wrog

Special Relativity continued from part 5, in which we learned about translating coordinates, the unit-later hyperbola, and velocity angles

Or you can start from Part 1

Today's topic is constant acceleration, which, in Special Relativity, is much weirder than you might expect.

The Unit-Up Hyperbola

Let's take the the unit-later hyperbola, reflect it through the (45°) upwards lightray, and see what we get. Same shape but curving upwards instead of futurewards, so that it's now the set of all events that are one unit proper distance upwards from the origin event $(\begin{matrix} 0 \\ 0 \end{matrix})$ where we have the Cherry Tree Explosion (because there has to be something dramatic happening there).

In other words, at any point along this curve, the segment joining it to the origin will be FTL-sloped (with proper length 1). Which means the tangent line at that point will be inverse-slope to that, and therefore STL.

Meaning this curve is a trajectory that someone can be following.

The velocity will be constantly changing, but since it's now STL at every event along the way, we can accumulate elapsed proper time from $(\begin{matrix} 0 \\ 1 \end{matrix})$ , the event the Stationary People see as being simultaneous with the origin.

…in exactly the same way we were accumulating proper distance along the unit-later hyperbola from $(\begin{matrix} 1 \\ 0 \end{matrix})$ , the event the Stationary People see as being co-located with the origin, in order to define the velocity-angle.

…every point along the way being somebody's [moving_Up], and the instantaneous velocity at that event being that of the corresponding [moving_later]

and then you realize the elapsed proper time to get to [moving_Up] and the velocity angle (proper distance to get to [moving_later]), the one being the reflection of the other, have to be the same.

…meaning the rate of velocity angle increase per unit proper time along the Unit-Up hyperbola is … well, um, … 1, actually…

…meaning the Unit-Above hyperbola is a constant acceleration trajectory, at least if we're defining "acceleration" as rate of change of velocity-angle per unit proper time…

Constant acceleration?

… which is actually a good question, since, in Special Relativity, "constant acceleration" can mean many things.

In particular, this trajectory is not, e.g., constant rate of change of velocity according to the Stationary People, i.e., as in (relative) velocity increasing by 1 per unit time on Stationary People clocks — that trajectory would be the parabola Newton would be expecting to see — since that would mean reaching lightspeed at $t = 1$ , double lightspeed at $t = 2$ , and so on. And, likewise, any smaller "constant acceleration" in this sense would still be hitting lightspeed after a finite amount of time.

Rate of change of relative Stationary People velocity per unit proper time, i.e., acceleration of the Stationary People as perceived by the passengers, also won't work for describing this, because, just as velocity angles can go arbitrarily high along the unit-later hyperbola, the unit-up hyperbola has arbitrarily large amounts of proper time available, and so, again, we would still see Stationary People exceeding lightspeed at some point.

And since this trajectory is invariant under translation to any (constant velocity) Moving People coordinates (i.e. has to look the same since everyone has to agree on proper distances, and so the translation takes the hyperbola to itself), it cannot be "constant acceleration" in either sense for any of the Moving People, either.

But the real question is how do the passengers, the ones actually following the trajectory, experience it?

What we can do is take any event along the unit-up hyperbola, consider the Moving People whose velocity matches the instantaneous passenger velocity at that moment, and see where that is in the coordinates of those Moving People. Since this translation takes the hyperbola to itself and tangent lines to tangent lines, we have to end up at $(\begin{matrix} 0 \\ 1 \end{matrix})$ , the one event along the hyperbola where the velocity is zero.

Or, rather, once we figure out what's going on at $(\begin{matrix} 0 \\ 1 \end{matrix})$ , the passenger experience will have to be the same at every other event along the hyperbola, since there's nothing special about any particular Moving People. Meaning this is indeed constant acceleration in the "passenger experience" sense and the notion of constant acceleration we care about.

But it would be nice to know what the constant is. And what's interesting about the place where the passenger velocity is (at least momentarily) zero, is that the velocities at all nearby (in time) events will be microsopic, meaning decidedly non-relativistic, meaning

velocity angle and velocity are essentially the same
(passenger) proper time and (Moving People) coordinate time are essentially the same

which means all of the various definitions of "acceleration" we get by mixing and matching these concepts all converge to Newton's as we reduce the time interval to zero. So this really is passengers experiencing an instantaneous velocity change of 1 per unit time.

… which obviously will not add up over time (e.g., no reaching lightspeed 1 unit later) because (once again) velocities do not add the way velocity angles do. However, velocity angle reaching 1 after 1 unit of time, 2 after 2 units of time, and so on, is not a big deal because none of those velocities are lightspeed.

All constant acceleration trajectories look like this

If we now scale down the unit-up hyperbola by a factor of $g$ to get the set of all events that are a proper distance of $1 / g$ up from the origin, all of the above holds true except in that proper time needed to increase the velocity angle by 1 gets scaled accordingly and hence the instantaneous acceleration everywhen is now $g$ . This will work for any value of $g$ , so now we know what all constant acceleration trajectories look like.

How to do a fleet of ships accelerating in synch

One interesting feature of all of these trajectories is that, at any point, the passengers will at least momentarily be sharing the viewpoint of the particular Moving People that see them as motionless, and the corresponding snapshot has to go through the origin. In other words, no matter where we are on the trajectory, the origin event, as far as the passengers are concerned, is always happening now and at a fixed distance $1 / g$ below where they currently are.

So if we now want to have Somebody Else be seen by our passengers as hovering some fixed distance above or below them, then, considering every snapshot through the origin, we see that this 2nd trajectory is likewise always some (other) fixed proper distance from the origin, therefore has to be a scaled version of the first trajectory, thus also a scaled unit-up hyperbola and thus also doing constant acceleration, but at a different rate because the (constant) distance to the origin will necessarily be different.

Repeat for all other ships in the fleet. So to get an entire fleet of accelerating ships staying together, from their own point of view, they all have to share a common origin event from which each ship is maintaining some fixed proper distance.

And the lower ones are are necessarily accelerating harder.

Thus, unlike in Newtonian mechanics, it is impossible in Special Relativity to create a (fake) gravitational field that is uniform in all directions and constant for all time — it can be time-independent and uniform in the directions transverse to the acceleration but that's the best we can do. The gravity has to weaken as you go up.

John Hancock Tower, Special Edition With Gravity

In case you were wondering, Earth-style gravity, $g = 9.806 m/s²$ , once you translate meters to seconds or seconds to meters, comes out to roughly 1 per year, or, more precisely, 1 per 353 days. Call this an acceleration (light-)year, or just "year", from now on.

Once we have enough accelerating ships flying in formation close enough together (with a sufficiently large conspiracy to keep them fueled and in good repair forever) — again, they all see each other as being motionless — we can imagine joining them with wires that won't stretch or break, and then build that all up into walls and stairs and whatnot.

This gives us a natural a coordinate system for The Tower since all of the hyperbolas — floor histories of accelerating floors — can be numbered by altitude above the origin event, and all of the snapshots — collections of events that everyone on all of the floors thinks are simultaneous — can be numbered by velocity angle 𝜙 (according to the Stationary People), which, as we've already noticed, is proportional to elapsed proper time since the acceleration is constant on each floor.

So now I'm standing on the 100th floor of the John Hancock Tower With Gravity, feeling my 1/year of standard earth gravity, and the Exploding Cherry Tree origin is in the lobby, one year downstairs from me. The floors are 1/100 of a year (3.53 days) apart so there's lots of room to hang artwork.

Upstairs from me, on the 101st floor, the gravity is roughly 1% less (really 100/101). Because the proper time between snapshots is proportional to altitude (gravitational time dilation), the residents of the 101st floor will be aging 1% faster than me. Downstairs on the 99th floor, the gravity is correspondingly stronger (100/99) and the residents age slower. If I can make it down to the 50th floor the residents will be aging half as fast; a veritable fountain of youth if one can stand the doubled gravity.

If I shine a light upstairs at someone on a higher floor, they'll see the frequency reduced (gravitational red-shift) by whatever the height ratio is, and similarly (gravitational blue-shift) for shining a light downwards. The actual calculation for this is complicated — it's more than just clocks ticking faster on the higher floors, even if the answer comes out unexpectedly simple, — but the essential idea is that while the floors are moving together at the moment a particular wave front is emitted, by the time it reaches its destination, the velocity of the destination floor will have increased, so, with source and destination not actually going the same speed, there will necessarily be some kind of frequency shift, red or blue according as the destination is above (moving away) or below (moving towards) the source.

And it's fun that we can get all of these special effects and I haven't had to derive or even tell you anything about General Relativity.

Because this is all taking place in completely flat spacetime.

Meanwhile, down in the lobby, we have the ultimate nightmare realm, where gravity is infinite and time has stopped. As far as the residents of the Tower are concerned, the Cherry Tree is, always has been, and always will be exploding.

But no one in the Tower will ever know this because the first light of the explosion never catches up to any of the floors above. (Yes, Virginia, you can outrun a light-ray by doing constant accelerating forever. All you need is an infinite array of friends to supply you with propulsion forever.)

In fact, everything happening in the "After +∞" region is all "beyond the future" and remains forevermore unknown to the residents of the Tower, though the After People can see all of the Tower happenings just fine.

Conversely, up on the 100th floor, I can peer down and view at least some of the prior history of the tree or other events in the "Before −∞" region, depending on how early I wake up, but none of the Before people will have any idea that the Tower even exists.

And the "Nope" region is completely inaccessible; none of those folks know about the Tower, nor can they be seen by the Tower. We could actually put a whole 'nother fleet of downward accelerating ships here, an Anti-Tower, if you like, but that would be a different coordinate system. The only way to reconcile the two would be to have one of them see the other as moving backwards in time. But it doesn't matter since they can't talk to each other anyway. (suffice it to say, things get super, super, wacky if we have any kind of FTL travel/communication available here).

To summarize, the upwards quadrant from Cherry Tree Explosion, bounded by the downwards lightray to it and the upwards lightray from it, is the largest extent over which the accelerated coordinate system makes any sense.

And yet the idea that there is any weird Infinite Stuff happening near the Cherry Tree is totally an illusion. The only way you get crushed to death is if you try to get on any of the accelerating ships down there, in which case it'll be the accelerating ships doing the crushing. The spacetime there is completely ordinary as switching to the more sensible Stationary People coordinates will demonstrate. (This kind of situation, is called a coordinate singularity, i.e., when the problem is a stupid coordinate grid as opposed to an actual singularity like you get at the center of a black hole, where there really is Infinite Stuff happening).

A surprisingly ineffective method for committing suicide

For Extra Special Fun, watch what happens if I jump out the window of my 100th floor apartment. I suddenly become weightless (I'm falling, of course). The Stationary and various Moving Peoples will see me retain whatever velocity I had when I jumped, and that's that. Over the next year, I see the lower floors continue accelerating upwards and, one by one, they pass me by, faster and faster.

But from the Tower residents' point of view I am falling past them, going faster and faster. Their last sight of me will be right before I hit the lobby, where I presumably get crushed to death by the infinite gravity. But this takes forever; they never actually see me hit, and what they do see gets insanely redshifted.

The moment I cross through that upward light ray from the Cherry Tree Explosion, which, for me, happens a year after I first jumped, I completely disappear from their universe. I can still see the Tower receding above me; any messages they beam downwards I will eventually get, but it's a decidedly one-way conversation.

That light ray is an event horizon. Once I cross through, I can never return, and they, in turn, will never have any idea what happened to me.

And if the description of how it appears to the folks in the tower while I'm crossing through looks an awful lot like what happens when distant observers see someone crossing the event horizon of a black hole, this is not a coincidence. It's the same math, mostly.

The one big difference here is there is no curvatuve.

In particular, there is no instant-death singularity taking up my entire future after I've crossed through. It's all just flat, empty space where I can hang out for as long as I like before arranging for Jean-Luc or whoever to rescue me so we can go have a beer…

…which I think we desperately need at this point.