss and Reinmann Minkowski, Lorentz, and Bolyai, we can see this progress of the dyamical nature that does not just end in classical definitions? You have to leave the sphere as someone said here yesterday.

By looking at the metric in the quark to quark distance we are given a view of this dynamical gravity field as "dimension?"

 

[Antonio Lao]

What I am doing in my research is to give a definition of a total dimensionality of reality based Level of Existence (LOE).

Total dimension, D= 3n+1, where n is the level. And the "1" is really the one dimension of time.

For n=0, D=1 (this is the level I'm using for my theory of quantized 1-dim space where spacetime and force are equivalent).
For n=1, D=4 (probably 4-dim spacetime of SR and GR)
For n=2, D=7
For n=3, D=10 (probably 10-dim of superstring theory)

 

[sol2]

I am looking for a link to Hyperphysics for demonstration of what you are saying. In the meantime I place this for consideration. Link will be placed here when ready.

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/imgqua/qhar.gif


The Kaluza-Klein compactification of strings can be done on more than one dimension at once. When n dimensions are compactified into circles, then this is called toroidal compactification, because the product of n copies of a circle is an n-torus, or Tn for short.

http://superstringtheory.com/experm/exper5a1.html

http://superstringtheory.com/forum/extraboard/messages12/792.html

I had a demonstration for one to see this, but the link is gone. In it's place, imagine the famous belt trick. Greg Egan is very helpful here. As well, Helicoid visualizations.

 

[Antonio Lao]

For n=0, D=1, the structure of a doubly twisted Moebius strip splits into a Hopf ring.
This ring is representable by vector products forming two distinct ring structures, $H^{+}$ and $H^{-}$

$$H^{+} = F_1 \times r_1 \cdot F_2 \times r_2$$
$$H^{-} = r_1 \times F_1 \cdot F_2 \times r_2$$