Physics And Dimensions

[Cheerful Charlie]

The science of dimensions and space have evolved radically since the days of Euclid and Aristotle. 3 dimensions. 4. 10, or eleven? Dimensions as a basic feature of reality or an emergent property of deeper underlying phenomena. Here is an essay on all of this I find informative.

How many dimensions are there, and what do they do to reality? | Aeon Essays

Relativity says we live in four dimensions. String theory says it’s 10. What are ‘dimensions’ and how do they affect reality?

aeon.co

 

[steve_bank]

Technically I see dimensions as arbitrary units of measure. We measure distance, change, and mass quantitativly with meters, seconds, and kilograms.

Some based mostly in scifi see time as an independent reality. To me it is a clock measured in ticks of a clock.

To quantify a point in space from a reference frame you need x,y,z coordinates in meters and time in seconds. Hence 4 'dimensions' to our observable reality.

If i have a rectangular obect I have three dimensions in meters. Length, width, and height.

 

[Cheerful Charlie]

Read the article I linked to. One can use Cartesian coordinates to model more than three dimensions. Known as manifolds.

 

[steve_bank]

I have been through this before.

I will bow out.

 

[bilby]

Greg Egan does an excellent job of imagining how the universe might appear with different dimensionality, or different topology.

If you haven't read his stuff, I can highly recommend it, though it can tie your brain in serious knots.

As far as actual reality is concerned, we appear to inhabit a universe with four macroscopic dimensions, three "spacelike" and one "timelike". These differ in the way that distances are calculated: To establish the distance between two points in space, we can simply take the square root of the sum of the squares of the distances in each dimension (Pythagoras Theorem). When we seek to know a distance in spacetime, we sum the squares of the distances in space, and subtract the square(s) of the distance(s) in time, then take the square root of that total.

Lots of interesting things fall out of this geometrical difference between space dimensions and time dimensions.

In Egan's Dichronauts he imagines a world where one space dimension has been replaced by a timeline dimension - inhabitants experience only one "arrow of time", but geometry behaves very strangely in the remaining three dimensions.

Mr Bank derides this kind of reference to SciFi in "real" scientific discussions, but ultimately it's just a thought experiment - no different in kind from Einstein's imagining what his tram ride would look like, were he moving at a large fraction of c.

String theorists suggest that there may be many more dimensions, too small to experience at even the atomic scale, but large enough for subatomic particles to move around in, giving rise to various quantum mechanical phenomena, and explaining such observations as the weakness of the gravitational force, by comparison to the other three fundamental forces.

Whether these hypotheses rise to the level of theory depends on who you ask. Personally I'd like to see some real world experiments designed that could test the various hypotheses, before granting them the title of 'theory'. But the concept of dimensions beyond our familiar four is not particularly outlandish. We just don't yet have a good way to test for their existence.

 

[Swammerdami]

However many "rolled-up" spatial dimensions there may be, our universe has four macroscopic dimensions: 3 spatial and 1 temporal. Max Tegmark concludes that only this (3,1) configuration leads to universes interesting enough for the inhabitants to contemplate such things!

Some superstring theories have more than one effective low-energy limit corresponding to classical spacetimes with different dimensionalities. We argue that all but the (3 + 1)-dimensional one might correspond to ‘dead worlds’, devoid of observers, in which case all such ensemble theories would actually predict that we should find ourselves inhabiting a (3 + 1)-dimensional spacetime. With more or less than one time dimension, the partial differential equations of nature would lack the hyperbolicity property that enables observers to make predictions. In a space with more than three dimensions, there can be no traditional atoms and perhaps no stable structures. A space with less than three dimensions allows no gravitational force and may be too simple and barren to contain observers

 

[lpetrich]

First off, a lot of people call "another dimension" some kind of parallel universe. So I'll leave out that kind of "extra dimension".

We say that space has three dimensions because it takes three values to specify the location of each point in it. In fact, one can use this insight to define generalizations to arbitrary numbers of dimensions.

So a surface is a two-dimensional space, a curve a one-dimensional space, and a point a zero-dimensional space.

We can also extend the concept of space into other domains, and that is often done with color. That's what makes "color space" a meaningful concept, like RGB (3D) and CMYK (4D).

An important property of familiar space is the distance between points, and one can generalize that to a distance function d(A,B) for points A and B:
d(B,A) = d(A,B)
d(A,A) = 0
d(A,B) <= d(A,C) + d(C,B)

For rectangular coordinates in flat space, the distance function is essentially Pythagoras's theorem:

\(\displaystyle{ d(x,y) = \sqrt{ \sum_i (x_i - y_i)^2 } }\)

For a continuous distance function, this can be generalized with small differentials of coordinate values:

\(\displaystyle{ d(x,x+\Delta x) = \sqrt{ \sum_{i,j} g_{ij} \Delta x_i \Delta x_j } }\)

or to use the notation often used, with implied summation,

\(\displaystyle{ d(x,x+\Delta x) = \sqrt{ g_{ij} \Delta x^i \Delta x^i } }\)

In differential geometry, a raised index means transforming like a tangent vector under coordinate transformations and a lowered index means transforming like a gradient.

This generalization of Pythagoras's theorem is how curved space-time is handled in general relativity.

 

[lpetrich]

Distance functions can also be applied to abstract spaces like color spaces.  Color difference mentions several distance functions that have been used for color spaces.

Along with distance functions, one can define angle functions for small differentials of coordinate values:

\( \displaystyle{ \cos a(\Delta x, \Delta y) = \frac{ g_{ij} \Delta x^i \Delta y^j }{ \sqrt{ g_{ij} \Delta x^i \Delta x^j } \sqrt{ g_{ij} \Delta y^i \Delta y^j } } } \)

It uses \( g_{ij} \) - the "metric tensor" that appeared in the distance value.


How does one measure curvature while staying inside a space? By something called "parallel transport", moving a vector on a closed loop. When it returns to its starting point, it will be displaced by an amount that is essentially (curvature inside the loop) * (area of the loop).

On a flat surface, if one tries that experiment, one gets zero change, meaning zero curvature.

Here is an example on a close approximation of a sphere. You start on the North Pole and go southward to the Equator, looking southward as you go. Then you go eastward or westward, looking southward, sideways relative to your motion. Then northward back to the North Pole, still looking southward, backward relative to your motion. When you return to the North Pole, your facing direction will be different from its original value.

 

[lpetrich]

In the two-dimensional, zero-curvature case, one can derive all of Euclid's axioms, including the fifth postulate. More recent mathematicians have shown that Euclid implicitly used some additional axioms, like a geometric figure not changing shape if one moves it. So they have come up with an improved set of axioms that capture all of Euclidean geometry: the fifth postulate with axioms of "absolute" or "neutral" geometry: Axioms of Neutral Geometry

If one omits the fifth postulate, then the neutral-geometry axioms imply constant curvature that may be nonzero. The fifth postulate is equivalent to zero curvature.

 

[lpetrich]

When Albert Einstein worked out special relativity, one of his teachers from his university days made a remarkable discovery about it. Space and Time (Hermann Minkowski) - Wikisource, the free online library written in 1909.

Gentlemen! The concepts about time and space, which I would like to develop before you today, have grown on experimental physical grounds. Herein lies their strength. Their tendency is radical. Henceforth, space for itself, and time for itself shall completely reduce to a mere shadow, and only some sort of union of the two shall preserve independence.

What did he discover?

For familiar 3-space, the distance d is given by d² = x1² + x2² + x3³ using rectangular coordinates x1, x2, x3.

What HM showed was that Lorentz boosts imply an overall space-time distance measure:

d² = x1² + x2² + x3³ - c²*t²

Time is a kind of space, but with a reverse contribution to distance. One finds five kinds of space-time intervals:

• Future timelike
• Future null
• Spacelike
• Past null
• Past timelike

Null = zero total length. If t is large enough, then d becomes imaginary, a time instead of a space distance.

 

[lpetrich]

Albert Einstein himself was not very impressed. He thought that HM's work was "superfluous learnedness", and he joked that "since the mathematicians have tackled the relativity theory, I myself no longer understand it any more." But he eventually came to embrace it as part of working with curved space-time in general relativity.

3.9 Constructing the Principles

 

[lpetrich]

As to the diagram from Max Tegmark's paper, here is what's behind it.

Suppose that there is more than one time coordinate. Then it's possible to go in circles in two of those coordinates:

t1 = (1/w)*cos(w*T)
t2 = (1/w)*sin(w*T)

where the metric involving those two time coordinates is
dT² = d(t1)² + d(t2)²

and where w is a constant.

So that means that space-time must have only one time coordinate, only one coordinate where objects are constrained to move.

This means that we must consider space dimensions.

 

[lpetrich]

For space dimensions, consider orbits in fields produced by elementary particles, particles that are monopole sources of those fields. Sources like mass for gravity and electric charge for electromagnetism.

The force law for such fields is 1/r^D-1 for distance r and D space dimensions. This is the same as for geometric dilution of radiation from a point source.

So we must consider a particle in the Newtonian limit that travels in a central force.  Bertrand's theorem states that only two power laws give closed orbits: r and 1/r². Related to that is what power laws give stable orbits. One starts with a particle in a circular orbit and one then perturbs that particle's motion. Will it do a stable oscillation or will it either fall in or fly away?

Equations of motion in polar coordinates for a central force f(r) for r the distance from the center and (theta) the position angle in the plane of the orbit, and ignoring the particle's mass,

\( \displaystyle{ \frac{d^2 r}{d t^2} - r \left( \frac{d \theta}{d t} \right)^2 = f(r) } \\ \displaystyle{ \frac{d^2 \theta}{d t^2} + 2 \frac{d r}{d t} \frac{d \theta}{d t} = 0 } \)

The second equation gives us conservation of angular momentum:

\( \displaystyle{ \frac{d^2 r}{d t^2} - \frac{h^2}{r^3} = f(r) } \\ \displaystyle{ r^2 \frac{d \theta}{d t} = h } \)

and making the position angle the independent variable and using the reciprocal of r: u = 1/r:

\( \displaystyle{ \frac{d^2 u}{d \theta^2} + u = - \frac{ f(1/u) }{h^2 u^2} } \)

Substituting a power law f(r) = - K r^-p gives us

\( \displaystyle{ \frac{d^2 u}{d \theta^2} + u = \frac{K}{h^2} u^{p-2} } \)

Let's now expand around a circular orbit, expanding the radius in a series in the position angle:

\( u = u_0 (1 + e \cos (g \theta) + \cdots ) \)

For a circular orbit,

\( \displaystyle{ \frac{K}{h^2} (u_0)^{3-p} = 1 } \)

\( g^2 = 3 - p \)

Since p = D - 1, for D space dimensions,

\( g^2 = 4 - D \)

Circular orbits will be stable if g² > 0, unstable if < 0, borderline stable if = 0. So circular orbits are stable for D <= 3, borderline stable for D = 4, and unstable for D >= 5. For D = 4, the possible angular momentum for a circular orbit is fixed by the force-law constant.

Extending the analysis to quantum mechanics, one finds that the solutions are well-behaved only for D <= 3. For D = 4, some solutions are concentrated at the origin, and for D >= 5, all solutions are.

 

[lpetrich]

So a universe that can have inhabitants must have one time dimension and at most three space dimensions. The possible complexity of the universe's inhabitants increases with increasing number of space dimensions, and it may be hard to have much of a universe for fewer than 3 space dimensions.

So the only kind of space-time that can allow us to come into existence has 3 space dimensions and 1 time dimension.

 

[lpetrich]

Going to Swammerdami's link, On the dimensionality of spacetime by Max Tegmark

Notes an additional problem with a small number of dimensions. In general relativity, the equation for gravity is
(a certain part of the space-time curvature) = (gravitational constant) * (energy-momentum tensor: mass-energy/momentum density/flux)

For two dimensions, the GR equations require an empty space-time, while in three dimensions, the space-time curvature is completely specified by the energy-momentum tensor, making space-time locally flat in empty parts of space, meaning no force of gravity.

I'm familiar with the mathematics behind these assertions, and the mathematics is rather complicated.

MT argues that more than one time dimension would not allow for a one-way flow of time, from considering what one would need for predicting one part of the Universe from another part of it. That's the same sort of thing as the circular path in time that I'd earlier shown here.