Until recently, I hadn't played "computer games" with any regularity since the mid-90s, when Doom and Nethack were procrastination tools for young cosmologists. (Both games are available for the iPhone, by the way – the more things change, the more they stay the same.) However, for better or worse, my household finds itself with a Playstation 4 and the attendant opportunities for 21st century parent-child bonding.
So I have been playing Destiny with my kids. Created by Bungie in 2014, Destiny did $500 million worth of business on the day it was released. But if you somehow avoided this cultural tsunami, Destiny is a "first person shooter" and players fight their way through the shattered remnants of human settlements on nearby planets in the solar system.
And it is fun, if somewhat repetitive. From the parental perspective it is more palatable than many similar titles, thanks to cartoonish enemies and a distinct lack of actual gore. I will admit that I haven't figured out the bewildering variety of aliens but it doesn't matter much: you shoot them all, and my offspring are on top of the finer details if it ever matters.
Every so often theoretical physics, gaming, and parenting overlap. Not that often, but it happens. For instance, in the final moments of The World's Grave level of Destiny, your Ghost (an annoying sidekick that follows you around, offering advice and resurrecting you after your frequent deaths) hacks into an alien library, announcing that it holds so much information that its curators must have found a way past the Bekenstein limit.
So what is the Bekenstein limit, asked the kids. Turns out, Dad's got this one: its a real thing in physics – the ultimate limit on the amount of information that can be stored in a finite volume.
Jacob Bekenstein, who died last month, deduced this limit by asking what happens to the entropy of stuff (e.g. a giant dying star) that collapses into a black hole. Possibly not the most obvious of things to ask, but it unlocks a hidden door to a vast storeroom of fundamental questions.
Entropy is a measure of disorder in a system, and is often synonymous with degradation and decay. Even if you are not sure what entropy is, you very possibly know it increases with the passage of time or, at best, stays constant, thanks to the Second Law of Thermodynamics. (This is why your house gets messier, not tidier, if you leave it to its own devices). But wherever there's entropy, there's information. Information is entropy's B-side, its secret identity: a disordered system is more complex than an orderly one, so more information is needed to describe it.
But does stuff – and its associated entropy – disappearing into a black hole provide a loophole to the Second Law? Starting from thought experiments like this in the early 1970s, Bekenstein realised that black holes themselves have entropy, and entropy does not vanish when a black hole is formed. Not only that, it seems that no object can have more entropy than a black hole of the same mass. If it did, turning that object into a black hole would engineer a violation of the Second Law. It is this ceiling on entropy that yields the Bekenstein limit on information density which, according to the makers of Destiny, was bypassed by the Hive.
Why the Hive would bother violating the Bekenstein limit is a different question. A sphere 1 meter in diameter holds 1070 bits of data at the Bekenstein limit, and that is a lot of data. Also unexplained is how your Ghost, an object roughly the size of your fist (at least when it is encased in an armoured gauntlet) carries this information away – presumably the Bekenstein limit doesn't bother it, either. (If you took every atom that makes up the planet Earth and attached all the data transferred on the internet in 2015 to each and every one of those atoms – about a zettabyte apparently – you would have to pack all that information into the abovementioned 1 metre sphere for it to hit the Bekenstein limit.)
The Bekenstein limit may seem almost simple (at least for something involving black holes and thermodynamics), but its consequences are still being understood. Entropy is not some sort of sauce that can be poured over a physical system; the entropy of a system is defined by how its internal components are organised. But a black hole has no internal components, and even if it did, anything inside the black hole is supposed to be hidden from observers on the outside. So if a black hole has entropy, where does it live?
Much like a game, this question leads to a new and tougher level for physics: quantum gravity. If black holes have microstates that encode the information that corresponds to their entropy, the microstates are presumably quantum mechanical, like all the other fundamental building blocks of the universe. On the other hand, a black hole is ruled by gravity, and quantum gravity is a boss fight on the path to a "theory of everything". One toolkit for tackling quantum gravity is string theory; and in the mid-90s, "stringy" calculations starting from a microscopic description of nature produced black hole entropy results like the ones Bekenstein and others found in the 1970s. This doesn't prove that the universe is made out of strings, but it is one why reason why physicists are excited by string theory.
Physicists have been playing with the connections between black holes, thermodynamics, gravity and quantum mechanics for over 40 years, and no-one knows where the adventure will end. Simply announcing that he was on the trail of a solution to the "information paradox" – the question of exactly what happens to information stored inside a black hole – during a lecture in Stockholm last month got Stephen Hawking worldwide news coverage, although the solution is at best a work in progress. (See Sabine Hossenfelder's Backreaction blog for commentary.)
One thing we do know is that while the phrase "Bekenstein limit" is a throwaway line in Destiny there is a huge amount of information hidden in those two words, with far more left to discover than we have already learnt. If you are trying to ask the questions that will lead to the "next big breakthrough" in fundamental physics, black holes and thermodynamics are a great place to look.
Coda: The full formula for the entropy of a black hole is due to both Bekenstein and Hawking. And while a black hole represents the upper limit on the entropy and information that can be stored in a given volume it seems that the microstates of a black hole all live on its surface. This leads to a proposal known as holography, suggesting that our apparently three-dimensional universe may, at some fundamental level, need only two dimensions. But that is a story for another day.