Spanreed Limitations

[ccstat]

Okay, so here is the thought experiment:

Shallan and Jasnah have paired spanreeds. Shallan is sitting in one place, and Jasnah can Elsecall to anywhere on Roshar before turning on her spanreed. Where on the continent can she go and still have the spanreeds write to each other?

 

The main question I'm asking here is how the relative rotational velocity of the planet at different locations will affect spanreed function. I find it most likely that the surge powering spanreeds' shared momentum operates spiritually in a way perceived to users as "moving relative to the ground under my feet." But what if it is actually based on absolute position in space? I'm a biologist but I enjoy physics, especially in the vein of XKCD's what-if articles. So here's my analysis. Did I miss anything important?

 

Short answer: Position-in-space referencing doesn't work at all. Planets are big and fast.

 

Spoilering for length:

Groundwork and assumptions:

• There is a distance limitation mentioned in the books, but clearly direct communication is possible between Kharbranth and both the Shattered Plains and the Davar home, to use only WoK instances. More appear in WoR. 
• Spanreeds don't work on boats. I presume this is because the movement of the boat dominates over any writing a passenger might attempt. (I turn on my receiving spanreed and it starts bobbing with the waves or moving through the house at several knots.)
• Roshar's day is about 19.3 earth-hours long, so it is spinning faster. But, it is also less massive and (presumably) smaller, so the linear velocity of Roshar's surface is probably comparable to Earth.
• The equator moves faster than the poles. Specifically, the speed of a point on the surface of Earth can be calculated in m/s as v=459*cos(latitude). 
• Points on opposite sides of the globe are necessarily moving in opposite directions. A point at the same latitude halfway between them is moving perpendicular to both. If we take θ=longitude_A-longitude_B, the relative motion of two points at the same latitude can be calculated with 2v*sin(θ/2). The angle of that motion relative to point A is θ/2, with an answer of 0° meaning down, 90° meaning West, and 180° meaning up.
• (The trig of how to combine different latitude and longitude is a bit more complicated, and I won't try right now. It took me a ridiculously long time to remember how to use trig identities to get to the above expressions).
• Veil put together a Roshar map with approximate lat/long lines. (Maresia also made a globe image based on slightly different assumptions.) Roshar appears to cover about 140° of longitude and 80° of latitude.

Calculations/Interpretation

• Potential latitude differences: Kharbranth very far south. It is ~80°S, giving a lateral velocity of 80 m/s. We don't know where house Davar is located,  but most of Jah Keved is about the same longitude as Kharbranth. It is reasonable to assume House Davar is between Northgrip (50°S) and Valath (65°S), but it could be closer to Vedenar (75°S). The respective velocities of those three cities are roughly 295 m/s, 195 m/s,  and 120 m/s. So, for a spanreed in any of those cities to communicate with each other, it would have to compete with a minimum of 40 m/s difference in lateral velocity.  That's 90mph=145km/h.
• Tolerable latitude difference: work backwards from the idea that you could tolerate a difference up to 2 m/s, which is a brisk walk (a very generous assumption). Speed changes slowest with latitude close to the equator (~1m/s per degree), but according to Veil's map (linked above) the northernmost point in Roshar is between 25-30°S. At 25°S, a change of 1° latitude is a change of 3.5 m/s. On Earth, 1° is ~110km; Roshar will be smaller, but probably not by too much. So even in the best case, you can only send spanreed messages up to ~50 km north or south of your current position.
• Potential longitude differences: Kharbranth and the Shallow Crypts are close to the same latitude (80°S) but are a bit over 45° of longitude apart. Even with the same lateral speed of 80 m/s at 80°S, they still have a dramatic 61 m/s difference in velocity, since they are moving in different directions. The spanreed in Kharbranth would end up pushing hard against the corner between the floor and the room's western wall, while its pair would be trying to escape through the floor/eastern wall of it its own room.
• Tolerable longitude differences: With the same 2 m/s allowance as above, we can calculate the allowed difference in longitude: 1.5°E/W @ 80°S; or 0.5°E/W @ 50°S; or 0.3°E/W @25°S. Those degrees of longitude at each point correspond to between 25-35km of surface distance. So even in the best case, you can only send spanreed messages up to 35km east or west of your current position. Predictably, at those distances the direction of force between the two spanreeds is almost directly E/W with no appreciable up/down component. 

Long story short, we know that spanreeds work at much greater distances than allowed in this model. Therefore, the spiritual "I'm moving this much, in this direction" model of spanreed function is much better than any position-in-space explanation.

 

Implications: spanreed-style conjoined fabrials will not be able to exploit Roshar's rotational velocity for energy harvesting, mining, or Roshar's emerging space program. They will not send anyone into orbit. (The amethyst type have incredible possibilities for untethered space elevators, etc, but those will likewise be based on a spiritual concept of motion and may be more difficult to model accurately without some a great many empirical observations.)

[Seloun]

The main issue is that you're giving one of the spanreeds a privileged frame of reference. In particular you are assuming that one of the spanreeds somehow assumes the same momentum as the other spanreed, but there's no reason to expect one of the spanreeds is distinguishable from the other. So what must be occurring is that momentum is being shared between the spanreeds. Initially this probably isn't a problem (the straight-line distance from the spanreeds remain constant despite the earth's rotation) as long as the connection is somehow compliant against rotation.

 

That said, the main problem is that of orientation, which breaks conservation of momentum all to heck without considering anything else (alternatively, the spanreeds would behave very strangely, essentially as a gyroscope; trying to move it e.g. left would cause it to generally try to drift in another direction as well, as the other spanreed tries to translate the change in linear momentum from its frame of reference).

[signspace13]

I had an interesting theory last night. The Parshendi use the cognitive realm to hear the music right, and that how they show emotion, by speaking by a different kind of song. Well i find it very likely that spanreeds use the cognitive realm to send their messages, what if a Parshendi was standing between two spanreeds? could he intercept the message, this would be a very interesting plot point, to be discovered by either Rlain or Eshonai.

[kari-no-sugata]

I think the simplest solution to the problem is that there's actually another "active" part: the holder for the spanreed acts as the frame of reference and all actions are relative to it. Otherwise, all spanreed boards would have to be orientated exactly the same even if they're very close by.

 

Or is there some indication that all spanreed boards do in fact need to be aligned in the same way?

[Stormfather-in-Law]

Or is there some indication that all spanreed boards do in fact need to be aligned in the same way?

Sanderson, Brandon (2010-08-31). The Way of Kings (p. 422). Tor Books. Kindle Edition:

Danlan turned the spanreed’s gemstone one notch, indicating that the request had been acknowledged. Then she checked the levels on the sides of the writing board—small vials of oil with bubbles at the center, which allowed her to make the board perfectly flat.

 

She has to level the board. I always wondered why, given that they were on other sides of the world (the thought of calculating angular velocity or whatnot like the OP is waaaay beyond). I would say in conclusion that this indicates Roshar itself is also flat.

[Swimmingly]

I would imagine that spanreed movement is calculated as compared to the center of Roshar. This should actually cause transfer of information from lower to higher altitudes to be enlarged slightly, but only a tiny amount as long as everyone is on the surface of Roshar, and it would scale, so you wouldn't be able to tell without examining a large sample of messages sent across altitude.