This equation does not emerge from a difference in time as a dimension, but as a difference in the way we treat time. The signage is different there because space is being measured in deltas and thus requires an offset. This is not requiring a negation of time, it's relating space to time through a negation. We could just as easily write an equation that determines the volume of a cube with a cut-out section 's' by L*W*H - sL*sW*sH.

That's not really true. The point of delta-s as I defined it above is that it is a 'distance' metric between two points that does not change when you move to a different reference frame. If you try to define the metric without the sign difference, you'll get a 'distance' that changes when you transform coordinates to a reference frame with a different velocity. That sign difference is necessary and fundamental, and shows up throughout special relativity.

The sign difference is necessary, but what I'm saying is that it's not a negative quantity being added. It's a positive quantity being subtracted. It's like saying that 3 feet to the right is the same as minus 3 feet to the left. There's no 'minus 3 feet', it's just 3 feet in a different direction. Consider what delta_t^2 means there and why it's present. (it's c^2 * delta_t^2, btw) The comparison of dr^2 and c^2dt^2 indicates whether the spacetime interval indicated is spacelike, timelike or lightlike. In other words, is the spacetime interval vector longer in space, longer in time or roughly equal.

But a point is not a real object. Even subatomic particles occupy more than a Planck length and are thus subject to movement through spatial expansion - at a bare minimum.

Expansion doesn't imply movement. Even if you have an object that's not a point particle, every point on that object can remain at a fixed position as space expands, so the object gets larger but experiences no change in spatial position.

In theoretical space, perhaps, but if expansion wasn't motion in real life then we'd never have realized that the universe is expanding. Remember that all positions are relative - iow imaginary. Until you can demonstrate a reference frame that is fixed (thus breaking relativity) you can't really say that anything is holding still.

If you consider the dimension of time as being spatial in nature then the movement is contiguous. That's what I'm saying. If you're talking about moving backwards in time through a wormhole then the movement is still contiguous, but requires at least one additional dimension for the wormhole to function.

This is easier to discuss with graphs:

In case A, an object moves through both space and time at less than the speed of light, as normal. In case B, an object moves through space without moving through time; this what I'm saying is impossible. Case C is the wormhole. I agree with you that movement along the red line is continuous. The red line is always moving through time as well, though, so at no point is the object moving 'horizontally', or even necessarily faster than the speed of light.

In retrospect, I was imprecise when I said 'that's not movement' - sorry about that. What I meant was that it's not movement directly through space from instance to instance like the horizontal movement in the diagram.

"Motion" through x without "motion" through t would appear as length in x. (a spacelike interval because dt = 0)

The problem here is that the common concept of motion depends on a change in position over time. When considering time spatially it's necessary to rotate such that an additional dimension is implied in order to observe the delta. I think we're on the same page with that, though.

Considering this, let's go back to the origin of dimension theory (ostensibly) and talk about the fly in the room. We can describe the fly's position at any point in time as x, y, z. If we elevate the observer by one dimension we can see the whole journey of the fly through the room as a contiguous spacetime object. From that perspective we could describe any point in the journey as x,y,z,t. If we move back down to a three dimensional space, but this time use x as a delta instead of t we still get sane data, but from a different perspective. A computer simulation of this would be interesting. In t,y,z space the fly can occupy more than one point at the same x, or can be a line along the t dimension.

It is a comparison, but it's interesting to me that using that comparison is the way you construct an invariant. My argument is that if time and space were truly interchangeable, you'd expect an invariant quantity to have dt show up interchangeably with dx, dy, and dz. If you try to construct the invariant interval that way, though, it doesn't work. (I know about the c^2; it's easier to use natural units, and in this instance it's conceptually simpler for people who are unfamiliar with relativity.)

This is an equation designed to compare space with time, though. If we were comparing xyt against z (using whatever terminology) then I see no reason why it wouldn't work. (In the 'rotated' fly-space, for instance.)

I can't call anything absolutely stationary, but my argument holds true in any inertial reference frame. I can certanly find an inertial reference frame and determine the position of objects within it without breaking relativity. I don't quite understand what you mean by theory, by the way; theory attempts to describe real life as well as it can. The expansion of the universe is extremely theoretical.

If there were no difference between theory and reality then there would be no need for theory. This is a tool that we use to understand things, yes. But this indicates that our understanding is not perfect. In this case we're imagining a universe that holds still for us to examine. You can create a reference frame, but what's actually holding still there? You've asserted that something can hold still in space while moving through time, but can you actually show evidence of this ever happening? In order for something to be shown as not moving we'd first have to demonstrate a reference frame that itself is not moving. This is against relativity. We can say that two objects are moving in parallel at the same speed, but that only means that they're holding still in reference to one another. It doesn't mean that either of them is not moving in space.

I agree more or less with what you're saying here. Using your setup, my point is that if you use t as a delta, the fly can only be at one location at once, but if you use x, y, or z, the fly can at multiple locations simultaneously - that suggests to me that there's something different about t.

It suggests to me that the fly interacts with the dimension of time the same way that we do. Just like a flatlander can't freely interact with the third dimension, we can't freely interact with the fourth. This does not effect the nature of the third dimension, though. It's simply a trait of the flatlander. However, it's worth considering that we regularly observe objects which are larger than a point. This suggests that 'motion' through space independent of time may simply be so common that we don't notice it. If we consider the fly as a point then it's not apparent, but if we remember that the fly is a collection of particles then even when t is the delta the fly is occupying more than one 'point'.

I'm pretty sure it wouldn't work. dr^2 - dt^2 is an invariant because of the equations for coordinate transformations between different reference frames, which should be the same in rotated fly-space. If you construct a similar equation to compare z-distance with xyt-distance, you'll get a quantity that can change from reference frame to reference frame, regardless of how you view the space it's measured in.

Language fails me at this point, but I can visualize this, both with the fly and with an entity that can move freely in t. I honestly believe that this is a limitation in the ways that this has been examined. The dr2 - dt2 equation is used to determine the spaciness or timeyness of the interval. If we're rotating ourselves in the way described we'd have to start talking about depth versus fruit-saladness, or something. In order to give that a fair shake we'd have to use reference frames that are oriented in the same dimensionality.

I'm not trying to claim that anything is holding still in the absolute sense that you mean - as you seem to agree, movement isn't meaningful in any absolute sense. I am claiming that something can hold still in a particular reference frame - that is, that its spatial coordinates in that reference frame could be constant. That much seems clear to me; in a reference frame attached to my desk, my monitor is stationary.

If I restrict my attention to inertial reference frames - that is, reference frames that aren't rotating, accelerating, or breaking the speed of light - I can then make my claim that objects can be stationary in space but not in time. In short, it doesn't matter that I have no absolute coordinates, because my argument holds true in any set of relative coordinates I choose, as long as those coordinates correspond to an inertial reference frame.

Don't we observe this behavior in QFT, though? Paticles that pop in and out of 3-space? An object that is 'stationary' in time would be encountered thuswise, wouldn't it?

Talking about the fly being in two spots at once was a little sloppy of me; the more precise thing to say would have been that the fly's movement could satisfy things like dx/dt = 0 or dx/dy = 0, but never things like dt/dx = 0, with dt in the numerator. According to special relativity, that much is true regardless of how the fly interacts with the fourth dimension.

As you say, you can view the fly as a collection of particles; your suggestion, then, is that there's one 'fly' particle, and at any particular time-step it moves through the positions of all the elementary particles in the fly? Even disregarding the issue of multiple elementary particle types, that's not movement in the sense that I was discussing - it's a jump, not a smooth progression.

I'm trying clumsily to use the fly as a volumetric particle. I'm mixing up the analogy a bit. Essentially I'm just pointing out that even elementary particles have volume, which could be viewed as a non-orthogonal intersection with 3-space. In other words, if you consider spacetime and draw a line that runs along the t-axis then you'd get a stationary point in 3-space. Since a point has no volume, this is a kind of persistent virtual particle, failing to meet the Planck length, it holds no locality. If you tilt the line so that it's not t-aligned then the point gains volume in 3-space since its intersection begins to become linear and can gain locality. Taking this to its logical extreme, we'd end up with a volumetric object which occupies less than a Planck time, another virtual particle.