Of bigger-inside houses, extra time in fairyland, and gravity.
A common feature of fantasy and science-fiction worlds is the “larger on the inside” space. Little huts with palaces inside, telephone booths with spaceships inside, small holsters that hold entire armories… certainly not all such stories have these, but many do.
Imagine one of these larger-on-the-inside houses; let’s be concrete and say it was 4 meters on a side on the outside but 8 meters on a side on the inside. Now, let’s remove the walls, so we just have the big-area-in-small-area space. What it looks like we’ll come back to in a moment, but for now lets assume the inside is a wood floor and the outside a lawn.
Now, imagine trying to walk the line between wood and grass. Place you left foot in the grass, your right foot on the wood, and start to walk forward. You swing your left foot a meter forward, then your right foot similarly a meter forward, then left, then right right…. At this point you’ve moved your left foot 2 meters forward—two meters of grass, which is ½ of the grass-side length of one of the removed walls. You’ve also moved your right foot 2 meters forward—two meters of wood, which is ¼ of the wood-side length of one of the removed walls. In other words, your right side is making progress more slowly than the left.
What happens when one foot makes progress less quickly than the other? This can be determined by the simple experiment of trying it out, walking with long strides with your left leg and short strides with your right. If you force yourself to move in a straight line, you’ll only manage a few steps before you experience pain of trying to put one foot several meters in front of the other. The only safe, sustainable way to do this is to walk in a circle, turning to the right.
Thus, if you are walking along the edge of this bigger-inside space, you’ll find yourself turning into the large space unless you actively work to not do so by trying to turn the other way. You won’t walk in a circle, though, because the moment both feet are on wood you’ll be walking straight again.
Now, let’s return to what this space would look like. Just as you have farther to go on the inside and thus make less progress, so to does light, meaning it, like you, will turn inward toward the bigger area when crossing the boundary. But this is exactly the same thing that happens when light crosses an air-glass boundary: light makes progress more slowly in air than it does in glass, causing it to bend in toward the glass. Thus a bigger-inside space will look like a block of glass.
Another common feature of some fantasy stories is transporting to another realm where the heroes can spend months or years and return only hours or days after they left. Typically these are separate realms, not connected in any direct way with the normal realm, but let’s imagine a model more like our wall-less house: a square of land that has twice as much time as the land around it. Could we see the boundary between normal-time and more-time land?
To understand this question, let’s consider space-time graphs.
In a space-time graph, anything staying stationary looks like a horizontal line; anything moving at a constant speed looks like a diagonal line. Thus, in the following graph the blue object is moving steadily to the west.
The speed of an object is related to the steepness of its slope in the graph; so in the following the green object is moving twice as fast as the blue one. The green object is also colliding with the red object and the blue object, at different times.
Now, let’s imagine our region with more time than other regions.
This region is much like our bigger-on-the-inside house: because there is more time inside it, straight paths make less progress forward in time inside it, meaning they bend to turn into it.
In other words, objects move more rapidly across the extra-time space. This make sense: with more time I can go farther.
So what does this region look like? Well, light travels in straight lines, meaning light will bend into the extra-time space. While it doesn’t do this in exactly the same way it does with extra space, the overall effect is visually similar: extra-time regions look much like blocks of glass.
Let’s take our extra time picture and tweak it a bit to have several extra-time regions, each with a bit more time than the next:
We see this approximation of acceleration as the object moves deeper into the extra-time space. The object is not aware of any such acceleration, of course: its just drifting at a constant speed in its own time. It is only us, observing from a fixed time, that sees it accelerate.
One of the properties of angle changes based on changing progress is “total internal reflection.” This happens when the angle change between the slower-progress space and the faster-progress space would put the path back into the slower-progress space. Often taught in optics as part of Snell’s Law, this applies to our more-time (and more-space) regions too:
You have to be moving pretty fast to get out of an extra-time region: too slow and you’ll simply bounce off the boundary11 I’ve never seen this escape velocity idea explored in fiction. Anyone can leave Narnia: they just have to be going really fast when they hit the wardrobe. .
Now, if you look at the above graph you’ll notice it looks a bit like the graphs of falling objects you might have seen in physics classes: the blue and green objects are a bit like the parabolas of objects moving in gravity. In fact, if we change the distinct regions of changing time to a continual gradient, we get exactly those objects-in-gravity:
Note that the ability we had with discrete regions of time density to have straight lines that did not exit the region (the red lines) has been lost: since the gradient is continuous, every location is a point of changing time density and hence a point of curving paths.
General Relativity describes gravity22 I tried explaining this idea five years ago but was never convinced I did a very good job, hence this second attempt. as being this phenomenon: heavy objects project an aura of extra time around them, causing things to fall towards them. Heavy things expand time.
Gravity expanding time, with heavy things having more time than light things and things closer to heavy things having more time than things far away might seem like a possible explanation for all kinds of phenomena: why fat people die young, why financial centers are almost all at low elevations, etc. It does not explain these things, because the amount of time expansion is very small. Moving a kilometer closer to the center of the earth only gives you about33 There imprecision is because the gain per kilometer decreases with altitude. 2–3 nanoseconds more time per day.
That such tiny time expansion can result in the rapid change in speeds we see with gravity is due to the speed of light. The math for bending when crossing boundaries depends in part on angles, and the angles need to be such that light gets drawn at a 45° angle, meaning every speed humans deal with are almost completely horizontal and tiny changes in the space-time angle of velocity results in big changes in speeds as we measure them.
As an aside, this is also not time dilation. In special relativity, faster things have more time than slower things (time dilation); in general relativity, heavier things have more time that lighter thing (relativistic gravity). These two factors partially cancel out for things like satellites that are moving fast (more time) but in less gravity (less time); for example, GPS satellites have 45 ms/day less time due to gravity but 7 ms/day more time due to orbital speed, for a total offset of 38 ms/day. and since GPS satelites are basically just extremely precise clocks with antennae to tell everyone what time it is where they are, their functioning is strong evidence that both kinds of time elasticity actually exist.