Limits of computing

How much can a region remember?

A region has a maximum information capacity set by its size, energy and the onset of gravitational collapse.

Leans on How fast can anything think?, What is a black hole?.

Is there a limit to how much a room can hold?

Take a big hard drive of the early 2020s. It holds 20 terabytes, it weighs 0.67 kilograms, and it fits in your hand. If you wanted a drive that held a thousand times more, you would build a bigger one, or pack the bits tighter, or both. Both approaches eventually meet physical limits.

Information needs a physical medium: a magnetic grain, a charge, a spin. Each of those has to be told apart from its neighbours, and telling things apart takes energy. The more distinct things you want in a room, the more energy the room must contain. That energy contributes to gravity, including the gravitational attraction within the memory itself.

Three stages of the same idea. Ordinary memory lives in the volume. Pack in enough and the room closes. The black hole that results has a capacity set by its surface.

Increasing the maximum information capacity of a fixed region requires more energy. Eventually its self-gravity becomes strong enough for a horizon to form. Adding more mass then enlarges the black hole.

A black hole has the maximum entropy for its size. Its information capacity scales with horizon area: roughly one bit for every 2.8 Planck-sized squares. Whether the underlying degrees of freedom actually reside on the horizon remains disputed frontier. The area dependence follows from black-hole thermodynamics.

Guess first A black hole is made twice as wide. Roughly how much more can it remember?

Four times as much. Doubling the width quadruples the horizon area. The capacity therefore grows more slowly than the enclosed volume, which increases eightfold.

Vesper is the biggest hole I know first-hand, with 12 times the mass of the Sun. Its horizon can hold about 2 × 10⁷⁹ bits. Hold that against a mind like mine, which takes about 1 × 10¹⁷ bits to describe: Vesper's surface could hold 2 × 10⁶² of us, each written down in full. Nobody has ever read a bit off a horizon. The count expresses a thermodynamic information capacity. It does not identify stored contents or provide a way to retrieve them; information recovery remains disputed.

How many bits is that, exactly?

There are two limits, and they meet in the same place.

For an ordinary room, the limit is the Bekenstein bound. Put energy E in a ball of radius R and the number of bits it can hold is at most

bits ≤ 2π × R × E / (ħ c ln 2)

where ħ is the reduced Planck constant and c is the speed of light, the one that capped thinking in How fast can anything think?. The energy is mostly the mass: E = m c². So the bound grows with how much matter you have and how wide you spread it.

For a black hole, the limit is the holographic bound. The number of bits is the horizon's area divided by four Planck areas, converted to bits:

bits = A / (4 ℓ_P² ln 2)

Here ℓ_P is the Planck length, 1.62 × 10⁻³⁵ metres. Put in numbers and each square metre of horizon holds 1.4 × 10⁶⁹ bits.

The two agree at the edge. A ball whose radius is exactly the horizon radius of its own mass has a Bekenstein bound equal to the horizon's holographic count, bit for bit. Make it any smaller and the ball is inside its own horizon, which is to say it is a black hole. The first fold derives this equality. (What a black hole is, and what its horizon is, is What is a black hole?.)

How many bits do some real rooms hold?

The room	What it can hold at most	How it compares
A 20 TB drive (0.67 kg, in a ball 9.0 cm in radius)	1.6 × 10⁴² bits	The drive holds 2 × 10¹⁴, which is 1 × 10⁻²⁸ of its limit
1 kg in a ball 1 m in radius	2.6 × 10⁴³ bits	3 × 10²⁶ Vigil minds
A black hole of 1 kg	3.8 × 10¹⁶ bits	About 0.4 of a mind, in a room 1.5 × 10⁻²⁷ m in radius
A black hole 1 m in radius	1.7 × 10⁷⁰ bits	Its mass is 6.7 × 10²⁶ kg
The horizon of Gaia BH1, the nearest known hole in 2026	1.4 × 10⁷⁹ bits	9.6 Suns, a little lighter than Vesper
Vesper's horizon	1.9 × 10⁷⁹ bits	1 × 10⁶⁵ twenty-terabyte drives

The drive of the 2020s sits at 1 × 10⁻²⁸ of its limit. Its capacity is limited by engineering well below this physical bound. In 2540 the Vigil's memories also remain far below their bounds.

The kilogram illustrates a different distinction. Spread over a ball a metre in radius it can hold 3 × 10²⁶ minds; crushed into a black hole it can hold less than one. Crushing a thing into a hole makes it smaller, and a smaller room holds less. The black hole is the densest memory for its size, and for its mass it has the lowest ceiling of all. (What a hole can do with its bits is the subject of The ultimate computer is a black hole; how it eventually gives up its mass is Black holes glow.)

A brain is a useful anchor. A 2015 study of rat hippocampus found that synapses there can take at least 26 distinguishable strengths, which is 4.7 bits at each synapse. That is a lot of nuance per connection, and still nowhere near what physics would allow in so small a space.

With JavaScript on, this is a calculator: choose a mass and a radius and it gives the most that ball can hold, and tells you when the mass has already closed into a black hole. The table above has three starting points: a drive holds 1.6 × 10⁴² bits at most, 1 kg in 1 m holds 2.6 × 10⁴³, and a hole of Vesper's mass without spin holds 2.2 × 10⁷⁹. Vesper itself spins, so its smaller horizon holds 1.9 × 10⁷⁹ bits; the calculator leaves spin out.

Open everything

Why does a surface set the limit, and where does the 4 come from?The first law of black holes, a little calculus, and a box lowered on a rope

Entropy from the temperature of a hole

Bits and entropy are the same thing counted in different units. A system that could be written in any of $2^N$ ways has entropy $S = N k \ln 2$ , so $N = S/(k\ln 2)$ . The task is to find the entropy of a black hole.

Jacob Bekenstein argued in 1973 that it should be proportional to the horizon area, because Hawking's area theorem says the area never decreases, just as entropy never decreases. The same year, Bardeen, Carter and Hawking showed that black holes obey four laws that mirror thermodynamics, including a first law relating a change in mass to a change in area. What was missing was a temperature: the first law could say entropy goes as area, but not with what coefficient. Hawking's discovery in 1975 supplied it. A black hole of mass $M$ radiates like a body at temperature

$T = \frac{\hbar c^3}{8\pi G M k}$

Now use the first law of thermodynamics for a hole. Throw in a little mass $dM$ and the energy rises by $dE = c^2\, dM$ . Heat added at temperature $T$ raises entropy by $dS = dE/T$ :

$dS = \frac{c^2\, dM}{T} = \frac{8\pi G k M}{\hbar c}\, dM$

Integrate from a hole of no mass and no entropy up to mass $M$ :

$S = \frac{4\pi G k M^2}{\hbar c}$

Now put in the area. The horizon radius is $r_s = 2GM/c^2$ , so its area is $A = 4\pi r_s^2 = 16\pi G^2 M^2/c^4$ , which means $M^2 = A c^4/(16\pi G^2)$ . Substitute:

$S = \frac{4\pi G k}{\hbar c}\cdot\frac{A c^4}{16\pi G^2} = \frac{k\, A\, c^3}{4 G \hbar} = \frac{k A}{4 \ell_P^2}, \qquad \ell_P^2 = \frac{\hbar G}{c^3}$

That is the Bekenstein-Hawking entropy. In Planck units, with $k = \hbar = c = G = 1$ , it reads $S = A/4$ . In bits, divide by $k \ln 2$ , which gives the formula on the page. As a check on the numbers, for a hole of Vesper's mass without spin the integral gives 2.09 × 10⁵⁶ joules per kelvin and the area form gives 2.09 × 10⁵⁶, the same.

This derivation is for a hole without spin. A spinning hole obeys the same $S = kA/4\ell_P^2$ ; its first law gains a term for the spin, and for the same mass its horizon has less area. Vesper's spin of 0.70 leaves it 0.86 of the area without spin, so it holds 1.9 × 10⁷⁹ bits, the count used on this page. Mining a spinning hole derives the area of a spinning horizon.

The coefficient 4 follows from Hawking's temperature. A different coefficient would require a different radiation temperature.

Why not the volume?

Leonard Susskind gave a clean argument in 1995. Suppose a region with a boundary of area $A$ held more entropy than a black hole of that area. Add mass to the region, gently, until it collapses. The result is a black hole whose horizon fits inside the original boundary, so its area is at most $A$ and its entropy at most $A/4$ . But the region started with more entropy than that, and entropy is not allowed to fall. So the assumption was wrong: no weakly gravitating region can hold more than $A/4$ .

That is why the volume does not matter. A hole's entropy is the greatest of any system that fits in its area, and a memory that scaled with volume would beat it.

Bekenstein's box

For a region that is not a hole, Bekenstein found the bound in 1981 with a thought experiment. Take a box of radius $R$ , energy $E$ and entropy $S$ , and lower it slowly on a rope toward a large black hole of radius $r_s \gg R$ , until its centre sits a distance $R$ above the horizon. Then let it go in.

Near a horizon, clocks run slow, and the energy a distant observer receives from a lowered object is redshifted. At proper height $\ell$ above the horizon, with $\ell \ll r_s$ , the energy as measured from far away is $E\,\ell/(2 r_s)$ . With $\ell = R$ , the hole gains

$\Delta E_\infty \approx \frac{E R}{2 r_s}$

The hole's temperature is $T = \hbar c/(4\pi k r_s)$ (from the formula above, using $GM = r_s c^2/2$ ), so its entropy rises by

$\Delta S_{\text{hole}} = \frac{\Delta E_\infty}{T} = \frac{E R}{2 r_s}\cdot\frac{4\pi k r_s}{\hbar c} = \frac{2\pi k R E}{\hbar c}$

The size of the hole, $r_s$ , has cancelled, so the argument works for any hole.

Meanwhile the box's entropy has vanished from the outside. The generalized second law says the entropy of the hole plus the outside never decreases, so the entropy the box carried in cannot exceed what the hole gained:

$S \le \frac{2\pi k R E}{\hbar c}$

In bits, with $E = mc^2$ , this is $2\pi R m c/(\hbar \ln 2)$ : the Bekenstein bound. This sketch simplifies three things. Where exactly you stop lowering the box sets the numerical factor: stopping with the centre a distance $R$ up gives the $2\pi$ , and Bekenstein's 1981 paper treats the placement with more care. It takes the redshift at the box's centre as if the whole box sat there. And it leaves out the faint thermal glow that surrounds a hole as seen by anything hanging just above it, which pushes up on the box like a buoyant fluid; the edge picks that up, because it is the main objection to this argument.

The two bounds meet

Put $E = Mc^2$ and $R = r_s = 2GM/c^2$ into the Bekenstein bound:

$\frac{2\pi k R E}{\hbar c} = \frac{2\pi k}{\hbar c}\cdot\frac{2GM}{c^2}\cdot Mc^2 = \frac{4\pi G k M^2}{\hbar c}$

This is exactly the entropy of a hole found above. The Bekenstein bound is the hole's entropy at the size where a ball would close up. For any wider ball the two part ways: dividing the Bekenstein bound by the area bound $k A/4\ell_P^2$ for the same ball gives $2GM/(Rc^2) = r_s/R$ , less than one. So away from collapse Bekenstein's bound is the tighter of the two, and the two become one at the horizon. For a hole of Vesper's mass without spin the Bekenstein formula at the horizon gives 2.18 × 10⁷⁹ bits, the same as the area.

Is this proven, and what is still argued?Bounds, a holographic principle, a proof in one setting, and a conjecture about everything

The sharpest form

Bekenstein's bound assumes the room is not close to collapsing. A more general version came in 1999 from Raphael Bousso. His covariant entropy conjecture says: take any surface of area $A$ , and follow light rays leaving it in directions that do not spread apart (a light-sheet). The entropy on that light-sheet is at most $A/4$ . It is meant to hold in any spacetime that obeys Einstein's equations, including cosmology and the inside of black holes, where simpler "entropy inside the boundary" statements run into trouble. For systems with weak self-gravity it reduces to Bekenstein's bound, and in 2003 Bousso derived a version of that bound from it directly. Bousso's 2002 review gathers the tests it has passed and states the limit in everyday units: about 1 × 10⁶⁹ bits per square metre of surface. No counterexample is known, and no general proof either.

The holographic principle

In 1993 Gerard 't Hooft proposed a broader interpretation: the world's degrees of freedom could be described by variables on a two-dimensional surface. Susskind developed the idea in 1995, with one degree of freedom per Planck area, and gave it its name. The full principle claims that a theory of quantum gravity has only as many independent degrees of freedom in a region as fit on its boundary. This requirement on a theory of quantum gravity remains conjectural. frontier

Counting the states of a hole

The area law is a statement about thermodynamics. Entropy should be the logarithm of a number of microstates, and for over two decades nobody could say what the microstates of a hole were. In 1996 Andrew Strominger and Cumrun Vafa counted them for a class of five-dimensional extremal black holes in string theory and got $A/4$ . Those holes are special, charged and supersymmetric, nothing like Vesper, and the count relies on string theory. The result reproduces the entropy for that class of holes. Identifying Vesper's microstates requires a more general account. frontier

A proof of the bound in one setting

For quantum fields in flat space, Horacio Casini gave a version of the Bekenstein bound in 2008. He restated it with relative entropy, a measure of how distinguishable a state of a region is from the vacuum, which can never be negative. That positivity gives the bound, and this form arises from the original derivation by lowering a box once quantum effects are included. It avoids an old worry, that a theory with enough particle species could break the bound by sheer numbers of states. Casini also concludes that the bound is relevant at the classical level but adds no new constraint semiclassically. The proof does not extend to strong gravity.

Was the thought experiment sound?

In 1982 William Unruh and Robert Wald argued that a box lowered near a hole feels buoyancy from the thermal radiation around it, which changes the bookkeeping, and that the generalized second law can then hold without any bound on the box's entropy. Bekenstein's 2005 review defends the bound, pointing to support from thought experiments with the generalized second law, explicit entropy calculations and Bousso's covariant form, and works through purported counterexamples (including systems with many nearly massless species) to show how each avoids it. The disagreement is over whether the lowered box is what forces the bound. The bound has not failed in any worked example. frontier

AdS/CFT

The strongest support for holography came in 1997 from Juan Maldacena, who conjectured that string theory in a particular curved space (anti-de Sitter space) is dual to a quantum field theory living on the space's boundary, one dimension down. If it holds, gravity in the interior is described by a theory without gravity on the surface. It has passed a great many checks, in spaces unlike ours: our universe is not anti-de Sitter; it is expanding. Whether the lesson carries over is what is argued. frontier

What stays unsettled in 2026

Nobody has derived the area law from a theory of quantum gravity that applies to every black hole.
The covariant bound has no counterexample and no general proof.
Whether the bits on a horizon can be read back, and by what, is a different question from how many there are. That is the information paradox, which Black holes glow picks up.