~Reggie Bain, Class of 2012, Physics and Mathematics student.
Consider the directions in which you can move: Back and Forth, Left and Right, Up and Down…simple enough. These three dimensions in which typical things can move are quite familiar to us, so it’s easy to visualize all objects in the universe existing in only these three ordinary spatial dimensions. We can also easily envision things existing in one and two dimensions. Points, lines, basic polygons, and planes are all mathematical constructs that reside in the one and two-dimensional universes. It thus seems logical to assume that we can conceptualize things existing in up to three dimensions, so what about four, five, or even more? When one first considers the prospect of extra dimensions of space beyond the three that we are intimately familiar with, the thought may seem utterly ridiculous. However, the idea of extra-dimensional space may not seem so outlandish if one is to contemplate the ancient Greek philosopher Plato’s Allegory of the Cave. In short, men are secured in every way imaginable inside of a cave without the ability to move, turn their head, etc. A fire is lit behind them such that all they can see is their shadows cast on the cave wall in front of them. If this cave dwelling existence is all they have known for their entire lives, they will essentially perceive that the universe exists in only the two dimensions present on the surface of the cave wall. They will likely not even be able to begin to imagine that there could exist in fact a third dimension of space.[1] While this example is somewhat crude, it provides an interesting illustration of the dilemma of pondering extra spatial dimensions.
In many modern theories of physics, dimensions beyond the three familiar ones are currently being investigated. One such theory is called “M-theory,” an extension of “Superstring theory” first introduced in the summer of 1995 by mathematical physicist Edward Witten from the Institute for Advanced Study in Princeton, NJ. This theory actually predicts that there may exist as many as ten spatial dimensions (when combined with one time dimension makes for a total of eleven dimensions). The theory also dictates that there may exist many other universes existing parallel to ours in these higher dimensions of space. In case you’re not familiar with these relatively new theories of physics, here’s a quick explanation of the basic ideas. Essentially, Superstring theory and its extension M-theory dictate that everything in the universe, at the most basic level, consists of tiny, vibrating strands of energy called “superstrings” or “strings.” This idea represents a paradigm shift from the former pictures of the quantum world that make point particles like electrons and quarks the fundamental constituents of matter. Unfortunately, if present in nature, these strings are predicted to be much smaller than even the smallest elementary particles known today. In terms of size, a string is to one atom as one atom is to our entire solar system.[2] This presents serious problems when trying to experimentally show the existence of these tiny vibrating strings, essentially making it impossible with today’s technology. Perhaps the most important consequence of this idea of strings is that they provide a possible “Theory of Everything” or “TOE” of theoretical physics. Basically, physicists found that if one could develop a mathematical model that could describe all of the physical phenomena of the universe, from the laws of gravity to those governing atoms and elementary particles. Superstring theory and M-theory are among the most exiting ideas in modern physics and are leading candidates, physicists believe, for a possible “Theory of Everything,” so to speak. Although it’s hard to imagine, M-theory essentially predicts that existing in dimensions that we cannot see (but are right next to us!) are universes that may be similar or very different to ours, possibly even governed by different laws of physics.[3] We may actually live in a universe with many more than three dimensions!
Another concept that we often assume clear-cut and simple is that of the nature of time. We experience everything in our lives over the passage of time. Time seems on the surface a constant, seemingly arbitrary idea that provides a reference frame for the order in which we experience events. It is in fact the case that time is much more intricate and “malleable” than may be obvious. Albert Einstein is perhaps responsible for the one of the most insightful realizations about the nature of time since the dawn of science. With the publications of his “Special” and “General” theories of relativity in 1905 and 1915 respectively, Einstein forever changed our understanding of the passage of time.[4] While he was not the first to suggest the idea, one of the most important postulates of Einstein’s “Special Theory of Relativity” is that the speed of light is a universal constant. This postulate of relativity, when applied to moving objects, dictates that as one travels at speeds closer and closer to the speed of light (which is about thirty million meters per second) they will experience time more slowly! This assertion has in fact been empirically proven using atomic clocks. Scientists took two extremely precise clocks, synchronized them, and put one in orbit around the earth, leaving the other on the ground. At the end of the trial, it was observed that the clock in orbit around the earth recorded less time than its earthbound counterpart, thus confirming Special Relativity’s prediction of the occurrence of what is called “Time Dilation.”
Ten years later Einstein published his “General Theory of Relativity” which threw out all assumptions that space and time were separate entities. He was able to show that space and time can in fact be interwoven into a “space-time” fabric. Einstein then claimed that the force of gravity is simply warps and curves in this fabric. Just as a bowling ball would stretch and bend a trampoline when placed on it, massive bodies such as planets, stars, and galaxies cause warps in space and time. Like the constancy of the speed of light, this characteristic of General Relativity dictates that time pass at different rates for objects at different positions within a gravitational field (the area of gravitational influence caused by an object like the sun). So, if one were to put incredibly precise clocks at the top and bottom of the Empire State Building, the clock at the bottom would record less time than would the clock at the top (although for this example there would be an infinitesimal difference).[5]
Yet another idea that some might assume to be simple (or perhaps just enigmatic) is that of infinity. When most people hear the word “infinity” they usually assume that it represents something unachievable or unattainable and don’t give the concept too much thought. After some consideration however, infinity can actually be discussed in some detail. Consider the counting numbers 1, 2, 3…and so on. It’s easy to understand that you can keep doing this forever. No matter how long you keep counting you’ll never actually get any closer to counting all of them. We thus say that there are infinitely many. Now consider what are referred to as the rational numbers, and let’s discuss only the positive ones for now. The positive rational numbers include 1, 1.1, 1.01, 1.001, 2.1, 3.001…etc. In this case, the positive rational numbers include the infinite number of counting numbers 1, 2, 3, etc. and they also include an infinite number of quantities between each of the counting numbers i.e. 1.01, 1.001, 1.0001 etc. We discussing the two separately we say that there are infinitely many counting numbers and that there are infinitely many rational numbers. However, we just showed that it is obvious that there exist far more rational numbers than counting number. This presents an interesting problem. Mathematicians have devised clever ways of dealing with various versions of infinity in branches of mathematics such as Real Analysis and Set Theory. Georg Cantor (1845-1918) was a pioneer of Set Theory and developed ingenious methods to deal which notions of infinite quantities, including complex ideas such as “ordinality” and “cardinality.”
Many aspects of our world possess fascinating complexities that are often veiled in obscurity or misunderstanding. Scientists and mathematicians alike have always sought to shed light on enthralling questions about the very nature of the universe in which we live. Ideas like those of space, time, and infinity are just a few aspects of the world that, while seemingly simple, can provoke deep and interesting questions about the most basic aspects of our existence.
[1] Webb, Stephen. Out of This World: Colliding Universes, Branes, Strings, and Other Wild Ideas of Modern Physics. 1 ed. New York: Springer, 2004. Print.
[2] The Theory of Everything by Tyler Cabot. Copyright © 2006 by Tyler Cabot. All rights reserved. Reprinted in The Best American Science Writing: 2007. First published in Esquire.
[3] Greene, Brian. The Fabric of the Cosmos: Space, Time, and the Texture of Reality. New York: Vintage, 2005.
[4] Isaacson, Walter. Einstein: His Life and Universe. New York: Simon & Schuster, 2007.
[5] Greene, Brian. The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory. New York: Vintage, 2000.
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