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A bit of a strange concept that's been floating around in my mind. We all know how grid-based movement works. Moving in a straight line along any direction of X, Y, and Z requires only one step/instance of movement to move one space from start while moving diagonally or along two axes requires more energy or more time/steps to achieve the same result of moving a space from start.

Grid for visual explanation, numbers representing how many steps or expenditures of energy is necessary to reach whichever point in space from point 0:

enter image description here

Reaching anything diagonally requires the same amount of energy/steps to move only one space away from point 0 than it does to move two spaces away from point 0 if you were to move in a straight line along purely one axis. This got me to thinking... what if the universe itself follows the same rules? Things occupy spaces, and such spaces must have a point in which they reside, and such points must be next to each other in such a way that it allows things to be next to each other or on top or below, etc, and if the universe itself has such a grid due to things needing to occupy spaces then surely it would be in anything's best interest to try and move as universally purely axially as possible in order to be as efficient as possible in its energy expenditure and time spent in movement.

Now, there's a very big problem that stops things from being able to move along the axes of the universe, much less even be able to detect it. Gravity and orbits. Things are ALWAYS spinning and moving in circles and orbiting and doing every single thing possible that would prevent things from moving in a completely straight line, always moving along more than one axis, making moving along a single axis of the universe a fever dream at best.

Light is also subject to gravity and diagonal movement, and so I began to wonder that, if a civilization managed to invent a technology that allows them to only move along one axis of the universe if the grid-based movement rules applied, would they be able to move faster than the light that is currently wasting energy and time moving along two axes and achieve something like FTL travel?

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    $\begingroup$ If space wasn't nicely isotropic, surely you'd be able to see odd effects like the brightness (or redshift) of moving stars changing. I'm fairly certain this would interfere badly with relativity. $\endgroup$
    – Starfish Prime
    Nov 3, 2021 at 10:21
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    $\begingroup$ It's changing the geometry of space to achieve movement. This sounds like an Alcubierre drive with a Tic-Tac-Toe/checkers explanation. (No denigration intended, just observation). $\endgroup$
    – Angry Muppet
    Nov 3, 2021 at 11:10
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    $\begingroup$ @JohnO "The real world spacetime is not a continuum. It's just the Planck scale" Ummm... have you solved then the problem of Lorenz contraction of a Planck length? Like, different observers measuring different values for that "minimal length"? $\endgroup$
    – Adrian Colomitchi
    Nov 3, 2021 at 15:30
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    $\begingroup$ @JohnO: The Planck length is not a quatum of space and the Planck time is not a quantum of time. I have no idea why you would think they were. $\endgroup$
    – AlexP
    Nov 3, 2021 at 16:20
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    $\begingroup$ Despite what the comments are arguing about this doesn't require space to be discrete-- it can be achieved by supplying spacetime an $L_1$ norm type structure (aka taxicab geometry). Unfortunately, the $L_1$ norm is not induced by any bilinear form, so we can't use the familiar setting of semi-Reimann geometry and must instead move our setting to more general Finsler manifolds. I don't know exactly how this would play out, but like @StarfishPrime said it definitely would be quite different from our own universe. $\endgroup$
    – el duderino
    Nov 4, 2021 at 14:00

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This thought experiment has interesting properties :

  • actually you can’t move diagonaly without moving one time ahead and then one time aside, because there is no common edge between the first one and the latter. For example, the square "2" has no common edge with the square "0", so you are forced to travel through a "1" to reach "2" from "0".
  • in a non-grid space (let’s say "like ours"), the energy to travel on a pure diagonal is equal to the square root of 2, not just 2, but it’s not a problem because you can do it, you don’t need a common edge. In your model, since you don’t have common edges in diagonal, you can’t.
  • in relativity theories, time has its own dimension: in your thought experiment, this means that you cannot travel at the same time in space AND in time because that would make you travel in diagonal. But in relativity, in a way, time IS space: if you travel far away, you also travel "late" away.
  • also in relativity theories, objects are not spinning, but they are travelling in straight lines: the space itself is distorted, and that makes thinking that objects are spinning. In your thought experiment, object could not actually spin in universe due to the rigid grid, so you will have to render forces another way. So, this implies, in your model, that gravity cannot make things accelerate or turn. So we can simply reject it.

So, with all these elements, I would say that you can’t travel faster than light, but you may just choose between travelling through space or through time at the speed of light, but not both at the same time, by steps of one. But actually this is enough, because 1 square of space = 1 square of time, so you have a perfect equivalence between them. That means that, at the speed of light, if you travel only in space, then time doesn’t flow FOR YOU ONLY. In an instant, you have reached the other part of the universe… But the people you left have aged equivalently to what you travelled. If you travel back, also in one instant, you will find your friends older… Except if they have travelled only through time, and not space, at the same speed, and one time forward, ONE TIME BACKWARD. Edit: you need to ignore the speed cone for that last thing to be possible (or you won’t be able to move backward in time…).

If you want to have deeper thoughts about it, you’ll have to make more hypotheses, like:

  • is it possible to travel backward in time like it is in space? (in our world, as far as I know, it is not)
  • is it possible to exchange time for space "freely"? (neither possible)
  • is it possible to travel in diagonal for "2", not for "square root of 2"?
  • what is your model for gravity?
  • do you have a special model for the other forces?
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  • $\begingroup$ I am not sure about your story about my friends being older when I reach my destination. It does not line up with the picture where we draw time in the vertical direction and space on the horizontal. If we start at (0,0) and I travel one unit through space to the right to (1,0) and then another unit to the left I just end up back at (0,0). So after 2 units of traveling I am back when I started. On the other hand after 2 units of traveling my friends are at (0,2). But of course we cannot interact with each other since they are at time 2 and I am at time 0. $\endgroup$
    – Daron
    Nov 9, 2021 at 9:22
  • $\begingroup$ I agree with you @Daron, I may have over-interpreted the grid-thing. I need another hypothesis for that, which is linked to the speed cone: it should not exist, so that you can move backward in time. Thanks for the comment. I’m gonna make a quick edit. $\endgroup$
    – Julien Busset
    Nov 9, 2021 at 19:59
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Any grid, square, hexagonal or whatever, wouldn't increase the speed of light. This is because you're applying real world physics to an imagined grid. The reality is that the grid shows a step, which is a fixed distance. So a diagonal step is still the same step as a sideways step. That means that light moving diagonally would still move at the same speed as sideways.

It is as you say. Reaching anything diagonally requires the same amount of energy/steps to move only one space away from point 0. Step is the same distance, energy usage is identical. No difference in moving straight or diagonally.

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  • $\begingroup$ Grid-based movement implies that you can only movement into squares/spaces that are next to the square/space within which you currently reside though. Moving in a single axial direction with enough energy to move you two spaces from point zero would allow you to move a distance equal to two spaces from the center of point zero to the center of point 2 in a single axial line, but move two axial directions, one up one left, diagonally up left, with the same energy then the total distance which you you traveled from the center point of point 0 to point 2 duo axially is equal to aprox 1.5 spaces. $\endgroup$
    – Lemming
    Nov 3, 2021 at 11:51
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    $\begingroup$ A better way to think about it might be to imagine that things can only occupy the intersections of the grid, and move along the lines connecting those nodes. $\endgroup$
    – BMF
    Nov 3, 2021 at 12:10
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Yes, you've invented hyperspace lanes.

Whatever made up science you have for it, basically if you fly in a particular place you go super fast. Hyperspace lanes where you can fly ftl are a popular sci fi concept, so this is certainly plausible.

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