Once Before Time - Martin Bojowald [40]
In physics before special relativity, the kinetic energy of a moving body is proportional to the square of its velocity. Every driver knows this, for the distance needed to stop a car by transforming its kinetic energy into heat in the brakes increases with the square of the velocity rather than linearly. Thus the consequences of a crash at a speed of 70 mph are, unfortunately, much more than just twice as devastating as those of a crash at a speed of 35 mph.
In special relativity, by contrast, energy behaves like this only at sufficiently small velocities. Otherwise, one has to take into account that mass as well as motion contributes to the energy, according to Einstein’s famous formula equating energy with mass multiplied by the speed of light squared. Moreover, energy and velocity (or, more precisely, momentum as the product of mass and velocity) play mutually transformable roles similar to what we have already seen for space and time. Energy as well as velocity thus enters the bottom line of energetic processes, such as stopping a car, with their squares.
Schrödinger’s equation makes use of the prerelativistic relationship and disregards the square of energy. Dirac’s new equation, on the other hand, does take into account the relativistic relation. But the square of a number, in contrast to the number itself, is independent of the sign: Minus one times minus one is one, just as is one times one. For every solution of the Schrödinger equation there are always two solutions to Dirac’s equation, differing in their energy signs and possibly in that of charges, too, but otherwise agreeing in properties such as the mass. One of the unshakable (and well validated) beliefs of theoretical physics is that every solution to a theory must either be inconsistent with some mathematical principle or correspond to something real. Dirac thus predicted the existence of a new world of phenomena of matter, just by mathematically combining special relativity and quantum mechanics: For every known particle, such as the electron, there should be an antiparticle of the same mass but opposite charge. When Dirac published his equation, such particles had not been observed; Dirac’s prediction thus came at high risk. (He reportedly hesitated to state it in clear terms.) But soon afterward, in 1933, Carl Anderson provided a direct detection of antimatter in the form of positrons (the antiparticles of electrons), a feat repeated later on for the other known particles. By now, corrections implied by the Dirac equation compared to Schrödinger’s can be measured very precisely, for instance in the frequencies of light emitted or absorbed by hydrogen.
Dirac arrived at this far-reaching view of a new world of matter by nothing more than a mathematical analysis of combining two theories. Nowadays we are faced with a similar, though mathematically much more challenging, problem of combining general relativity and quantum theory. What new worlds will such a combination reveal? Special relativity determines the motion of particles in space-time, while general relativity describes the behavior of space-time itself. It is then natural to expect a consistent combination of general relativity and quantum physics to reveal not new matter but new regions of space-time, or new parts of the universe.
To understand space-time and the universe completely—including