Relativity take two (1 of n)

So, here's another go at explaining Special Relativity. I remain annoyed at how few people really get it, even amongst avid SF consumers, entirely too accustomed to generations of SF writers papering over FTL issues with technobabble, 'cause we need that galactic empire, don'tcha know.

Also, it's long past time to dump poorly motivated 1930s pedagogy that real physicists abandoned long ago (e.g.,. "Your mass increases as you go faster", "What? why?", "Fuck you, it just does" [spoiler alert: just No; forget you ever heard that. And if, in 2026, anyone is still trying to teach it that way, someone needs to sit them down for A Talk]).

All we need is basic geometry you knew or could have learned about in 6th grade plus a bit of algebra (up to Pythagorean Theorem). I think I can get by without using a single square-root sign.

But there will be fewer handwaves this time. Because we have to be clear why things have to Not Be The Way You Expected and not leave wiggle room. Here goes:

The speed of light as a constant

Shine a flashlight off of a moving boat. How fast does the light go?

• It could be like throwing a baseball, where speed of the boat matters.
  
  
• Or it could be like the waves that spread out when you jostle the boat or have it move, where replacing the boat with a possibly-moving helicopter tossing out stationary gravel makes zero difference—the wave fronts have a velocity with respect to the water and only the current/lack-thereof in the river/lake matters.

This is part of why they spent so much time from the 1600s onward arguing over whether light was a "particle" or a "wave" and they couldn't make headway because light goes so fast its speed was impossible to measure…

…until the 19th century, when the tech got good enough and the theory advanced to being able to say what light is (thanks, Maxwell), and we (sort of) get an answer:  "It's like the waves in the water, but (surprise) there is no water". What we have is a number for "speed of light in a total vacuum", derived completely from constants, leaving no way to have it depend on any boat or water-current velocity, never mind there's nothing there that can have any kind of current in it.

Or, rather, the moment you try to imagine there being something there, you should then be able to infer the speed of any current in it by looking at the wave patterns you get from sources moving in various directions, even if this "water" is otherwise totally undetectable.

Except, they tried this. Result:  The speed of the "current" is always zero (thanks, Michaelson and Morley).

Which then gets extra-special weird if you now arrange for there to be two boats in the same place, one stationary and one moving in some direction and somehow the magic "water" is moving in sync with both boats at the same time. How does this even work?

Answer:  It can't. This is the point where Isaac Newton's head explodes.

The only way out is if one or both of the boats is having their distance and time measurements fucked with. But to get a handle on that, we need to go back to first principles on what it means to measure distance and time.

There's also the question of what it means for a boat to be "moving" when you don't have water anymore. Actually, let's start there:

I am Stationary

Let's make our lives as simple as possible by being somewhere out in interstellar space, far away from any distinguishing stars, planets, asteroids, or other landmarks of any kind, and where there's no gravity to speak of. Turn the engines off so that there's nothing accelerating us. Do shit with gyroscopes to make sure we are not spinning — gets rid of all of the weird (centrifugal, etc.) forces that would cause. (I vaguely recall Einstein spending multiple pages on what an "inertial reference frame" is, but this is all you get from me)

What we're left with is Newton's First Law: No forces on us means we are either "at rest" or moving at constant velocity in some direction. Except we can't tell the difference if all we have to look at is a bunch of distant stars that are all moving randomly. There would have to be Somebody Else local looking at us to say whether/how we're moving with respect to them. And even given that, we could still say, "No, we are stationary and you are the one who is moving."

If we can't tell the difference, perhaps it shouldn't matter. So given all of this (i.e., that I feel no forces, therefore none of the scenarios where I would know that I'm moving are in effect), I'll just declare myself Stationary and go from there.

Secondly, if I only trust measurements made by Other People Who Are Also Stationary, then I will not be relying on assumptions about how moving around affects measurements; i.e., if something is dicking with those numbers, I don't have to know in advance how that works. So, for Other People, it suffices to be able to tell whether they are also stationary; if not, then we just ignore them for now.

Measuring Time

Time is easy. Use your clock.

What is a clock? Some physical process — a totally reliable ISO Standard Gerbil In A Cage, or a spinning cesium-137 nucleus in a particular mode, we don't care, just pick something — takes a known amount of time to do a thing, making a "tick". Add some standardized mechanism to count ticks that everyone can build. As long as basic physical laws are the same everywhere (a not unreasonable assumption that may not actually be true but has sure worked really, really, really well so far), this should work. So I can now time everything happening where I am.

Measuring Distance

The trick here is to get as much mileage (pun intended) as we can out of the speed of light being this totally universal constant, which we now assume because that's what the experiments seem to be telling us..

If it's a constant, then it's like a conversion factor:

299,792,458 meters is one second;

9,460,536,207,068,016 meters is one (Gregorian) year.

Most authors would trouble to say "light-second" or "light-year" here, but (I think) that's actually less helpful. Light goes 1 second per second, 1 year per year, or 1 meter per meter, or Just 1, Period (or "Full Stop", as my Brit friends would say). If it's already clear from context what we're measuring, the "light-" prefix is just unnecessary extra noise.

(I don't want to go so far as to say time and distance are the same thing; they're really not, even if it is natural to use the same units. Just like we can naturally measure distances on the surface of a sphere in degrees or radians, but we wouldn't want to say that distance and angle are the same thing; that would be confusing.)

Seems like we should now be able to use clocks and light to measure distance:

I have (1) a clock in my lap, (2) a video projector that can project an image of that clock onto a screen some distance away from me, and (3) sufficiently good eyesight to read what's on the screen

• Light needs time to get there and back, so the screen image will be delayed. If the clock image I see is, say, exactly 2 microseconds behind the clock in my lap, then the light must have traveled a total distance of 2 microseconds (≈ 2000 feet), one microsecond going out and one microsecond coming back.
  
  
• Even if the screen is moving, we can still know where it was at the exact moment that the light-ray bounced off of it:  at exactly one microsecond (distance) away from me at exactly one microsecond ago.

Oh look, I just measured a distance. Go me.

What "Stationary" means

If I can now look at the screen over some period of time and see that distance does not change, i.e., that clock on the screen steadfastly remains 2 microseconds behind the clock in my lap, then I can know for certain that the screen cannot be moving. At least not in the radial (towards/away-from me) direction.

But since I'm not spinning, then transverse motions are also covered, i.e., if, I'm not having to be move my eyes or turn my head to track it, it can't be moving in those directions either, and we're done. I now have somebody located Elsewhere who I can trust.

My new friend can now put her own clock — identical construction, operating principles, and time units — next to her screen where I can see it, and I, in turn, can set up a screen next to me and return the favor.

• She has to see the same delay — light rays repeatedly covering the same distance back and forth, take the same amount of time, — so we have to agree on how far apart we are.
  
  
• If one of our clocks were to tick faster than the other, we'd have to agree on whose it is. Except that if we each have identical equipment and there's nothing else around us to distinguish where we are (e.g., if I were next to a black hole and she weren't, that would definitely be A Problem, so let's suppose there's nothing like that happening), meaning if flipping the entire universe 180° around the midpoint between us does nothing important except exchanging our identities, then how can swapping our names change the clock speeds? So this can't happen. Yay, symmetry. Which then means…
  
  
• The difference between our clock times being fixed, we can now synchronize our clocks. I suppose at this point we could have a political problem where we can't come to an agreement, but there'd still be a fixed difference, and so I will know what to subtract from her time readings to translate them to the time standard I want to use, so we may as well stipulate/assume her clock is synchronized with mine and be done with it.

Now repeat this for all of my other stationary friends. Now imagine an entire network of stationary friends located in all of the places I care about. I now know All Distances, to everyone and also between everyone because we talk to each other (over radios or whatever). This gives us a fishnet of spatial measurements, enough to derive some kind of coordinate system from (or I could take a WW2 Tokyo approach and use people's names to identify locations; whatever works).

And all of those in-synch clocks can time everything happening elsewhere on my network.

Finally, one last (temporary) simplification:

1-Dimensional Universe

Hi, we all live in the John Hancock Tower, where you never have to leave the building, where going anywhere means riding the elevator or going up and down stairs if that's more to your liking (since we're out in interstellar space with no gravity, stairs shouldn't be quite so onerous). Hey, it's got condos, offices, restaurants, movie theatres, everything you could ever want, right? (Yes, once upon a time, it really was considered Cool and Futuristic to construct buildings around the idea that people would want to live like this.)

One dimensional — we can add back the other dimensions later — means the floor number (z) is the only position coordinate we need. The aforementioned fishnet is now just an up-and-down line of stationary people, one on every floor, each of whom has a clock synched with mine.

Which then allows us to fit everything that happens on a 2-dimensional page, a spacetime grid with space going up-and-down and time scrolling left-to-right (yes this is flipped from how I've have previously done this and how most physics books do this; cope). Every horizontal line is a history of all of the events that happen on a particular floor of The Tower. Every vertical line is a snapshot of the universe/Tower at a particular time.

Some set of events will happen. I will record what happens on (or near) my floor with the time it occurs. All of my friends will do likewise. Everything goes onto the grid. And, at the End of Time, we get a picture of What Happened as far as We The Stationary People are concerned.

In the figure, we can see what a typical distance measurement looks like. I point my projector up (or down), send out a light ray at 'Sent', the light bounces off of a screen at 'Upper (Lower) Bounce' at some particular distance (n) above (below) me, and comes back to me at 'Received'. Both bounces and 'Halfway Tick' are all in the same snapshot, so they're all taking place at the same time. Easy.

What Nice Units We Have

One consequence of using the same units for distance and time — whether it's feet (= nanoseconds), microseconds (= kilofeet), seconds, etc. — is that all light rays, whether traveling upwards or downwards, will be going at 45° (traversing one distance unit per time unit) on our grid.

Why care about this? Because so much becomes more obvious when you scale time vs. distance the right way.

E.g., imagine surveying your backyard measuring north-south distances in meters and east-west distances in Astronomical Units (1AU=150 million km), i.e., compressing the east-west scale by a factor of 150 billion for basically no reason. That would be stupid, right?

But that's essentially what we were doing before 1905 and why it took 200+ years to figure out Newton had screwed something up.

And now it's time to…

Meet the Moving People

(to be continued)

Relativity (2 of n): Meet the Moving People

My revamped explanation of Special Relativity, trying to keep it completely geometric and not invoking Walls of Math, continued from here where I have now completely beaten to death the concept of "Stationary" and it's time to

Meet the Moving People

Actually, we're going to be particular about who we associate with. We want Moving People who will likewise be able to say that they don't think of themselves as moving, meaning they can't be accelerating or spinning, either. Which leaves having them coast at some constant velocity that is non-zero (because otherwise they'd be stationary and One of Us) and upwards (just to pick a direction).

We also want them moving slower than lightspeed (STL), for Reasons.

Why FTL Sucks, part 1

At this point we are not saying that faster-than-lightspeed (FTL) People cannot exist. It's just that we have no idea how they'll be measuring distances. They outrun all light signals, so the whole business of setting up a screen to reflect off of will never work; no reflections will ever reach them.

Not that they can't have some New Method for measuring distance, but whatever it is, it'll be Different, they won't be able to not know it's different, and, beyond that, there's little we can say about it, so I'm not going to try. So let's just focus on the folks we know do exist.

Also, if, later on (spoiler alert), it turns out we don't actually need FTL people for anything, so much the better.

The Stationary People are Not Special

If the Moving Person and their friends do everything we did as above to establish their own coordinates for everything that happens, they should get a consistent world view out of it, even if it turns out to be different from ours in some unexpected way.

In other words, feel free to imagine them building an entire Moving John Hancock Tower, complete with its own (moving) floors, someone on every floor who the original Moving Person likewise thinks of as stationary, everybody with their own clocks, set in such a away that they think they're all synchronized.

Which means they've got their own spacetime grid, complete with its own floor histories and snapshots. And we just have to plot what that has to look like on our own (stationary) grid.

The "horizontal" grid lines are easy: They're just the trajectories of the various people on the moving floors.

• Constant velocity trajectories are linear, so we can say that at least of the original Moving Person.
  
  
• Moving clocks repeat some physical process which should keep working the same way everywhen along a trajectory, so their tick events have to be evenly spaced.
  
  
• Having repeated distance measurements turn out the same — taking a particular shape made out of events and being able to repeat it arbitrarily later along a trajectory — means all of the trajectories are linear, parallel, and have the same tick spacings.

Note that slope (= rise over run) and velocity (= distance over time) are the same thing in this representation (one reason to prefer time going left-to-right). STL means they're less steep than the (45°) light rays. And parallel means all Moving People velocities are the same.

Or, to summarize,

The Moving People's trajectories / floor histories are all lines with the same slope $v$ (velocity)

Their "vertical" snapshot grid lines are where shit gets weird.

Watch what happens when Moving People try to measure/verify distances. Someone aims their projector, either upwards or downwards or both (two projectors, yay), light ray (yellow) heads out, bounces off of a screen, Upper Bounce and/or Lower Bounce, depending, then comes back, and now they, too, have a clock image they can compare with what's sitting in their lap.

The three events Sent, Upper (or Lower) Bounce, Received must form three corners of a rectangle since the light rays are all 45°.

The Moving People don't think they're moving, so they have no reason to think of the light as traveling different distances going out and back. Which means the time the Moving People infer for the bounce events must be whatever their clock is saying halfway between Sent and Received, the event at the center of the rectangle.

The yellow rectangle, being a rectangle, has axes of symmetry, light rays coming out of its center that I have helpfully drawn in magenta. That is, reflecting everything through one of them (doesn't matter which) maps the rectangle onto itself while swapping the diagonals, therefore each diagonal makes the same angle with that light ray; it just swings the other way.

Or, equivalently, the slopes of the diagonals must be inverses. Recall slope = rise over run, and mirroring everything through 45° swaps rises with runs, so if one diagonal is sloped $v$, the other must be sloped $1/v$.

No matter how big or small the rectangle is, if it's centered on the Halfway Tick, then that other diagonal has to coincide with the $1/v$-sloped (blue) line through Halfway Tick, so that must be where all events deemed simultaneous with Halfway Tick end up. Same goes for all other clock ticks and all other Moving People trajectories. Therefore,

The Moving People's snapshots are all lines with slope $1/v$

In particular, these lines cannot be vertical — as Newton and everyone else had assumed — unless the velocity is zero. In other words, the Stationary People and Moving People will disagree on which pairs of events are simultaneous.

You can also see that compressing the (left-right) time scale by a factor of a hundred million in order to make ordinary, everyday velocities like one meter per second visibly non-horizontal will also take a snapshot line that was already off from vertical by only one part in a hundred million and reduces that to one part in ten quadrillion; it's no wonder Newton et al thought it was vertical.

In other news:

The Moving People's distance units in their snapshots are spaced the same as the time units in their histories

This follows from we the Stationary People seeing both diagonals of the rectangle having the same length while the Moving People infer the same number of (distance/time) units along each diagonal (because they also take speed of light to be 1).

Notice that if we had made Newton's assumption that their snapshots are vertical like ours, then their clock ticks would have to be spaced farther apart to line up with ours because their trajectories are sloped. So we won't take anything for granted here. Their unit spacing is whatever it is; we'll work it out later.

Thus, now we know their grid (plotted onto the our Stationary People spacetime grid), up to some scaling factor, looks like this:

(a bunch of rhombuses) which by itself is enough to prove all manner of fun facts. For example,

The Moving People and The Stationary People agree on what their relative velocity is.

(When you see what goes "wrong" later, you'll be glad we checked this. Also if you're thinking this is obvious, try coming up with your own argument before looking at mine.)

Start at some rendezvous between a Stationary and a Moving Person. The Stationary People pick some amount of time to wait (${\mathrm{\Delta t}}_s$), doesn't matter how long, then see how far the Moving Person has gone (${\mathrm{\Delta z}}_s$) during that time, then do the division ${\mathrm{\Delta z}}_s/{\mathrm{\Delta t}}_s$. Easy.

The Moving People do likewise, but using their own grid lines. If they pick the right ${\mathrm{\Delta t}}_m$, then we get this diagram:

We then notice that the two gray triangles have the same angles, hence are similar, hence lengths of corresponding sides have the same ratio.

The triangle on the right is smaller than the upper triangle by some factor v, and we get that same number. whether we divide the two short sides, the two medium sides, or the two hypotenoi^*.

^*Yeah, I know this is not the actual plural of 'hypotenuse'; in a just universe it would be.

And yes, for the Moving People, the segment lengths aren't actually ${\mathrm{\Delta t}}_m$ and ${\mathrm{\Delta z}}_m$ because of the as-yet-unknown (distance/time) unit spacings, but, fortunately, since the spacings are the same, they cancel out here, and so the Moving People get the same answer doing ${\mathrm{\Delta z}}_m/{\mathrm{\Delta t}}_m$.

Now that you know how to do that, you can probably figure out how to show that

Moving people also see light rays going at velocity 1

(exercise!)

Let's try something slightly more complicated:

There is an intermediate trajectory that looks the same to both the Moving and Stationary People

…by which I mean there can be Intermediate People moving in the same direction as the Moving People, but enough slower so that the Moving People see them moving downwards at the same speed that we the Stationary People see them moving upwards.

You'd think this would be obvious. Except, as it turns out, …

The intermediate velocity is not actually half of the Moving People velocity.

We can, from the previous diagram, consider the range of possible intermediate trajectories, starting with our own (Stationary, horizontal) one, and then gradually increasing the slope until it matches the Moving People's velocity (v).

Stationary People will care about where the trajectory crosses the (vertical) blue arrow to get their Δz and then divide, thus getting a velocity $v_s$, ranging from 0 to v.

Moving People do likewise except, because their notion of what's simultaneous is different, they will be concerned with where the trajectory intersects the slanty downward blue arrow. The velocity they see, $v_m$, will correspondingly range from $v$ to 0.

Subtracting $v_s-v_m$ gives us something that goes from $-v$ to $+v$ and crosses through zero exactly once somewhere. That crossing is the trajectory we want.

Also, the trajectories where either of us is seeing a velocity of $v/2$ (crossing one of the blue arrows exactly half way up the triangle) are clearly not the same, thus neither is the one we want, and that the actual Intermediate velocity generally needs to be larger, giving us our first warning that

Velocities do not combine the way you'd think they do

Presumably, we'd already know this because slopes do not add. (Hint: 45° is slope 1; combining two 45° angles gives you 90° which is infinite slope, whereas slope 2 is around 63.435°).

Yes, this isn't entirely constructive since we haven't told you what the intermediate velocity is. It's complicated — well okay, it's not that hard, but you need to solve an equation and it's quadratic (exercise!). Fortunately, for what follows, all we need is that the trajectory exists and the velocity is no faster than what the Moving People were originally doing and is therefore STL.

We then invoke our previous result — that every pair of observers has to agree on their mutual relative velocity — to get:

The Intermediate People see the Moving and Stationary People going the same speed in opposite directions

So we now have a viewpoint where The Moving People are slower and The Stationary People become the The Anti-Moving People. We can now take advantage of symmetry to show …

… something completely and totally cool that I will now defer to the next installment