Descriptions of the first moments in time describe the expansion of SpaceTime with the consequential cooling and creation of particles. This standard cosmology places the macrospace dimensions that we experience in a separate category from the microspaces. The microspaces are assumed to be static over this period of time. No description of how a Calabi-Yau manifold evolved in these first moments is given. One can imagine such an evolution of microspaces being similar to the macrospaces, that they expanded but differently than the macrospaces. This is a description of an evolution of a 10 dimension spacetime based on a simple superstring model.
Particles in this description are called momentons. Since the momentum is the only property common to all the fundamental particles, it is useful to think of them bits of momentum rather than particles. Particles brings up a vision of points to mind, whereas bits of momentum are visualized as vectors, items in motion. String properties are more closely related to vectors than points. This simple model does not include the wrapping number for a string. Strings are assumed to only wrap once around on a manifold.
The initial universe is a 10-sphere with a radii of the Planck length. The initial total momentum is zero, with the sum of vectors canceling out. This 10-sphere is embedded in an 11-space. Initially, all ten dimensions start to expand equally. Although the net momentum is zero, there are a huge number of momentons traveling in this space. Since the space dimensions are so tiny, all these particles have huge momentums, hence energy, therefore relative masses. Except that mass as a concept does not have much meaning in this symmetric space. Mass is derivative of the momentum 4 vector. The momentum values are multiples of the minimum momentum possible. The minimum momentum is that with the longest possible wavelength that fits on a 10 sphere.
A wavelength fitting on a sphere can be visualized as a line encircling a globe, making a circle. At the equator it's a great circle. Envisioning many great circles on this sphere, they can be imagined as all the momentons that exist. They interact with each other, mostly causing the momentons to not only orbit the sphere but to also rotate about, causing their orbits to precess. Precession of an orbit is similar to a coin spinning an a table. The primary interaction of the momentons in the ten-sphere is creation, destruction of these momentons as well as the change in precession. As the 10 sphere increases in volume, the allowed energies of the momentons and their allowed motions increases in number. The number of momentons increases, while the average momenton's value decreases. The possible lengths are getting longer increasing to fill the expanding 10 space. Longer wavelengths corresponds to lower momentums.
One of the reasons for the hypersphere fails to shrink rather than grow is that the existing momentons must have shorter wavelengths to fit back in the smaller dimension. Momentons must made bigger in momentum, to make their wavelengths shorter in size. Entropy prevents the recollapse.
As the 10-sphere expands all momentons look and interact alike. All symmetry is maintained. As the space expands to a certain radii, something occurs which slows down the expansion in three of the 10 dimensions, the something is described below. They stop expanding, the other seven continue to expand. The uniform sphere turns into a hyper-toroid. The Calabi-Yau manifold transforms from one type to another. Calabi-Yau is six dimensions. So one must use the three smaller dimensions plus 3 of the remaining 6 spatial dimensions. Time is the last dimension of the 10. The momentons in the 3 smaller now non-expanding dimensions no longer can interact equally with the remaining seven dimensions (six spatial and one time). They still interact, but some of the previous rotational modes are no longer available. The momentons trapped in the micro 3 space have properties associated with the strong force. Quarks, gluons and color are associated with the interactions that are left available for the momentons within the 3 dimensional microspace. The momentons in the remaining 7 space still have symmetry, but a reduced number of possible modes. The electroweak interactions remain possible. The strong - electroweak no longer are possible. Strong particles no longer can transform equally with weak ones. The space expands, the average value of momentons continues to decrease, the number of momentons increase.
What is the something that occurred that stopped the expansion in three dimensions? It could be speculated that the momentons in this strong force microspace are locked into a three body solution. For a three body solution, in macrospace with three masses of equal size, the motions are chaotic and unstable. For three equal sized momentons in a microspace, the number of available modes are very constrained. This removes the chaotic nature and allows a stable configuration to be available. Once a stable configuration is established, these modes stop decay into lower momentums, the entropy for three dimensions, halts. The spatial expansion for these three dimensions also halts. The remaining seven dimensions can still interact together allowing them to still expand.
As the diameter of the 10 toroid expands, the coupling constants are changing values. The interaction between the macro seven spacetime and the micro three space are continually changing. At a certain point, 2 more dimensions stop expanding. The 10 dimension toroid that previously had one hole now has two holes. Symmetry of between weak and electric forces are now broken. The hyper toroid now has two distinct microdimensions. The momentons stuck in the new two nonexpanding dimensions consist of particles associated with the weak forces. The particles of the strong and weak force move about freely in the remaining macro 5 space. Although motions of what we would associate with electric charge are present, the mass of the electron would appear larger than at present. The 5 macro spaces continue to expand. Two dimples form in the macro 5-sphere, opposite of each other, symmetry is breaking once again. The 5 surface is maintaining a constant 5 area as 4 of the dimensions continue to expand. The dimples deepen until they touch. Symmetry is broken again, creating the electric forces. The electron mass sized is now fixed across all of the remaining macro 3 space. Gravitational forces are much greater at this point in time than now. The warping of space time in the newly formed e-space is barely greater than in the macrospace. The original solution by Kaluza demonstrating electromagnetic theory with gravitational theory applies for the macro 4 spacetime and the one microspace for electrons, e. Theorist added 5 more columns and rows to the matrix. These entries were seen as mathematical tools for gauge theories to describe the strong and weak forces. These 5 and the one for e make the 6 microdimensions of the Calabi-Yau manifold. The dimensions and the genus of the manifold are now unchanging relative to the momentons traveling on this 6-surface.
The momentons described here have length and direction as a description of classical momentum require. They also have a wavelength from quantum theory. So one must imagine these lines not only encircling a radius of a dimension, one must imagine it acting like a vibrating string in order to see the wavelength. This is one reason that one wrap around the manifold is assumed, different portions along the momentons length would be near each other as they wrapped, causing destructive interference. Of course, a wrap with a harmonic mode which constructively interfere and would be allowed. A double wrap with twice the momentum fits nicely on manifold.
Back to expansion. The macro 4 spacetime continues to expand. The microspace now consist of 6 spatial dimensions with 3 distinct sizes. The original 10 sphere is now a 10 toroid with 3 holes in it. The 3 sizes allows similar momenton motions that differ primarily in the size of the momenton that fits into the space. The electron is the simplest momenton path available a path with a half wrap. This solution can be made about each of the toroids holes, one for each of the three available sizes. The major difference is in the value of the momenton, it is greater as the momenton dimension is forced to be shorter in the smaller microspaces. This larger momenton value is measured as a larger mass.
The continued expansion of the macro 3 space continues today. The macro space expansion did not create any further holes since the electric force one formed. The rate of expansion is expected to be at the speed of light, but with the macro 3 space so large, that as a percentage its increase is insignificant. The force of gravity therefore continues to weaken over time as compared to the microspaces. Gravity is intimately connected to space and time. The measured gravitational constant is expected to be a measure of the coupling between the space and time dimensions. The available modes for the momentons has increased immensely. Many of the initial symmetries of the 10 sphere are gone. Symmetries don't allow for any memory of the previous instants. But once some modes are frozen out, memory is created. Momentons are locked into specific modes, no longer free to move in all the previously available paths.
Gauge theories assumed geometric solutions with an interior geometry, but assumed that it only provided a means for a mathematical formalism. Sean Carroll provides an excellent set of lecture notes on General Relativity that referred to this aspect. This description assumes for superstring models that these dimensions are as real as the exterior ones used for General Relativity. Brian Greene's work on showing how Calabi-Yau manifolds can transform from one genus into another, supports this concept that the manifold actually evolved over time.
One feature that is similar to the standard model is the time where the different forces break from their symmetry. These occur at the same point in the universe's evolution where the temperature in the standard model states that they should occur. The size of the space should match the temperature at which the strong space particles start freezing out. The protons and neutrons are first created, with the strong space dimensions. One can view this in a high energy experiment. When a high energy experiment is performed, the momentums of the high energy particles have shrunken their sizes so that they could possibly ricochet into one of the microspaces. This shrinkage in dimension can be viewed as either due to relativistic effects, or as an increase in momentum which decreases the uncertainty in x. A ricochet of a particle into a microspace means that it can only interact in ways that the microspace allows, it has became a particle of the microspace. The symmetry between macrospace and the microspace is recreated in high energy interactions. A momenton particle once again has the energy that allows it to once again can fit into the microspace.
The actual strong interaction occurring in a high energy interactions vs. the interactions in the big bang is different in some details. To be fully symmetric between high energy and big bang requires that the weak interactions are unavailable. Weak sized dimensions didn't exist as a separate entity when the strong dimensions stopped expanding. The dimension for weak particles would still have been part of the 7 dimension macrospace. In the big bang, the weak interactions are not unique and have energies much greater than they now have. Strictly strong force interactions would match, but the decay particles are different between present high energy interactions and big bang interactions.
Coupling constants are characteristic of the ratio of sizes of the dimensions that are being considered. Since the big bang expanded the spaces over time, the coupling constants decrease over time. The sizes that stopped expanding have fixed coupling constants. Whether macrospace is still expanding or has already stopped should be evident in whether gravity has changed over its history. The fact that the background radiation is evidence of expansion is clear, when that expansion occurred and when it stopped is not. The macrospace may be at a fixed size, held in place by the ultra low energy photons that would need to be made more energetic. The expansion though can only communicated at the speed of light, and if it has stopped would not seen for billions of years. This relativistic effect makes the present time look dynamic with respect to the expansion.
The microspace sizes are fixed across the universe, even though they are no longer in causal contact with each other. This is another reason they can no longer change. The speed of light locks out the possibility for further change. The universe can only collapse once light that travels the circumference of the universe has reached its starting point.
Now for the something that was referenced earlier. What causes the initial expansion and why do some dimensions stop expanding? The size of a dimension is dependent on the total momentum traveling in the dimensions of the microspace. More momentum means a smaller space. Less trapped momentum means a larger spatial size. This is based on the uncertainty principle, p*x = h. The strong dimensions are the smallest and have the largest average value for their momenton's. If the momentons could decay out of the strong space, the strong dimension could grow further. The momentons though are locked into the dimension, unable to decay, the only escape mechanism. The amount of momentum contained in kinetic energy present in the macro spaces are many orders of magnitude less than a microspace. All the momentons in macrospace are the zero mass particles. As the number of photons decrease, the space grows larger. The initial expansion would therefore be caused by an instantaneous moment at a point where the momentum fell to zero.
In the present universe, the characteristic path the particle travels changes as it goes up in momentum. A photon approaching Compton's wavelength starts to have a size that fits into the e space dimension. A photon that fits into the e space is seen as another electron. A second Compton wavelength at the W space dimension should also be evident. The size being the fine structure constant smaller, requires the photon being 137 times more energetic. This makes the photon so short that it would create many positron-electron pairs before it approached a particle in which it could actually interact in a Compton like interaction.
The view of the big bang in this superstring model is somewhat unique with reference to the microdimension sizes. It should be recalled that while standard superstring theory states that all strings are on the order of the Planck length as well as the dimensions of the Calabi-Yau manifold on which they vibrate, this model assumes the these dimensions are much larger. In this model, strings are bits of momentum with an associated length. The constraints for this momentum length is the size of the dimensions on which they travel. This constraint is from quantum theory, p*x = h. The energy, hence potential mass of the momentum element, momenton, is inversely proportional to its length. Based on the momentum 4-vector, all momentons must travel at the speed of light along their vector of travel.
Last Updated on Aug 11, 2001 by Bob Rutkiewicz