Thermodynamic Characterization of Planar Shapes and Curves, and the Query of Temperature in Black Holes

Abstract

The purpose of this research is to characterize shapes in thermodynamic terms, namely, in terms of total energy, dissipative energy, entropy, and temperature. As case studies, polygons and some well-known curves were taken, and they were characterized using physical terms. The relation between entropy and curvature was elucidated, and the black hole surface gravity and temperature were criticized and reinterpreted from this point of view. Particular energy attributions were evaluated by comparing the position of any edge of a polygon (i.e. its angle with the horizontal axis) with a broken crystal surface. Energies of all edges were added up at all positions between 0˚ - 360˚. In regular polygons, the total energy decreases with the increase of the number of edges. Entropy increases in the reverse order, and the increase of the number of edges increases entropy. It implies that the circle has the lowest energy but the highest surface entropy. In curves (circle, sine-curve, spiral, and exponential curve), the total energy, dissipative energy, and entropy all depend on amplitude and also on specific variables. Black hole entropy expressed in terms of the surface area is a configurational entropy and not thermal entropy; therefore, it does not involve a varying temperature term. The surface gravity of a black hole is connected to acceleration and thus to curvature. To relate it with the temperature needs to be reinterpreted, because, surface gravity behaves like an attractive force not exactly like temperature. Hawking radiation is still possible, but the black hole does not get warmer as it evaporates. Material loss from the black hole gets faster as its radius decreases due to the curvature effect, i.e. by a mechanism similar to the Gibbs-Thomson effect.