Haven’t done this in a while. Also, these are just notes.

Can’t ignore units. For instance, in

**E = m c**^{2}

**E** is measured in joules or electron-volts or some shit, **m**ass is in grams or kilograms, and **c**, the speed of light, is in meters or centimeters per second.

Next comes the funkiness. I don’t care to do any actual math right now, so forget what the values might be and just look at the relationships in terms of the units:

to pull some units in at random. I skipped a step or two evaluating **c**^{2}, as (cm/s) squared turns into cm squared per second squared … shouldn’t be too hard to figure out.

Note, already, that there is a weird relationship between energy, mass, distance, and time. The algebra gets a bit hairy, but you can solve the above relationship in terms of any single unit. I’ll do it for you:

You get the picture. There’s some true weirdness here.

Distance and time are special, one might think. An object can have a certain value for mass or a certain value for energy, but can an object *have* distance? Can an object *have* time?

We percieve time and distance only with relation to ourselves. An object can be ten inches *away*, but it can’t be ten inches. Like, as a *property*. Likewise, an object can’t be forty-five seconds. It *can*, however, be forty-five seconds *ago*…. This seems to imply that time and location, duration and distance can only be perceived as a delta. What looks like unaccelerated motion — a steady-state affair — is actually a rate of change of distance (although there is an “imaginary” component — the potential component of motion tangential to the observer rather than perpendicular).

What seems linear obviously isn’t. Our brains take away all the irrelevant crap and leave us with a linearized process that is more relevant to matters at hand. But the details are there if you look.

Imagine two point-sources of visible light. They are, to the best of your ability to resolve, just dots. They are identical in every detail except one is a quarter the brightness of the other. Assuming every other measure of the dots is identical, your brain would jump to the conclusion that one dot is just twice as far away as the other and move on.

Fair enough. But imagine that you also are a tiny dot, with no arms or legs or volume or ability to move or touch or otherwise explore your environment. You can’t even tell which direction you are facing. From where you sit, there is no tangible difference between the point-sources of light, and there never will be. One of them is, however, mysteriously dimmer than the other. You look harder, and you find another thousand of these identical things, all varying only by brightness. So you come up with an arbitrary property called distance, or, more likely, some property that resolves itself to be the square of what we call distance because that makes all the math easier, and you move on….

…until you notice that these things are interacting, and the extent to which they affect one another seems to be inversely proportional to the property you’ve assigned them — except these property values don’t match the ones *you’ve* given them. They all seem to have different values with respect to one another that has little relationship to the ones you’ve given them. But other than that, everything is still consistent with the math you’ve already developed….

It’s not so funky. Consider one of the primary paradoxes of relativity. Two electrons at rest with respect to each other repel each other proportional to the strength of their charges and inversely proportional to the distance between them. However, if they are both moving with respect to me, I see them as *attracting* one another proportional to the strength of their charges and inversely proportional to the square of the distance between them, because charges in motion generate an attractive magnetic field. I actually observe the gap between them closing, but *they* observe the distance between them increasing….

It really only makes mathematical sense if the “distance” dimension of the universe is maxed out at “1” and all math involving a distance property is inverted and squared. Then “distance” math merely becomes a matter of phase, where items in contact are “in phase” and items that are maximally distant are “180-degrees out-of-phase”.

All of this math works fairly well for time, too. If you’ve ever played with the railroad-car-at-the-speed-of-light thought experiments, you can see how events can be in-phase or out-of-phase depending on motion and acceleration with respect to an arbitrary observer.

The math — especially the math where energy, mass, 1/Δd^{2}, and 1/Δt^{2} are converted into one another — makes tons more sense when you close your eyes. Conversion from one property to another is just a rotation in phase-space when all the properties are normalized with respect to one another. See what this does to the math:

Assuming **t** to be the traditional measure of time, we can express phase-Time **T** as **1/Δt**^{2}. Likewise, we make distance-ish property phase-Distance **D** from **1/Δd**^{2}. This remanufactures **c**^{2}, the speed of light squared, into some linear scaling constant in terms of **T/M**.

That’s not very hairy, is it? Things in contact are in phase **D****-wise, simultaneous things are in phase ****T**-wise, there are no infinities (or if there are, they’ve been shoved away to the other end of the equation where they don’t get in the way).

We can do this same trick for any open-ended property, actually. If we assume that there is a maximum energy value and work in terms of phase relationships, those nasty infinities go completely away when it’s time to run the numbers. This appears to be valid for every other property in QED. Why not time and distance?

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