wrog

My explanation of Special Relativity, trying to keep it almost completely geometric, continued from part 2 wherein I describe

what the Moving People spacetime grid looks like when mapped out by the Stationary People,
how all we have left to do is figure out their unit spacing, and
how there can exist Intermediate People who see the Moving and the Stationary People moving in opposite directions at the same speed.

We now take a moment to introduce, out of left field, a new definition that will turn out to be very useful:

Give a pair of events, separated by a distance $Δz$ and elapsed time $Δt$ , the interval between them is defined as the quantity ${Δz}^{2} - {Δt}^{2}$

which looks like a squared distance except for an annoying minus sign,
which seems to depend on whose coordinates we're using (Stationary People vs. Moving People vs. somebody else), but we will now show that:

For any given pair of distinct events, the interval between them is an invariant

… by which we mean every observer that we care about who calculates this in their own coordinates will get the same number.

We'll start by showing that Moving and Anti-Moving (née Stationary) People have to agree on this.

Recall where these numbers ( $Δz, Δt$ ) come from. We have one event pick out a history line while the other picks out a snapshot, these have to intersect at some common point/event, counting units along the history line gives you the elapsed time while counting units along the snapshot gives you the spacing. And you'd still get the same numbers if you switch which event picks out the history vs. the snapshot (since you'd just be going around the other side of the rectangle/parallelogram).

So the Moving People get ( ${Δz}_{m}$ , ${Δt}_{m}$ ) and the Anti-Moving People get ( ${Δz}_{s}$ , ${Δt}_{s}$ ).

Normally, with two arbitrary Peoples moving at arbitrary velocities, we'd be stuck at this point, but we're actually doing two sneaky things:

We are using the point of view of the Intermediate People, using their spacetime chart, where the velocities are equal and opposite. The angles that the Anti-Moving People's history and snapshot lines make vs. a downward light ray are the same as for Moving People's lines vs. an upward light ray.
We have the Moving People follow history from one event while the Anti-Moving people follow history from the other event.

Which gives us two extra right angles that you, perhaps, weren't expecting to see.

Which means we have two right triangles sharing the same hypotenuse (dotted red line between the two common points).

Which means we have two different ways to invoke the Pythagorean Theorem to get that length,

Which means we have two different sums of squares that have to be equal.

The one remaining potential problem is that what we're squaring are segment lengths on a page that we'd then have to divide by the (as yet unknown) time/distance unit spacings to get the corresponding Δz and Δt values.

But, fortunately, in this viewpoint, since the Moving and Anti-Moving People are carrying identical clocks at the same speed in opposite directions, all of these unit spacings have to be the same. So we can cancel them out, leaving us with

{Δz}_{s}^{2} + {Δt}_{m}^{2} = {Δz}_{m}^{2} + {Δt}_{s}^{2}

Subtracting ${Δt}_{s}^{2} + {Δt}_{m}^{2}$ from both sides gets what we want

{Δz}_{s}^{2} - {Δt}_{s}^{2} = {Δz}_{m}^{2} - {Δt}_{m}^{2}

This being purely a statement about Moving and Stationary People measurements, we do not have to care what the Intermediate People measurements actually are. For this to work, it suffices that the Intermediate People merely exist.

And for that. we only need the Moving vs. Stationary People relative velocity to be STL.

And, since there's nothing special about this particular choice of Moving People, that means everyone going STL according to The Stationary People will agree with The Stationary People on the interval value for this pair of events.

Or any given pair of events since there was nothing special about this particular pair.

One easy consequence of this:

Since agreeing on the value of ${Δz}^{2} - {Δt}^{2}$ includes agreeing on its sign, that means agreeing on whether $| Δz |$ is less than, equal to, or greater than $| Δt |$ and hence whether the trajectory between those events is STL, lightspeed, or FTL.

Therefore, everyone going STL according to The Stationary People must see each other's trajectories as STL and agree that all other trajectories are not. Which, among other things, means it doesn't matter which of them was originally picked to be the Stationary People; we still unambiguously get the same group of People Who Are Going STL with respect to each other, which is then the group of observers we care about.

Why FTL Sucks, part 2

If you want to postulate FTL People existing and imagine that one of them might accidentally get chosen as the initial Stationary People, then it will suffice to make a small addition to our list of requirements (i.e., to be in the set of observers I care about) to immediately disqualify such folks: "Must be able to measure the length of your own arms using light rays and reflectors," or maybe, "Must be receiving light from all directions." (Yes, there need to be distant stars everywhere for this argument to work; this is not a problem in our universe.)

And, yes, having all STL people see each other as moving STL is very weird.

For example, if you have people going opposite directions, each at 2/3 lightspeed, their relative velocity still has to be STL and therefore cannot be 4/3 lightspeed. It's actually 12/13 lightspeed, if you must know. Once again:

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

In fact, it turns out, the only time they do add/subtract the way you'd think is either when one of the velocities is zero or when they're equal and opposite (and thus sum to zero).

About proper time

The proper time is the time lapse between two events that are occuring in the same place. There's a sense in which this kind of time lapse is more "real" than other kinds of Δt, since in this case you or whichever of your friends happens to be there is sitting right there directly experiencing, up close and personal, both events and All of The Physics happening between them.

But this concept also makes sense for any two events that have a STL (i.e., negative) interval between them, because that means we can have somebody right there, coasting at that velocity, who then sees both events occurring in the same place.

Same place means Δz is zero for that observer, so the proper time squared is minus the interval and so this is a number all observers will agree on. Note that this includes agreement on the sign since everybody agrees on which time direction is the future and hence which event is happening first. (If you have trouble with this, imagine everybody has both a clock and a steam kettle with escaping steam.)

This (finally) gives us the Moving People time/distance unit spacing, i.e., if you find two events on a Moving People trajectory whose interval is 1 (calculated in your own or anybody else's coordinates), then you have two events that are 1 unit apart in proper time and therefore 1 unit of elapsed time according to the Moving People (because they're seeing them happening in the same place).

Which means that when a Moving Person advances forward in time by 1 unit according to us Stationary People, they're also moving a positive distance $v < 1$ , which then makes that interval $- (1 - v^{2})$ , meaning the proper time is less than 1, and therefore the point at which they will have experienced a full unit of time is farther into the future. And the closer they are to lightspeed (1), the farther it gets. This is time dilation at work: For anything that we see as moving, its clocks ticks more slowly than ours (according to us).

Except, since there's nothing special about us, it must also be that, according to them, it's our clocks that are ticking more slowly. Which may seem like a contradiction. But it's not, because we're not agreeing on what's simultaneous and we're counting time in different directions.

Probably the easiest way to see this is to look at it from the Intermediate point of view.

We meet. We each tick off a unit of time. But once we've done that, we're not in the same place anymore, and each of our snapshots is tilted back into the other person's past. So this actually does work:

For extra fun, if someone were to hitch a ride upwards with Moving Person at the meeting point, then jump off right at the moment where the Stationary People think 1 unit of time has elapsed, with sufficient velocity downwards so as return to the meeting point (i.e., where the Stationary People consider the meeting point to be) promptly at the stroke of 2 (units of Stationary People proper time), then we can see that:

Both of the traveler's upwards and downwards journeys have the same interval, i.e., on both legs, a particular distance (length of the upwards blue arrow) gets traversed in 1 unit of time, at least according to the Stationary People, but, then, everybody else's calculation of the interval has to match what the Stationary People get.
The traveler is experiencing (much) less than 1 unit of proper time on each leg.
The traveler will be (much) less than 2 units older when they return to their starting point.

This is the resolution of the famous "Twin Paradox", in which one member of a pair of identical twins journeys and the other stays behind, and it's only a paradox if you think it's somehow against the rules for identical twins to age differently, but by now it ought to be clear that Time is not this Absolute Thing anymore.

If you need actual numbers: The figure is to scale if both the Moving and Stationary People have a relative velocity of 12/13 lightspeed and the Intermediate People are seeing them each go at 2/3 lightspeed in opposite directions.

For the sake of not writing more fractions than we have to, let's have the distance/time unit be 13 years. Then, the jumping off point is, for the Moving People, 5 years after the initial meeting, but, for the Stationary People, it'll be 13 years after and 12 (light-)years away from (above) the initial meeting (13²−12²=5², so it's still 5 years proper time when the Stationary people calculate it). The traveler's return journey is 12/13 lightspeed downwards according to the Stationary People and 62/63 lightspeed downwards according to the Intermediate people (so even though the return jouney looks a whole lot longer on the Intermediate People chart, there's a whole lot more time dilation).

Upon returning (bottom event), the traveler will be 10 years older, as opposed to the Stationary People being 26 years (2 units) older.

About proper distance

Two distinct events that are simultaneous have a proper distance between them. One can debate exactly how "real" this is, but, being the positive square root of the (positive) interval (i.e., |Δz| for whoever sees Δt as zero), it is, at least, a number that everyone must agree on.

But this concept also makes sense for any two events that have a FTL (i.e., positive) interval between them, because we can invert that FTL velocity to get something STL that is how fast someone needs to be coasting in order to see those events being simultaneous.

Yes this is different from proper time in that we make no attempt to get agreement on signs, hence why we take the trouble to say positive square root and say $| Δz | instead of Δz . That's because, even though our 1-dimensional universe does have a universally agreed "up" and so we$ could define a signed proper distance if we wanted, we would then lose in higher dimensions. That is, adding back the x and y directions makes Δz into a vector and then one can do purely spatial rotations, making nonsense of any notion of spatial "sign".

Why FTL Sucks, part 3

Any FTL pair of events will be seen as simultaneous by somebody. Also people going slightly slower or slightly faster than the velocity you need to see them as simultaneous will disagree about which event comes "first", a distinction which won't matter if there's no way to communicate or send physical objects between the events,…

…but if you can, then we have a Problem. Whatever technobabble method we the Stationary People can use to get stuff from one event to another across an FTL interval should be doable in the same way by any of the STL Moving People to similarly get stuff moved across any other FTL interval that has the same proper distance including the backards version of that first interval (i.e. swapping origin with destination), or any number of other choices that connect the destination to somewhere/when before its origin and voilà! instant, easy time loop.

Or to put it another way, the moment you can take the stargate / ansible / warp drive / thing that can make jump points in arbitrary empty space / whatever and put it on a ship going at any STL velocity in any direction and have it still work the way it did in the Stationary Place — which it should because there's a whole lot of Physics that depends on this — that's all you need to wreck causality.

And yes, nothing says the universe can't be acausal (it's mainly humans that like causality), it still means anyone who's developing FTL travel/communication (or putting it in a story) is inevitably going to need a plan for dealing with an acausal / Bill&Ted universe.

Why FTL Sucks, Stargate SG-1/Atlantis Edition

The stargates make instantaneous connections across hundreds of light-years, routinely. So, e.g., the Stationary people can have a pair of stargates separated by 99.999999 (light-)years, one dials the other, you can go out, then close the connection, immediately dial back the other way, and return with virtually no time having passed at the starting place.

You could, if you want, imagine that the far-end events are all offset in time by some amount, but then that would mean for one of the directions you'd have to be going backwards in time, which is not supposed to be possible without the Stupid Solar Flare Plot Device games they play when they actually want to do time travel. So we have to assume the opening and closing connection events at both ends are synchronized/simultaneous (not that this actually saves us).

Now take the same pair of gates, reduce the separation to 99.99999700000002 years (the difference is roughly Earth-Jupiter, so no one will care) and have them set in motion going exactly 0.02/100.000001≈0.0002 lightspeed or roughly 60 km/s in a particular direction that we will call "up".

Meaning we now have Moving People in charge of the gates.

Their snapshots will be slanted futurewards (slope = 100.000001/0.02 = 5000.00005) if you follow them upwards, or pastwards if you follow them downwards.
If you follow a snapshot up, it'll take 100.000001 years (Stationary People distance) to reach the other gate's trajectory (it's moving away, remember), and you're also going forward in (Stationary People) time by 0.02 years ≈ 175 hours
(Upper gate's position (z) being 99.99999700000002 + 0.02×5000.00005 = 100.000001, so this works out. I could have just made you solve for where the snapshot intersects the upper gate's trajectory, but I figured I'd just give it to you. You're welcome.).
The interval is then (100.000001)²−(0.02)²=(99.999999)², so the proper distance, i.e., the distance as measured by the Moving People between the two (stationary in their view) gates, is (surprise!) 99.999999 years.

Which means those gates can operate exactly as before, even though they're moving, because the gates themselves can't tell that they're moving. So, if you go in one end, you come out the other end 99.999999 (light-)years away at exactly the same (Moving People) time, but according to the Stationary People time, you're coming out 100.000001 years up and forwards in time by 175 hours.

Or those same distances/times backwards when you do the return/downward transit.

Now arrange for a second pair of Anti-Moving gates, i.e., with all of the same parameters but moving downwards, timing/placing them so that someone coming out of an upwards journey through the Anti-Moving gates can immediately dive into the upper Moving gate and arrive back where they started 350 hours (a little under 15 days) in the past.

Instant, cheap time machine. Look Ma! No solar flares! And way, way less work than that McKay-Carter Intergalactic Gate Bridge they built to get from Milky Way to Pegasus that spanned 3 million light years and used up hundreds of gates.

Are we done yet?

Next up: The Storrow Drive Theorem or How We Put Back the Other Dimensions