The United Federation of Paradox

In Star Trek, the Federation is a post-capitalist utopia where citizens act out of a desire to better themselves or civilization rather than attain monetary wealth. It's not entirely clear how this utopia came about, but we're often told humanity transcended its violent, greedy impulses through cultural evolution. A more cynical view is that the advent of replicators eliminated most scarcity, and with it any need to be violent or greedy.

I would like to offer an alternative hypothesis: Every episode, Starfleet ships employ technology that permits time travel, so the Federation should be able to Seven Days its way out of any mistake on the path to utopia. You see, the physics of the 20th century—special relativity—tell us that any method of FTL (whether warp drive, subspace communication, or galactic spore network) is also a method of time travel. FTL permits time travel because reality has no rigid, universal stage on which all events play out. Instead, space, time, and the events that occupy space and time are all linked together by a consistent set of interrelationships.

Galileo made this argument while trying to convince others that a spinning, moving Earth wouldn't throw everything out of whack. What we now call Galilean relativity says the laws of motion don't depend on your (inertial) frame of reference. A frame of reference is just a perspective from which to observe the universe. If you're sitting in a chair reading this, you and the chair constitute a frame; if you're hurtling through interstellar space (at a constant speed) in a starship, that's another frame. Also, go you.

Galilean relativity means that as long as you occupy an inertial frame, you never notice anything funny that doesn't accord with the laws of motion. Whether you're in a turbolift or a shuttlecraft, if your velocity is constant, a tossed ball will land where you expect it and you won't feel any mysterious forces pushing on you. The upshot is that no frame of reference is privileged or what's "really" happening. All are equally valid.

The tricky part is translating one reference frame to another. Walking down the aisle of a plane, everyone on the plane can treat you as moving only a few miles per hour. Everyone on the ground, however, needs a way to combine your velocity and the plane's. This feat is accomplished via a transformation, which is just a mathematical tool for moving between reference frames. In Galilean relativity, that transformation is easy and basically commonsense: to an observer on the ground, your speed = plane speed + walking speed.

It is these transformations—which spell out equally valid and consistent ways of interpreting reality from different frames of reference—that allow for time travel. To see how, we have to move from Galilean relativity to Einstein's special relativity.

Special relativity is a generalization of the Galilean variety. There are two postulates that end up having deep consequences:

(1) The laws of physics don't depend on your frame of reference.

This is an expansion of Galileo's rules to include electromagnetism.

(2) The speed of light (c) is a law of physics.

This postulate is implicitly included in the first one, because Maxwell's equations for electromagnetism predict a speed of light. It's the revolutionary part in all this, though, so Einstein spelled it out explicitly.

By itself, a law that dictates a speed is not terribly noteworthy. Any wave equation specifies the speed at which the wave travels. We usually think of waves as traveling through a medium, in which case Galilean relativity might apply. To an outside observer, the total wave speed = medium speed + wave equation speed. Physicists assumed this applied to light as well and proposed a luminiferous aether to serve as a reference frame and medium.

The trouble was, the properties required by a luminiferous aether (given how light behaved) seemed ludicrous and unphysical, and when measured, c always seemed to be the same. Additionally, and famously, the Michelson-Morley experiment failed to detect any sign of the aether. The alternative, according to Einstein, is that c is not defined relative to a frame of reference; instead, the speed of light is a law of physics and the same for all inertial observers.

But this violates the rules of the Galilean transformation, because it means you can't add velocities when light is involved. If a Klingon runs at you firing a laser pistol (canon in some of the TOS era), Galileo says the laser's speed = Klingon running speed + c. Einstein says the speed is always only c, for both you and the Klingon. And that means we need a new transformation that is, as before, equally valid and consistent for all inertial frames of reference. For special relativity, that's called the Lorentz transformation.

Rather than just show you the Lorentz transformation (it involves c and some square roots and reduces to the Galilean transformation at everyday speeds), I want to provide a visual explanation for how all observers can measure the same c. Memory Alpha says Vulcan is 16 light years from Earth. So let's imagine there's a starbase between the two planets, 8 light years from each. If the starbase emits a radio signal traveling at c, it reaches both Earth and Vulcan 8 years later. How do we represent this graphically?

Credit: Paramount/CBS for the Trek stuff and NASA for the Earth stuff.

The x-axis (horizontal) is distance in light years and the t-axis (vertical) is time in years. If our reference frame is the starbase and the planets are not moving relative to it, then they move upward in time without moving left or right through space. The radio signals, on the other hand, move 1 light year per year, so they travel 45 degrees out from the starbase. Where the radio signal and the world line of a planet intersect is the location in spacetime (at the planet, 8 years in the planet's future) where the signal reaches the planet.

Now let's say the Enterprise is at the starbase and starts heading toward Vulcan at sublight impulse speeds. What does that look like?

Credit: Paramount/CBS for the Trek stuff and NASA for the Earth stuff.

Because impulse is slower than light, its path is tilted more toward the vertical than the radio signal; more time is required to go the same distance. Since we’re dealing with special relativity, there is an inertial reference frame following along with the Enterprise, and from that frame we have to measure the same c. According to the graph, this doesn't seem possible. It sure looks like the radio signal hasn’t gotten as far away from the Enterprise as it has the starbase (horizontal distance) in the same amount of time (vertical distance).

So here's where we need to perform a coordinate transformation that takes us from the reference frame of the starbase to the reference frame of the Enterprise. For a frame centered on one inertial object, the object's position doesn't change in time. For the starbase, that means its path through spacetime follows the vertical—or time—axis. So then let's define a new time axis (t') for the Enterprise which follows its diagonal path. If c is the same in all references frames, that means we also need a new space axis (x'), which has the same angular separation from the radio signal as t’.

Credit: Paramount/CBS for the Trek stuff and NASA for the Earth stuff.

Because x' and t' are tilted toward the radio signal by the same amount, the signal still moves 1 light year per year in this new reference frame; the ratio doesn't change. This has weird consequences, though. For starters, reconciling a constant c seems to have involved squishing space and time together. But it gets worse.

In the starbase reference frame, lines parallel to the x-axis are single moments in time. Any event on such a parallel line happens simultaneously for all observers sharing that frame. For the Enterprise frame, simultaneous events happen on lines parallel to the x' axis, which is a diagonal line that cuts through time in the starbase frame. This means events that are simultaneous in the Enterprise frame happen at different times for observers in the starbase frame, and vice versa.

For example, if you draw a line parallel to the x'-axis through the moment when the radio signal reaches Vulcan, you see that the event of the signal reaching Earth is ahead of that line; it happens later in the Enterprise's frame, despite the two planets being equidistant from the starbase. This is (a) the relativity of simultaneity, (b) patently ridiculous, (c) absolutely true, and (d) the feature we want to exploit to travel through time and create a problem-free utopia.

Normally (in special relativity), observers disagreeing on the order of events doesn't matter. If observers are limited to light speed or less, by the time they're able to meet up and discuss the discrepancies, all the events they disagree about are in everybody's past. FTL lets you circumvent this restriction.

So here's how to resolve every 42-minute Star Trek plot in 3 easy steps. The scenario presented here is set up for graphical simplicity; it smooths over a few wrinkles and might not perfectly align with Star Trek technology. (Then again, neither does Star Trek technology.)

Step 1: A space-ooze-energy monster attacks the Defiant, but it turns out the creature is just misunderstood. To restock on redshirts, Worf activates the Lorentz Protocol! Via subspace, the Defiant sends a message to Deep Space Nine.

Credit: Paramount/CBS

If subspace communication is instantaneous (which it looks close enough to being in most episodes), then Worf just finds the Bajoran system along the x-axis and puts the message there. Because no time passes, the message arrives along the x-axis.

Step 2: On DS9, Sisko gives the message to O'Brien, who hops into a runabout and flies away from the Defiant at impulse (some speed close to c).

Credit: Paramount/CBS

In our diagram, we're now switching to the runabout's moving reference frame. Its speed relative to the Defiant establishes a new frame of reference.

Step 3: The runabout sends a warning about the interdimensional slug to the Defiant's location in space via subspace.

Credit: Paramount/CBS

Because we are in a new reference frame moving relative to the Defiant, an "instantaneous" subspace message no longer appears somewhere on the horizontal line but along the runabout's x'-axis, which intersects the Defiant's spacetime location in its past.

Ultimately, the speed of the runabout and its distance from the Defiant determine, via a pretty simple triangle, how far into the Defiant's past the subspace warning goes. Arrange things correctly and Worf gets the warning before ever running into the crystalline spider-snake.

But of course, now Worf's gone and killed his own grandfather (who he may have been?—time travel!). That is, if he receives the warning before sending out the message to request a warning, then he avoids the cybernetic mind worm attack and never needs to send out a message in the first place. Paradox!

This is the central reason why physicists think FTL communication or travel is a non-starter. Other aspects of special relativity prohibit reaching c, but there’s nothing about naturally faster-than-light processes. They do, however, invariably lead to issues with causality.

There's a saying about this. Pick two: special relativity, FTL, or causality.

As we've just seen, special relativity + FTL means you lose a coherent narrative leading from the past to the future. You can preserve causality with FTL but only if you abandon the rules of special relativity. Or you can live in the universe we seem to inhabit, which has relativity and causality but loses all that FTL fun.

Of course, when asked to pick two, Star Trek usually just picks one: FTL. Most time travel stories in Trek are rife with causality issues that are usually intentionally ignored, except by having characters say things like, "Oh yeah I totally flunked temporal mechanics at Starfleet Academy, haha!" And relativity is almost entirely absent, because there's rarely any mention of time dilation or length contraction or all the other whacky things that happen when you get close to c.

Nevertheless, the United Federation of Planets is a utopia, and it must have gotten there somehow... or will get there... or will have already gotten there. (Oh boy. Consult Dr. Streetmentioner's book for tense corrections.) Or maybe not—after all, utopia does mean no-place.

Nightfall

(Spoilers for a 76 year old Isaac Asimov story, which you can read here for some reason.)

"Nightfall" is one of my favorite Asimov stories. It's set on an alien planet in a system with six suns, arranged so that at least one is always up. Consequently, the people of this planet never know night. What drives the action is the discovery of a moon (invisible due to the constant sunlight) that astronomers predict will eclipse a sun when all the others have set. The effect would be sudden, inescapable darkness, which they fear will drive people mad (and may have led to past catastrophes).

This story has been on my mind since a little before our solar eclipse. I had heard repeatedly that a total solar eclipse is an event unlike any other, that everyone should try to experience one at some point during their lives. But although some say totality can be drop-to-your-knees-and-weep life-changing, there is as far as I know no evidence of totality-induced civilization-wide collapse. Of course, we experience night daily, so sudden darkness is not as extraordinary for us.

What is it about a total solar eclipse that inspires such numinous feeling, then? Having stood within the moon's umbral gloom for slightly more than two minutes, I can offer my own perspective.

On Sunday, August 20, I traveled to Greenville, South Carolina with a friend and his family, who had family in the area willing to put us up for two nights. The drive down to Greenville from Maryland took about 11 hours. 11 hours of tedium and traffic for 2 minutes of totality—an easy choice for most, whatever that choice may be.

That evening, I passed out eclipse glasses to those who needed them and jury-rigged a solar filter onto my binoculars with index cards and masking tape. As the resident astronomy expert, I had been told by multiple eclipse veterans that it was my responsibility to do dry runs of totality so the uninitiated would be prepared for the moment. Instead we watched Game of Thrones and considered our return travel plans in light of the awful traffic coming down.

The day of, August 21, we found a nearby baseball diamond and set up our equipment about fifteen minutes before the start of the partial phase. A partial solar eclipse is a weird and cool but ultimately very detached phenomenon. You can't (or shouldn't) look directly at the sun, so watching the moon's shadow creep across its face requires filters or those eclipse glasses you've heard way too much about by now.

Through them, there was only the waning orange disc of the sun and blackness—the black of the moon, the black of sky, the black of anything else we might try to look at. Witnessing a partial eclipse was like looking through an insufficiently detailed virtual reality environment. On top of that, up until about 80% obscuration, there was very little change in our surroundings to indicate that anything was up.

But the orange disc inexorably slid into a crescent, which served as a visceral countdown to the main event: totality.

At about fifteen minutes before second contact, with the sun a thin wedge, we began to notice that it was substantially cooler out and strangely dim. The sun was still a blazing fireball in a bright blue sky, but the whole scene was a few shades darker, as if seen through sunglasses. Unfortunately, we didn't have an opportunity to see much in the way of strange shadows where we were.

As the moon reduced the sun to an arc of light, I watched through my binoculars until the orange shriveled to nothing, leaving only black. Then I looked up and experienced totality.

There's something of a twist in "Nightfall," which is that it's not night that drives people mad. In the story, they had been preparing for it. In fact, a minute or two in a totally dark room was akin to an amusement park ride for us—thrilling and hair-raising, maybe too much for some, but ultimately pretty safe.

What drove them mad was a phenomenon they were utterly unprepared for, which shattered their conception of the world and forced them to pick up the pieces.

When night finally fell, the stars came out. Except in myth, their world had consisted entirely of one planet and its attendant suns. But each pinprick of light against the black was another sun, another possible world. Each twinkling tear in the curtain of night let them peek into a much, much larger universe, one too big for their minds to bear. So they went mad instead.

I knew intellectually—from descriptions and pictures—what totality was going to be like. None of that prepared me for the moment itself, when the whole solar system was laid out before me.

Night fell and the stars came out, yes. And birds and bugs acted up. And the sun disappeared.

But here's what stuck with me. I don't remember all that many stars and it was never truly dark out. After gaping at the eclipsed sun for a moment, I saw Jupiter to the east and Venus to the west. They flanked the sun, and I could draw a straight line through all three of them. That line is the ecliptic plane, the disc of our solar system. But in the middle, instead of a sun, there was a hole in the sky—the moon. It, too, lay in that plane, along with me staring up at it all.

With the moon intercepting the light of day, the sun's faint outer atmosphere became visible. For most of our lives, the sun is a featureless glare we have to avoid. We only glimpse it during sunrise and sunset. But even then, the beauty of dawn and twilight is in the intermingling of sun and sky; it's never just you and the sun.

But the corona is the crown of the sun. By eye alone I could see exquisite detail and structure in the threaded, incandescent layers that were hidden from me a moment before. All this made the sun very real—not an untouchable brilliance, not a puddle of mixing reds, not a perfect orange disc against the black, but a giant ball of plasma reaching out to me. And it sat in the middle of a vast solar solar system of planets, with me on a tiny blue one hurtling around it.

Then it was over. The eclipse didn't fade away like a half-remembered dream. It just ended. There were a few seconds of twinkling at the edge of the black and then daylight returned, and the sun and planets and solar system were gone.

The initial seed for Asimov's "Nightfall," so the story goes, was a conversation between him and his editor, John W. Campbell. There's a line in a Ralph Waldo Emerson essay that reads, "If the stars should appear one night in a thousand years, how would men believe and adore, and preserve for many generations the remembrance of the city of God!" Campbell gave this line to Asimov essentially as a prompt, telling him he thought "men would go mad" instead.

And indeed, that's what happens. The short story ends with the main characters holed up in a fortified observatory, watching a crimson glow on the horizon that is not the return of the sun, but a city aflame.

So Asimov and Campbell are pretty cynical about our capacity to cope with a terrifyingly large world. By nature I share that perspective, and a gander at my Twitter feed seems to confirm the validity of such cynicism. While we are still stuck on this pale blue dot, the complexity of the world has grown dramatically in the last couple centuries.

We find ourselves unable to confront the reality of global warming, to the extent that some of us deny it while most of us pretend everything will work out somehow. Our societies have become increasingly interconnected and pluralistic, leading many to retreat into xenophobia that is at best ugly and at worst fiery and violent. Given all that, it doesn't seem unreasonable to imagine that a revelation as world-expanding as "Nightfall"'s might just unhinge us permanently and end our little experiment with civilization.

After totality ended, we stuck around for a bit chatting with others who had come to our baseball diamond, then eventually made our way back to my friend's family's place. There, I was told that a neighboring family had questions for the astronomer on location. Apparently they meant me.

I wandered over and met with a five year old and his mom and dad. The mom asked questions about the eclipse—the why of shadow bands and of different eclipse paths. Then the kid launched into questions about dwarf planets. He wanted to see all of them, so I showed him pictures of Pluto and Charon taken from New Horizons and Ceres from Dawn, and then explained that because dwarf planets are so small and so far away, we needed to build bigger telescopes and faster probes before we could see the rest of them. After that I managed to satisfy his curiosity with some moons, including my favorite Enceladus (about which I've been writing a post since my planetary science course in 2015).

The mom wanted to make sure I didn't dumb down my explanations for her son. The dad wanted to know if there was alien life out there (either on some moon in the solar system or on an exoplanet light years away) and when we were going to Mars.

I talked with the young family for about half an hour, answering questions and trying to feed their enthusiasm with as much knowledge as I could. Talking with strangers is not an activity that comes naturally to me (understatement), but after two semesters as a teaching assistant leading discussions and labs, I have come to enjoy this type of interaction.

What I find particularly heartening about being an ambassador for astronomy is the sheer wonder and curiosity we can have for the enormous, mind-blowing universe our telescopes have revealed. People are drawn to strange new worlds and the idea that we might someday have a home beyond Earth. Maybe fear and madness are natural and understandable reactions to a world too big to wrap our heads around, but they're not the only possible responses. How do we cultivate such wonder? How do we embrace curiosity so that it extends beyond pretty pictures and to all the unbearable complexity we are faced with?

I don't know the answer to that question. Maybe it takes witnessing once in a lifetime astronomical marvels. (Helpfully, if you missed this one, the US has another in seven years.) In the meantime, maybe read some imaginative, thoughtful, mind-expanding science fiction. For the foreseeable future, that's as close as we can get to a larger world.

The War on Stars

This post contains spoilers for both Star Wars: Episode VII The Force Awakens and my academic semester. Read on at your own peril.

As always, I must begin by apologizing for not having posted in months. My academic load this semester, combined with my work schedule, was probably about the limit of what I could handle and didn't leave me with a lot of time left over for blogging (or sleeping, for that matter). To remedy that, during winter break I'm going to try to find time to write about the classes I took, maybe posting every week or so. We're starting off today with my observational astronomy course.

But we're getting there through Star Wars. To begin, I enjoyed the movie a great deal (all three times). I also go into a Star Wars movie turning off the part of my brain that cares about scientific plausibility or consistency. In fact, I'm partial to the idea that Star Wars is science fantasy rather than science fiction, whatever that distinction may signify. Yet looking at media through a scientific lens is a fun way for me to analyze it, and it might even be educational. We'll see.

So, of course, TFA has a galaxy's worth of scientific errors, but there's one visual in particular I'd like to take a look at, because I think it gets at something important in astronomy. When the First Order fires the weapon from Starkiller Base at the New Republic, Finn on Takodana (Maz Kanata's planet) sees the beam split up and strike different planets in the Hosnian system. This is an impossible image, given the assumption that Starkiller Base, Takodana, and the Hosnian system all orbit different stars. The reason this image is so impossible is because, as the great Douglas Adams informed us, space is big, really big.

Now, I'm not thinking about the fact that light travels at a finite speed and there wouldn't have been time for the image to show up in Takodana's atmosphere. This is a universe with faster than light travel, so let's just mumble something about hyperspace and ignore that. Imagine that it did take years for the light of the beam to stretch across the lightyears; it still wouldn't look like it does.

The problem is that you can see multiple beams at all, that they can be resolved as striking different places. In astronomical terms, the angular separation between the beams is absurdly large. This point can be made with a simple trigonometric argument. If we imagine two lines connecting Finn's eyes and the planets struck by the beams, and another line connecting those two planets, we can make a little triangle.

What we're looking for is the angle between lines C and A. For our purposes, the relative lengths of A and C don't matter and we can just call one of those lines the distance between Takodana and the Hosnian system. Trig gives us the formula sin θ = B/C. But in astronomy we make use of the small-angle approximation a lot, which says that for very small θ, the sine of θ is approximately θ. So then we have θ = B/C.

The significant part of this formula is that, for astronomical purposes, staring up at the sky only gives us θ, not B (the size of the thing we’re looking at) or C (the distance to the thing we’re looking at). This means, without other factors, we can’t tell if we’re looking at a big object far away or a small object nearby.

Digging around Wookieepedia and starwars.com, it seems that Takodana is supposed to be in the Mid Rim of the galaxy and Hosnian Prime in the Core. If we assume that this galaxy is about the same size as ours (not necessarily a great assumption, but published maps show something like a spiral galaxy), then halfway out of the Core gets us a distance of 25,000 lightyears. We don't know the distances between the planets in the system, but if we make the very generous assumption that they are as far apart as Earth and Neptune, we get a distance of 4 lighthours. Plugging those numbers into the above formula (B=4 lighthours, C=25,000 lightyears), our angular separation is 2x10^-8 radians, which converts to 4 milliarcseconds (mas). 1 mas is 1/1000 of an arcsecond, which is 1/60 of an arcminute, which is 1/60 of a degree. By comparison, the moon has an angular size of 31 arcminutes, over 400,000 times bigger.

So the beams wouldn't appear that far apart. In fact, you wouldn't be able to tell them apart at all. Okay, but why am I fussing about this? Because it gets into some interesting aspects of observational astronomy having to do with the wave nature of light. Specifically, when light waves enter an aperture, they diffract around the edges and form interference patterns. It's inevitable and must be taken into account no matter what type of observation you're doing.

When light diffracts through a perfectly circular aperture, it forms the following interference pattern, called an Airy disk.

"Airy-pattern" by Sakurambo at English Wikipedia 桜ん坊

That is, if you were to shine a laser pointer through a circular hole, instead of a dot on the other side, you would get the above pattern. However, trying this with a store-bought laser pointer and a hole punch won’t get you much, because the pattern is very sensitive to the wavelength of light used and the size of the hole.

In the case of the Starkiller beam, the aperture we're talking about is your pupil. The human pupil can change in size based on lighting conditions, but a good average diameter is 5 mm. The wavelength of the beam's light is based on its color. The red light of the Starkiller beam is at the long end of the visible spectrum, so let's call it 700 nm. These two variables play into the size and spread of the interference fringes.

In the 19th century, Lord Rayleigh proposed a criterion for determining the limits of image resolution. He said that if two images are closer together than the first minimum of the interference pattern, then you can't resolve them as two objects. This is arbitrary, but not entirely made up. If you add together the intensities of two interference patterns separated by less than that minimum, this is the difficult to interpret graph you get. Are you looking at one object or two?

The pattern of the Airy disc is described by a Bessel function, which is a special function invented to be the solution to some common differential equations. The first minimum of the Airy disc is the point where the function goes to 0 for the first time and happens at an angular distance of θ = 1.22λ/D, where λ is the wavelength of light, D is the diameter of the aperture, and 1.22 is a rounded-off figure for a number that goes on forever, because Bessel functions aren't very nice functions.

In fact, my observational astronomy professor explained that if we're going to use 1.22, we might as well memorize a few more digits because that number only comes up with perfectly circular apertures anyway, and 1.22 is not much greater than 1, so you're not gaining much precision as it is. In most cases, making the approximation that θ = λ/D works well enough. The interesting thing to note about this criterion is that fine angular resolution results from small wavelength or large aperture. This is why radio telescopes are much bigger than optical telescopes. Radio telescopes are looking at very large wavelengths (centimeters to meters compared to hundreds of nanometers), so to be able to resolve images, they need much larger apertures.

Since I just made up the wavelength of our beam and I'm assuming the pupil is exactly 5 mm, let's leave off the .22. In that case, our minimum angular resolution is 700 nm/5 mm = 1.40x10^-4 radians, which comes out to 29 arcseconds. This limit is ~7000 times higher than our estimated angular separation of 4 mas for the Starkiller beams. To our eyes, the split beams would look like one beam.

...if they looked like anything at all. If you remember, Finn also saw the beams during the daytime. And as you may also remember, the only celestial object we tend to see during the day is the Sun (and the moon depending on its phase, and occasionally some planets and stars near sunrise and sunset). We intuitively know why this is: the Sun washes out dimmer objects. Even the reflected light of the Sun in the atmosphere is bright enough to wash out dim objects.

But why should that be? If the point where a star is has star and atmosphere, shouldn't it be a smidgen brighter than atmosphere alone? And shouldn't we be able to tell the difference? It turns out we can't, and the reason why is preserved in an ancient system for judging the brightness of stars that has persisted to this day with a few modifications.

The Greek astronomer Hipparchus set about cataloging the fixed stars a little more than two thousand years ago, managing to compile the position and brightness of several hundred of them. He called the brightest ones “stars of the first magnitude,” the second brightest “stars of the second magnitude,” and so on down to the dimmest stars visible to his naked eye, which he placed at magnitude six. Many an astronomy student today curses Hipparchus for giving lower numbers to brighter stars, but the system has stuck nonetheless.

In the 19th century, the English astronomer Norman Pogson realized that with a little fudging, it looked like 1st magnitude stars were 100 times brighter than 6th magnitude stars. You can divide this up a little further and discover that a magnitude jump of 1 represents a change in brightness of about 2.5 (2.5⁵ ~ 100). But to our eyes, 1st magnitude stars don't seem to be 100 times brighter than 6th magnitude stars. They're not necessarily 6 times brighter either, but that's much closer to what we perceive than the physical reality. That's because human eyes don't respond to light in a linear fashion, but on a logarithmic or power scale instead (the details are messy and beyond my understanding).

If one star is twice as bright as another star, the above relation tells us that the magnitude difference is less than 1. In other words, Hipparchus might not even have noticed. The gist is that very small changes in brightness don't register to us if they are below a threshold called the just-noticeable difference. So while star+atmosphere is slightly brighter than atmosphere alone, it's not enough of a difference for our eyes to notice. And if the Starkiller beams shine with the brightness of a star (which seems about right given that Starkiller Base seems to explode into a star), then we wouldn't be able to see the beams at all during the day, let alone tell them apart.

But this isn't a problem just for human eyes. We don't point our telescopes at the sky during the day for the same reason. Modern telescopes pipe their images down to CCDs, digital devices that convert photons into electrons and count them up at each pixel. We can tell we've found something in a CCD if there's a signal that is significantly more intense than the background. But the background is noisy, and if the fluctuations from noise are greater than the difference between the background and the signal, then we can't tell if we've actually found anything at all.

Returning to Hipparchus for a moment, early astronomers noticed that brighter, lower magnitude stars appeared bigger than dimmer stars. We now know that the biggest stars are about a thousand times wider than our Sun. Yet we don’t see any stars in the sky that are a thousand times bigger than any other stars. In fact, it turns out the star with the largest angular size is R Doradus at 0.057 arcseconds. This is still tiny, with the moon about 30,000 times wider. But it doesn’t seem plausible that we could line up 30,000 stars as we see them in the night sky across the face of the moon.

The answer goes back to diffraction. To the naked eye, all stars are too small to have a resolvable disk. Instead, while the width of the central peak of a diffraction pattern is a function only of the aperture size and wavelength, the intensity across that width depends on the overall brightness of the object. As such, brighter stars appear bigger to our eyes, because diffraction means the whole Airy pattern is brighter, and that pattern is not point-like. Thus the size of a star in the night sky is not directly related to its physical size except insofar as bigger usually means brighter. We are not seeing the physical disk of the star itself, only the illusory Airy disk that results from diffraction.

Anyway, I think that’s enough nerding out over Star Wars and astronomy for now, what with me passing the 2000-word mark. I'll have more to say about observational astronomy later because I want to touch on image processing, which was a big chunk of the class. Next up will probably be quantum physics, though, because that’s a demon I’ve yet to exorcise.

On the Particular Qualities of Good SF

Science fiction is in the subtitle of this blog, but I haven’t really talked about it beyond flaunting my nerd cred. It’s been on my mind lately, however. So here’s a post in which I pontificate on what I think makes good SF. Feel free to tune out if you came here for more than just my opinions dressed up as theory.

I’m a little late on the review bandwagon, but this post is more or less inspired by my thoughts on star: trek Into darkness. If you haven’t seen the movie yet, you should cover your eyes while you read this part because here’s the spoiler to the earth-shattering final twist of the movie: Kirk doesn’t die. I know, I know—shocking. How the writers managed to keep a lid on that one is anyone’s guess.

Anyway, why doesn’t Kirk die? It turns out the reason he doesn’t die is because the brilliant physician Bones McCoy has made a monumental discovery in medicine that will change every life in the Federation forever and surely be the focal point of the next Star Trek movie. Yes, Bones has managed to discover the secret to immortality in the blood of a man created with 200 year old technology.

At least one thing I said in the preceding paragraph is true. (There will be another Star Trek movie.) But let’s put aside the snark for a moment. Why does this grate me so? Because it’s a missed opportunity. If Bones really had discovered immortality, and it really did change the Federation in some way, then that would make for some pretty interesting science fiction. Instead it will likely never be mentioned again, because it was only a gimmick to create suspense at the end of the movie.

But who cares about discussing immortality, right? It’s never going to happen, or life is only meaningful because of death, or it’s just some nerd boy’s fantasy, right? Well, yes, immortality isn’t real, but that would seem to make it excellent fodder for fiction. Good fiction is traditionally supposed to explore the human condition, and that’s a nearly endless fount from which to plumb. There’s love, hatred, war, jealousy, and all that other good human condition stuff. But those elements of the human condition are the low-hanging fruit; they’ve been picked. What else is being human about?

If there are examples we can point to that are solidly within the domain of the human condition, then the next place we can look is outward at the edge cases. Edge is a good word here. After all, how do you identify objects you can see? You look to the edge of the thing, see where it stops, trace its shape. If you want to know what something is, find out where it stops and draw a line.

I’m reminded of a topic from linear algebra that might make for a useful analogy or might just confuse people even more. Stick with me. Take a matrix with, say, 10 columns. If this matrix is full of unknowns, then there’s a method for figuring out every single way to add those unknowns together and come out with 0.

This is called the nullspace of a matrix. In essence, it tells you everything a matrix is not. The nullspace of a matrix is described by a number of dimensions. Let’s say we have a matrix with a nullspace of 6 dimensions. One of the neat things about linear algebra is that the number of columns (10) minus the dimensions of the nullspace (6) is always equal to the dimensions of the row space (4). What’s the row space? Well, in short, it’s everything a matrix could be. What does that all mean? It means that you can find out what something is by figuring out what it’s not.

Which takes us back to science fiction. Good science fiction tells us about the human condition specifically by telling us about something other than the human condition. It tells us about things near the human condition, at the edge, making up the border. And by doing so, it lets us create a rough outline of what the human condition really looks like.

So, then, magic is science fiction, right? I mean, it’s definitely not human, which means stories about it tell us about what is human, right? Ah, uh, no. The key here is that you have to look at the edges of a thing where you think it might be. The reason is that there’s generally a lot more that a thing isn’t than a thing is.

If you’ll allow me, let’s return to the linear algebra example. If the nullspace of a matrix is infinite and the row space is finite, then I can start calling out random numbers and have a pretty good chance of giving you a number in the nullspace rather than the row space. But what does this tell me about the matrix? Basically nothing. However, when you’re done finding the nullspace of a matrix, you’re left with a formula that tells you how to figure out what’s in it. The formula lets you extract useful information from the problem.

A formula is basically just a set of rules to follow. And it’s these rules that get to the heart of good science fiction. They let you find the line between what’s human and what’s not. Technology in fiction might follow rules, but it also might not. That’s the difference between your hard SF and something like Star Wars. Star Wars has lasers and spaceships, but they don’t follow any rules, which means they’re not telling you anything about the boundaries of the human condition.

And it’s also the difference between good science fiction and the most recent Star Trek movie. Because we’re never going to see Khan’s immortal blood again, it’s not following any rules; it’s essentially magic. If it had been explored as a topic, we could have learned something from it.

Now, magic systems in high fantasy might have intricate rules, but if those rules are describing something completely alien, they’re not telling you about what it means to be or not be human. In linear algebra, it’s akin to knowing the formula for a different matrix. It tells you something, but it doesn’t tell you what you want to know.

By the way, I’m not bashing Star Wars or fantasy. Both of them can be good fiction, and both of them can tell you things about the human condition. But they’re really only doing so the old-fashioned way—by looking at what we know for certain is human. They’re not doing it by exploring the edge cases.

But we need to explore the edge cases; we need to find the rules. Why? Because eventually we’re going to pick all the low-hanging fruit and we’re not going to have anything new to say about the human condition. The only way we’ll be able to keep learning about ourselves is by finding our boundaries and pressing up against them. And the best way to do that is by writing good science fiction.

TANSTAAFL

I’m still going to do a post about transferring energy from gasoline to lawnmower blades, but a thought occurred to me while I was obsessively rereading my last post that I think might make for a good post in and of itself.

I’ve got a copy of The Way Things Work on my bookcase, and sitting right beside it is a copy of The New Way Things Work. I gobbled up the former book as a kid; it was my first introduction to the idea that electrons release photons whenever they jump down shells, to the concept of computer logic gates, how fission and fusion work, and a whole host of other scientific ideas. I know the book is primarily about machines, but for some reason it was the microscopic stuff that stuck with me more than the mechanical bits.

Anywho, there’s a page about transformers in the book, and I still remember thinking when I read it that transformers were somehow cheating.

You're the man, David Macaulay. Seriously.

You wrap wire around a magnet, wrap more wire around the other end of the magnet, and voila, you’ve increased the electricity pumping through the wire. So why is energy in such short supply? Are there not enough magnets?

I didn’t have a clear understanding of what the conservation of energy meant as a kid (and it can certainly be argued that I don’t now, either, given that I have no idea how Noether’s theorem about symmetry leads to conservation laws), but it still struck me as somehow wrong. And it struck me as wrong for a very long time, right up until about this semester, when I learned what a transformer is really doing.

I think this speaks to the fact that we (the general public) have a very fuzzy idea of what energy is. We know that climbing an electric fence that says “DANGER: 10,000 VOLTS”...

Please don't sue me, Steven Spielberg.

...is somehow more dangerous than licking a 9-volt battery. We also know that “it’s not the voltage that kills you; it’s the amperage.” So both of these are somehow electrical quantities that correspond to bigger and more powerful, but are they related to energy? Related, yes, but they aren’t synonymous with energy.

It turns out that a volt is a measure of energy per charge. You can think of it as the amount of pressure an electron is under. An amp, on the other hand, is the amount of charge moving past some point per second. If you multiply these two quantities together, the charge cancels out and you get energy per second. This quantity is known as the watt, which most people recognize as being what the power company delivers.

But more importantly, this gives us a way to measure energy over time. Because energy is conserved, we know that the amount of energy pumped into a transformer must be equal to the amount of energy pumped out (assuming some ideal transformer with no real world problems). What a transformer does, however, is increase the voltage of the current flowing through it. That is to say, it increases the amount of energy packed into each charge. In order to ensure that the same amount of energy passes through a transformer, less charge must flow out.

(It’s also true, however, that charge is a conserved quantity, so you might think just as much charge has to come out as goes in. But that’s not what conservation really means. It just means that the amount of charge does not change with time; it says nothing of where that charge must go. What’s really happening is that electrons go into the transformer, dump their energy into the transformer’s magnetic field, and then circle back to wherever they came from. The energy they dumped into the magnetic field, however, is then transferred to new electrons at the other end of the transformer. The number of electrons never changes, and the amount of energy never changes, but there are fewer electrons coming out one end than going in the other.)

Say we have a current carrying 1 watt of power at 1 volt and 1 ampere through a transformer. After 1 second, 1 watt pumps 1 joule of energy into the transformer. Then we turn the current off. We expect that 1 joule is going to come out of the transformer a second later. If the transformer increases the voltage of the current to 10 volts, then we have 10 joules per coulomb coming out of the transformer. But since we only have 1 joule to begin with, this means we can only send out 1/10 of a coulomb of charge. Our transformer, then, has turned our 1 volt, 1 ampere current into a 10 volt, 0.1 ampere current. The power is still 1 watt, however, which means that the amount of energy delivered over time is constant.

As the title of the post says, there ain’t no such thing as a free lunch. Transformers don’t cheat; they just do some clever bookkeeping. (I’ve tried to imagine a mechanical analog to a transformer but haven’t been able to come up with anything that doesn’t seem tortured and contrived. Perhaps this is one of the places where the world of electricity and magnetism doesn’t mirror the real world, or perhaps I’m simply having a failure of imagination.)

Until next time, folks.

On the Physical Principles of Graminoid Elimination

On Facebook this past Saturday I commented that I had “mastered E&M and multivariable calculus, but can’t seem to figure out how to start a lawnmower.” Leaving aside the image of a pasty nerd being carried off by an out of control garden tool, I think it’s interesting that the relatively basic physics introduction I’ve been given over the past two semesters is sufficient to explain almost completely how a lawnmower works. So I’m going to share that neatness with my dear and devoted reader(s).

It starts with the starter cord (no surprises there). Pulling the cord almost literally spins the engine’s crankshaft into motion. The crankshaft pulls the piston, which lets fuel and air into the cylinder. The crank continues to turn, the piston compresses the mixture and then, well, then Faraday’s law happens.

There are some chemicals that ignite on contact with air. There are other chemicals that ignite when they are compressed to any degree. Gasoline is not one of those chemicals. Gas is volatile, but not too volatile, which makes it a good method of storing energy. In order to get energy out of gas, you have to mix it with air, compress it, and then raise its temperature all at once.

I don’t want to get too much into the process of combustion, partly because that’s more chemistry than physics, and partly because I don’t understand it very well. But the very basic principle is something Dr. Dave Goldberg explained to me a few years ago when I asked why matter and anti-matter annihilate. His answer: because they can. Essentially, because matter and anti-matter are oppositely charged, it’s very easy for them to disappear and be replaced by electrically neutral photons with the same amount of energy.

The same principle applies to combustion. Oxygen doesn’t have enough electrons, and gasoline has electrons to give. So when you mix the two together, gasoline gives up electrons, breaks down into simpler compounds, and releases energy, because it can. That energy heats and expands the mixture, pushing the piston out and turning the crankshaft, which subsequently turns the lawnmower’s blade.

Why this releases energy is complicated. The simple explanation is that more energy is stored in the system than it takes to release that energy. The usual analogy is to a ball sitting at the top of a hill. When you kick a ball, you transfer kinetic energy from your foot to the ball, and the ball moves away from you until that kinetic energy has been lost to friction and air resistance.

But if you kick a ball so that it rolls down a hill, the energy you provide is enough to “release” the gravitational potential energy stored in the ball, so that by the time it gets to the bottom of the hill it has way more kinetic energy than you provided from your foot alone. So gasoline, because of its structure (yes, that’s a copout), has a lot of potential energy just waiting to be released if you give it a big enough push.

The key is that gasoline is less likely to give up the electrons that form its chemical structure than it is to give up the electrons that bond one molecule of gas to another, so you need more energy to get at those deeper electrons. This energy is provided in the form of a spark that raises the temperature of the mixture.

Where does the spark come from? The spark plug, of course. A spark plug is essentially just a capacitor that is designed to fail. As I explained a couple posts back, charging a capacitor creates an electric field. If the electric field gets too strong, the insulating material between the capacitor plates ionizes. That is, its electrons are stripped away, which turns it into a conductor, causing all those stored electrons in the capacitor to flood across the gap at high speed, producing lightning. This lightning is a spark. But where does a tiny gas-powered lawnmower get the necessary voltage to produce lightning?

This guy:



Courtesy Wikipedia

Lawnmowers use a device known as a magneto. As the engine spins, it rotates a permanent magnet inside a coil of wire. This changing magnetic field induces a current in the coil. But with every turn of the engine, that current is interrupted by a contact breaker, so the current drops down to zero.

Or it would, if it weren’t running through a coil of wire. This coil acts like an inductor, and inductors create electromagnetic fields whose strength is determined by how quickly the local electromagnetic field is changing. So even if there is only a very small current through an inductor, if that current is immediately cut off then the change in current is very large, which in turn produces a very strong electromagnetic field.

But we're not done there. After the magneto comes this:



What would I do without you, Wikipedia?

There’s a second stage, a transformer, which increases the voltage even more, enough to induce a spark in the spark plug. The other pole of the permanent magnet has its own set of wires coiled about it, but it has far more windings than the first set. When the magnetic field produced by the inductor changes due to the broken contact, this induces an electric field in the secondary coil.

But the interesting thing about Faraday’s law is that it talks about changing magnetic flux rather than just a changing magnetic field. Flux is the flow of magnetic field lines through an area. The larger the area, the greater the flux. And the area we’re dealing with here is the circle formed by a loop of wire. But if you have a hundred loops of wire, you have a hundred times the area as far as Faraday’s law is concerned, which means the voltage induced is a hundred times larger.

So by exploiting Faraday’s law to create a rapidly changing magnetic field through a very large area, pulling on a starter cord can induce a large enough voltage to create lightning, which ignites the fuel in the lawnmower’s engine. Pretty cool.

I also wanted to look at converting the chemical energy of gasoline into the rotational energy of the lawnmower blade, but I think this post is long enough already. So tune in next time and there might be a discussion of energy density, thermodynamics, and angular momentum.

Out of Phase

With the semester winding down, I think it’s a good time to discuss a slightly annoying idiosyncrasy I’ve observed this spring. E&M has, as a co-requisite, vector calculus. My school expects these two classes to be taken together, and E&M involves a fair amount of vector calc. But despite this connection, or perhaps because of it, my math and physics have been, you might say, out of phase.

What do I mean by out of phase? I think a thousand words can sum it up best…

As you can see here, Lt. Cmdr. Geordi La Forge’s hand is out of phase with the engineering console. They’re in the same place at the same time, and yet somehow they’re not actually interacting. The solution, of course, is a concentrated burst of anyons. Yep.

Or maybe I mean something a little more like this:

  

We just got through a unit on AC circuits, which are significantly more complex (heh) than DC circuits. The main reason is that the three basic elements of a circuit all respond differently to alternating current. Resistors don’t much care. They reduce the voltage across them, but (ideally) their response isn’t dependent on the incoming voltage. Thus, the voltage across a resistor looks like whatever current is coming from the AC source.

A capacitor, on the other hand, has its maximum voltage when it’s fully charged. And when it’s fully charged, the current across it is zero. Thus, the voltage across a capacitor is at its peak when the blue line crosses zero.

And inductors resist changes in current, which means an inductor will always have a voltage that is positive when the source is decreasing, and negative when the source is increasing.

Why did I go through all of that? Well, for two reasons. First, the differences between those circuits give rise to the different voltage graphs above, and those differences are known as phase shifts. Resistors are “in phase” with their voltage source, whereas inductors and capacitors are “out of phase.” That is, they don’t line up.

When two waves that are in phase with each other combine, they produce one wave that is bigger than either of the two waves. Conversely, out of phase waves that hit cancel each other out. This is known as interference, and maximizing the acoustics of a room is all about making sure sound waves are in phase when they get to the audience.

Secondly, a complete description of the wave-like behavior of AC circuits involves complex numbers—the square roots of negative numbers—a topic which has only been given the briefest of overviews in any math class I’ve ever taken. Consequently, the entirety of the complex aspect of AC circuits was covered in a single line from my professor: “And if we increase this factor over here, you see that we get imaginary numbers.” So, you see, my math and physics classes are out of phase.

But it doesn’t stop there. Near the beginning of the semester, we learned that the electric field can be thought of as the change in voltage across space. The text and our professor briefly mentioned that this is referred to as a gradient and involves partial derivatives. But at that moment in math, we were learning how to plot three-dimensional vectors.

Later on, we learned about Gauss’ law for electricity, which lets you figure out the amount of charge enclosed by a surface if you know what the electric fields coming out of that surface look like (and vice versa). When you first learn how to use Gauss’ law, you’re only asked to apply it to surfaces that are highly symmetric, which makes the math a lot easier. But it turns out that Gauss’ law is a very general principle in vector calculus involving surface integrals. And what, pray tell, were we learning about in math at that moment? Partial derivatives, of course.

Now, as the semester draws to a close, our professor has shown us Maxwell’s equations in all their glory. All their integral glory, anyhow. We’ve seen them in one form or another already, and we’ve used three of them. The one we haven’t used is Gauss’ law for magnetism, which is like the electricity law except it says there’s no such thing as magnetic charge and there’s never any net flow of magnetic field lines through a surface. Why is there a law saying the answer to your question is no? Because, our professor explained, the law is more useful in its differential form, where it uses a vector calculus principle known as divergence. We’ll be getting to divergence next week in math. Right now we’re studying surface integrals.

Maybe I'm misinterpreting all this. Maybe learning about the idea in physics first primes us for the material in math later. But I'm not sure I buy that argument. I feel as if we'd be better served by having to take vector calculus first, or by having E&M and calculus classes that were more, well, in phase with each other.

Anywho, so when are we going to learn the math behind complex numbers and phases? Apparently it involves phasor arithmetic and phasor diagrams. I don't know about you, but I encountered phasor diagrams waaaay before I started taking math classes...

Knot Another Punny Title

So, this is more of an update post than anything else. Work and school have been pretty busy. I'm also writing (well, editing at this point) a short story, which is where the title of my post comes from. How's that work?

For reasons that escape me, I'm writing a short story that heavily features a branch of mathematics known as knot theory. Wait, there's a branch of mathematics about knots? Why yes, yes there is.

Math is about numbers, duh, but it's also about geometry. Many of the ancient Greek mathematicians did math purely through geometry, in fact, because they didn't have algebra to represent general forms or calculus to deal with infinitesimals. Anywho, there was a point in the 19th century at which studying geometry morphed into studying surfaces, and this led to ideas such as differential geometry and topology. Differential geometry is the math behind Einstein's general theory of relativity, and topology tells us that doughnuts and coffee mugs are the same thing.

Doughnuts and coffee mugs are the same thing? Apparently, yes, because in topology, objects are homeomorphic if you can transform (stretch, squeeze, rotate, twist) one into the other without creating any new shapes or holes. There's a little gif on wikipedia showing that this isn't quite as crazy as it sounds.

Now, there's a sub-branch of topology known as knot theory, which studies circles embedded in space (or spheres embedded in 4-space, etc.). It turns out that some circles are just circles (called the unknot), whereas others are true knots that cannot be transformed back into circles without cutting the knot. Take a rubber band, for example. It's just a circle.

No matter how many times you twist it around and tie it into knots, it's still just a circle that can (theoretically) be undone.

But if you cut the rubber band...

...tie it into a knot, and then reconnect the severed ends (and pretend it's not being held together with tape), then you've created a true knot that cannot be transformed back into the unknot.

You can take that one new knot and twist it all around into different-looking knots, but knot theory says that, as long as you don't cut the rubber band again, it is still fundamentally the same knot, the same way a coffee mug and a doughnut can be fundamentally the same topological space.

So, how'd I write a short story about that? Well, it turns out knot theory has uses outside of rubber bands. In fact, Kelvin kind of got some of the credit for starting this whole knot theory business when he hypothesized that atoms were just knots (vortices) in the aether. But then Michelson and Morley kind of threw a wrench in that whole thing.

There are some modern applications of knots, however. DNA gets itself tied up into knots, and knot theory can help explain how it undoes those knots. Quantum field theory can also be described in a knot-like way, and there might also be quantum computers based on knots.

My story extrapolates this all well into the realm of science fiction, to the point that it would probably piss off the 3 mathematicians who study knots and read short science fiction. But I think it goes in interesting directions. I tie in (ha, not intentional) notions of Buddhist and Celtic endless knots, the Gordian knot legend, and the knot-based number system used by the Inca, known as Quipu. Fun stuff, I hope. I currently have two beta readers attempting to determine whether or not it's ridiculously boring. We'll see.

(Hm. I guess that was a little more than just an update.)