4: Space & Time

General issues:

  • What is a frame of reference, exactly? If it's manri1, then what's the rest of {manri}? Are there default frames of reference?

Space

2D Euclidean

The flat plane should be addressible with two dimensions. We have four words:

  • berti: north
  • snanu: south
  • stici: west
  • stuna: east

These words all have a nice Euclid-like property; they are all defined as something like "x1 is translated in [the opposite of] one of the basis vectors relative to x2 in frame of reference x3", where a frame of reference is something like an origin and choice of three axes forming a unit basis.

These words are all relative:

da berti de di <=> de snanu da di
da stici de di <=> de stuna da di

There is a note that {berti} should accord with right-hand rules; given a 3D space and a vector, we should use the right-hand rule to address the 2D bivector which spans the obvious subspace (that is, the flat plane which is perpendicular/tangent to the given vector.)

These words should generalize to any locally-Euclidean manifold, including physical approximations. The obvious desire is to have them work on Earth, where a frame of reference is a choice of (north) pole, and the right-hand rule then gives the traditional meanings of "north", etc.

Other Words

  • The cmavo: be'a, ne'u, du'a, vu'a

3D Euclidean

Similarly, flat space should be addressible with three dimensions. We should have six words:

  • crane: forward
  • trixe: backward
  • zunle: leftward
  • pritu: rightward
  • gapru: upward
  • cnita: downward

With axioms of relativity:

da crane de di <=> de trixe da di
da zunle de di <=> de pritu da di
da gapru de di <=> de cnita da di

Other Words

  • The cmavo: ca'u, ti'a, zu'a, ri'u, ga'u, ni'a
  • dizlo?

Spacetime

Spacetime might have to be defined with a compound of some sort.

There are four tenses. They have no frames of reference; instead, they are defined using relativity, by considering regions of Minkowski spacetime which are (naturally) invariant (up to Lorentz transformations between inertial viewers). This invariance means that we don't have "x1 is temporally related to x2 according to frame of reference x3", but instead "x1 is spatiotemporally related to x2 (regardless of frame of reference)".

  • balvi: futureward
  • purci: pastward
  • cabna: here and now
  • xlane: elseward

We'll construct events in a moment, but first consider points. If we only acted on points, then the causal structure implies that either one point causally occurs before the other, or the two points are clearly disjoint, or the points are nearby each other and possibly equivalent or equal depending on topological axioms. We'd like for this to be a strict disjunction, and it is — for points. We can at least note that spacetime diagrams can be turned upside down:

da balvi de <=> de purci da

Lorentz transformations preserve inaccessibility:

da xlane de <=> de xlane da

Equivalence is symmetric:

da cabna de <=> de cabna da

Finally, physical causation is always of the spatiotemporal variety:

da rinka de ... => da balvi de

Events

So, now let's crack open the table of event abstractors. We'll start with very simple cases: connected regions of spacetime with boundaries. Such regions can overlap and also be large enough to span multiple portions of a spacetime diagram, which complicates our reasoning. We have some hints from baseline definitions; {balvi} and {purci} both indicate that they are aorist in that they do not exclude each other, and {cabna} does not exclude either of them either.

But now we get into nuanced questions. Suppose {da balvi de} and {da purci de}. Does it follow that {da cabna de}? Well, no. But if {da} is connected, then either {da cabna de} or {da xlane de} (or both), because there has to be some path from past to future, either through the present or elsewhere.

Open Questions

But what is rinka3? Conditions?

CLL 10.7 says VIhA series {vi'i}, {vi'a}, {vi'u}, {vi'e} correspond to "cognitive" or "essential dimensionality", and specifically it sounds like a generalization of manifolds and projections. A {vi'i} action is 1D, but it occurs in 3+1D spacetime; {vi'i} indicates that the data of the action is linear. Even curves (CLL gives walking on a mountain) can still have low-dimensional data.

CLL 10.7 says that FAhA could be augmented with "pastward" and "futureward" directions.

Fourth tense PU is {xa'ei}, VA is {xa'e}, SELBRI is {xlane}, all experimental.

{ta'e} requires agentivity. The other three can be explained in terms of the event covering its spacetime interval.

CLL 10.23 clearly enunciates two sets of equivalent bridi families. The first are using BAI tags:

X .i BAI bo Y <=> BAI gi Y gi X <=> X BAI le nu Y

The other, spelled out in CLL 10.16, is used for TENSE, which are PU, ZI, FAhA, VA, or ZAhO. In all of those cases:

Y .i TENSE bo X <=> TENSE gi X gi Y <=> X TENSE le nu Y

I think that, in both cases, the final form {X do'e le nu Y} is most primitive. This implies that {le nu} or similar is primitive.

But also, CLL 10.17 uses {.i ... bo} to glue logical connectives to BAI and TENSE, with e.g. {.ibabo} akin to {.ijebo}. So, uh?

Story Time

Conversationally, Lojban bridi refer to events. The story-time convention connects those events spatiotemporally. The utterance {broda .i brode} implies something like {lo nu broda ku purci lo nu brode}.