Difference Equations and Modern Physics

Martin Bojowald of Scientific American, in a recent article on the physical theory of "Loop Quantum Gravity," noted that "theoretical physicists commonly describe the world using differential equations, which specify the rate of change of physical variables, such as density, at each point in the spacetime continuum. But when spacetime is grainy, we instead use so-called difference equations, which break up the continuum into discrete intervals."

What is Loop Gravity?

Why is it important that space and time are broken up into discrete chunks like this? What benefit does that have upon theories of gravitation?

The problem has faced physicists for several decades: How to reconcile the force of gravity with the nearly complete Standard Model of physics, which incorporates electromagnetism, the strong force and the weak force.

Often lost among the vast sea of "potential" theories floating around the physics community, loop quantum gravity attempts to find a solution to this problem by treating gravity as if it were not continuous, but as if it were divided up into incredibly small "chunks" (so small that it would be almost undetectible by any known method, somewhere around the order of 10^-35 meters).

This method mirrors the rise of quantum mechanics in the beginning of the twentieth century, when physicists finally began to realize that other entities which were believed to be continuous, such as light, were broken into smaller parts ("quanta").

What Difference Do Discrete Quantities Make?

When light was first quantized in 1900 by Max Planck, it was done so for purely mathematical purposes, after the realization that the equations which could be used to describe light in a discrete form led to entirely different conclusions than the differential equations used to describe continuous light, and as a result, certain problems (specifically the problem of "black-body radiation") could finally be solved.

Something very similar is hoped for in loop quantum gravity. By breaking up the very fabric of space-time into "atoms," differential equations which require continuity are no longer helpful; they must be replaced by "difference" equations, which give similar answers in many circumstances to differential equations, but in extreme circumstances lead to entirely different results.

For example: Difference equations are being applied to the "big bang" which is said to have been the beginning of the universe around 15 billion years ago from a "singularity" (an infinitely dense speck of space-time - a giant black hole). Under traditional equations, there was no problem with this understanding, but in loop quantum gravity, there is a change: when particles of space-time become densely packed, they exhibit different features; suddenly, the attractive force of gravity actually becomes repulsive, and there is a mathematical expansion.

Using these equations, physicists have shown that the big bang might have truly been nothing more than a "big bounce," stemming from the contraction of a previous universe, which expanded and then contracted into a near-singularity itself.

The consequences of this, while purely hypothetical at the moment, are rather astounding. The universe, in a sense, could be seen to be infinite in age, undergoing a constant series of expansions and contractions!

This is just one of the many possible theories of gravitation currently being studied by physicists, however, and there is much work to be done before it can even begin to be confirmed, but at the same time it is certainly very interesting, and demonstrates a very important mathematical principle - the difference between differential calculus and difference calculus - and why such things matter.

References:

Bojowald, Martin. "Big Bang or Big Bounce?: New Theory on the Universe's Birth." Scientific American. Oct. 2008.

Loop Gravity might change how we view black holes

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