by Stephen Wolfram — filed to Zero-G · Reflections
Credit: Paramount Pictures

Quick, How Might the Alien Spacecraft Work?

[This post is about the movie Arrival; there are no movie spoilers here.]

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Connecting with Hollywood

“It’s an interesting script,” said someone on our PR team. It’s pretty common for us to get requests from movie makers about showing our graphics or posters or books in movies. But the request this time was different: could we urgently help make realistic screen displays for a big Hollywood science fiction movie that was just about to start shooting?

Well, in our company unusual issues eventually land in my inbox, and so it was with this one. Now it so happens that through some combination of relaxation and professional interest I’ve probably seen basically every mainstream science fiction movie that’s appeared over the past few decades. But just based on the working title (“Story of Your Life”) I wasn’t even clear that this movie was science fiction, or what it was at all.

But then I heard that it was about first contact with aliens, and so I said “sure, I’ll read the script.” And, yes, it was an interesting script. Complicated, but interesting. I couldn’t tell if the actual movie would be mostly science fiction or mostly a love story. But there were definitely interesting science-related themes in it — albeit mixed with things that didn’t seem to make sense, and a liberal sprinkling of minor science gaffes.

When I watch science fiction movies I have to say I quite often cringe, thinking, “someone’s spent $100 million on this movie — and yet they’ve made some gratuitous science mistake that could have been fixed in an instant if they’d just asked the right person.” So I decided that even though it was a very busy time for me, I should get involved in what’s now called Arrival and personally try to give it the best science I could.

There are, I think, several reasons Hollywood movies often don’t get as much science input as they should. The first is that movie makers usually just aren’t sensitive to the “science texture” of their movies. They can tell if things are out of whack at a human level, but they typically can’t tell if something is scientifically off. Sometimes they’ll get as far as calling a local university for help, but too often they’re sent to a hyper-specialized academic who’ll not-very-usefully tell them their whole story is wrong. Of course, to be fair, science content usually doesn’t make or break movies. But I think having good science content — like, say, good set design — can help elevate a good movie to greatness.

As a company we’ve had a certain amount of experience working with Hollywood, for example writing all the math for six seasons of the television show Numb3rs. I hadn’t personally been involved — though I have quite a few science friends who’ve helped with movies. There’s Jack Horner, who worked on Jurassic Park, and ended up (as he tells it) pretty much having all his paleontology theories in the movie, including ones that turned out to be wrong. And then there’s Kip Thorne (famous for the recent triumph of detecting gravitational waves), who as a second career in his 80s was the original driving force behind Interstellar — and who made the original black-hole visual effects with Mathematica. From an earlier era there was Marvin Minsky who consulted on AI for 2001: A Space Odyssey, and Ed Fredkin who ended up as the model for the rather eccentric Dr. Falken in WarGames. And recently there was Manjul Bhargava, who for a decade shepherded what became The Man Who Knew Infinity, eventually carefully “watching the math” in weeks of editing sessions.

All of these people had gotten involved with movies much earlier in their production. But I figured that getting involved when the movie was about to start shooting at least had the advantage that one knew the movie was actually going to get made (and yes, there’s often a remarkably high noise-to-signal ratio about such things in Hollywood). It also meant that my role was clear: all I could do was try to uptick and smooth out the science; it wasn’t even worth thinking about changing anything significant in the plot.

The inspiration for the movie had come from an interesting 1998 short story by Ted Chiang. But it was a conceptually complicated story, riffing off a fairly technical idea in mathematical physics — and I wasn’t alone in wondering how anyone could possibly make a movie out of it. Still, there it was, a 120-page script that basically did it, with some science from the original story, and quite a lot added, mostly still in a rather “lorem ipsum” state. And so I went to work, making comments, suggesting fixes, and so on.

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A Few Weeks Later…

Cut to a few weeks later. My son Christopher and I arrive on the set of Arrival in Montreal. The latest X-Men movie is filming at a huge facility next door. Arrival is at a more modest facility. We get there when they’re in the middle of filming a scene inside a helicopter. We can’t see the actors, but we’re watching on the “video village” monitor, along with a couple of producers and other people.

The first line I hear is “I’ve prepared a list of questions [for the aliens], starting with some binary sequences…” And I’m like “Wow, I suggested saying that! This is great!” But then there’s another take. And a word changes. And then there are more takes. And, yes, the dialogue sounds smoother. But the meaning isn’t right. And I’m realizing: this is more difficult than I thought. Lots of tradeoffs. Lots of complexity. (Happily, in the final movie, it ends up being a blend, with the right meaning, and sounding good.)

After a while there’s a break in filming. We talk to Amy Adams, who plays a linguist assigned to communicate with the aliens. She’s spent some time shadowing a local linguistics professor, and is keen to talk about the question of how much the language one uses determines how one thinks — which is a topic that as a computer-language designer I’ve long been interested in. But what the producers really want is for me to talk to Jeremy Renner, who plays a physicist in the movie. He’s feeling out of sorts right then — so off we go to look at the “science tent” set they’ve built and think about what visuals will work with it.

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Writing Code

The script made it clear that there were going to be lots of opportunities for interesting visuals. But much as I might have found it fun, I just didn’t personally have the time to work on creating them. Fortunately, though, my son Christopher — who is a very fast and creative programmer — was interested in doing it. We’d hoped to just be able to ship him off to the set for a week or two, but it was decided he was still too young, so he started off working remotely.

His basic strategy was simple: just ask “if we were doing this for real, what analysis and computations would we be doing?” We’ve got a list of alien landing sites; what’s the pattern? We’ve got geometric data on the shape of the spacecraft; what’s its significance? We’ve got alien “handwriting”; what does it mean?

The movie-makers were giving Christopher raw data, just like in real life, and he was trying to analyze it. And he was turning each question that was asked into all sorts of Wolfram Language code and visualizations.

Christopher was well aware that code shown in movies often doesn’t make sense (a favorite, regardless of context, seems to be the source code for nmap.c in Linux). But he wanted to create code that would make sense, and would actually do the analyses that would be going on in the movie.

In the final movie, the screen visuals are a mixture of ones Christopher created, ones derived from what he created, and ones that were put in separately. Occasionally one can see code. For instance, there’s a nice shot of rearranging alien “handwriting,” in which one sees a Wolfram Language notebook with rather elegant Wolfram Language code in it. And, yes, those lines of code actually do the transformation that’s in the notebook. It’s real stuff, with real computations being done.

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A Theory of Interstellar Travel

When I first started looking at the script for the movie, I quickly realized that to make coherent suggestions I really needed to come up with a concrete theory for the science of what might be going on. Unfortunately there wasn’t much time — and in the end I basically had just one evening to invent how interstellar space travel might work. Here’s the beginning of what I wrote for the movie makers about what I came up with that evening (to avoid spoilers I’m not showing more):

Obviously all these physics details weren’t directly needed in the movie. But thinking them through was really useful in making consistent suggestions about the script. And they led to all sorts of science fiction-y ideas for dialogue. Here are a few of the ones that (probably for the better) didn’t make it into the final script. “The whole ship goes through space like one giant quantum particle.” “The aliens must directly manipulate the spacetime network at the Planck scale.” “There’s spacetime turbulence around the skin of the ship.” “It’s like the skin of the ship has an infinite number of types of atoms, not just the 115 elements we know” (that was going to be related to shining a monochromatic laser at the ship and seeing it come back looking like a rainbow). It’s fun for an “actual scientist” like me to come up with stuff like this. It’s kind of liberating. Especially since every one of these science fiction-y pieces of dialogue can lead one into a long, serious, physics discussion.

For the movie, I wanted to have a particular theory for interstellar travel. And who knows, maybe one day in the distant future it’ll turn out to be correct. But as of now, we certainly don’t know. In fact, for all we know, there’s just some simple “hack” in existing physics that’ll immediately make interstellar travel possible. For example, there’s even some work I did back in 1982 that implies that with standard quantum field theory one should, almost paradoxically, be able to continually extract “zero point energy” from the vacuum. And over the years, this basic mechanism has become what’s probably the most quoted potential propulsion source for interstellar travel, even if I myself don’t actually believe in it. (I think it takes idealizations of materials much too far.)

Maybe (as has been popular recently) there’s a much more prosaic way to propel at least a tiny spacecraft, by pushing it at least to nearby stars with radiation pressure from a laser. Or maybe there’s some way to do “black hole engineering” to set up appropriate distortions in spacetime, even in the standard Einsteinian theory of gravity. It’s important to realize that even if (when?) we know the fundamental theory of physics, we still may not immediately be able to determine, for example, whether faster-than-light travel is possible in our universe. Is there some way to set up some configuration of quantum fields and black holes and whatever so that things behave just so? Computational irreducibility (related to undecidability, Gödel’s Theorem, the Halting Problem, etc.) tells one that there’s no upper bound on just how elaborate and difficult-to-set-up the configuration might need to be. And in the end one could use up all the computation that can be done in the history of the universe — and more — trying to invent the structure that’s needed, and never know for sure if it’s impossible.

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What Are Physicists Like?

When we’re visiting the set, we eventually meet up with Jeremy Renner. We find him sitting on the steps of his trailer smoking a cigarette, looking every bit the gritty action-adventurer that I realize I’ve seen him as in a bunch of movies. I wonder about the most efficient way to communicate what physicists are like. I figure I should just start talking about physics. So I start explaining the physics theories that are relevant to the movie. We’re talking about space and time and quantum mechanics and faster-than-light travel and so on. I’m sprinkling in a few stories I heard from Richard Feynman about “doing physics in the field” on the Manhattan Project. It’s an energetic discussion, and I’m wondering what mannerisms I’m displaying — that might or might not be typical of physicists. (I can’t help remembering Oliver Sacks telling me how uncanny it was for him to see how many of his mannerisms Robin Williams had picked up for Awakenings after only a little exposure, so I’m wondering what Jeremy is going to pick up from me in these few hours.)

Jeremy is keen to understand how the science relates to the arc of the story for the movie, and what the aliens as well as humans must be feeling at different points. I try to talk about what it’s like to figure stuff out in science. Then I realize the best thing is to actually show it a bit, by doing some Wolfram Language live coding. And it turns out that the way the script is written right then, Jeremy is actually supposed to be on camera using Wolfram Language himself (just like — I’m happy to say — so many real-life physicists do).

Christopher shows some of the code he’s written for the movie, and how the controls to make the dynamics work. Then we start talking about how one sets about figuring out the code. We do some preliminaries. Then we’re off and running, doing live coding. And here’s the first example we make — based on the digits of pi that we’d been discussing in relation to SETI or Contact (the book version) or something:

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What to Say to the Aliens

Arrival is partly about interstellar travel. But it’s much more about how we’d communicate with the aliens once they’ve showed up here. I’ve actually thought a lot about alien intelligence. But mostly I’ve thought about it in a more difficult case than in Arrival — where there are no aliens or spaceships in evidence, and where the only thing we have is some thin stream of data, say from a radio transmission, and where it’s difficult even to know if what we’ve got should be considered evidence of “intelligence” at all (remember, for example, that it often seems that even the weather can be complex enough to seem like it “has a mind of its own”).

But in Arrival, the aliens are right here. So then how should we start communicating with them? We need something universal that doesn’t depend on the details of human language or human history. Well, OK, if you’re right there with the aliens, there are physical objects to point to. (Yes, that assumes the aliens have some notion of discrete objects, rather than just a continuum, but by the time they’ve got spaceships and so on, that seems like a decently safe bet.) But what if you want to be more abstract?

Well, then there’s always mathematics. But is mathematics actually universal? Does anyone who builds spaceships necessarily have to know about prime numbers, or integrals, or Fourier series? It’s certainly true that in our human development of technology, those are things we’ve needed to understand. But are there other (and perhaps better) paths to technology? I think so.

For me, the most general form of abstraction that seems relevant to the actual operation of our universe is what we get by looking at the computational universe of possible programs. Mathematics as we’ve practiced it does show up there. But so do an infinite diversity of other abstract collections of rules. And what I realized a while back is that many of these are very relevant — and actually very good — for producing technology.

So, OK, if we look across the computational universe of possible programs, what might we pick out as reasonable universals to start an abstract discussion with aliens who’ve come to visit us?

Once one can point to discrete objects, one has the potential to start talking about numbers, first in unary, then perhaps in binary. Here’s the beginning of a notebook I made about this for the movie. The words and code are for human consumption; for the aliens there’d just be “flash cards” of the main graphics:

OK, so after basic numbers, and maybe some arithmetic, what’s next? It’s interesting to realize that even what we’ve discussed so far doesn’t reflect the history of human mathematics: despite how fundamental they are (as well as their appearance in very old traditions like the I Ching) binary numbers only got popular quite recently — long after lots of much-harder-to-explain mathematical ideas.

So, OK, we don’t need to follow the history of human mathematics or science — or, for that matter, the order in which it’s taught to humans. But we need to find things that can be understood very directly — without outside knowledge or words. Things that for example we’d recognize if we just unearthed them without context in some archeological dig.

Well, it so happens that there’s a class of computational systems that I’ve studied for decades that I think fit the bill remarkably well: cellular automata. They’re based on simple rules that are easy to display visually. And they work by repeatedly applying these rules, and often generating complex patterns — that we now know can be used as the basis for all sorts of interesting technology.

From looking at cellular automata one can actually start to build up a whole world view, or, as I called the book I wrote about such things, A New Kind of Science. But what if we want to communicate more traditional ideas in human science and mathematics? What should we do then?

Maybe we could start by showing 2D geometrical figures. Gauss suggested back around 1820 that one could carve a picture of the standard visual for the Pythagorean theorem out of the Siberian forest, for aliens to see.

It’s easy to get into trouble, though. We might think of showing Platonic solids. And, yes, 3D printouts should work. But 2D perspective renderings depend in a lot of detail on our particular visual systems. Networks are even worse: how are we to know that those lines joining nodes represent abstract connections?

One might think about logic: perhaps start showing the true theorems of logic. But how would one present them? Somehow one has to have a symbolic representation: textual, expression trees, or something. From what we know now about computational knowledge, logic isn’t a particularly good global starting point for representing general concepts. But in the 1950s this wasn’t clear, and there was a charming book (my copy of which wound up on the set of Arrival) that tried to build up a whole way to communicate with aliens using logic:

But what about things with numbers? In Contact (the movie), prime numbers are key. Well, despite their importance in the history of human mathematics, primes actually don’t figure much in today’s technology, and when they do (like in public-key cryptosystems) it usually seems somehow incidental that they’re what’s used.

In a radio signal, primes might at first seem like good “evidence for intelligence.” But of course primes can be generated by programs — and actually by fairly simple ones, including for example cellular automata. And so if one sees a sequence of primes, it’s not immediate evidence that there’s a whole elaborate civilization behind it; it might just come from a simple program that somehow “arose naturally.”

One can easily illustrate primes visually (not least as numbers of objects that can’t be arranged in non-trivial rectangles). But going further with them seems to require concepts that can’t be represented so directly.

It’s awfully easy to fall into implicitly assuming a lot of human context. Pioneer 10 — the human artifact that’s gone further into interstellar space than any other (currently about 11 billion miles, which is about 0.05 percent of the distance to α Centauri) — provides one of my favorite examples. There’s a plaque on that spacecraft that includes a representation of the wavelength of the 21-centimeter spectral line of hydrogen. Now the most obvious way to represent that would probably just be a line 21 cm long. But back in 1972 Carl Sagan and others decided to do something “more scientific,” and instead made a schematic diagram of the quantum mechanical process leading to the spectral line. The problem is that this diagram relies on conventions from human textbooks — like using arrows to represent quantum spins — that really have nothing to do with the underlying concepts and are incredibly specific to the details of how science happened to develop for us humans.

But back to Arrival. To ask a question like “what is your purpose on Earth?” one has to go a lot further than just talking about things like binary sequences or cellular automata. It’s a very interesting problem, and one that’s strangely analogous to something that’s becoming very important right now in the world: communicating with AIs, and defining what goals or purposes they should have (notably “be nice to the humans”).

In a sense, AIs are a little like alien intelligences, right now, here on Earth. The only intelligence we really understand so far is human intelligence. But inevitably every example we see of it shares all the details of the human condition and of human history. So what is intelligence like when it doesn’t share those details?

Well, one of the things that’s emerged from basic science I’ve done is that there isn’t really a bright line between the “intelligent” and the merely “computational.” Things like cellular automata — or the weather — are doing things just as complex as our brains. But even if in some sense they’re “thinking.” they’re not doing so in human-like ways. They don’t share our context and our details.

But if we’re going to “communicate” about things like purpose, we’ve got to find some way to align things. In the AI case, I’ve in fact been working on creating what I call a “symbolic discourse language” that’s a way of expressing concepts that are important to us humans, and communicating them to AIs. There are short-term practical applications, like setting up smart contracts. And there are long-term goals, like defining some analog of a “constitution” for how AIs should generally behave.

Well, in communicating with aliens, we’ve got to build up a common “universal” language that allows us to express concepts that are important to us. That’s not going to be easy. Human natural languages are based on the particulars of the human condition and the history of human civilization. And my symbolic discourse language is really just trying to capture things that are important to humans — not what might be important to aliens.

Of course, in Arrival, we already know that the aliens share some things with us. After all, like the monolith in 2001: A Space Odyssey, even from their shape we recognize the aliens’ spaceships as artifacts. They don’t seem like weird meteorites or something; they seem like something that was made “on purpose.”

But what purpose? Well, purpose is not really something that can be defined abstractly. It’s really something that can be defined only relative to a whole historical and cultural framework. So to ask aliens what their purpose is, we first have to have them understand the historical and cultural framework in which we operate.

Somehow I wonder about the day when we’ll have developed our AIs to the point where we can start asking them what their purpose is. At some level I think it’s going to be disappointing. Because, as I’ve said, I don’t think there’s any meaningful abstract definition of purpose. So there’s nothing “surprising” the AI will tell us. What it considers its purpose will just be a reflection of its detailed history and context. Which in the case of the AI — as its ultimate creators — we happen to have considerable control over.

For aliens, of course, it’s a different story. But that’s part of what Arrival is about.

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The Movie Process

I’ve spent a lot of my life doing big projects — and I’m always curious how big projects of any kind are organized. When I see a movie I’m one of those people who sits through to the end of the credits. So it was pretty interesting for me to see the project of making a movie a little closer up in Arrival.

In terms of scale, making a movie like Arrival is a project of about the same size as releasing a major new version of the Wolfram Language. And it’s clear there are some similarities — as well as lots of differences.

Both involve all sorts of ideas and creativity. Both involve pulling together lots of different kinds of skills. Both have to have everything fit together to make a coherent product in the end.

In some ways I think movie makers have it easier than us software developers. After all, they just have to make one thing that people can watch. In software — and particularly in language design — we have to make something that different people can use in an infinite diversity of different ways, including ones we can’t directly foresee. Of course, in software you always get to make new versions that incrementally improve things; in movies you just get one shot.

And in terms of human resources, there are definitely ways software has it easier than a movie like Arrival. Well-managed software development tends to have a somewhat steady rhythm, so one can have consistent work going on, with consistent teams, for years. In making a movie like Arrival one’s usually bringing in a whole sequence of people — who might never even have met before — each for a very short time. To me, it’s amazing this can work at all. But I guess over the years many of the tasks in the movie industry have become standardized enough that someone can be there for a week or two and do something, then successfully hand it on to another person.

I’ve led a few dozen major software releases in my life. And one might think that by now I’d have got to the point where doing a software release would just be a calm and straightforward process. But it never is. Perhaps it’s because we’re always trying to do majorly new and innovative things. Or perhaps it’s just the nature of such projects. But I’ve found that to get the project done to the quality level I want always requires a remarkable degree of personal intensity. Yes, at least in the case of our company, there are always extremely talented people working on the project. But somehow there are always things to do that nobody expected, and it takes a lot of energy, focus and pushing to get them all together.

At times, I’ve imagined that the process might be a little like making a movie. And in fact in the early years of Mathematica, for example, we even used to have “software credits” that looked very much like movie credits — except that the categories of contributors were things that often had to be made up by me (“lead package developers,” “expression formatting,” “lead font designer” …). But after a decade or so, recognizing the patchwork of contributions to different versions just became too complex, and so we had to give up on software credits. Still, for a while I thought we’d try having “wrap parties,” just like for movies. But somehow when the scheduled party came around, there was always some critical software issue that had come up, and the key contributors couldn’t come to the party because they were off fixing it.

Software development — or at least language development — also has some structural similarities to movie making. One starts from a script — an overall specification of what one wants the finished product to be like. Then one actually tries to build it. Then, inevitably, at the end when one looks at what one has, one realizes one has to change the specification. In movies like Arrival, that’s post-production. In software, it’s more an iteration of the development process.

It was interesting to me to see how the script and the suggestions I made for it propagated through the making of Arrival. It reminded me quite a lot of how I, at least, do software design: everything kept on getting simpler. I’d suggest some detailed way to fix a piece of dialogue. “You shouldn’t say [the Amy Adams character] flunked calculus; she’s way too analytical for that.” “You shouldn’t say the spacecraft came a million light years; that’s outside the galaxy; say a trillion miles instead.” The changes would get made. But then things would get simpler, and the core idea would get communicated in some more minimal way. I didn’t see all the steps (though that would have been interesting). But the results reminded me quite a lot of the process of software design I’ve done so many times — cut out any complexity one can, and make everything as clear and minimal as possible.

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Can You Write a Whiteboard?

My contributions to Arrival were mostly concentrated around the time the movie was shooting early in the summer of 2015. And for almost a year all I heard was that the movie was “in post-production.” But then suddenly in May of this year I get an email: could I urgently write a bunch of relevant physics on a whiteboard for the movie?

There was a scene with Amy Adams in front of a whiteboard, and somehow what was written on the whiteboard when the scene was shot was basic high-school-level physics — not the kind of top-of-the-line physics one would expect from people like the Jeremy Renner character in the movie.

Somewhat amusingly, I don’t think I’ve ever written much on a whiteboard before. I’ve used computers for essentially all my work and presentations for more than 30 years, and before that the prevailing technologies were blackboards and overhead projector transparencies. Still, I duly got a whiteboard set up in my office, and got to work writing (in my now-very-rarely-used handwriting) some things I imagined a good physicist might think of if they were trying to understand an interstellar spacecraft that had just showed up.

Here’s what I came up with. The big spaces on the whiteboard were there to make it easier to composite in Amy Adams (and particularly her hair) moving around in front of the whiteboard. (In the end, the whiteboard got rewritten yet again for the final movie, so what’s here isn’t in detail what’s in the movie.)

In writing the whiteboard, I imagined it as a place where the Jeremy Renner character or his colleagues would record notable ideas about the spacecraft, and formulas related to them. And after a little while, I ended up with quite a tale of physics fact and speculation.

Here’s a key:

1. Maybe the spacecraft has its strange (here, poorly drawn) rattleback-like shape because it spins as it travels, generating gravitational waves in spacetime in the process.

2. Maybe the shape of the spacecraft is somehow optimized for producing a maximal intensity of some pattern of gravitational radiation.

3. This is Einstein’s original formula for the strength of gravitational radiation emitted by a changing mass distribution. Q_ij is the quadrupole moment of the distribution, computed from the integral shown.

4. There are higher-order terms, that depend on higher-order multipole moments, computed by these integrals of the spacecraft mass density ρ(Ω) weighted by spherical harmonics.

5. The gravitational waves would lead to a perturbation in the structure of spacetime, represented by the 4-dimensional tensor h_μν.

6. Maybe the spacecraft somehow “swims” through spacetime, propelled by the effects of these gravitational waves.

7. Maybe around the skin of the spacecraft, there’s “gravitational turbulence” in the structure of spacetime, with power-law correlations like the turbulence one sees around objects moving in fluids. (Or maybe the spacecraft just “boils spacetime” around it…)

8. This is the Papapetrou equation for how a spin tensor evolves in General Relativity, as a function of proper time τ.

9. The equation of geodesic motion describing how things move in (potentially curved) spacetime. Γ is the Christoffel symbol determined by the structure of spacetime. And, yes, one can just go ahead and solve such equations using NDSolve in the Wolfram Language.

10. Einstein’s equation for the gravitational field produced by a moving mass (the field determines the motion of the mass, which in turn reacts back to change the field).

11. A different idea is that the spacecraft might somehow have negative mass, or at least negative pressure. A photon gas has pressure 1/3 ρ; the most common version of dark energy would have pressure -ρ.

12. The equation for the energy–momentum tensor, that specifies the combination of mass, pressure and velocity that appears in relativistic computations for perfect fluids.

13. Maybe the spacecraft represents a “bubble” in which the structure of spacetime is different. (The arrow pointed to a schematic spacecraft shape pre-drawn on the whiteboard.)

14. Is there anything special about the Christoffel symbols (“coefficients of the connection on the tangent fiber bundle”) for the shape of the spacecraft, as computed from its spatial metric tensor?

15. A gravitational wave can be described as a perturbation in the metric of spacetime relative to flat background Minkowski space where Special Relativity operates.

16. The equation for the propagation of a gravitational wave, taking into account the first few “nonlinear” effects of the wave on itself.

17. The relativistic Boltzmann equation describing motion (“transport”) and collision in a gas of Bose–Einstein particles like gravitons.

18. A far-out idea: maybe there’s a way of making a “laser” using gravitons rather than photons, and maybe that’s how the spacecraft works.

19. Lasers are a quantum phenomenon. This is a Feynman diagram of self-interaction of gravitons in a cavity. (Photons don’t have these kinds of direct “nonlinear” self interactions.)

20. How might one make a mirror for gravitons? Maybe one can make a metamaterial with a carefully constructed microscopic structure all the way down to the Planck scale.

21. Lasers involve coherent states made from superpositions of infinite numbers of photons, as formed by infinitely nested creation operators applied to the quantum field theoretic vacuum.

22. There’s a Feynman diagram for that: this is a Bethe–Salpeter-type self-consistent equation for a graviton bound state (which we don’t know exists) that might be relevant to a graviton laser.

23. Basic nonlinear interactions of gravitons in a perturbative approximation to quantum gravity.

24. A possible correction term for the Einstein–Hilbert action of General Relativity from quantum effects.

Eek, I can see how these explanation might seem like they’re in an alien language themselves! Still, they’re actually fairly tame compared to “full physics-speak.” But let me explain a bit of the “physics story” on the whiteboard.

It starts from an obvious feature of the spacecraft: it’s rather unusual, asymmetrical shape. It looks a bit like one of those rattleback tops that one can start spinning one way, but then it changes direction. So I thought: maybe the spacecraft spins around. Well, any massive (non-spherical) object spinning around will produce gravitational waves. Usually they’re absurdly too weak to detect, but if the object is sufficiently massive or spins sufficiently rapidly, they can be substantial. And indeed, late last year, after a 30-year odyssey, gravitational waves from two black holes spinning around and merging were detected — and they were sufficiently intense to detect from a third of the way across the universe. (Accelerating masses effectively generate gravitational waves like accelerating electric charges generate electromagnetic waves.)

OK, so let’s imagine the spacecraft somehow spins rapidly enough to generate lots of gravitational waves. And what if we could somehow confine those gravitational waves in a small region, maybe even by using the motion of the spacecraft itself? Well, then the waves would interfere with themselves. But what if the waves got coherently amplified, like in a laser? Well, then the waves would get stronger, and they’d inevitably start having a big effect on the motion of the spacecraft — like perhaps pushing it through spacetime.

But why should the gravitational waves get amplified? In an ordinary laser that uses photons (“particles of light”), one basically needs to continually make new photons by pumping energy into a material. Photons are so-called Bose–Einstein particles (“bosons”) which means that they tend to all “do the same thing” — which is why the light in a laser comes out as a coherent wave. (Electrons are fermions, which means that they try never to do the same thing, leading to the Exclusion Principle that’s crucial in making matter stable, etc.)

Just as light waves can be thought of as made up of photons, so gravitational waves can most likely be thought of as made up of gravitons (though, to be fair, we don’t yet have any fully consistent theory of gravitons). Photons don’t interact directly with each other — basically because photons interact with things like electrons that have electric charge, but photons themselves don’t have electric charge. Gravitons, on the other hand, do interact directly with each other — basically because they interact with things that have any kind of energy, and they themselves can have energy.

These kinds of nonlinear interactions can have wild effects. For example, gluons in QCD have nonlinear interactions that have the effect of keeping them permanently confined inside the particles like protons that they keep “glued” together. It’s not at all clear what nonlinear interactions of gravitons might do. The idea here is that perhaps they’d lead to some kind of self-sustaining “graviton laser.”

The formulas at the top of the whiteboard are basically about the generation and effects of gravitational waves. The ones at the bottom are mostly about gravitons and their interactions. The formulas at the top are basically all associated with Einstein’s General Theory of Relativity (which for 100 years has been the theory of gravity used in physics). The formulas at the bottom give a mixture of classical and quantum approaches to gravitons and their interactions. The diagrams are so-called Feynman diagrams, in which wavy lines schematically represent gravitons propagating through spacetime.

I have no real idea if a “graviton laser” is possible, or how it would work. But in an ordinary photon laser, the photons always effectively bounce around inside some kind of cavity whose walls act as mirrors. Unfortunately, however, we don’t know how to make a graviton mirror — just like we don’t know any way of making something that will shield a gravitational field (well, dark matter sort of would, if it actually exists). For the whiteboard, I made the speculation that perhaps there’s some weird way of making a “metamaterial” down at the Planck scale of 10–34 meters (where quantum effects in gravity basically have to become important) that could act as a graviton mirror. (Another possibility is that a graviton laser could work more like a free-electron laser without a cavity as such.)

Now, remember, my idea with the whiteboard was to write what I thought a typical good physicist, say plucked from a government lab, might think about if confronted with the situation in the movie. It’s more “conventional” than the theory I personally came up with for how to make an interstellar spacecraft. But that’s because my theory depends on a bunch of my own ideas about how fundamental physics works, that aren’t yet mainstream in the physics community.

What’s the correct theory of interstellar travel? Needless to say, I don’t know. I’d be amazed if either the main theory I invented for the movie or the theory on the whiteboard were correct as they stand. But who knows? And of course it’d be extremely helpful if some aliens showed up in interstellar spaceships to even show us that interstellar travel is possible…

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What Is Your Purpose on Earth?

If aliens show up on Earth, one of the obvious big questions is: why are you here? What is your purpose? It’s something the characters in Arrival talk about a lot. And when Christopher and I were visiting the set we were asked to make a list of possible answers, that could be put on a whiteboard or a clipboard. Here’s what we came up with:

As I mentioned before, the whole notion of purpose is something that’s very tied into cultural and other context. And it’s interesting to think about what purposes one would have put on this list at different times in human history. It’s also interesting to imagine what purposes humans — or AIs — might give for doing things in the future. Perhaps I’m too pessimistic but I rather expect that for future humans, AIs, and aliens, the answer will very often be something out there in the computational universe of possibilities — that we today aren’t even close to having words or concepts for.

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And Now It’s a Movie…

The movie came together really well, the early responses look great…and it’s fun to see things like this (yes, that’s Christopher’s code):

It’s been interesting and stimulating to be involved with Arrival. It’s let me understand a little more about just what’s involved in creating all those movies I see — and what it takes to merge science with compelling fiction. It’s also led me to ask some science questions beyond any I’ve asked before — but that relate to all sorts of things I’m interested in.

But through all of this, I can’t help wondering: “What if it was real, and aliens did arrive on Earth?” I’d like to think that being involved with Arrival has made me a little more prepared for that. And certainly if their spaceships do happen to look like giant black rattlebacks, we’ll even already have some nice Wolfram Language code for that…

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About the story

This story was written by Stephen Wolfram. He is the creator of Mathematica, Wolfram|Alpha and the Wolfram Language; the author of A New Kind of Science; and the founder and CEO of Wolfram Research. Over the course of nearly four decades, he has been a pioneer in the development and application of computational thinking—and has been responsible for many discoveries, inventions and innovations in science, technology and business. Originally published on Stephen Wolfram’s blog.

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