1.2 Black Hole Entropy and Bekenstein Bound: The Universe as a Finite-Capacity Hard Drive

In the previous section, we pointed out the infinite divergences caused by the continuum hypothesis and proposed the necessity of natural cutoff for physical reality at the Planck scale. If ultraviolet divergence is merely theoretical “ugliness,” then the discovery of black hole thermodynamics provides solid, even mandatory physical evidence for this discrete ontology.

1.2.1 Bekenstein’s Insight: Does Information Have Volume?

In the early 1970s, when physicists were still debating whether black holes were merely mathematical singularities of general relativity, Jacob Bekenstein proposed a seemingly naive but highly subversive question: If I pour a cup of hot tea (with entropy) into a black hole, does the total entropy of the universe decrease? Because the black hole swallows everything, including information.

If the second law of thermodynamics is universal, black holes themselves must possess entropy.

Through thought experiments, Bekenstein discovered that black hole entropy $S_{BH}$ should not be proportional to its volume (like ordinary thermodynamic systems), but proportional to the surface area $A$ of its horizon. Subsequently, Stephen Hawking confirmed this relationship through semiclassical calculations and gave that famous formula:

$$S_{BH}=\frac{k_B{c}^3}{4G\hslash }A=\frac{A}{4l_P^2}$$

where $l_P=\sqrt{G\hslash /c^3}\approx 1.6\times 10^{-35}$ meters is the Planck length.

This formula is one of the most beautiful equations in the history of physics, unifying thermodynamics ($k_B$), relativity ($c$), gravity ($G$), and quantum mechanics ($\hslash$). But for our discussion, its most important significance lies in revealing the geometric nature of information.

1.2.2 Planck Pixels and Finite Capacity

Let us carefully examine the meaning of this formula. Entropy in information theory corresponds to the number of information bits ($S=N\ln 2$). The Bekenstein-Hawking formula tells us that every $4l_P^2$ area (four Planck areas) on the black hole horizon can store exactly 1 bit of information.

This is a startling conclusion: Information is not a continuously distributed fluid, but discretely “paved” on the surface of spacetime.

Furthermore, Bekenstein proposed the Bekenstein Bound: For any spherical spatial region containing energy $E$ and radius $R$, the maximum entropy (i.e., maximum information) $S_{max}$ it can contain is finite and satisfies:

$$S\le \frac{2\pi k_{B}RE}{\hslash c}$$

When this region collapses into a black hole, entropy reaches the maximum value $S=A/4l_P^2$.

This means: For any finite volume of space in the universe, no matter how we compress matter or energy into it, the total number of quantum states $W=e^S$ it can contain is strictly a finite integer.

1.2.3 The End of Continuum

If space were continuous, then even a tiny needle tip could theoretically contain infinite information (because we could infinitely subdivide coordinates). But the black hole entropy formula directly negates this. It shows that if we pile too much information in a region, space itself will “crash” (become a black hole) due to gravitational collapse, thus locking the information limit.

This provides the strongest physical support for our discrete ontology:

1. Space is not a container, but a storage medium: The geometric area of space directly corresponds to storage capacity (hard drive size). Planck length $l_P$ is the smallest magnetic domain (Bit) of this cosmic hard drive.

2. Holographic Principle: Since the maximum information in a volume is determined by its surface area, this suggests that the “bulk” information of three-dimensional space can actually be losslessly encoded on a two-dimensional boundary. Like holograms, this dimensional reduction encoding is mathematically possible only in discrete systems (continuum cardinalities differ, preventing one-to-one mapping).

1.2.4 Conclusion: The Universe as a Finite State Machine

Synthesizing the above derivations, we must accept an extremely profound conclusion: Our universe, at any given moment, for any finite observation horizon, contains a finite total amount of information.

If state space is finite and evolution rules are unitary (information conservation), then the universe is essentially equivalent to a finite state machine or a quantum cellular automaton (QCA) running on a huge but finite lattice.

Infinity not only does not exist physically, but is also redundant in information theory. Black holes are not merely celestial bodies; they are the “memory overflow” protection mechanism of this giant cosmic computer, reminding us that the granularity of physical reality has a bottom line.

In the next section, we will shift our gaze from macroscopic black holes to microscopic qubits, exploring how “matter” itself emerges from pure information.