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Special relativity and Middle-Earth

18 December, 2022 in expository, math.MP | Tags: special relativity

This post is an unofficial sequel to one of my first blog posts from
2007, which was entitled "Quantum mechanics and Tomb Raider".

One of the oldest and most famous allegories is Plato's allegory of
the cave. This allegory centers around a group of people chained to a
wall in a cave that cannot see themselves or each other, but only the
two-dimensional shadows of themselves cast on the wall in front of
them by some light source they cannot directly see. Because of this,
they identify reality with this two-dimensional representation, and
have significant conceptual difficulties in trying to view themselves
(or the world as a whole) as three-dimensional, until they are freed
from the cave and able to venture into the sunlight.

There is a similar conceptual difficulty when trying to understand
Einstein's theory of special relativity (and more so for general
relativity, but let us focus on special relativity for now). We are
very much accustomed to thinking of reality as a three-dimensional
space endowed with a Euclidean geometry that we traverse through in
time, but in order to have the clearest view of the universe of
special relativity it is better to think of reality instead as a
four-dimensional spacetime that is endowed instead with a Minkowski
geometry, which mathematically is similar to a (four-dimensional)
Euclidean space but with a crucial change of sign in the underlying
metric. Indeed, whereas the distance {ds} between two points in
Euclidean space {{\bf R}^3} is given by the three-dimensional
Pythagorean theorem

               \displaystyle ds^2 = dx^2 + dy^2 + dz^2

under some standard Cartesian coordinate system {(x,y,z)} of that
space, and the distance {ds} in a four-dimensional Euclidean space
{{\bf R}^4} would be similarly given by

           \displaystyle ds^2 = dx^2 + dy^2 + dz^2 + du^2

under a standard four-dimensional Cartesian coordinate system
{(x,y,z,u)}, the spacetime interval {ds} in Minkowski space is given
by

         \displaystyle ds^2 = dx^2 + dy^2 + dz^2 - c^2 dt^2

(though in many texts the opposite sign convention {ds^2 = -dx^2 -dy^
2 - dz^2 + c^2dt^2} is preferred) in spacetime coordinates
{(x,y,z,t)}, where {c} is the speed of light. The geometry of
Minkowski space is then quite similar algebraically to the geometry
of Euclidean space (with the sign change replacing the traditional
trigonometric functions {\sin, \cos, \tan}, etc. by their hyperbolic
counterparts {\sinh, \cosh, \tanh}, and with various factors
involving "{c}" inserted in the formulae), but also has some
qualitative differences to Euclidean space, most notably a causality
structure connected to light cones that has no obvious counterpart in
Euclidean space.

That said, the analogy between Minkowski space and four-dimensional
Euclidean space is strong enough that it serves as a useful
conceptual aid when first learning special relativity; for instance
the excellent introductory text "Spacetime physics" by Taylor and
Wheeler very much adopts this view. On the other hand, this analogy
doesn't directly address the conceptual problem mentioned earlier of
viewing reality as a four-dimensional spacetime in the first place,
rather than as a three-dimensional space that objects move around in
as time progresses. Of course, part of the issue is that we aren't
good at directly visualizing four dimensions in the first place. This
latter problem can at least be easily addressed by removing one or
two spatial dimensions from this framework - and indeed many
relativity texts start with the simplified setting of only having one
spatial dimension, so that spacetime becomes two-dimensional and can
be depicted with relative ease by spacetime diagrams - but still
there is conceptual resistance to the idea of treating time as
another spatial dimension, since we clearly cannot "move around" in
time as freely as we can in space, nor do we seem able to easily
"rotate" between the spatial and temporal axes, the way that we can
between the three coordinate axes of Euclidean space.

With this in mind, I thought it might be worth attempting a
Plato-type allegory to reconcile the spatial and spacetime views of
reality, in a way that can be used to describe (analogues of) some of
the less intuitive features of relativity, such as time dilation,
length contraction, and the relativity of simultaneity. I have
(somewhat whimsically) decided to place this allegory in a
Tolkienesque fantasy world (similarly to how my previous allegory to
describe quantum mechanics was phrased in a world based on the
computer game "Tomb Raider"). This is something of an experiment, and
(like any other analogy) the allegory will not be able to perfectly
capture every aspect of the phenomenon it is trying to represent, so
any feedback to improve the allegory would be appreciated.

                           -- 1. Treefolk --

Tolkien's Middle-Earth contains, in addition to humans, many
fantastical creatures. Tolkien's book "The Hobbit" introduces the
trolls, who can move around freely at night but become petrified into
stone during the day; and his book "The Two Towers" (the second of
his three-volume work "The Lord of the Rings") introduces the Ents,
who are large walking sentient tree-like creatures.

In this Tolkienesque fantasy world of our allegory (readers, by the
way, are welcome to suggest a name for this world), there are two
intelligent species. On the one hand one has the humans, who can move
around during the day much as humans in our world do, but must sleep
at night without exception (one can invent whatever reason one likes
for this, but it is not relevant to the rest of the allegory). On the
other hand, inspired by the trolls and Ents of Tolkien, in this world
we will have the treefolk, who in this world are intelligent
creatures resembling a tree trunk (possibly with some additional
branches or additional appendages, but these will not play a central
role in the allegory). They are rooted to a fixed location in space,
but during the night they have some limited ability to (slowly) twist
their trunk around. On the other hand, during the day, they turn into
non-sentient stone columns, frozen in whatever shape they last
twisted themselves into. Thus the humans never see the treefolk
during their active period, and vice versa; but we will assume that
they are still somehow able to communicate asynchronously with each
other through a common written language (more on this later).

    Remark 1 In Middle-Earth there are also the Huorns, who are
    briefly mentioned in "The Two Towers" as intelligent trees kin to
    the Ents, but are not described in much detail. Being something
    of a blank slate, these would have been a convenient name to give
    these fantasy creatures; however, given that the works of Tolkien
    will not be public domain for a few more decades, I'll refrain
    from using the Huorns explicitly, and instead use the more
    generic term "treefolk".

When a treefolk makes its trunk vertical (or at least straight), it
is roughly cylindrical in shape, and has horizontal "rings" on its
exterior at intervals of precisely one inch apart; so for instance
one can easily calculate the height of a treefolk in inches by
counting how many rings it has. One could think of a treefolk's trunk
geometrically as a sequence of horizontal disks stacked on top of
each other, with each disk being an inch in height and basically of
constant radius horizontally, and separated by the aforementioned
rings. Because my artistic abilities are close to non-existent, I
will draw a treefolk schematically (and two-dimensionally), as a
vertical rectangle, with the rings drawn as horizontal lines (and the
disks being the thin horizontal rectangles between the rings):

[tree]

But treefolks can tilt their trunk at an angle; for instance, if a
treefolk tilts its trunk to be at a 30 degree angle from the
vertical, then now the top of each ring is only {\cos 30^\circ = \
frac{\sqrt{3}}{2} \approx 0.866} inches higher than the top of the
preceding ring, rather than a full inch higher, though it is also
displaced in space by a distance of {\sin 30^\circ = \frac{1}{2}}
inches, all in accordance with the laws of trigonometry. It is also
possible for treefolks to (slowly) twist their trunk into more
crooked shapes, for instance in the picture below the treefolk has
its trunk vertical in its bottom half, but at a {30^\circ} angle in
its top half. (This will necessarily cause some compression or
stretching of the rings at the turnaround point, so that those rings
might no longer be exactly one inch apart; we will ignore this issue
as we will only be analyzing the treefolk's rings at "inertial"
locations where the trunk is locally straight and it is possible for
the rings to stay perfectly "rigid".)

[tilted]

Let us also say that there is a hard limit of {45^\circ} as to how
far a treefolk can tilt its trunk; thus for instance it is not
possible for a treefolk to place its trunk at a 60 degree angle from
the vertical. (This is analogous to how matter is not able to travel
faster than the speed of light in our world.) In fact, treefolks
prefer to stay very close to being vertical, and only tilt at
significant deviations from the vertical in rare circumstances.

Now we turn to the nature of the treefolk's sentience, which is
rather unusual. Namely - only one disk of the treefolk is conscious
at any given time! As soon as the sun sets, a treefolk returns from
stone to a living creature, and the lowest disk of that treefolk
awakens and is able to sense its environment, as well as move the
trunk above it. However, every minute, with the regularity of
clockwork, the treefolk's consciousness and memories transfer
themselves to the next higher disk; the previous disk becomes
petrifed into stone and no longer mobile or receiving sensory input
(somewhat analogous to the rare human disease of fibrodysplasia
ossificans progressiva, in which the body becomes increasingly
ossified and unable to move). As the night progresses, the locus of
the treefolk's consciousness moves steadily upwards and more and more
of the treefolk turns to stone, until it reaches the end of its
trunk, at which point the treefolk turns completely into a stone
column until the next night, at which point the process starts again.
(In particular, no treefolk has ever been tall enough to retain its
consciousness all the way to the next sunrise.) Treefolk are aware of
this process, and in particular can count intervals of time by
keeping track of how many times its consciousness has had to jump
from one disk to the next; they use rings as a measure of time. For
instance, if a treefolk experiences ten shifts of consciousness
between one event and the next, the treefolk will know that ten
minutes have elapsed between the two events; in their language, they
would say that the second event occurred ten rings after the first.

The second unusual feature of the treefolk's sentience is that at any
given time, the treefolk can sense the portions of all nearby objects
that are in the same plane as the disk, but not portions that are
above or below this plane; in particular, some objects may be
completely "invisible" to the treefolk of they are completely above
or completely below the treefolk's current plane of "vision". Exactly
how the treefolk senses its environment is not of central importance,
but one could imagine either some sort of visual organ on each disk
that is activated during the minute in which that disk is conscious,
but which has a limited field of view (similar one that a knight
might experience when wearing a helmet with only a narrow horizontal
slit in their visor to see through), or perhaps some sort of
horizontal echolocation ability. (Or, since we are in a fantasy
setting, we can simply attribute this sensory ability to "magic".)
For instance, the picture below that (very crudely) depicts a
treefolk standing vertically in an environment, fifty minutes after
it first awakens, so that the disk that is fifty inches off the
ground is currently sentient. The treefolk can sense any other object
that is also fifty inches from the ground; for instance, it can "see"
a slice of a bush to the left, and a slice of a boulder to the right,
but cannot see the sign at all. (Let's assume that this somewhat
magical "vision" can penetrate through objects to some extent (much
as "x-ray vision" would work in comic books), so it can get some idea
for instance that the section of boulder it sees is somewhat wider
than the slice of bush that it sees.) As the minutes pass and the
treefolk's consciousness moves to higher and higher rungs, the bush
will fluctuate in size and then disappear from the treefolk's point
of "view", and the boulder will also gradually shrink in size until
disappearing several rings after the bush disappeared.

[vision]

If the treefolk's trunk is tilted at an angle, then its visual plane
of view tilts similarly, and so the objects that it can see, and
their relative positions and sizes, change somewhat. For instance, in
the picture below, the bush, boulder, and sign remain in the same
location, but the treefolk's trunk has tilted; as such, it now senses
a small slice of the sign (that will shortly disappear), and a (now
smaller) slice of the boulder (that will grow for a couple rings
before ultimately shrinking away to nothingness), but the bush has
already vanished from view several rings previously.

[tilted-vision-1]

At any given time, the treefolk only senses a two-dimensional slice
of its surroundings, much like how the prisoners in Plato's cave only
see the two-dimensional shadows on the cave wall. As such, treefolks
do not view the world around them as three-dimensional; to them, it
is a two-dimensional world that slowly changes once every ring even
if the three-dimensional world is completely static, similarly to how
flipping the pages of an otherwise static flip book can give the
illusion of movement. In particular, they do not have a concept in
their language for "height", but only for horizontal notions of
spatial measurement, such as width; for instance, if a tall treefolk
is next to a shorter treefolk that is 100 inches tall, with both
treefolk vertical, it will think of that shorter treefolk as "living
for 100 rings" rather than being 100 inches in height, since from the
tall treefolk's perspective, the shorter treefolk would be visible
for 100 rings, and then disappear. These treefolk would also see that
their rings line up: every time a ring passes for one treefolk, the
portion of the other treefolk that is in view also advances by one
ring. So treefolk, who usually stay close to vertical for most of
their lives, have come to view rings as being universal measurements
of time. They also do not view themselves as three-dimensional
objects; somewhat like the characters in Edwin Abbott classic book "
Flatland", they think of themselves as two-dimensional disks, with
each ring slightly changing the nature of that disk, much as humans
feel their bodies changing slightly with each birthday. While they
can twist the portion of their trunk above their currently conscious
disk at various angles, they do not think of this twisting in
three-dimensional terms; they think of it as willing their
two-dimensional disk-shaped self into motion in a horizontal
direction of their choosing.

Treefolk cannot communicate directly with other treefolk (and in
particular one treefolk is not aware of which ring of another
treefolk is currently conscious); but they can modify the appearance
of their exterior on their currently conscious ring (or on rings
above that ring, but not on the petrified rings below) for other
treefolk to read. Two treefolks standing vertically side by side will
then be able to communicate with each other by a kind of transient
text messaging system, since they awaken at the same time, and at any
given later moment, their conscious rings will be at the same height
and each treefolk be able to read the messages that the other
treefolk leaves for them, although a message that one treefolk leaves
for another for one ring will vanish when these treefolk both shift
their consciousnesses to the next ring. A human coming across these
treefolks the following day would be able to view these messages
(similar to how one can review a chat log in a text messaging app,
though with the oldest messages at the bottom); they could also leave
messages for the treefolk by placing text on some sort of sign that
the treefolk can then read one line at a time (from bottom to top) on
a subsequent night as their consciousness ascends through its rings.
(Here we will assume that at some point in the past the humans have
somehow learned the treefolk's written language.) But from the point
of view of the treefolk, their messages seem as impermanent to them
as spoken words are to us: they last for a minute and then they are
gone.

[godot]

             -- 2. Time contraction and width dilation --

In recent years, treefolk scientists (or scholars/sages/wise ones, if
one wishes to adhere as much as possible to the fantasy setting),
studying the effect of significant tilting on other treefolk,
discovered a strange phenomenon which they might term "time
contraction" (similar to time dilation in special relativity, but
with the opposite sign): if a treefolk test subject tilts at a
significant angle, then it begins to "age" more rapidly in the sense
that test subject will be seen to pass by more rings than the
observer treefolk that remains vertical. For instance, with the test
subject tilted at a {30^\circ} angle, as 100 rings pass by for the
vertical observer, {100 / \cos 30^\circ \approx 115} rings can be
counted on the tilted treefolk. This is obvious to human observers,
who can readily explain the situation when they come across it during
the day, in terms of trigonometry:

[length-2]

This leads to the following "twin paradox": if two identical treefolk
awaken at the same time, but one stays vertical while the other tilts
away and then returns, then when they rejoin their rings will become
out of sync, with the twisted treefolk being conscious at a given
height several minutes after the vertical treefolk was conscious at
that height. As such, communication now comes with a lag: a message
left by the vertical treefolk at a given ring will take several
minutes to be seen by the twisted treefolk, and the twisted treefolk
would similarly have to leave its messages on a higher ring than it
is currently conscious at in order to be seen by the vertical
treefolk. Again, a human who comes across this situation in the day
can readily explain the phenomenon geometrically, as the twisted
treefolk takes longer (in terms of rings) to reach the same location
as the vertical treefolk):

[twin-1]

These treefolk scientists also observe a companion to the time
contraction phenomenon, namely that of width dilation (the analogue
of length contraction; a treefolk who is tilted at an angle will be
seen by other (vertical) treefolk observers as having their shape
distorted from a disk to an ellipse, with the width in the direction
of the tilt being elongated (much like the slices of a carrot become
longer and less circular when sliced diagonally). For instance, in
the picture at the beginning of this section, the width of the tilted
treefolk has increased by a factor of {1 / \cos 30^\circ \approx
1.15}, or about fifteen percent. Once again, this is a phenomenon
that humans, with their ability to visualize horizontal and vertical
dimensions simultaneously, can readily explain via trigonometry
(suppressing the rings on the tilted treefolk to reduce clutter):

[width]

The treefolk scientists were able to measure these effects more
quantitatively. As they cannot directly sense any non-horizontal
notions of space, they cannot directly compute the angle at which a
given treefolk deviates from the vertical; but they can measure how
much a treefolk "moves" in their two-dimensional plane of vision.
Let's say that the humans use the metric system of length measurement
and have taught it (through some well-placed horizontal rulers
perhaps) to the treefolk, who are able to use this system to measure
horizontal displacements in units of centimeters. (They are unable to
directly observe the inch-long height of their rings, as that is a
purely vertical measurement, and so cannot use inches to directly
measure horizontal displacements.) A treefolk that is tilted at an
angle will then be seen to be "moving" at some number of centimeters
per ring; with each ring that the vertical observer passes through,
the tilted treefolk would appear to have shifted its position by that
number of centimeters. After many experiments, the treefolk
scientists eventually hit upon the following empirical law: if a
treefolk is "moving" at {v} centimeters per ring, then it will
experience a time contraction of {\sqrt{1+\frac{v^2}{c^2}}} and a
width dilation of {\sqrt{1+\frac{v^2}{c^2}}}, where {c} is a physical
constant that they compute to be about {2.54} centimeters per ring.
(Compare with special relativity, in which an object moving at {v}
meters per second experiences a time dilation of {1/\sqrt{1-\frac{v^
2}{c^2}}} and a length contraction of {1/\sqrt{1-\frac{v^2}{c^2}}},
where the physical constant {c} is now about {3.0 \times 10^8} meters
per second.) However, they are unable to come up with a satisfactory
explanation for this arbitrary-seeming law; it bears some resemblance
to the Pythagorean theorem, which they would be familiar with from
horizontal plane geometry, but until they view rings as a third
spatial dimension rather than as a unit of time, they would struggle
to describe this empirically observed time contraction and width
dilation in purely geometric terms. But again, the analysis is simple
to a human observer, who notices that the tilted treefolk is
spatially displaced by {tv} centimeters whenever the vertical tree
advances by {t} rings (or inches), at which point the computation is
straightforward from Pythagoras (and the mysterious constant {t} is
explained as being the number of centimeters in an inch):

[velocity-1]

At some point, these scientists might discover (either through actual
experiment, or thought-experiment) what we would call the principle
of relativity: the laws of geometry for a tilted treefolk are
identical to that of a vertical treefolk. For instance, as mentioned
previously, if a tilted treefolk appears to be moving at {v}
centimeters per second from the vantage point of a vertical treefolk,
then the vertical treefolk will observe the tilted treefolk as
experiencing a time contraction of {\sqrt{1+\frac{v^2}{c^2}}} and a
width dilation of {\sqrt{1+\frac{v^2}{c^2}}}, but from the tilted
treefolk's point of view, it is the vertical treefolk which is moving
at {v} centimeters per second (in the opposite direction), and it
will be the vertical treefolk that experiences the time contraction
of {\sqrt{1+\frac{v^2}{c^2}}} and width dilation of {\sqrt{1+\frac{v^
2}{c^2}}}. In particular, both treefolk will think that the other one
is aging more rapidly, as each treefolk will see slightly more than
one ring of the other pass by every time they pass a ring of their
own. However, this is not a paradox, due to the relativity of
horizontality (the analogue in this allegory to relativity of
simultaneity in special relativity); two locations in space that are
simultaneously visible to one treefolk (due to them lying on the same
plane as one of the disks of that treefolk) need not be
simultaneously visible to the other, if they are tilted at different
angles. Again, this would be obvious to humans who can see the
higher-dimensional picture: compare the planes of sight of the tilted
treefolk in the figure below with the planes of sight of the vertical
treefolk as depicted in the first figure of this section.

[length-flipped]

Given two locations in {A} and {B} in (three-dimensional space),
therefore, one treefolk may view the second location {B} as displaced
in space from the first location {A} by {dx} centimeters in one
direction (say east-west) and {dy} centimeters in an orthogonal
direction (say north-south), while also being displaced by time by
{dt} rings; but a treefolk tilted at a different angle may come up
with different measures {dx', dy'} of the spatial displacement as
well as a different measure {dt'} of the ring displacement, due to
the effects of time contraction, width dilation, non-relativity of
horizontality, and the relative "motion" between the two treefolk.
However, to an external human observer, it is clear from two
applications of Pythagoras's theorem that there is an invariant

\displaystyle dx^2 + dy^2 + c^2 dt^2 = (dx')^2 + (dy')^2 + c^2 (dt')^
                                 2:

See the figure below, where the {y} dimension has been suppressed for
simplicity.

[location]

From the principle of relativity, this invariance strongly suggests
the laws of geometry should be invariant under transformations that
preserve the interval {dx^2 + dy^2 + c^2 dt^2}. Humans would refer to
such transformations as three-dimensional rigid motions, and the
invariance of geometry under these motions would be an obvious fact
to them; but it would be a highly unintuitive hypothesis for a
treefolk used to viewing their environment as two dimensional space
evolving one ring at a time.

Humans could also explain to the treefolk that their calculations
would be simplified if they used the same unit of measurement for
both horizontal length and vertical length, for instance using the
inch to measure horizontal distances as well as the vertical height
of their rings. This would normalize {c} to be one, and is somewhat
analogous to the use of Planck units in physics.

                 -- 3. The analogy with relativity --

In this allegory, the treefolk are extremely limited in their ability
to sense and interact with their environment, in comparison to the
humans who can move (and look) rather freely in all three spatial
dimensions, and who can easily explain the empirical scientific
efforts of the treefolk to understand their environment in terms of
three-dimensional geometry. But in the real four-dimensional
spacetime that we live in, it is us who are the treefolk; we inhabit
a worldline tracing through this spacetime, similar to the trunk of a
treefolk, but at any given moment our consciousness only occupies a
slice of that worldline, transferred from one slice to the next as we
pass from moment to moment; the slices that we have already
experienced are frozen in place, and it is only the present and
future slices that we have some ability to still control. Thus, we
experience the world as a three-dimensional body moving in time, as
opposed to a "static" four-dimensional object. We can still map out
these experiences in terms of four-dimensional spacetime diagrams (or
diagrams in fewer dimensions, if we are able to omit some spatial
directions for simplicity); this is analogous to how the humans in
this world are easily able to map out the experiences of these
treefolk using three-dimensional spatial diagrams (or the
two-dimensional versions of them depicted here in which we suppress
one of the two horizontal dimensions for simplicity). Even so, it
takes a non-trivial amount of conceptual effort to identify these
diagrams with reality, since we are so accustomed to the dynamic
three-dimensional perspective. But one can try to adopt the perhaps
this allegory can help in some cases to make this conceptual leap,
and be able to think more like humans than like treefolk.

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8 comments

Comments feed for this article

18 December, 2022 at 8:44 pm

flyingtext

[fed]

Good and thought provoking article. :)

Reply

19 December, 2022 at 12:26 am

Tobias Fritz

[d7c]

Is there a good way to design the internal physics of this world such
that the maximal treefolk trunk tilt of 45deg is explained naturally?
Since purely from the perspective of the Riemannian geometry, it
would seem more natural to forego this analogy with special
relativity and allow the trunks to tilt arbitrarily much, even
downwards. That gives a nice setting to contemplate time travel
paradoxes. This kind of world features in Greg Egan's series of
novels [Orthogonal](https://www.gregegan.net/ORTHOGONAL/00/PM.html).
His website explains the physics of a Riemannian block universe in a
lot of detail, and the time contraction phenomenon even plays a
central role in the plot: "Only one solution seems tenable: if a
spacecraft can be sent on a journey at sufficiently high speed, its
trip will last many generations for those on board, but it will
return after just a few years have passed at home. The travellers
will have a chance to discover the science their planet urgently
needs, and bring it back in time to avert disaster."

In any case, I think I will have treefolk in mind now every time I
draw worldlines in a spacetime diagram :)

Reply

   
    19 December, 2022 at 11:12 am

    Terence Tao

    [3c7]

    Hmm, you're right that the 45 degree tilt has no particular
    significance in this Euclidean spacetime, as opposed to Minkowski
    spacetime where this tilt is an invariant of the geometry (the
    second postulate of special relativity). Geometrically it comes
    down to the fact that the hyperbola/hyperboloid is a conic
    section with asymptotes, whereas the circle/sphere is a conic
    section without asymptotes. So there is a tension here in trying
    to make the allegory faithful to Euclidean spacetime geometry and
    making it replicate the features of Minkowski spacetime. As you
    say though, if one doesn't impose a limit on the tilt then the
    geometry can become quite acausal and it becomes less valuable as
    a source of intuition, so this is the best compromise I could
    think of to reconcile the two competing aims here.

    Reply

19 December, 2022 at 5:58 am

Fabian

[]

I think your allegory perfectly explains what spacetime is, but is
there any explanation as to why that is the case?

Reply

   
    19 December, 2022 at 11:16 am

    Terence Tao

    [3c7]

    I think it's still an open question in physics why we ended up
    with a spacetime with a 3+1 signature at physically observable
    scales; one could in principle lay out a theory of relativity
    with any signature, so one cannot answer this question purely
    from relativistic first principles. Perhaps a future theory of
    quantum gravity may shed some light on this. (I believe string
    theory had some proposals in this direction, though there does
    not appear to be a consensus among physicists as to how plausible
    they are.)

    Reply

19 December, 2022 at 7:36 am

Anonymous

[]

Could the treefolk be foxtail palms?

Reply

19 December, 2022 at 8:58 am

Jan Decat (@deepconvonet)

[def]

I can't help but ask: what fantasy books do you like?

Reply

19 December, 2022 at 1:57 pm

James T. Biscuit

[ac5]

Now consider the case where at the dawn of time, there were two
treefold. One ring tall each. They wink at each other and create a
new treefolk who is the sum of the heights of the parents. This new
treefolk is 2 rings tall. The treefolk winks again and suddenly
another treefolk appears at 3 rings tall. And again! 5 rings tall.
And so on. Scholars referred to this at the Tribonacci epoch.

In an alternative universe the winks followed prime rules, but the
treefolk could never work how high they could go, or the relationship
between their relatives. Scholars referred to this as the Primeval
epoch.

Reply


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