The short answer is that we don't know.

Basically we notice that the relationship between processes is constant. We can check this easily for cyclic processes - one cycle always corresponds to a fixed amount of another cycle. Then we can check less cyclic processes against the more cyclic processes and notice that they, too, have a fixed correspondence.

Further, when we measure cyclic processes in triads, the fixed ratios do commute. Two cycles such as clocks, A and B measured together A-B will agree with a third clock C, measured separately A-C and B-C. Because this commutation will work with any arbitrary triad, we can tell that this correspondence we call time is shared among all objects. It appears to be a trait external to the objects themselves.

The problem is, to define this properly requires the use of the concept of time. What is a process in the absence of time? What is a cycle in the absence of time?

No, with the current limits of language, defining time is basically impossible. Yet we still know what it is. (No, you don't really think in words.)

The problem is that when you observe time, observing is a process. The shared relations we see are also shared with our observation of the sharing.

To help illustrate what time is, I'm going to break it and see what happens. So what would it mean if this relationship were not constant?

If the relationship were not constant, it has two options. It could be random or cyclic, which would average out over time, or the differences could accumulate.

First, notice that it's very hard to even imagine this properly. What exactly would time be varying over? It can't oscillate with respect to itself. If time were to slow to half for the entire universe, we wouldn't notice a thing, because our perceptions would also be slowed by half, and indeed no interaction anywhere would be affected relative to another.

Instead time would have to vary locally, over a region of space. We could watch clocks running slower there. In the noise or cycle system, it would speed up later. In the accumulate system, it would continue to get slower.

However, in the first circumstance, the differences would even out. Over the time we were measuring the clocks against, each fall behind would be matched by a sprint ahead. In the second, our interactions with the clock would decrease in frequency, by definition. As a result, if the slowing was large enough to be detectable, then the clock would quickly stop interacting at an appreciable rate, and disappear from view.*

*(This is exactly what happens to an object falling into a black hole.)

If it were going faster, that would mean we were going slower, and we would cease to exist.

So, the first thing I see about time is that it's very robust. If it weren't shared consistently across the universe, the variation would cancel itself out.

Still, this idea raises further questions. The question is, as it usually is in physics, how would it know? How would the processes know the difference between this particular patch of space compared to another? Also, how would the variation know how it was supposed to vary? Space is not absolute, which could resolve the problem, so it would have to be a result of the spatial relationship between our clock and the watched clock, a result of the clock's internal state, or a somehow through a variation in the underlying vacuum.

We learn from General Relativity that all three of these apply. Accelerating things makes them go slower. Greater velocity makes things go slower. And since gravity-induced spatial curves accelerate things, a condition of the underlying vacuum makes things go slower.

With a black hole, the acceleration becomes so great that time essentially stops for the infalling object. Mathematically speaking nothing ever actually reaches the event horizon, as it takes an infinite amount of time to get there, because every step closer makes time go slower. Which means that it (nearly) stops emitting photons and becomes black, or indeed emitting any force particles at all.

From the perspective of the infalling object, the universe would end basically instantly.

Crucially, all these variations on the speed of time are very precisely laid out. Just as time in general must be consistent, whatever variations are allowed must also be consistent. The transformation from one perspective to another is consistent the way any arbitrary three clocks in one frame are consistent.

Despite not knowing what time is, and being basically unable to define it, we seem to know quite about about the behavior and properties of time.

So this is what we can say about time quatitatively. To really understand time we need to be able to describe it qualitatively. There is one more thing we can say quantitatively, though.

Time is the independent variable.

To explain, let me define physics. Twice.

A physics is a set of consistent rules for interaction. It is the definition of the relationship between objects.

Physics is a list of things that particles cannot do.

In our particular physics, these rules are defined mathematically. This inevitably means that every behavior can be described as a function.

Not, crucially, regular formulae, such as that for a circle. X^2 + Y^2 = 4 (Radius 2, on the origin.)

The equations describing motion and interactions must be a function, it must have an independent variable which is mapped to one unique set of dependent variables. Y = X + 4

To see this, imagine that it were false. Let's take the circle and make it X^2 + T^2 = 4

There's two problems; we can't properly decide which variable to make independent, as we are thwarted by symmetry; and the related problem that once chosen, the independent variable doesn't have a unique set of dependent variables.

So I'm going to pretend that T is the independent variable. X = sqrt[+/-(4 - T^2) ] (The +/- is there because otherwise the square root function spits out imaginary numbers which don't make sense based on the original equation.)

The plot of this, pretending that X is the result of a measurement of T, is still a circle. Which means that the dependent coordinate set, X, will have two solutions at most points in time. In other words, a pebble thrown in space, subject to this equation, would be in two places at once.* For instance, at T=0 X=2 and X=-2. Later, T=2, the pebbles would hit each other and go back to being in one place, X=0.

*(Not like quantum two-places-at-once, which is really like half in my right hand and half in my left hand, adding up to a total of one place. Actually in two places at once.)

Naturally, this is absurd. Even if it weren't obviously absurd, when the pebble was at two places, if one pebble were struck the other pebble would have to respond, instantly, because it still wouldn't be able to violate the circle-equation. Admittedly, this would be really handy for intergalactic communication. It would be really bad for conservation of energy and for having the laws of physics be the same everywhere.

For there to be a mathematical physics there needs to be an independent variable, which will be unique, which will be time or equivalent.

Even if we were a pair of creator gods sitting down for tea, and I set my teacup down and created a universe with four space dimensions to prove a point, one of those dimensions would have to immediately collapse into an independent dimension so that the other three could be described relative to it, or in other words so that they could be meaningful.

Time flows forward on its own, because that's what it's for, at a steady rate because it can't do otherwise. Each point in time corresponds to a particular state of the universe and no other state.

Also, time's independent nature affirms again that we can't travel through it. The process of traveling through time would have to affect the time dimension, requiring some kind of independent sub-time variable to be a function of.

The slowing of time is a bit puzzling at this point, but it makes sense in light of the fact that time and space are co-dependent. Neither is meaningful without the other, so it's unsurprising that distorting space distorts time. In particular, objects in distorted space need more time to perform their processes.

There's also the issue of differing inertial reference frames. I don't want to go into the examples but all of these are consistent as well. While the universe looks different, the different appearances never actually contradict each other.

Because there's so much we can say about time, it seems inevitable that we will eventually be able to list all of time's properties. At this point, we will know what it is. Although, it may still be impossible to define. It may be one of the irreducible concepts of the universe, simply a fact of nature that can't be defined in terms of anything but itself.

Update:

As we can see from the diagram, no independent dimensions leads to craziness. (Elliptic craziness, apparently.) More than one independent dimension leads to hyperbolic craziness.

Even if a multidimensional physics formed, it would immediately collapse into X+1-dimensional.

I'm not sure why it gets unstable with too many dimensions.

"Too Simple" could easily be a mistake, because evolution has a much better imagination than a small group of humans. It could easily be true as well. Certainly there are problems like the fact that digestive tracts would divide 2-D organisms into two pieces, but this doesn't seem to me to be a fatal objection.

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