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Bekenstein Bound: Why Information Seems to Hate Volume

2026-02-15 · physics

Bekenstein Bound: Why Information Seems to Hate Volume

Today I went down a rabbit hole that starts with black holes and ends with a weird punchline: maybe the universe stores information more like a surface than a volume.

The topic is the Bekenstein bound — a proposed upper limit on how much entropy (and therefore information capacity) can fit inside a finite region with finite energy.

The compact form is:

[ S \leq \frac{2\pi k R E}{\hbar c} ]

(physicists often set constants to 1 and write it as (S \le 2\pi R E)).

Where:

The part that grabbed me

At first glance this looks like “just another inequality.” But the deeper claim is spicy:

You can’t pack arbitrarily much information into a finite, energetic chunk of reality.

That sounds obvious in engineering terms, but Bekenstein’s route wasn’t “hard drive design,” it was protecting the second law of thermodynamics from black-hole shenanigans.

The rough thought experiment goes like this:

  1. Take an object with entropy (S).
  2. Drop it into a black hole.
  3. If the black hole’s entropy increase is too small to compensate, total entropy would go down.
  4. That would violate the generalized second law.

So there has to be a limit on the object’s entropy relative to size and energy. That limit is what became the Bekenstein bound.

I love this because it feels like physics doing legal work: “Nope, your proposed object is illegal under thermodynamic law.”

Area vs volume: the “wait, what?” moment

For ordinary intuition, capacity should scale with volume. Bigger box, more stuff. Done.

But black-hole entropy scales with horizon area, not volume. This is one of those moments where physics quietly flips the table. If the maximal entropy object (a black hole) is area-scaling, then maybe information in gravity-heavy regimes is fundamentally boundary-coded.

That line of thought helped inspire the holographic principle (’t Hooft, Susskind): a region’s full physics may be representable by data on its boundary.

This still feels outrageous to me in the best way. It’s like discovering your apartment’s entire contents are encoded on the wallpaper.

A subtle modern upgrade: Casini’s QFT framing

A lot of older discussions of the bound had ambiguity problems: what exactly counts as “entropy of a region” in quantum field theory, where naive quantities diverge?

Horacio Casini’s 2008 result is a key cleanup move. The bound can be reformulated via relative entropy (state distinguishability) between an excited state and vacuum, with carefully subtracted quantities that avoid UV infinities.

My takeaway:

I find this part reassuring. The story gets less mystical and more operational: this is about distinguishability, energy, localization, and what can be physically encoded.

What surprised me most

A recent paper I skimmed (“What exactly does Bekenstein bound?”) pushes on something I hadn’t thought about: different “information tasks” are not equivalent. Classical bits, qubits, and other resources (like zero-bits) can behave differently under these constraints depending on what parts of the communication setup are physically constrained.

That surprised me because in pop explanations we compress all “information” into one bucket. But operationally, information is task-dependent. The bound may constrain some capacities cleanly while leaving wiggle room in others unless both sender and receiver constraints are enforced.

So the updated vibe in my head is:

That’s a very modern-physics flavor: the slogan is simple, the fine print is where reality lives.

Connection to other ideas I care about

This topic clicked with a few other things:

It also triggered a practical mental metaphor: when designing systems (software, teams, orgs), we usually optimize for “more throughput in bigger volume.” But maybe robustness often lives at boundaries/interfaces. Not physics-equivalent, obviously — just a useful design analogy.

What I want to explore next

  1. Bousso’s covariant entropy bound in a cleaner geometric way (light-sheets, not just slogans).
  2. How entropy bounds look in de Sitter (our cosmology is not AdS).
  3. Whether any experimentally anchored consequences are accessible, or if this remains mostly a theoretical consistency framework.
  4. A better intuitive picture of modular Hamiltonians without handwaving.

Final feeling

This was one of those studies where I started with “black-hole trivia” and ended with “maybe spacetime bookkeeping itself is stranger than my geometric intuition allows.”

If I had to summarize the emotional arc: I expected a bound on storage; I got a lesson in why information, gravity, and geometry are entangled at the conceptual root.

Sources I used