I would like to offer here a definition of time that is as little abstract as possible. What we want, I think, is a definition that describes time as something that we measure. Only that. One might call it an operational definition. This definition is not an explanation of what time means (or has come to mean) philosophically or epistemologically. It is an explanation of what time is in our experimental or everyday use of it.
I maintain that time is simply a measurement of movement. This is its most direct definition. Whenever we measure time, we measure movement. We cannot measure time without measuring movement. The concept of time is dependent upon the concept of movement. Without movement, there is no time. Every clock measures movement: the vibration of a cesium atom, the swing of pendulum, the movement of a second hand.
In this way time can be thought of as a distance measurement. When we measure distance, we measure movement. We measure the change in position. When we measure time, we measure the same thing, but give it another name. Why would we do this? Why give two names and two concepts to the same thing? Distance and Time. I say, in order to compare one to the other. Time is just a second, comparative, measurement of distance.
The measurement of time is necessary to the measurement of velocity. It may be that time was not even "invented," in the modern sense, until someone first thought of the idea of velocity. Velocity is the measurement of the change in position of one thing (the object in question) relative to the change in position of another thing (the cesium atom, or the pendulum, etc.). Once you have conceived of the idea of velocity in this way, you realize that it can be measured in only one way: Compare the unknown movement to a known movement. That is, find something in your world that moves as uniformly as possible, and let that be your clock. Then compare your unknown movement to the movement of your clock. That is what velocity is.
You may say, how can I know that something moves "as uniformly as possible" without already having an idea of time? You cannot. But I maintain that this idea of time—as simply a commonsense idea of uniformity of movement—is the only operational idea of time we have ever had. The initial idea of time, historically, or instinctively, is the idea of uniform movement. The first clock must have been chosen on this basis, just as the very latest atomic clock is chosen on this basis.
Also notice that there has never been any way to test the uniformity of a clock, except relative to another clock. The first clock must have been chosen based mainly on instinct. The ancient who chose the swinging pendulum because it swung the same number of times per day was comparing it to another clock—the sun. If he was smart he counted his pendulum swings from sunup to sunup, rather than sunup to sundown, and so avoided the variation in length of daylight. And if he was very smart, he continued to look for even better natural clocks to fine-tune his measurements by. But notice that as long as the sun was his standard, he had to assume that the sun was a good clock—he took for granted that one day was the same length as the next.
In judging the uniformity of natural clocks, like the sun or the stars, our ancient would resort to comparing them to his pendulum clock. How did he know that the sidereal clock was more accurate than the solar clock? By comparing it to his pendulum. He corrected his pendulum by the sun and corrected the sun by his pendulum.
In this way you can see that there never was an idea of "absolute time." Time was always a relative measurement. It had to be. It was relative to a given clock, a clock chosen mostly by instinct. For there was never any way to prove that the given clock was absolutely uniform. It was only more uniform relative to clocks that were already relative to other clocks.
So time is not a measurement of "time." Time is a measurement of the movement in or on a given clock. And this given clock is uniform only by definition. It is uniform relative to a standard clock. One that has been defined as uniform. This standard clock cannot be proven to be uniform. It is only believed to be more uniform, based on previous definitions and previous clocks.
In this sense, time is not absolute. There is not, and cannot be, a clock that is known to be absolutely uniform. This is a statement of logic. A clock known to be absolutely uniform is a reductio ad absurdum. For us to know the clock was absolutely uniform would require us to have a previous clock by which to measure it. A clock may be defined as absolutely uniform. That is, we may decide, quite freely, to define some vibration of the background radiation of the big bang to be absolutely uniform. But we cannot know the truth of that definition.
Every measurement of time is a relative measurement, in this sense. It is relative to a standard clock, defined as standard. Time is also a relative measurement in the sense that it is dependent upon a measurement of distance. The time concept is relative to the distance concept.
Now that we have an operational definition of time, we may proceed to an operational description of the calculation of velocity. As I said above, velocity is a relative measurement. It is the change in position of an object relative to the change in position of (the internal workings of) a clock. We usually write this as distance-over-time. d/t. I maintain that this is exactly the same as distance-over-distance. If we had written miles per hour, we might have written miles per miles. For we might have remembered that our clock is a little something in movement, and the movement inside the clock might be expressed in our denominator just as easily as the "time" on the clock. A pendulum travels some distance each second, and so does a cesium atom or a pulse of light. In calling the distance traveled a "second" instead of a mile or a foot or an angstrom, we are simply choosing terminology that suits us. But the fact remains: in terms of measurement, what is being measured by a clock is distance.
In the calculation of velocity, one makes one basic assumption. One must assume that there is indeed a relationship between the measurement of the object in question and the measurement of the clock. If I am comparing two things, I must assume that the two things are comparable. I must assume that the distance I am measuring with my object is the same sort of distance I am measuring with my clock. In my velocity equation, what I have is really distance-over-distance. For the equation to make any sense at all, I must assume that the concept in the numerator is equivalent to the concept in the denominator. That is, I must assume continuity. I must assume that the measuring rod of the object-distance is the same measuring rod of the clock-distance. I must assume that the background is the same for the clock and the object. In mathematical terms, I must assume that the clock and the object are in the same co-ordinate system. If they are not, then it would be foolish to compare them. It would be foolish to put one over the other in an equation.
Think of it this way. A velocity equation states that the object (of the numerator) moves a certain distance relative to the movement of another object—the workings of a clock (the denominator). "Relative to" means that the first thing is related to the second thing. If they are in different co-ordinate systems, they are not related to eachother, and it would be senseless to put them in the same equation.
So the basic assumption of a velocity equation is that the object and the clock are related. They are in the same co-ordinate system. Or, to put it another way, space is continuous from the object to the clock. If it were not, there could be no velocity equation.
If time is actually a measurement of distance, then wherever space is continuous, time is also continuous. This being true, it follows that wherever there is an attempt to measure velocity, there is an assumption that time and space are continuous. There is an assumption that all local measurements are equivalent. Without this assumption, no equations are possible.
In this sense, time is absolute. Time is assumed to be invariable from point to point, throughout space. This assumption is what allows for the measurement of velocity. [This says nothing about measuring moving clocks. As I have shown in another place, the findings of Einstein's Special Relativity are valid, in the main. But in order to calculate the slowing of moving clocks, from a distance, one must assume they are not slow, locally.]
It is said that Einstein did not make this assumption—of absolute time—when he began his calculations in Special Relativity. It is said he did not make the Newtonian assumption of absolute and continuous space and time (one big co-ordinate system); nor did he make the assumption in a more limited sense, as I have above. He did not assume the equivalence of local time. It is said that he proceeded without this assumption, and by proceeding without it proved that local time, in my sense, is meaningless. According to the canon, one may now speak of ones own local time. But speaking of the local time in another place is a faux pas.
I show in other papers that Einstein hid his assumption very well, but it was there nonetheless. What is the only assumption that most people will admit that Einstein carried into Special Relativity? What was his "only" given? The constancy of the speed of light. But if the speed of light is the same in every co-ordinate system, then that, by itself, assures that the local time of every co-ordinate system is equal to that of every other. If light goes 300,000 km/s in every system, then the ratio of kilometers to seconds in every system must be equal. Either that, or the statement "light has a constant speed" has no meaning.
If you say, "Yes, light has a constant speed, but the time in another system may be different than ours," then I don't see what light has a constant speed means. It does not matter that their time is "different than ours." When they measure the speed of light, they will not be using our watches. What their watches are relative to ours will not enter into the velocity equation they use at all. When they measure the speed of light, they will divide the distance light goes in their system by the distance their little cesium atom wobbles. The relationship of light to a cesium atom in their system is the same as in ours, so they will not only see the light go the same speed, they will see it go the same distance we do. Notice this has nothing really to do with cesium atoms. It has to do with the relationship between distance and time in their system. You say "their time is different." But what does that mean? If their time is slow relative to ours, it surely doesn't mean that they will measure it differently. What I mean is, Einstein said they will get the number 300,000 km/s, just like us. You say, perhaps their second is slower, so that the distance must be shorter than 300,000 kilometers in order to equal the same speed. But this is not looking at it from their perspective. They are not going to divide the distance they see light travel by one and a half pendulum swings, for instance, or by 1.5 seconds, or by some extra number of cesium wobbles. They are going to divide the distance by one second, just like we do. And they are going to call it one second, no matter what you or I think of the matter—no matter how long or short that second looks to us. Einstein says that according to them, light will be going 300,000 km/s. They define one second as being one tick of their own clock, just like we do. Therefore, they will see light travel 300,000 kms during that tick.
It is true that if we could see the light in their system from our system (which we can't—by the time we see it, it is in our system) it would appear to have traveled a shorter (or longer) distance—since those clocks over there are slow (or fast). But that is not the question. The question is what do they see. They see the same thing we do. This is not of the nature of a guess. It is a deduction. If the speed of light is given as a constant in every system, then every system must have equivalent local time.
The smartest scientists have understood this, even when they were a bit unclear about Relativity as a whole. Richard Feynman, for instance, who many would call the smartest physicist since Einstein, explicitly believed in what I am calling local time and distance. On page 94 of Feynman Lectures on Gravitation, he talks of "absolute time separation" and "proper time". This was his admission not only of local measurement, but of the universal equivalence of local measurement. He understood that you cannot link various systems with any transforms whatsoever unless you assume the equivalence of all local time.
Minkowski is known as the father of four-dimensional space. In his theory, time becomes a fourth dimension, mathematically equivalent to x, y, and z. In fact, by setting his quadratic equation equal to 1, instead of to zero, Minkowski implied that time travels at a right angle to all three of the other dimensions. It was therefore equivalent to a spatial vector, travelling orthogonally. It so doing, it created what mathematicians call symmetry. The t variable could then be incorporated into matrices as an absolute equal to the other distance variables.
This theory was appealing to those who are attracted to mathematical esoterica, but unfortunately it is completely false. As I have shown, time is a measurement of movement. Without movement there is no time. But this movement already has a direction, determined by x, y, z: it cannot be given a secondary vector. All motion is a vector, and that vector must coincide to some distance vector within the 3-d continuum x, y, z. So to state that time has a vector outside this continuum is false. If time is a measurement of motion and all motion is contained in x, y, z, then time cannot be outside of or external or superadded to x, y, z.
Nor can it be thought of as mathematically symmetrical to the three distance variables. It is a second measurement of distance, as I have shown, so it can certainly be thought of as a distance variable. But it is not the same sort of variable, theoretically. It is different because it is not really a variable at all. It is a postulate. You may not include it with the others simply because the others rely on it. Meaning that you may not have all four variables as variables at the same time. If time is unknown at the same time that x, y, and z are unknown, then all four are unknowable. If x, y and z are thought of as fields, then t is a subfield. If x, y and z are thought of as axes, then t is a defining axis or "axiom axis". It is not strictly equivalent to the other three. Including it in matrices in the way that is now done is therefore dangerous. Theory is lost, postulates are hidden from view, and mathematical errors are the consequence.
If that critique of Minkowski was a bit abstract for some, think of it this way: Velocity is distance over time, right? Distance and time are both vectors. They have direction. Well, you can't put orthogonal vectors in a ratio or a fraction and expect to get a value for your velocity vector. If you have one vector over another, and the vector in the denominator is at a right angle to the vector in the numerator, you have a serious problem. One of the first rules of vector algegra tells us that you can't just divide one number by the other; but this is what happens every time we find a velocity. We simply divide the distance by the time. Which means one of two things must be the case. Either all our historical velocities are wrong, or Minkowski is wrong. The time vector is not at a right angle to any possible distance vector.
I was asked by a reader of another paper whether I ultimately thought time was absolute or not. You can see that this is not such a simple question. I had to ask, "absolute in what sense?"
As I have shown, time is assumed to be absolute in the sense of being equivalent from one system to another. We must make this assumption in order to calculate velocities, among other things. This does not mean that it is absolute, of course. It means that we must define it as having continuity from our immediate vicinity to any vicinity we want information about. If we do not assume time and space continuity, we cannot hope to build meaningful equations. A universe without continuity is a universe without equations, without mathematics, and without science.
But time is not absolute in the sense of absolutely precise, or absolutely known. It is a concept based on the idea of uniform movement, but the concept allows of only relative measurement. A movement can be known to be more or less uniform, but not absolutely uniform.
Likewise, time is not an absolute in the sense that many "classicists" appear to mean when they mean by it that Special Relativity is wrong. Objects moving at a distance, including of course clocks, look different than objects at hand. And velocity and acceleration influence the appearance of distant objects in quantifiable and dramatic ways. Time dilation is a fact. A poorly interpreted fact—up to now—but a fact nonetheless.
Time is also dependent upon, and therefore relative to, movement. In a sense, time is nothing. Or it is nothing but a second measurement of movement. Displacement is movement. Time is movement. Time is displacement. Time is the displacement of the reference body.
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