ABSTRACT
Following this idea we interpret T-lines as the boundaries of domains in which an extremal for a given process or physical phenomenon appears or as the lines indicating the locality of a physical process. We apply a metrical approach instead of a topological one. It is known that many physical processes are described by harmonic or more general functions v(x,y) indicating, for example, the temperature in a point (x. y) of a thermal phenomena, the velocity of the movement of a point in hydrodynamics, the tension in the theory of elasticity and electric processes etc. For a given function v(x,y) we introduce a complex function w(z) = u(x.y) + iv(x,y). If 7(0) is the real axis, then the 7(0)-line of the function w(z) is a level set of iy(x,y)1 in which the temperature, the movement velocity or the tension is equal to zero. Similarly, if ^(t) is the straight line {w.lmw = 0}, then v(x,y) = t on a 7(t)-line. Hence a 7(£)-line separates the domains in which the mentioned physical magnitudes are correspondingly greater or less than t. Consequently, while investigating 7(t)-lines for the "critical'' values of t, we, actually, investigate the boundaries of the domains where the considered physical process becomes extremal or critical. In other words, a physical phenomenon undergoes a "catastrophe" on 7(£)-lines. It is clear that: the domains of high tension are the most possible places for the cracking of the material
5.1. T-lines in Physics 145
in the theory of elasticity, the domains of high voltage are the most possible places for electric discharges in electrodynamics and the domains of high temperature are where substance turns to plasma. 7(t)-lines for t = 0°C and t = 100°C can be considered as lines separating the domains of ice, water and steam for thermal processes in water. It is evident that for a critical value t the smallness of the length of critical 7(£)-lines in comparison with the whole boundary of the given process implies that the extremal behavior is observed only in some small parts of the field, i.e. the phenomenon is localized in a certain sense. Also, the smallness of the boundary of domains of high tension and temperature can lead to the conclusion that, for example, the cracks inside the material are localized and cannot spread down to the boundary and that steam, magma and plasma conceived in the interior of the earth will not spread outside.
