Once Before Time - Martin Bojowald [138]
But for a unique solution it is not sufficient to know only its rate of change; one also needs a firm and fixed starting point whence the changes start. If we had a good model for the development of stock prices, telling us by a mathematical equation how fast the rates go up or down at any given time, we would have to put in the starting values at one time. Similarly, in physics, such a start could be an initial condition (one would say that the studied quantity begins with a certain value at a chosen moment in time) or a boundary condition (one would say that the quantity takes given values at the boundary of a spatial region). Since relativistic physics knows space-time only as a single object, not space and time in separation, one can combine these two kinds of conditions into a single one. (The structure of differential equations suggests a distinction also in the relativistic case. But distinguishing space and time does not play a role in this chapter, in contrast to our earlier discussion of time’s arrow.)
Initial or boundary conditions can be seen as the theoretical equivalent of an experimenter’s decisions in constructing and performing an experiment. The theory itself, by contrast, should, at least approximately, correspond to the behavior of Nature under the natural laws imposed on her. An experiment is always a special situation in nature specified by the experimental setup (for instance, a pendulum) and the initial configuration (the position in which one lets the pendulum swing freely). Such a selection is theoretically described by boundary and initial conditions. In other words, the theory used is selected by choosing a specific natural phenomenon, and the conditions for its solutions stem from the specific realization of the phenomenon within its general possibilities.
In cosmology, the first point—the general phenomenon of the universe—ultimately leads to quantum gravity as the theory for its description. But then there is only one specific realization of the phenomenon: our universe. A selection of initial or boundary conditions should not at all be necessary; instead one should, by cold logic and in consistent keeping with the general circumstances, expect a unique solution of the theory without fixing any further conditions whatsoever. The problem now is that in quantum gravity, as in other areas of physics, we are dealing with differential or difference equations—again, equations for change, starting from a fixed initial point to be chosen in addition. These equations find all these multifaceted applications in physics because they represent such a powerful method that hardly any other mathematical construction can eclipse them. They also determine what happens in cosmology, as has been impressively confirmed by observations, but as usual they require the specification of a starting point for solutions of the universe in addition to the equations themselves.
Will the aim of explaining the uniqueness of our universe remain a fantasy? The statement of our universe’s uniqueness may fall dangerously close to a tautology, but the desire to derive it by physical methods is surely reasonable. We observe the world in its details and can never see the universe as a whole, but a large part of physics consists of extrapolations, sometimes exceedingly daring ones. Why should we not try to abstract our small-scale knowledge of the cosmos to obtain a theory extrapolated to the whole universe, possibly proving the uniqueness of the world? This undertaking