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Space 9 / Relativity — "Meanwhile, Back on Earth…" 
24th-May-2017 06:25 pm
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(Yes, more Space! Part 1 is here but you don't need to go back that far to follow this)

Having at least covered (see Part 8) the question of why the speed of light is constant, the next order of business is the weird and wacky consequences. I'll skip straight to the one that nobody seems to get:

If you're writing your Galactic Empire story and you find yourself needing to say, "Meanwhile, back on Earth,…" that, right there, means you are doing it all wrong.

This can be explained, but first, we need some building blocks:

A Digression on Events and Light-Rays

An event is some instantaneous thing that happens somewhere.

  • On April 11, 1945, just off Okinawa,
    the USS Enterprise gets hit by a kamikaze.

  • On 287 พשּׁ₽ủ⿓, 389284th Year of the BL🐡R🐢G,
    somewhere near Gamma Leporis (~30 light-years from Earth)
    construction is completed on The Giant Commemorative Mirror.

  • On June ▒, 200▒, in a particular hospital room in Bellevue,
    my son is born.

I could do this all day. The entire history of the universe is basically one huge grab-bag of events.

We typically identify events by saying when and where they happened, but that is perhaps a bit misleading since actual numbers for time and place turn out to be negotiable given that observers are free to set up their own coordinate systems. Surveying your back yard, you can calculate latitude and longitude for all of your fence posts, but a prequisite and key piece of that puzzle is the direction you decided to call "North" (South Pole Station inhabitants can have particular fun with this). Change that and the latitude/longitude numbers all change,… but that doesn't cause the fence posts to move.

There's a similar choice with time. If you, sitting in your spaceship observing things, fire your engines, then you're changing the direction that you're calling "The Future". A series of events that were originally all going to take place near your ship is instead going to occur at progressively farther distances from you. But, again, despite that, the events themselves, The Things That Happen, our fence posts, remain wherever/whenever they are.

In particular, if the Things That Happen send messages to each other, then everyone observing them has to agree on who's sending what to whom. Because light rays carry information, whatever happens at the source has consequences at the destination, and no switching of viewpoints should be able to affect that.

So if the kamikaze blowing up on the deck of the Enterprise sends light out into space, which travels for 30 light-years, then encounters the Gamma Leporis mirror, and bounces back to the aforementioned hospital room, where yours truly had the foresight to set up a telescope pointed in exactly the right direction, then we can be in the birthing room watching/experiencing the battle of Okinawa in real time, with all that implies

(..."Oh cool! Enterprise control tower just got blown up. Okay, hon, time for another push…" and I'm pretty sure this would have had consequences if I'd actually done this.)

Building a Map

What you might not have expected is that just this much, i.e., having the universe be a grab-bag of events tied together by light-rays, is, by itself, enough for any single observer to map out everything in his/her vicinity.

Here's one way to do it:

  • We have bugs everywhere, helpfully planted by the NSA. I suppose if we want to be slightly more realistic, it'll suffice that there be a bug present at just the events where we need them, and they just get there somehow (and exactly how, we probably don't want to know).

  • I have a clock — a device based on some well-understood physical process, e.g., an incredibly reliable gerbil running on his little gerbilwheel specially contrived to make a "tick" every so often in a uniform way and sufficiently quickly that I can get whatever timing accuracy I need.

  • I have a transmitter that the bugs are all tuned to listen to. Every time my clock ticks, my current timestamp (= number of ticks since I started my clock) heads out into the universe at the speed of light.

  • The bugs have two jobs:
    1. Remembering what time it is, or, rather, what timestamp they most recently received from me. Since I'm constantly sending these out, they'll constantly be getting new ones. That way, they don't need clocks of their own.
    2. Reporting back to mommy. Every time a bug notices Something Interesting in its immediate vicinity, it compiles a (brief) report ("ship go boom" or "baby born") with my latest timestamp, and then broadcasts it in all directions.

  • Finally, I can have a very sensitive and very directional receiver.

So I sit there for a while collecting bug reports. The trumpet sounds and Time comes to an End, or maybe I just get bored and decide I'm done with all this. And then I can compile all of the reports I've received from everywhere and plot when and where all events occur onto one big, happy space-time diagram:

The way you read this is Time goes bottom to top. The black vertical axis in the middle is me and all of my clock ticks, evenly spaced because my gerbil is so incredibly reliable. Easy.

The horizontal axis is one of the spatial dimensions, call it "X", and there's some choice here about which way to point it. For the sake of argument I'll use some event that I care about and point it that way, meaning when the event finally occurs, it'll be on that line somewhere.

The speed of light being constant, once we send a signal, we know exactly how far away it'll be at any given time. So we can draw, in blue, lines representing all of the timestamp signals heading outwards, and they all have to have the same slope (because the slope is how fast they're going and that number is always the same).

I've taken the liberty of bolding one of the lines. All of the events where there are bugs receiving that particular timestamp have to be on that particular blue line.

Similarly, in red, I've drawn the possible paths of incoming signals from the bugs, and, again, helpfully bolded one of them. If I receive a signal at a particular time from a particular direction, then it had to have come from some event along that red line.

Once I receive a report and extract the timestamp, that, along with the time direction of receipt is enough to nail down exactly where the event is on this diagram.

(… It is a bit unfortunate that this 2-dimensional picture can only represent happenings in a 1-dimensional universe, but that's really all we need for now. If you want, you can imagine a Y axis pointing straight into the page/screen. The bent blue line becomes a blue cone, while the red line remains a single ray intersecting it at exactly one point, so this all still works. If you're totally insane, you can try putting back the Z axis as well, coming orthogonally out of reality, though if you were already able to do that you probably didn't need me to be explaining this stuff…)

What's interesting here is where the left-right axis comes from. Since I'm not moving — yeah, I know, hold that thought,— light rays will cover the same distance going out to an event and back. Even if the event is happening on some moving object, or perhaps the bug doing the report is itself moving around, or both, we simply don't care, because the bug receiving the timestamp, the event itself, and the bug's reporting of it, is all over and done with in such a short span of time that any errors resulting from the relative movements will be teentsy.

Therefore if the timestamp received at event E was 1006 and the corresponding report came back at 1026, then, obviously, the event E
  • happens midway between the two times, at 1016, and

  • at a distance of 10 ticks away from me, because if the signal is taking 20 ticks to get back to me, it must have taken 10 ticks to go out and another 10 ticks to come back.

Yes, I'm using "tick" as a unit of distance:  it's how far light travels in a clock tick. The speed of light is then 1, i.e., one tick per tick, and that's one less multiplication we have to do, so yay. This is also why we have the blue and red lnes being sloped 45°, in case you were wondering; life is so much more convenient when you use the right units.

We can also have other events happening at the same time (1016) at different distances away. But in order to infer that something happened at time 1016, the timestamp has to be just as much earlier as the receipt time is afterwards. Which then dictates that all of those events are on the same horizontal line as you'd expect.

Notice how I'm using my clock to measure distance. This is in fact the only good way to measure distance; actual, physical rulers are far too prone to being stretched, bent, or broken. Yes, things can happen to light beams, too, but we'll generally know what's happening and be able to compensate for it.

But the real point is so that we can do all of our measurements without having to go anywhere. All measurements come to us if we live long enough.

Now for the fun part…

Another Point of View

… in which Somebody Else, say, you, tries to reconstruct what I've done.

In what follows, you are somewhere off the screen, with your own clock built in its own peculiar way (you have an extraordinarily dependable hamster) ticking away at whatever rate along a vertical line, broadcasting your own timestamps, none of which I'll be bothering to show.

Your own "tick" is whatever it is, but for for the sake of clarity I'll assume your distance unit likewise matches up with your time unit so that you, too, can have light rays sloped at 45°.

What remains is to modify the bugs ever-so-slightly so that they're smart enough to distinguish your timestamp signals from mine and nice enough to include both timestamps in whatever reports they send. And then I ignore your timestamps, you ignore mine, and all proceeds as before.

At some point the trumpet sounds, or you get bored, and now it's your turn to reassemble the puzzle pieces.

There being NSA bugs in my clock, that's as good a place as any to start and you'll have all the information you need to plot where and when all of my clock ticks are taking place. You thus discover that I'm moving at some particular velocity (because different people can have different velocities and my future will be in a different place from yours; who knew?)

The velocity is constant because we're all just coasting along with engines turned off, nobody is spinning, and we're out in deep space where there's no gravity to worry about. And whatever direction it is I'm moving, we'll take that to be your X direction so that everything can stay on the page.

Once that's done, given that we have to agree on which events are joined by light rays, then for any other event, like E, no matter what place time numbers you assign to it, it remains the case that a (blue) timestamp message travels from me to E and a (red) bug report gets sent back; you have to have those light-rays linking E to the same ticks on my clock as I do. Which means once you've nailed down all of my clock ticks, E is nailed down as well.

The same goes for all of the other events I know about, including all of the ones on that now-not-so-horizontal line where I have t=1016.

Recall that the only assumption that went into this is that I have a constant nonzero velocity in your X direction, which necessarily tilts my time axis from your vertical. The rest of my grid of light-rays and events folds up sideways like an empty wine-bottle carton being stepped on, and there's only one way it can fold once you know how fast I'm going. Which means there's no way around concluding that my X axis, i.e., all of the events I consider to be simultaneous with my own here now, must be tilted from your own horizontal one…

and, it's not too hard to show, by the same angle. Meaning if, according to your grid, my 1026 clock tick is 500 light-years away from my 1016 clock tick — the sort of scale you need for a Galactic Empire story, — then, for you, E is happening 500 years later than my 1016 clock tick (rather than simultaneously, as it is for me).

It also turns out to be fairly simple to find an observer for whom E is happening 500 years earlier than my 1016 clock tick.

We don't even need to know exactly where you are or how fast your clock is ticking; which is why I haven't bothered to plot any of that.

Everything thus far is an unavoidable consequence of events being connected by light-rays and the speed of light being constant…

…and this much is enough to screw the pooch for any kind of Faster Than Light travel/communication.

(though exactly how will need to wait for Part 10)

This entry was originally posted at http://wrog.dreamwidth.org/61663.html. Please comment there using OpenID.
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