The Hidden Reality_ Parallel Universes and the Deep Laws of the Cosmos - Brian Greene [61]
This little episode represents the rule rather than the exception. Textbook problems are very special, being carefully designed so that they’re completely solvable with reasonable effort. But modify textbook problems just a bit, changing this assumption or dropping that simplification, and they can quickly become intricate or intractable. That is, they can quickly become as difficult as analyzing typical real-world situations.
The fact is, the vast majority of phenomena, from the motion of planets to the interactions of particles, are just too complex to be described mathematically with complete precision. Instead, the task of the theoretical physicist is to figure out which complications in a given context can be discarded, yielding a manageable mathematical formulation that still captures essential details. In predicting the course of the earth you’d better include the effects of the sun’s gravity; if you include the moon’s too, all the better, but the mathematical complexity rises significantly. (In the nineteenth century, the French mathematician Charles-Eugène Delaunay published two 900-page volumes related to intricacies of the sun-earth-moon gravitational dance.) If you try to go further and account fully for the influence of all the other planets, the analysis becomes overwhelming. Luckily, for many applications, you can safely disregard all but the sun’s influence, since the effect of other bodies in the solar system on earth’s motion is nominal. Such approximations illustrate my earlier assertion that the art of physics lies in deciding what to ignore.
But as practicing physicists know well, approximation is not just a potent means for progress; on occasion it also brings peril. Complications of minimal importance for answering one question can sometimes have a surprisingly significant impact in answering another. A single drop of rain will hardly affect the weight of a boulder. But if the boulder is teetering high on a cliff’s edge, that drop of rain could very well coax it to fall, initiating an avalanche. An approximation that disregards the raindrop would miss a crucial detail.
In the mid-1990s, string theorists discovered something akin to a raindrop. They found that various mathematical approximations, widely used to analyze string theory, were overlooking some vital physics. As more precise mathematical methods were developed and applied, string theorists could finally step beyond the approximations; when they did, numerous unanticipated features of the theory came into focus. And among these were new types of parallel universes; one variety in particular may be the most experimentally accessible of all.
Beyond Approximations
Every major established discipline of theoretical physics—such as classical mechanics, electromagnetism, quantum mechanics, and general relativity—is defined by a central equation, or set of equations. (You don’t need to know these equations, but I’ve listed some of them in the notes.)1 The challenge is that in all but the simplest situations, the equations are extraordinarily difficult to solve. For this reason, physicists routinely use simplifications—like ignoring Pluto’s gravity or treating the sun as perfectly round—that make the mathematics easier and bring approximate solutions within reach.
For a long time, research in string theory has faced even bigger challenges. Just finding the central equations proved so difficult that physicists could develop only approximate versions. And even the approximate equations were so intricate that physicists had to make simplifying assumptions to find solutions, thus basing research on approximations of approximations. During the 1990s, however, the situation vastly improved. In a series