For decades, physicists have grappled with a fundamental question: how disordered are black holes on the inside? The problem isn’t just that we can’t see inside them—it’s that the very concept of disorder breaks down when applied to these extreme regions of spacetime. Recent breakthroughs in mathematics have finally allowed scientists to calculate black hole entropy, revealing a surprising connection between what is and what we can know about the universe.
The History of Entropy
The idea of entropy originated in the 19th century, with physicists like Ludwig Boltzmann struggling to explain why engines always lose energy as waste heat. Boltzmann realized that entropy measures the number of microscopic arrangements that produce the same macroscopic outcome. Imagine a room full of gas molecules: they can be arranged in countless ways, but only a few would gather all the molecules into one corner. Entropy quantifies this hidden chaos.
This concept was later extended into quantum mechanics by John von Neumann in the 1930s. In the quantum world, particles don’t have fixed properties but rather probabilities of being measured. Von Neumann showed that entropy could quantify this inherent uncertainty, including how entangled systems—where two regions are deeply connected—affect our knowledge of the whole.
The key difference is that Boltzmann’s entropy describes what is physically happening, while von Neumann’s describes what we can know.
The Black Hole Paradox
In the 1970s, Jacob Bekenstein challenged Stephen Hawking, arguing that black holes must have entropy to avoid violating the second law of thermodynamics (which states that the universe’s total entropy must always increase). Hawking initially dismissed this, as black holes were thought to have no internal structure. However, Hawking later discovered Hawking radiation, proving that black holes have a temperature—and therefore, entropy.
This raised a new question: if black holes have entropy, what is the underlying microscopic structure that creates it? Some physicists theorize that it could be an arrangement of particles, entangled quantum information, or even more abstract building blocks of spacetime itself.
Breaking Through Mathematical Barriers
For decades, researchers struggled to make progress. The problem was that quantum mechanics treats space-time as static, while general relativity says it bends and flexes in response to matter and energy. This discrepancy made calculations impossible, often leading to meaningless infinities.
In 2023, a team led by Ed Witten at the Institute for Advanced Study (IAS) flipped the script. They wove gravity into the quantum calculations from the ground up, allowing space-time to participate in the quantum churn. This stabilized the calculations and eliminated the infinities.
The Shocking Convergence
Using Witten’s new mathematics, Gautam Satishchandran and his colleagues at Princeton University calculated the von Neumann entropy of a black hole. The results were astonishing: the entropy calculated using thermodynamic arguments (Bekenstein-Hawking) was exactly equal to the von Neumann entropy, which measures what we can observe.
This implies that the external surface of a black hole is perfectly entangled with its interior, meaning that we don’t need to peer inside to understand its full structure. This discovery is akin to deducing the contents of a chaotic room simply by observing the door—a powerful convergence between reality and observation.
Implications for the Cosmos
The implications extend beyond black holes. The same principles apply to the cosmological horizon, the furthest distance we can observe due to the universe’s expansion. The Hawking-Gibbs equation, which describes the entropy of an expanding universe, also matches the von Neumann entropy.
This suggests that gravity itself might exhibit quantum-like behavior, where different observers access different parts of the universe and shape what they can measure. As Satishchandran notes, “The line between what’s real and what’s observable is growing thinner.”
In conclusion, these breakthroughs suggest that entropy isn’t just a measure of disorder but a fundamental property that connects spacetime to quantum observation. The universe may be governed by the limits of what we can know, rather than by hidden structures beyond our reach.
