Fatal Attraction of Singularity

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Wikipedia tells us:

  • In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as (boundedness, continuity) or differentiability

An example of a singularity is the gravitational potential from a unit point mass at the origin:

  • \phi (x)=\frac{1}{\vert x\vert} for  x\neq 0,

which is singular (infinite) at x=0. A singularity is like a mine bomb with catastrophical effect if stepped upon and thus has to be avoided.

The way to handle singularities mathematically is to regularize or smooth and e.g. replace the gravitational potential \phi (x) by the regularized potential \phi_\epsilon (x) defined by

  • \phi_\epsilon (x)=\frac{1}{\vert x\vert} for  \vert x\vert >\epsilon,
  • \phi_\epsilon (x)=\frac{1}{\epsilon} for  \vert x\vert \le \epsilon,

where \epsilon is a small positive number, which corresponds to replacing the unit point mass  by a uniform sphere of radius \epsilon again of unit mass.

By regularization a singularity can be tamed and analyzed. Without regularization a singularity carries a mystery which can give rise to fantasies as shown in the following two examples:

Aerodynamics: Theory of Flight

  1. Kutta-Zhukovsky circulation theory of lift of an airfoil.
  2. Prandtl boundary layer theory for drag in slightly viscous flow.

Kutta-Zhukovsky theory assumes the airfoil to have a sharp trailing edge and lift is obtained by circulation around the wing section determined so as to eliminate the singularity of potential flow at a sharp trailing edge with a pressure singularity of the form \frac{1}{\vert x\vert} (with the trailing edge at x=0).

Prandtl boundary layer theory postulates that drag results from a vanishingly thin boundary layer, where the fluid velocity changes from non-zero free flow velocity to a no-slip zero velocity at a solid boundary, that is from a flow velocity discontinuity.

In the new theory of flight presented in The Secret of Flight it is shown that the classical theory of flight, which is Kutta-Zhukovsky theory for lift and Prandtl theory for drag, is incorrect in the sense that it does not describe  the actual physics of flight.

Nevertheless, Kutta-Zhukovsky-Prandtl theory has served as the theory of flight propagated in text books since its conception in the beginning of the last century into our time.  The reason this incorrect theory has survived for so long is that it is based on singularities which are closed to inspection, and science like government without inspection can go astray.

Black Holes

The singularity par excellence is a black hole:

  • There is general consensus that supermassive black holes exist in the centers of most galaxies. In particular, there is strong evidence of a black hole of more than 4 million solar masses at the center of our galaxy, the Milky Way.
  • A black hole the weight of a car would have a diameter of about 10−24 m and take a nanosecond to evaporate, during which time it would briefly have a luminosity more than 200 times that of the sun. 
But maybe the center of our galaxy is not a black hole but instead a smooth distribution of dark matter which is the opposite of a singularity.

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