Most, if not all, accepted systems of measurement take time to be a fundamental quantity. Time, however, has been something difficult to define. Why do we take it to be a fundamental quantity? There is no reason why it must be done this way. If we order our physics on a somewhat different scheme, then some simple dimensional analysis makes time much easier to understand.

Rather than take time as something fundamental, I suggest that we look at time as a derived quantity. Specifically, we derive it from mass, length, and force. This new system looks like something of a hybrid of the SI system, in which mass, length, and time are fundamental (it measures them in units of kilograms, meters, and seconds), and the English system of measure, in which weight (force), length and time are fundamental (pounds, feet, and seconds).

We start with the basic equation

1) F = dp/dt

which is Newton’s second law of motion. When mass is constant, this reduces to the familiar F=ma. Either way, let us change F to the fundamental quantity f. This is done by reducing the vector F to a scalar of equal magnitude, f. We then break down the right side of the equation to its fundamental quantities to yield

2) f = ml/t^2

where m = mass, l =  length, and t = time. Assuming force to be a fundamental quantity is perfectly legitimate, the English system of measure takes force units of pounds to be fundamental, and derives its mass unit of slugs. What I am proposing takes both mass and force to be fundamental, along with length.

To distinguish t as a derived quantity, we will write it as T. Solving for Time, we have

3) T = sqrt(ml/f)

The above equation suggests that time is a scalar quantity that results from the interplay between matter (note that ml is one-dimensional matter) and force. Intuitively, this makes sense; massive objects resist being pushed, and this resistance is manifested as time. If objects had no mass, they would offer no resistance, and their movements would take place instantaneously. The amount of time manifested by matter is related to its mass, and inversely related to the amount of force applied. If there were no matter, there would be no time. Time is a material phenomenon.

Though we derived this definition of time from a law of motion, matter does not need to be in motion for time to exist. Matter does not even need to be in a state of change. From the study of materials, we know that all solids possess elasticity; they deform to some greater or lesser degree under stress. The deformation is not always linear, though when it is (or nearly so), we can describe the system with the spring equation:

4) F = -kx

in which k is the material’s spring constant, x is a measure of the material’s deformation, and F is the force exerted by the material. Note that the equation of motion was built around an externally aplied force, whereas this equation identifies an internal force. We can convert this to an equation relating deformation to an external force simply by removing the negative sign. 

In the non-linear case, a material’s spring constant may vary with the amount of deformation, thereby complicating the equation. But the basic relationship is unchanged; the greater the force applied, the greater the deformation. Assuming a static force and unmoving matter, the spring equation can be written in fundamentals quantities as

5) F = (ml/lt^2)l

Replacing F with f and t with T and solving for the derived quantity of time yields

3) T = sqrt(ml/f)

Finally, we have the case of material moving at constant velocity when damping or friction forces are present. (And in any actual case, damping and/or friction are always present.) Force is needed to overcome friction and damping, and again we can rearrange fundamental quantities to yield

3) T = sqrt(ml/f)

More complicated scenarios can be modelled as linear combinations of position, acceleration, and velocity. An accelerating object undergoing deformation and encountering friction is experiencing a force described by

6) F = ma + cv + kx

where v is the velocity and c is the damping or friction factor experienced at this velocity. Because this is a simple linear combination, time T continues to have the same dimensions as above.

What of a material object that is not accelerating, and not subject to deforming forces or friction? It should be clear that no external forces are acting on such an object. If conditions were such that we could ignore the motion of the electrons of this object, its Brownian motions, its thermal motions at the molecular scale, and so on, then no forces at all will be acting on this object and time, effectively, will have ceased. Let us call this isolated condition, in which there is an absence of all forces, “pritine.” The object will undergo no changes until it makes contact with another object or force, and at that time it will be identical to the condition it was in when it first entered the pristine state. While in a pristine state, time for the object will effectively have stopped.  From this we see that force is a necessary component of time. Time is meaningless without force, and zero force causes our time equation (equation 3) to become undefined.

An object moving at constant velocity in this pristine state will transit between two points within some given time, as measured by some system not in contact with the object. This fact might seem to defeat the claim that time, for this object, has stopped. But the only way to know how far along the object is during transit would be to interact with it in some way. This cannot be done without disturbing the object, without a force being applied to matter. From such interactions come time. Without such interactions, no time happens. The object might also be said to be effectively outside our universe.

The concept of a pristine state is put forward as a thought experiment, and is not meant to accord with anything in actual reality. All matter that we know of is, in fact, under the influence of one or more forces. When force is applied to matter, the result is not just time, but also change. There is no action without an equal and opposite reaction and matter reacts, however microscopically, to every force that acts on it. Thus, everything that exists in time is undergoing change. Everything is in the process of becoming something else. Moreover, it is becoming something new; the Second Law of Thermodynamics prevents systems from reverting back to a previous state; all motion must be forward. (Note that our derived quantity of time is always positive.)

If everything is in the process of becoming something else, then anything can be used as a clock. Naturally, some changes make for more convenient clocks than others; the erosion of a block of granite would not be useful in measuring the speed of an automobile. But such erosion might be useful in describing very long spans of human history. Time has no existence or meaning independent of these physical, material processes.

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