The entropy of black holes and information

Since the 1970s, the study of black holes has profoundly transformed the understanding of entropy, a fundamental concept in thermodynamics, by tying it to the most mysterious astrophysical objects in the universe. The revelation that these strange regions of spacetime can possess entropy, measuring the amount of information they contain, has not only opened new perspectives on the nature of information in the universe but also shed light on deep paradoxes between general relativity and quantum mechanics. Understanding how the “event horizon,” this invisible boundary that delineates the black hole, harbors this entropy has become a central issue in theoretical physics. Recent advances in quantum field theory and quantum gravity today offer promising insights into the fundamental role of this entropy in maintaining physical coherence and information conservation in a universe dominated by extreme gravity.

At the heart of this scientific endeavor, the discovery of Hawking radiation in 1974 shattered paradigms by demonstrating that black holes are not totally opaque entities but emit thermal radiation, characterized by a well-defined temperature linked to their surface gravity. This temperature, thin but non-zero, makes plausible the association of entropy proportional to the area of the event horizon according to the famous formula of Bekenstein-Hawking. An entropy that, contrary to usual intuition, rests on proportionality to area and not volume, suggesting an essential role of the surface in the stored information, a concept that contemporary physics connects to the holographic principle.

• Black holes possess entropy proportional to the area of their horizon, not to their volume, thus defying classical thermodynamics concepts.
• Hawking radiation implies that black holes have a non-zero temperature, despite their absorbing nature.
• The information paradox raises the question of whether information is conserved or lost during collapse into a black hole.
• Advanced theories of quantum gravity seek to describe the microscopic structure of black hole entropy.
• Models like string theory provide solid avenues for interpreting the entropy of black holes in a microscopic dimension.

The origin of the notion of entropy for black holes and its historical foundations

The concept of entropy applied to black holes stems from an initial questioning of energy conservation and physical properties during the merging of these cosmic objects. Stephen Hawking, while studying the possible collision of two black holes, made a remarkable observation: when two black holes merge, the surface area of the resulting black hole can never be less than the sum of the surface areas of the two initial black holes. This property directly evokes the second principle of thermodynamics, which states that the entropy of an isolated system never decreases. This analogy opened the door to the idea that black holes might possess a defined entropy linked to their surface, while they were traditionally considered objects devoid of any form of emission and thus without temperature or entropy.

The crucial contribution of Jacob Bekenstein was to anchor this concept on a quantum basis by associating entropy with a number of bits of information contained within the black hole. By considering the effects of quantum mechanics, particularly the effective size of elementary particles, Bekenstein demonstrated that a black hole must necessarily absorb a minimum increase in surface area proportional to a fraction of the Planck area when it captures this information. This heuristic approach resulted in formalizing a relation between the event horizon area and the stored information, which was a major breakthrough toward understanding black hole thermodynamics.

These developments laid the groundwork for a new scientific discipline, black hole thermodynamics, which establishes strong bridges between gravity, thermodynamics, and quantum field theory. It notably relies on formal analogies such as that which expresses the variation of a system’s internal energy as a function of its temperature and entropy. This formulation, extended to black holes, was the starting point for formulating the relationship between mass, surface area, and surface gravity, a fundamental element for quantifying this entropy.

Hawking radiation and the revelation of black hole temperature

Before 1974, black holes were envisioned as perfectly black objects, without any emission. The demonstration by Stephen Hawking marked a radical turning point by showing, based on quantum mechanics applied to the gravitational field near the event horizon, that black holes emit thermal radiation. This radiation, called Hawking radiation, gives them a temperature inversely proportional to their mass, related to their surface gravity.

In more precise terms, this radiation comes from pairs of virtual particles created at the edge of the event horizon. One of the particles can fall inside the black hole, while the other escapes, producing radiation perceived by an outside observer. This thermal emission thus allows associating a temperature T and, by extension, an entropy with this system.

The Bekenstein-Hawking formula expresses this entropy S as a function of the surface A of the event horizon, of the Boltzmann constant k_B, and fundamental constants such as the reduced Planck constant and gravity:

Parameter	Meaning	Approximate value
Entropy (S)	Measure of the stored information	Proportional to the area of the horizon
Surface A	Area of the event horizon	Variable according to the black hole
kB	Boltzmann constant	1.38 × 10⁻²³ J/K
ℏ (reduced Planck constant)	Fundamental in quantum mechanics	1.05 × 10⁻³⁴ J·s

This discovery was essential to clarify the previous inconsistencies between general relativity and thermodynamics applied to black holes. Indeed, it officially confirmed that entropy was linked to an observable physical characteristic, the temperature of radiation, and that the black hole could be treated as a thermodynamic system in the strict sense.

One of the most debated paradoxes related to this radiation is the information paradox. Indeed, if a black hole radiates and gradually evaporates, how can the information contained in the absorbed matter be preserved, thus preventing a violation of the fundamental principle of information conservation in quantum mechanics? This paradox remains at the center of current research in theoretical physics.

The information paradox and debates on conservation in black holes

The information paradox results from a deep tension between several major branches of physics, notably general relativity and quantum mechanics. General relativity describes the black hole as an object whose event horizon creates an insurmountable barrier, which implies an apparent loss of information for an outside observer as soon as matter crosses this invisible boundary.

However, quantum mechanics imposes that information, representing the complete state of a physical system, must be strictly preserved over time. The tension arises precisely when a black hole, through Hawking radiation, appears to lose mass and evaporate, suggesting that the information associated with the absorbed matter would be irretrievably lost.

The conservation of information is crucial for the consistency of modern physical laws. Numerous theories have been proposed to resolve this paradox, including:

• The existence of a holographic correspondence which states that all information within a volume is encoded on its surface, referred to as the holographic principle.
• Modifications of conventional physics near the Planck scale, this limit induced by quantum gravity, where the effects of quantum mechanics and gravitation combine in complex ways.
• Advanced models suggesting that information escapes the black hole through processes still poorly understood in advanced quantum physics.

In this context, thermodynamics applied to black holes offers a valuable framework for examining how entropy and information evolve, notably through the study of famous paradoxes in physics and their technically complex resolutions. This research remains at the heart of current issues to bridge the gap between general relativity and quantum mechanics, particularly within the framework of quantum gravity.

Quantum gravity, string theory, and the microscopic structure of black hole entropy

One of the most fundamental questions today is to understand the intrinsic nature of black hole entropy: to what physical microstates does it correspond? The classical formula shows that entropy is proportional to the square area of Planck length, which points towards a quantum description of gravity.

In 1995, major advances were made by Andrew Strominger and Cumrun Vafa who, within the framework of string theory, were able to reproduce the entropy of so-called supersymmetric black holes by precisely counting the microscopic states arising from D-branes, the fundamental components of this theory. This work constitutes a first solid proof that quantum gravity can offer a statistical understanding of black hole entropy.

At the same time, loop quantum gravity proposes an alternative vision, decomposing the horizon surface into discrete units related to the Planck length. These approaches have allowed for the assignment of a plausible physical interpretation to black hole entropy for specific types. However, their generalization to a broader spectrum of black holes remains a significant scientific challenge.

This confrontation between different theoretical approaches attests to the complexity and richness of the problem. Research continues to rely on powerful tools from quantum mechanics to refine our understanding of the structure of information and the physical laws at work at these extreme scales.

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The implications of black hole entropy on our understanding of information in the universe

Black hole entropy now extends beyond purely theoretical considerations to influence fundamental concepts regarding the very nature of information in the universe. The proposition that all information related to a spatial volume could be encoded on its surface, in accordance with the holographic principle, revolutionizes our approach to the structure of spacetime and information conservation.

The unique specificities of black holes, notably their entropy dependent on the area of their horizon, have led to a reconsideration of classical notions of space, time, and information. In contemporary physics, these ideas fuel research involving quantum models of spacetime where the concepts of entropy, information, and quantum gravity are intimately linked.

To illustrate these relationships, the issue of “lost information” in black holes embodies the theoretical challenge to reconcile the laws of quantum mechanics with those of relativity. The evolution of our understanding of this entropy could well provide essential keys to construct a unified theory between these two pillars of modern physics.

The spectacular advances of recent decades allow us to contemplate the future with the hope that the mysteries surrounding black holes, entropy, and information will soon find concrete answers. In the meantime, this research fuels numerous exciting debates within the international scientific community.