By R.F.J. van Linden
From various points of view a fractallike universe is described. Unlike in
usual fractals, recurring patterns correlate with the number of dimensions in the observation, i.e., zooming
out and in occurs by adding or removing dimensions rather than changing scale. In this way, the forces of
Nature together with their fermions and bosons are ordered hierarchically through their number of dimensions.
The first sections shows some examples of physical phenomena
that reappear in spaces with one dimension less or more. The sections describe
1, 2, 3 or 4 dimensional worlds that are closed in the next higher dimension. In
section 5, this will be put in a broader context,
covering all of Nature's forces and particles.
1. Closed universes in n dimensionsThe descriptions that follow all take a closed universe as a starting point, i.e., if you would travel long enough in one direction, you would eventually return to the point where your journey started. That concept will be explained beginning with a simple 1D world.
Imagine some pointshaped being living on a circle. His limited 1D vision makes
the circle appear to him as a straight line because to actually see the curvature of
the circle he would need to have 2D vision.
Also for our own 3D world, closed in the 4^{th} dimension, this means
that there should be "linked", points, lines or surfaces. A potential
example regarding linked surfaces in particular will be given later in this text.
2. Closed 4D worlds: Black holes and gravityIt is here assumed that massive objects, specifically black holes, and their gravity field represent the 4D world. The fourth dimension is proper time (following Euclidean relativity concepts [1]) while it is closed in a 5^{th} dimension. When nearing a black hole these dimensions will show increasing curvature. An
object that falls towards the black hole's Schwarzschild radius follows a
geodesic path along these curved dimensions. An observer at infinity, using a
flat 5D coordinate system will observe that the velocity vector of the falling
object will be subject to a 3dimensional rotation. An observer, using a flat 3D coordinate system, will observe this as an initial acceleration in space that will eventually decelerate again until the particle fully stops when it reaches the Schwarzschild radius. It finally has zero velocity in both space and (proper) time like predicted in general relativity, and seems to stay forever at the Schwarzschild radius. The graph below shows the three different speed components of such an object that falls radially to a black hole from infinity with zero starting velocity in space. The speed components are expressed as a function of the radial coordinate distance r in a flat 5D coordinate system as used by an observer at infinity:
3. Curvature in a closed universeThe example above shows that, even in a flat 3D coordinate system, an observer at infinity is still indirectly able to perceive the curvature of spacetime in the neighborhood of the black hole because he observes the acceleration of the falling object. That curvature is given in classical general relativity as the ratio between infinitesimal coordinate distance dR and radial distance dr : The ratio becomes 1 at infinity, where curvature of space is assumed to be zero as a a result of the absence of mass. But if the 4D spacetime of our universe is closed in the 5^{th} dimension, curvature on a cosmological scale
must exists everywhere, even with a total absence of mass, and at infinity
dR/dr will not be equal to 1 (the classical formula would
need an extra component to account for this). Near a massive object (for the Flatlander that is the center of the horn torus) the curvature deviates from 1 (dR/dr >1) according to the Flatlander who uses his observation point with dR/dr=1 (i.e., zero curvature) as a reference. But it will also deviate from 1 when observing deep space (where dR/dr <1). Consequently, the velocity vector of an object in that far distance will show components in space and the 5^{th} dimension, similar to the situation where it is nearing a black hole. After all, the object will only appear to be at rest relative to an observer at those places where dR/dr =1. Its observed spatial speed goes up both when dR/dr <1 and dR/dr >1.
Universe with accelerated expansion
Missing mass
The inside of black holes It means that whatever falls into a black hole will appear again instantly at the edge of the universe. It defines the edge of the universe as a gigantic "white hole" which should therefore radiate energy (cosmic background radiation?). The universe then resembles a 5D, multihole torus, or ntorus.
4. Closed 3D worlds once more: Photon absorption and emission by electrical chargesIt is here assumed that electrical charges and their electric field represent the 3D world, closed in the 4^{th} dimension. Close to the charge its dimensions will be curved in a similar way as the 5D gravity dimensions are curved nearby a black hole. Consistency between ndimensional models then implies that the absorption of a photon by an electrical charge will also involve a rotation of the photon's velocity vector, which in free space has constant magnitude c in space while its temporal component is zero. This time the rotation will be towards the 4^{th} dimension (proper time), eventually leading to a full stop in 3D at the moment it reaches the charge. As an example formula for the velocity vector components (the actual formula must be determined empirically) we could bluntly replace mass by charge, resulting in the following speed components expressed as a function of the radial coordinate distance r to the charge in a flat 4D coordinate system as used by an observer at infinity:
In this example, one Coulomb of charge would thus slow down the photon's velocity in space for about 1% at a distance of about 10^{5}m. So far, no such effect was ever recorded through experiment but putting a net charge of one Coulomb in such a tiny volume is a challenge on its own and may very well prohibit an easy setup of such an experiment. The force involved in this example would be in the order of 2x10^{19 }N! The "spontaneous" emission of a photon by a charge could have its origin in the simultaneous absorption of a photon by another charge, linking charged particles through the 4^{th} dimension in a similar way as the black holes and the edgeofuniverse were linked through the 5^{th} dimension. This would implicate an instantaneous transformation of information in 3D, which on its turn brings the EPR experiment to mind where something similar happens with two "linked" photons. The seemingly instantaneous information transfer in this experiment indicates that, again, a higher dimension may be involved in the link and that the two photons might actually be one and the same, looked at from two opposite sides.
5. Ordering Nature's particles and fields in ndimensional worlds
What looks like a fermion from "below" (e.g. 3D) looks like a boson from "above" (e.g. 4D). The tables below gives an overview of how particles and fields may be observed from various dimensional "levels". When observed as a boson, the particle will follow a path equivalent to a null geodesic for that dimensional level and its speed will be measured as c, otherwise its path is timelike with speed <c.
When we look at mass particles like e.g. an electron, we see it as the fermion for the electromagnetic field. If we would have been Hyperspacelanders with 4D spatial vision we would always see these electrons move with velocity c in our 4D space and they would behave like bosons (of gravity) [3]. It's like dimensional zooming out, not by changing magnification but by changing the number of dimensions that are observed.
^{(1) }: The nuclear forces consist of a 2 or 3 dimensional subset of our 4D spacetime. They may however rotate in 4D and thus occupy any of the 4 dimensions at a given moment. ^{(2) }: If the strong nuclear force is indeed the field to be associated with a 3D Euclidean spacetime and bosons and fermions are mutually dual between dimensional levels then the bosons of the electromagnetic field must be the dual of the fermions of the strong nuclear field, i.e., a photon might just be another variation of a quark. ^{(3) }: Black holes are actually 5D fermions and are the gravity "charge"
Dimensional zooming out and in
Note that the n^{th} spatial dimension does not disappear! It gets fully contracted, curled up if you like [4], in the lowerdimensional space (hence the massive rollers) and becomes the proper time dimension for the observer with (n1)dimensional vision. Also for us, Spacelanders, the proper time dimension is fully contracted into our 3D space, which explains why we can "see" relativistic nonsimultaneities in moving objects and also have no problem observing time dilation in moving clocks. But it also says that all bosons in 4D remain visible for us as fermions in 3D. A similar effect takes place in the world of the Flatlander: all photons of our 3D space will still exist as fermions in his 2D space.
6. The everlasting Big BangThe list of fields may continue with fields that have 6 or more dimensions but have so far not been observed. There may also be a 1dimensional field with hitherto unknown properties. I dare to suggest that even 0dimensional and negativelynumbereddimensional fields exist. After all, who are we to say that our familiar 4 dimensions are at the bottom of the list? The fact that we number them 1  4 doesn't mean a thing. We could have numbered them 2.356  2.359 just as well. If we can't see the fifth dimension why would we be able to see dimension 0 or dimension 12 ?
Imagine now a being that is able to observe dimensions 1, 0, 1 and 2 (so also four in total). For that being, the picture
is complete for his four forces, making our weak nuclear force look like electromagnetism in
the eyes of this "shifteddimensional" being. Similarly, what we call gravity
may be any other kind of field for another shifteddimensional being.
And what if individual dimensions were created a fraction after each other in climbing order? That process might still be ongoing with the creation of
yet more higher dimensions. Our Big Bang as a snapshot in an everlasting Bang that started much earlier already?
Bibliography
Updated September 17th 2015, © Copyright 20042015 R.F.J. van Linden
