Insights into Bekenstein entropy with an intuitive mathematical definitions: A look into Thermodynamics of Black-holes

Jacob Bekenstein
(Wikipedia)

Preamble

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

Black holes are astronomical objects that are not directly observable due to their mass condensed in a small area. The primary object we will use is something called Planck length $L_{p}$ and it implies physically possible smallest patch of the space-time, this is associated with the state of the black holes on their horizon. We won't define the Planck length here in detail but with the knowledge of fundamental physics constants and dimensional analysis we mentioned, one can get a constant value for this length. 

Definition: Finite entropy $S_{f}$ of an object is associated with the number of states $\Omega$ a system can attain.

If we combine this definition for a black hole entropy : 

Definition Finite entropy of a black-hole $S_{f}^{BH}$ is  associated with the number of its states $\Omega$, number of elements on it's surface area of $A$. The elements are discretised with  small patches $a_{p}=L_{p}^{2}$. Then intuitively,  $\Omega$ yields to $A$ divided by $a_{p}$.

  

Bekenstein entropy is not thermodynamic entropy alone and family of Bekenstein entropies

The unit analysis tells us that $A$ has a dimension of length square.  We intentionally omit any equality in the above definition upon $S_{f}^{BH}$ because, in practice Bekenstein Entropy is not thermodynamic entropy alone. The formulation usually presented as BE in general uses equality for the above approach. However this is not strictly thermodynamical alone, that's why we specify definitions as finite entropy and only express the relationship as association. Similarly any other constants as it can yield to different Bekenstein entropies such as introduction of new constants would yield to family of Bekenstein entropies.

Why surface area defines states of a black-hole?

This is an amazing question and Bekenstein's main contribution is to associate this to number of states of a black-hole on event horizon, i.e., point of of no return layer whereby ordinary matter can't return. The justification is that all other properties of black hole defines this surface. Here is the intuitive definition of states of black-hole.

Definition A surface area $\mathscr{A}$ is formed by the set of physical properties forming an ensembles. such as charge density, angular momentum. These ensembles indirectly samples thermodynamics ensembles. 

Even though intuition is there, this question might still be an open question further.

Conclusion

We provided the primary idea that Bekenstein tried to convey in his 1973 paper intuitively. However,  we identify its thermodynamic limit is an open research area. Thermodynamic limit implies that taking infinite limit of both area and the discretised areas, even though it sounds that the values might converge to infinity, simultaneous limit would converge to a finite value for a physical matter. 

Primary Papers

• 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)

Primary Book

• Wald, Quantum field theory in curved space times (1994) 

Please cite as follows:

 @misc{suezen23ibe, 

     title = {Insights into Bekenstein entropy with an intuitive mathematical definitions}, 

     howpublished = {\url{https://science-memo.blogspot.com/2023/02/bekenstein-entropy.html}, 

     author = {Mehmet Süzen},

     year = {2023}

  }

Postscript A: 

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.

Postscript B: 

What is Holographic principle? its origins from Bekenstein Entropy perspective

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.

Heavy-matter-wave and ultra-sensitive interferometry: An opportunity for quantum-gravity becoming an evidence based research

    Solar Eclipse of 1919
(wikipedia)

Preamble 

   

Cool ideas in theoretical physics are ofter opaque for general reader whether if they are backed up with any experimental evidence in the real world. The success of LIGO (Laser Interferometer Gravitational-wave Observatory) definitely proven the value of interferometry for advancement of cool ideas of theoretical physics supported by real world measurable evidence. An other type of interferometry that could be used in testing multiple-different ideas from theoretical physics is called matter-wave interferometry or atom interferometry: It's been around decades but the new developments and increased sensitivity with measurement on heavy atomic system-waves will pave the technical capabilities to test multiple ideas of theoretical physics. 

Basic mathematical principle of interferometry

Usually interferometry is explained with device and experimental setting details that could be confusing. However,  one could explain the very principle without introducing any experimental setup.  The basic idea of of interferometry is that if a simple wave, such as $\omega(t)=\sin\Theta(t)$, is first split into two waves and reflected over the same distance, one with shifted with a constant phase, in the vacuum without any interactions. A linear combination of the returned waves $\omega_{1}(t)=\sin \Theta(t)$ and  $\omega_{2}(t)=\sin( \Theta(t) + \pi))$, will yield to zero, i.e.,  an interference pattern generated by $\omega_{1}(t)+\omega_{2}(t)=0$. This very basic principle can be used to detect interactions and characteristics of those interactions wave encounter over the time it travels to reflect and come back. Of course, the basic wave used in many interferometry experiments is the laser light and interaction we measure could be gravitational wave that interacts with the laser light i.e., LIGO's set-up.

Detection of matter-waves : What is heavy and ultra-sensitivity?

Each atomic system exhibits some quantum wave properties, i.e., matter waves. It implies a given molecular system have some wave signatures-characteristics which could be extracted in the experimental setting. Instead of laser light, one could use atomic system that is reflected similar to the basic principle. However, the primary difference is that increasing mass requires orders of magnitude more sensitive wave detectors for atomic interferometers. Currently heavy means usually above ~$10^{9}$ Da (comparing to Helium-4 which  is about ~4 Da), these new heavy atomic interferometers might be able to detect gravitational-interactions within quantum-wave level due to precisions achieved ultra-sensitive. This sounds trivial but experimental connection to theories of quantum-gravity, one of the unsolved puzzles in theoretical-physics appears to be a potential break-through. One prominent example in this direction is entropic gravity and wave-function collapse theories.  

Conclusion

Recent developments in heavy matter-wave interferometry could be leveraged for testing quantum-gravity arguments and theoretical suggestions. We try to bring this idea into general attention without resorting in describing experimental details. 

Further Reading & Notes

• Dalton, mass-unit used in matter-wave interferometry. 
• Atom Interferometry by Prof. Pritchard YouTube.
• Newton-Schrödinger equation.
• Papers of Kingsley R. W. Jones are also very novel in this direction. 
• A roadmap for universal high-mass matter- wave interferometry  Kilka et. al. AVS Quantum Sci. 4, 020502 (2022). doi
• Current capabilities as of 2022, atom interferometers can reach up to ~300 kDa.
• Testing Entropic gravity, arXiv. 
• NASA early stage ideas workshops : web-archive

Testing emergent gravity: Gravitational Lensing to atom interferometer

Paranal Telescopes in Chile. (ESO/H.H. Heyer)

In this post, I would like to briefly discuss emergent gravity, an idea, that gravity itself is an artefact of more fundamental description, such as entropy. Despite the fact that we have an experimental evidence of gravitational waves, thus gravity really exist and physically detectable: The recent results of Laser Interferometer Gravitational-Wave Observatory (LIGO) programme in detecting gravitation waves was a landmark experimental evidence supporting Einstein's General Relativity theory.

Emergent Gravity: Verlinde's thesis

Eric Verlinde has proposed a controversial hypothesis in 2010 that gravity is originated from an entropic force [here]. He has shown that both Newtonian and Einstein's gravitational equations are artefacts of entropic force and in 2016 he proposed a similar approach in explaining galactic motions without the need of using dark matter [here].

Testing Emergent Gravity: Gravitational Lensing

Margot M. Brouwer and her co-workers have published a work [here], using weak gravitational lensing data from ESO telescopes. This was the first evidence for Verlinde's theory which attracted a lot of media attention because of implications in our understanding of the nature of gravity.

Testing Entropic Gravity Directly: Atom Interferometer

Despite this initial test of emergent gravity, there is still a lack of direct experimental evidence for Verlinde's initial hypothesis that force laws are artefacts of entropic force. Recently another approach is proposed to test this entropic force argument, [here], using mater-wave interferometry via utilising part of Newton-Schroedinger. The core idea is there is a direct relationship between the gravitational constant G and the atomic system's quantum state. If this is experimentally feasible, maybe with next generation atom interferometer systems, this could be a direct test for Verlinde's entropic force argument.