FMBR Editorial: Mar, 2002
The Mysterious Zero/Infinity
William C. Gough
A new deck of 52 cards usually has two jokers. Likewise there are two jokers that bedevil physics -- zero and infinity. They represent powerful adversaries at either end of the realm of numbers that we use in modern science. Yet, zero and infinity are two sides of the same coin -- equal and opposite, yin and yang. "Multiply zero by anything and you get zero. Multiply infinity by anything and you get infinity. Dividing a number by zero yields infinity; dividing a number by infinity yields zero. Adding zero to a number leaves the number unchanged. Adding a number to infinity leaves infinity unchanged." Yet, the biggest questions in science, philosophy, and religion are about nothingness and eternity, the void and the infinite, zero and infinity.
Zero is behind all of the big puzzles in physics. In thermodynamics a zero became an uncrossable barrier: the coldest temperature possible. In Einstein's theory of general relativity, a zero became a black hole, a monstrous star that swallows entire suns and can lead us into new worlds. The infinite density of the black hole represents a division by zero. The big bang creation from the void is a division by zero. In quantum mechanics, the infinite energy of the vacuum is a division by zero and is responsible for a bizarre source of energy -- a phantom force exerted by nothing at all. Yet dividing by zero destroys the fabric of mathematics and the framework of logic -- and threatens to undermine the very basis of science.
The biggest challenge to todays physicists is how to reconcile general relativity and quantum mechanics. However, these two pillars of modern science were bound to be incompatible. "The universe of general relativity is a smooth rubber sheet. It is continuous and flowing, never sharp, never pointy. Quantum mechanics, on the other hand, describes a jerky and discontinuous universe. What the two theories have in common -- and what they clash over -- is zero." "The infinite zero of a black hole -- mass crammed into zero space, curving space infinitely -- punches a hole in the smooth rubber sheet. The equations of general relativity cannot deal with the sharpness of zero. In a black hole, space and time are meaningless."
"Quantum mechanics has a similar problem, a problem related to the zero-point energy. The laws of quantum mechanics treat particles such as the electron as points; that is, they take up no space at all. The electron is a zero-dimensional object, and its very zerolike nature ensures that scientists don't even know the electron's mass or charge." But, how could physicists not know something that has been measured? The answer lies with zero. According to the rules of quantum mechanics, the zero-dimensional electron has infinite mass and infinite charge. As with the zero-point energy of the quantum vacuum, "scientists learned to ignore the infinite mass and charge of the electron. They do this by not going all the way to zero distance from the electron when they calculate the electron's true mass and charge; they stop short of zero at an arbitrary distance. Once a scientist chooses a suitably close distance, all the calculations using the "true" mass and charge agree with one another." This is known as renormalization -- the physicist Dr. Richard Feynman called it "a dippy process."
The leading approach to unifying quantum theory and general relativity is string theory. In string theory each elemental particle is composed of a single string and all strings are identical. The "stuff" of all matter and all forces is the same. Differences between the particles arise because their respective strings undergo different resonant vibrational patterns -- giving them unique fingerprints. Hence, what appear to be different elementary particles are actually different notes on a fundamental string. In string theory zero has been banished from the universe; there is no such thing as zero distance or zero time. Hence, all the infinity problems of quantum mechanics are solved.
But, there is a price that we must pay to banish zero and infinity. The size of a typical string in string theory is the Planck length, i.e., about 10-33 centimeters. This is over a thousand trillion times smaller that what the most advanced particle detection equipment can observe. Are these unifying theories, that describe the centers of black holes and explain the singularity of the big bang, becoming so far removed from experiment that we will never be able to determine their correctness? The models of the universe that string theorists and cosmologists develop might be mathematically precise, beautiful and consistent and might appear to explain the nature of the universe -- and yet be utterly wrong. Scientific models/theories, philosophies, and religions will continue to exist and be refined. However, because of zero and infinity, we can never have "proof". All that science can know is that the cosmos was spawned from nothing, and will return to the nothing from whence it came.
William C. Gough, Mar 2002
Charles Seife, Zero: The Biography of a Dangerous Idea, NY: Viking, 2000, pp.191-209.
Brian Green, The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory,NY: Vintage Books, 1999.