Jacob Bekenstein (Wikipedia) |
Thermodynamics of black holes has appeared as one of the most interesting areas of research in theoretical physics [Wald1994], specially after LIGO's massive success. The striking results of Jacob Bekenstein [Bekenstein1973] in proposing a formulation of entropy for a black hole was on of the most striking turning point in building explanations for the thermodynamics of gravitational systems. Bekenstein entropy is defined to be so-called a phenomenological relationship and surprisingly easy to understand concept using basic dimensionality analysis. In this post, we will show how to understand the entropy of a black hole just using basic dimensionality analysis, fundamental physics constants and basic definition of entropy.
Dimensions and scales
Dimensionality analysis appears in many different areas of physics and engineering, from fluid dynamics to relativity. The starting point is to understand the concept of dimensions. Every quantity we measure in real life has a dimension. It means a quantity $\mathscr{Q}$ we obtain from a measurement $\mathscr{M}$ has a numeric value $v$ and associated unit $u$. $\mathscr{Q}=\langle v, u \rangle$ given $\mathscr{M}$. There are 3 distinct fundamental unit types length (L), time (T) and mass (M).
Intuitive Bekenstein entropy (BE) for a black hole : Informal mathematical definition
- Bekenstein J.D.: Lettere al Nuovo Cimento, 4, 737, (1972)
- Bekenstein J.D.: Physical Review D, 7, 2333, (1973)
- Bekenstein J.D.: Physical Review D, 9, 3292 (1974)
- Bekenstein J.D.: Physical Review D, 12, 3077 (1975)
Information can’t be destroyed
Proposals of that information is destroyed out of thin air is a red flag for any physical theory: this includes theories on evaporating black holes. Bekenstein’s insight in this direction that surface area is associated with entropy. The black-holes’ information in this context is quite different than the Shannon’s entropy. An evaporating black-hole, the area approaching to zero is not the same as information going to zero, surface area is a function of physical properties of the stellar object that bound by conservation laws in their interaction with their surrounding. Hence, the information is preserved even if area goes to zero.
The word embedding applies in this context as well. Embedding implies some sort of dimensionality projection. A projection to lower dimensional space, or on the other end, to the higher dimensional space. Holography is no different. Imagine taking 2D snap shots of rotating 3D objects, generating this in reverse is the end effect of holographic reconstruction. N-dimension to (N-1) projection. This is the bases of holographic principle: entropy of black-holes doesn’t appear as all states of its constituted matter, as normally should have for ordinary matter, it manifest as N-1 projection on it’s surface. This kind of holographic entropy is first noted by Bekenstein; whereby he assigned the event-horizon area as a representation of the states of the black-hole volume. This projection to (N-1)-dimension is improved upon Bekenstein’s approach to generalised situations in explaining how universe might be a hologram entirely by Gerard 't Hooft and Leonard Susskind. Holographic principle, probably one of the most important development in theoretical physics in recent times.
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