Visiting flatland has helped understand the nature of space-time. Returning there once again will present the concepts of a string theory. In this case, showing how a property in normal space looks like a different property in a flattened universe.
We three-dimensional beings understand about photons and electrons, energy and matter. Photons are massless particles that move at the speed of light while electrons have mass and can move at any desired speed except the speed of light. The principles of conservation apply equally to both types of particles. That is, mass-energy and momentum is conserved. Quantum theory allows a photon or an electron to have only one particular wavelength for a given value of momentum or energy of these particles. But photons have momentum without a rest mass. P=mv doesn't seem to apply. Photons and electrons have a continuum of momentum values associated with them. The energy in photon is also falls along a continuum. These photons are identical except for their energy. The electron on the other hand belongs in a family of leptons. Only three different rest masses of otherwise identical particles exist.
A way to overcome this is to create flatland by curling up one of our three dimensions, z, down to a subatomic size. If we took a universe where the three dimensions all had photons traveling along them, including z, we would have an interesting dilemma to solve. Photons with wavelengths greater than our new z size would not fit into our curled down dimension. The largest wavelength traveling along z couldn't be longer than z. Boundary value solutions of photons traveling in closed space are very similar as photons in a black box. Only particles with wavelengths that are integral multiples of the size of z may travel along z.
Another aspect that is created is the view that the photon is now motionless. The lowest energy photon fills the entire length of the z dimension and is a standing wave. The photon didn't actually stop, it is just that it gets all the way around the universe before its tail end leaves. Those flatland resident viewing it from the x or y dimension would interpret it as we do massed particles. A flatland resident would apply a different physics to this z traveling photon. X and y traveling photons are interchangeable in their properties, but not with z. Z photons come in discrete energy values, x and y photons energies fall in a continuum.
X and Y photons move at the speed of light, z photons seem unmoving. An x photon can easily be bounced into a new vector from x to y or y to a vector half way between x and y. An x photon only seems to convert to something unique once its energy is high enough or its wavelength is short enough. In this case, a photon creates a particle with a particular rest mass, with the remaining energy converted to a velocity for the particle.
From our non-flatland macrospace view, we see a photon being bounced from the x direction into one with some z and some x. The previous value along x was at the speed of light, but the remaining velocity along x is now less. If one looks close one sees that the photon is now following a spiral around the curled up z. This photon is still travelling at the speed of light, but because the z dimension is so small, only the x velocity is perceptible and is the x velocity clearly moving at less than the speed of light. One can try to increase the photons energy along the x vector but it will always have some vector along z and so that the x velocity is always less than the speed of light.
The major difference between this flatland view and the general theory P & F's view is that there is more than one curled up dimension. Superstring theorist's state that there are six microspaces. If there were only one, the lepton family would have an infinite number of family members. But the other dimensions create other constraints.
An electron is an element of momentum that travels along a minimum of two unequally sized microspaces, curled up with a torus topology. An up or down vector along one microdimension determines if it is electron or positron. Whether the vector spirals left or right determines if it's spin is up or down.
Last Updated on February 22, 1998 by Bob Rutkiewicz