Ni’a has provided us with a challenge by insisting on using “exact” numbers in their narrative in their book Outsider.
Their fictional version of themself can get the exact numbers of whatever they are looking at, just by looking at it. This is pure magic, of course, and even when I wrote about it I lampshaded the whole thing by saying “It’s also what Phage can do.” Essentially.
And Ni’a themself has lampshaded their numbers a bit further by saying “Rattling off the exact numbers as they entered my head calmed me down, even though they’d be wrong by the time I got done speaking them. Rounding them off helps a little to cover the discrepancy, but my psyche will continue to update me for a few moments afterward even if I cut myself off.“
But, in their first chapter, titled “Outsider”, they rattle off the distance between the Sunspot and its parent ship the Terra Supreme at the moment that they are speaking to Phage. And we are seriously fudging this number.
We aim to, in time, refine it, as we come to understand Lorentz Calculations better, and as we figure out how to incorporate constant acceleration into them. And if somebody wants to do all that for us and just give us an exact number at some point, we’ll be grateful.
But here is what we have done so far:
The first thing we did was establish how long the Sunspot has experienced its own existence, from when it was created from the shipyards of the Terra Supreme and started accelerating away from it, 130,296 years. Importantly, the Terra Supreme has been constantly accelerating away from that point in space/time at the same rate, and from their perspective it has been 130,296 years as well.
But relativity does weird things with that.
To get a picture of what we’re grappling here, we recommend that you watch Henry Reich’s course on Special Relativity at minutephysics on YouTube. (url: https://youtu.be/1rLWVZVWfdY)
Before watching that course, we simply remembered what we were taught in school, and drew up a light cone diagram of the travel of the two ships, and just sort of tried to visually estimate how long it would take for a signal of light to travel from one ship to the other starting at 130,296 years after they’d left each other’s presence. Because Ni’a wanted to rattle off that specific number. And here is what we created:
Oops. That says 130,298. Where did 130,296 come from? (Plurality, it does weird things to your memory at all levels, and mistakes like this are way too common for us, even when we write things down.)
If you are familiar with these kinds of calculations and visualizations, you can probably see flaws in this that even we still can’t see. We are not physicists. We are hobbyists who are trying to learn this on the fly so that we look kind of like we know what we’re doing. Most readers will just accept the number we give and move on. But we want to be close enough to the actual number so that our readers who are mathematicians and physicists won’t end up feeling like they have to write huge blog posts like this one to rant about how wrong we are.
Please! Let us do that work! (Just, help if you can with comments below. Thank you.)
Anyway, we wanted a placeholder number fast so that we could jot it down and keep writing, which we would then create later when we finally found an equation we could just plug numbers into and get answers from. At this point, we contacted our old friend from fifth grade, Nick Scholtz, and asked him for help, which he graciously gave us by helping us find the video above. But that took a little while, so our place holder number stayed in our document until today, and was actually online and publicly readable for a couple of days.
Squinting at our diagram and waving our hands at it a bit, we decided that a reasonable placeholder estimate might be something like twice the time the two ships have spent traveling away from each other, which looked like 260,592 years.
Ah-hahaha. Ah-haHAhahaha. HAHAHAHAHAAHAHAHA – no.
We were in a hurry. We wanted to let Ni’a focus on writing their story, not stopping to spend a week on physics.
It was basically just essentially tracing a path from one ship to the other down their lines of travel, and vaguely rounding it off. And then kind of thinking that between time dilation and constant travel at near speed of light, it should balance out to something like that.
But after watching the above videos, we decided that a better estimate, though still too simplistic, was to use a Lorentz calculation, which is specifically designed to find numbers like this. We’re still not taking into account the constant accelerations of the ships, nor actually calculating their relative velocity at the moment that Ni’a is talking. We knew at this point that the relative velocities of both ships were something close to 99.99% the speed of light, and that acceleration at this point was so minimal as to be ignorable (in our minds at least – we could still be very wrong about that).
So what we did was we went to this Lorentz calculator right here:
And we plugged the year that the story is taking place into the top (130,296 – and it should have been 130,298 and we need to do some more editing now), and the speed of light with some speed shaved off (the hundreds, tens, and ones digits just dropped to zero), and got a number that we’re calling “close enough for now”. Which is 1,793,187 light years. We think.
We think we’re interpreting that calculator’s results correctly. We know the number is still wrong. But we expect it to be much closer than our first estimate.
At some point, we will get the ships’ actual relative velocities as we’ve set them up (according to their accelerations and the time since they started traveling apart), and plug that into the calculator, and get a closer number. But their relative accelerations are distorted by time dilation as well, and need to have a Lorentz calculation applied to them to begin with, to get the resultant relative velocities. And then that needs to be taken into account as Ni’a’s hypothetical pulse of light travels from one ship to the other starting at the point where they start speaking their sentence. And there’s a point at which it really isn’t worth getting more precise. There’s a point where it’s just too much work for the reward of being accurate.
So, anyway, you may see that number change a few times in the future. Especially if someone does the work for us and puts a more correct version of it in the comments here. And if we ever put together the right set of precisely accurate calculations to get it, we’ll share them, too. The actual equations.