Physical Paradox - Paradoxes Relating To Unphysical Mathematical Idealizations

Paradoxes Relating To Unphysical Mathematical Idealizations

A common paradox occurs with mathematical idealizations such as point sources which describe physical phenomena well at distant or global scales but break down at the point itself. These paradoxes are sometimes seen as relating to Zeno's paradoxes which all deal with the physical manifestations of mathematical properties of continuity, infinitesimals, and infinities often associated with space and time. For example, the electric field associated with a point charge is infinite at the location of the point charge. A consequence of this apparent paradox is that the electric field of a point-charge can only be described in a limiting sense by a carefully constructed Dirac delta function. This mathematically inelegant but physically useful concept allows for the efficient calculation of the associated physical conditions while conveniently sidestepping the philosophical issue of what actually occurs at the infinitesimally-defined point: a question that physics is as yet unable to answer. Fortunately, a consistent theory of quantum electrodynamics removes the need for infinitesimal point charges altogether.

A similar situation occurs in general relativity with the gravitational singularity associated with the Schwarzschild solution that describes the geometry of a black hole. The curvature of spacetime at the singularity is infinite which is another way of stating that the theory does not describe the physical conditions at this point. It is hoped that the solution to this paradox will be found with a consistent theory of quantum gravity, something which has thus far remained elusive. A consequence of this paradox is that the associated singularity that occurred at the supposed starting point of the universe (see Big Bang) is not adequately described by physics. Before a theoretical extrapolation of a singularity can occur, quantum mechanical effects become important in an era known as the Planck time. Without a consistent theory, there can be no meaningful statement about the physical conditions associated with the universe before this point.

Another paradox due to mathematical idealization is D'Alembert's paradox of fluid mechanics. When the forces associated with two-dimensional, incompressible, irrotational, inviscid steady flow across a body are calculated, there is no drag. This is in contradiction with observations of such flows, but as it turns out a fluid that rigorously satisfies all the conditions is a physical impossibility. The mathematical model breaks down at the surface of the body, and new solutions involving boundary layers have to be considered to correctly model the drag effects.