Entropy and Geometric Objects

Different notions of entropy can be identified in different communities [1]: (i) the thermodynamic sense, (ii) the information sense, (iii) the statistical sense, (iv) the disorder sense, and (v) the homogeneity sense. Especially the “disorder sense” and the “homogeneity sense” relate to and require the notion of space and time. One of the few prominent examples relating entropy to geometry and to space is the Bekenstein-Hawking entropy of a Black Hole. Although being developed for the description of a physics object – a black hole – having a mass, a momentum, a temperature, a charge etc. absolutely no information about these attributes of this object can eventually be found in the final formula. In contrast, the Bekenstein-Hawking entropy in its dimensionless form [2] is a positive quantity only comprising geometric attributes like an area A- which is the area of the event horizon of the black hole- , a length L_P – which is the Planck length - and a factor ¼. A purely geometric approach towards this formula will be presented. The approach is based on a continuous 3D extension of the Heaviside function [3] drawing on the phase-field concept of diffuse interfaces [4]. Entropy enters into the local, statistical description of contrast resp. gradient distributions in the transition region of the extended Heaviside function definition. The Bekenstein- Hawking formula structure can eventually be derived based on such geometric-statistic considerations.

1. Haglund, J.; Jeppsson, F.; Strömdahl, H.: “Different Senses of Entropy—Implications for Education.” Entropy 2010, 12, 490-515.
2. Bekenstein, J. D. (2008): Scholarpedia, 3(10):7375.     doi:10.4249/scholarpedia.7375
3. see e.g. : Weisstein, Eric W. "Heaviside Step Function." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/HeavisideStepFunction.html
4. see e.g. Provatas, N., Elder, K.: Phase-Field Methods in Materials Science and Engineering, Wiley VCH, Weinheim (2010),ISBN: 978-3-527-40747-7