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):

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”. Curvature of the trunk in this allegory is the analogue of acceleration in our spacetime universe.)

treefolks prefer to stay very close to being vertical, and only tilt at significant deviations from the vertical in rare circumstances; it is only in recent years that they have started experimenting with more extreme angles of tilt. Let us 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.) [Removed this hypothesis as being unnatural for the underlying Euclidean geometry – T.]

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.

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.

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.

— 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:

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):

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):

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 ${c}$ is explained as being the number of centimeters in an inch):

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.

Similarly, the twin paradox discussed earlier continues to hold even when the “inertial” treefolk is not vertical:

[Strictly speaking one would need to move the treefolk to start at the exact same location, rather than merely being very close to each other, to deal with the slight synchronization discrepancy at the very bottom of the two twins in this image.]

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.

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.