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No change in time per distance

Speed can be zero, that is, the change in spatial position per unit of duration can be zero. Can the change in temporal position per unit of distance be zero, too? Let’s see.

First, the denominator cannot be zero. We cannot simply invert a zero speed because that would lead to a zero denominator, which is disallowed mathematically. The denominator is non-zero no matter what the measured quantity is.

Second, the units in the denominator are the reference for what the numerator is measured against. It’s as if the units keep ticking away while the numerator is measured. Since time is often in the denominator, the seconds, minutes, hours, etc. seem to be ticking away no matter what the value of the numerator is.

Third, in this case the distance units are in the denominator. The context is that distance units are ticking away while the duration is measured.

Here’s an example of what this means. Suppose you’re on a train going at a steady speed. The click clack of the train reminds you that it’s making distance. In your mind the click clack measures the distance away from your departure and closer to your destination.

Suppose a train comes up beside yours and goes at the same speed. You aren’t moving relative to the other train. But in units of distance what is the change in time? Since your motion is synchronized, there is no relative change in temporal position between the two trains. The relative change in time is zero, while the distance ticks off, click clack click clack.

Yes, a change in time per unit of distance can be zero.

The flow of time and space

Marcus Aurelius wrote, “Time is like a river made up of the events which happen, and a violent stream; for as soon as a thing has been seen, it is carried away, and another comes in its place, and this will be carried away too.” Time flows, and keeps flowing day and night, whether anything is happening or not.

But a river flows in space as well as time. Heraclitus is reported to have said, “No man ever steps in the same river twice”. This is because a river moves, and the water that was here is now there. The spatial position of water in a river keeps changing. Space is like a river, too.

There is a sense that time is always moving because clocks are always moving. But on the world’s roads vehicles are always moving somewhere, and odometers are always measuring distances. So in a similar sense space is always moving.

Clocks sometimes stop or are stopped intentionally. If we stop measuring time, time does not stop. The same is true with distance. If we stop measuring distance, space does not stop. We have not reached the edge of the universe.

We cannot measure time without movement so movement must have some special relation to time. But with a ruler we move our eyes from one place to another even if we don’t move the ruler. With longer distances we must move the ruler. A rolling distance measuring wheel can move indefinitely. If we measure distance with light as surveyors or astronomers do, the light moves. We cannot measure distance without movement as well.

As with duration, so with distance in a parallel manner. As with time, so with space. Time flows? So does space.

Is time three-dimensional?

This post is a companion with the post “Is space one-dimensional?“. As we can compare the alleged one-dimensionality of time with how we think about space, so we can compare the three-dimensionality of space with how we think about time. In both cases the comparison is instructional. Space and time are parallel in both cases but in the latter case which we are examining here, the answer to the question is Yes.

A world line is the path of an object through spacetime. It has both spatial and temporal components, so its simplest representation is with one dimension of space and one dimension of time on a two-dimensional surface as a graph. But then we say that space has two other dimensions which are not represented in such a case.

These two other dimensions of space are movements in different directions, so there are components of distance in a total of three dimensions. Does it take time for an object to traverse these components of distance in the other two dimensions? Yes, it takes time to move in each dimension, that is, an object has a measurable component of duration in each of the dimensions.

So there are three components of distance in three dimensions, and there are three components of duration in three dimensions. Since we speak of these three dimensions as space (because of the three components of distance), should we not also speak of time as having three dimensions (because of the three components of duration)? Yes, we should.

Why haven’t we seen the three dimensions of time? Time has been associated with movement and space with stasis. But the movement of an object involves change in both space and time. The movement of an object is measured by its distance and its duration. For example, speed is a ratio of change in spatial position (distance) over change in temporal position (duration).

The conception of spacetime that comes from relativity theory is ready-made for this recognition of the three-dimensions of time. Space and time are parallel and intertwined and so might be considered together, as a six-dimensional spacetime.

Is space one-dimensional?

While the answer is No, space is three-dimensional, it is instructional to compare space with time as people speak about it.

The philosopher JME McTaggart made a well-known distinction between an A-series, which is “the series of positions [in time] running from the far past through the near past to the present, and then from the present to the near future and the far future,” and a B-series, which is the “series of positions [in time] which runs from earlier to later” (Mind 17 (1908), p.458).

In general usage, a world line is “the sequential path of personal human events (with time and place as dimensions) that marks the history of a person”. So a personal world line shows places where we were before the present, the place we are presently located, and (perhaps) places we may be in the future. It also shows the dates we were at each location. From this it may be determined the times at and between locations, and the distances too.

A world line exhibits a temporal B-series and, if the present time is indicated, an A-series as well. But a world line also exhibits spatial versions of A-series and B-series: the series of positions in space “running from the far past through the near past to the present, and then from the present to the near future and the far future” as a spatial A-series and the series of positions in space “which runs from earlier to later” as a spatial B-series. A world line shows that the path we take through space is analogous to the path we take through time. Since we “know” that time is one-dimensional, space must be one-dimensional, too.

The problem is that time is confused with change. McTaggart wrote, “It would, I suppose, be universally admitted that time involves change”. No, that should not be admitted. Both space and time involve change and stasis. For example, speed is the change in spatial position divided by the change in temporal position. Both space and time may involve stasis as well. Even an object that does not change spatial position has parallel cases in which an object does not change temporal position.

This may happen in several senses. The strongest sense is that of relativity theory, in which time at a certain speed (that of light) or in certain cases (depending on forces) does not result in measurable duration and so time stops. Another sense is that when we measure duration (as with any other measurement) there is a beginning to that measurement so if the measurement of time has not begun, then there is no change in time. For example, before a stopwatch is started, no duration is measured. A third sense in which time stops is that time is measured by cycles and at the end of each cycle, time has returned to its starting point and so has not changed position.

Neither space or time are one-dimensional.

Time in spacetime

Consider a worldline in one dimension of space and one dimension of time that tracks the position of a point that moves from position 20 to 10 to 15. This could represent the movement of a point in the E-W dimension. Another worldline could track the movement of the same point in the N-S dimension. All would agree that the two diagrams together represent two dimensions of space. But the case with time is completely analogous; the two diagrams together represent two dimensions of time.

To see this consider someone traveling on city streets arrayed in a grid oriented N-S and E-W with two stop watches. To keep it simple say they are traveling only north and east. They use one stop watch when they travel north and the other stop watch when they travel east. So we would have two travel times: one going north and the other going east, which would correspond to two dimensions of travel distance. As we would all agree that the travel distances are associated with two dimensions, we should agree that the two travel times are associated with two dimensions.

One objection might be that the dimensions here are all “spatial” rather than “temporal”. But the travel times are measured in units of time independently of the travel distances (which might not even be known). It seems arbitrary to say that there are two dimensions in the case of travel distances but not in the case of travel times.

There is a tendency to associate dimensionality with space rather than time (although one strange dimension is granted to time). But dimensionality is a mathematical concept that can be applied to many things, as multivariate analysis shows. As we apply concepts of scalar and vector to spatial quantities, so we can apply these to temporal quantities. Both space and time are multi-dimensional.

Space, time and causality

If we drop a stone into a calm body of water, it sends out circular waves. As the waves move outward, the clock is ticking and we say the dropped stone caused the waves, which are an irreversible process in time. But we would also say the waves are moving in space, so why isn’t it an irreversible process in space? It is, we just don’t ordinarily speak that way.

What is the difference between the “now” and the “here”? The now is the present, which seems to move with us in time. But the here moves with us in space, like a webcam that follows us everywhere.

Is the past where we were or what we were? It’s both. Is the future where we will be or what we will be? Again, it’s both. Events in times past can cause events in the present time, and events in places past can cause events in the present place. There is an exact parallel.

Causality is transmitted through time and space. We’ve heard this in relativity theory but we don’t need relativity to realize it’s true. The world line of an object in space and time is subject to causality in space and time.

But just as space has three dimensions (directions), so does time. So causality has what — six dimensions? No, causality has three dimensions because they are the same three dimensions. This is no different from saying that force has three dimensions, which are the same three dimensions as space. Dimensions are a property that applies to vector quantities.

We’re so used to thinking that dimensions are spatial but they are just as much temporal — and dynamical (having to do with force and torque) and a property of every other vector quantity in physics. Dimensions are an abstraction that applies to many physical quantities.

A dual of the second law of thermodynamics

This is a continuation of the series of posts on the duality of space and time. Consider an isolated system of particles over a period of time. The system covers a specific distance in space and a specific duration in time. Consider only one dimension of space and one dimension of time with an origin point.

The second law of thermodynamics says that the entropy of the distribution of particles at each timepoint over the space tends to increase with increasing duration. Call this the s-entropy since the distribution is over space. What about the distribution of particles at each point of space over the time period? Call this the t-entropy (time entropy).

Consider different scenarios. If the system is at equilibrium, there will be no change over time and the distribution will be constant, which would be the minimum t-entropy, that is, zero. If the system is near equilibrium, there will be little change over time and the distribution will be near constant, which would be a low t-entropy. If the system is far from equilibrium, i.e., the particles are bunched up together, the system will change toward equilibrium.

The tendency is for small intervals of time with many particles to end up with fewer particles, and small intervals of time with few particles to end up with more particles. If the origin is near the concentration of particles, the t-entropy of the distributions of particles in time over space will tend to decrease. If the origin is away from the concentration of particles, the t-entropy of the distributions of particles in time over space will tend to increase.

Mechanics with multiple dimensions of time

As (spatial) velocity and acceleration are vectors, so are their temporal analogues. This perspective makes sense because of the multiple dimensions of time. There is an implicit recognition that time has directionality since time is commonly considered as any real number, and not a non-negative real number, which it would be if time were merely a magnitude. This “reverse time” is an example of time’s directionality — which however has nothing to do with reverse causal sequences.

The (spatial) length (especially of an object) is a magnitude that is used to represent physical space. Similarly, the duration (or length of time) is a magnitude that is used to represent physical time. We speak of the location or position in space of an object or a place. Similarly, we speak of the point in time or temporal position of an action or event.

A point in space or time is “that which has no part” (Euclid) whose location is represented by a position vector. A point itself is an abstraction that is zero dimensional but makes up all multiple dimensions abstract ‘spaces’ (which may represent space or time or whatever). If s is the distance of a point from a specified origin point in space, then its position may be represented by a position vector whose magnitude equals s. If t is the duration of a point in time from a specified origin point in time, then its position may be represented by a temporal position vector whose magnitude equals t.

The movement of a point through space in time may be represented by a vector function of temporal position t whose value is the spatial position at each temporal position t. The movement of a point through time in space may be represented by a vector function of spatial position s whose value is the temporal position at each spatial position s.

During the time interval (duration) Δt = t2 – t1, the position vector of an object changes from r1 = r(t1) to r2 = r(t2), with a displacement vector Δr = r2r1 (boldface represents vectors). The rate of change of the displacement vector is the average (time) velocity vector over the time interval, vavg = Δr / Δt. The rate of change of the average velocity vector is the average acceleration vector aavg = Δv / Δt.

Similarly, while traversing the space interval (length) Δs = s2 – s1, the position vector of an object changes from p1 = p(s1) to p2 = p(s2). The rate of change of the displacement vector is the average space velocity vector over the length of space, uavg = Δp / Δs. The rate of change of the average space velocity vector is the average space acceleration vector bavg = Δu / Δs.

Instantaneous velocity is considered to be measured over a differential of time (duration), dt. In that case the instantaneous (temporal) velocity is defined as v(t) = ds/dt and the instantaneous (temporal) acceleration as a(t) = dv/dt = d2s/dt2.

Similarly, the coincidental spatial velocity may be measured over a differential of space (length), ds. The coincidental spatial velocity is defined as u(s) = dt/ds and the coincidental spatial acceleration as b(s) = du/ds = d2t/ds2.

Measures of speed and velocity

The speed of an object is the ratio of distance (or length) traveled and the duration of travel. It is derived from the distance traveled during a given duration. It is expressed as the measured distance divided by the given duration, that is, distance relative to duration in units of distance over duration, e.g., m/s, km/hr, etc.

For example, the speeds of vehicles passing a fixed point along a roadway may be measured over a given duration by loop detectors and other fixed-location speed detection equipment. These are called spot speeds. The (arithmetic) average of such speeds is called the time mean speed since they are measured during a given period of time. Accordingly, each speed could be called a time speed.

But there is another, complementary way of determining speed. One can select a distance and measure the duration of travel while traversing that distance. Then the measured duration should be in the numerator to show the duration relative to distance, with units s/m, hr/km, etc. Unless the speed is constant, this is not the inverse of the time speed because the distances and durations will not match. It is called the pace, which means the change in time per change in position.

For example, probe vehicles may be in the traffic stream which measure their distance during a set period of time. Or these may be sampled using automatic vehicle location (AVL) data. The harmonic average of such speeds is called the space mean speed since it is measured over a given segment length. Accordingly, each speed could be called a space speed.

Why the harmonic average? Consider each space speed as an inverse speed: put the measured duration of travel in the numerator and the segment length in the denominator, so that the given segment length provides the units for this pace.

Now the average speed may be related to the average pace as follows: invert each speed to put the duration in the numerator and the length in the denominator, take their (arithmetic) average, and invert again to get the average speed. This is the harmonic mean of the space speeds.

Velocity is a vector of speed with the direction of movement. A time velocity may be defined as a velocity whose magnitude is a time speed, and a space velocity as a velocity whose magnitude is a space speed. If its magnitude is a pace, the components are duration divided by length, which is not velocity. It could be called lenticity. Why might one use lenticity instead of velocity? If the duration is measured for a given length, the lenticity gives the appropriate measure: duration relative to length.

What does the direction of the lenticity mean? Since it measures duration (relative to a given length), its direction is the temporal direction of movement. This shows again that the same three dimensions may be associated with time (duration) as well as space (length).

Homogeneity and isotropy of time

The homogeneity and isotropy of space are well-known. The homogeneity of time is partly known but is confused by an “arrow of time” concept that is not applicable to space and time. The isotropy of time is unknown (and usually denied) also because of confusion with an inapplicable “arrow of time” concept.

I previously wrote about the Multiple dimensions of time. As space has three dimensions, so there are three dimensions of time, and they are the same three dimensions.

As space is homogeneous in each dimension, so is time. For example, it does not matter whether an experiment takes place “here” or 10 minutes north and 5 minutes east of “here” (if they are both inertial reference frames). The translational invariance of time is exactly like the translational invariance of space.

As space is isotropic, i.e., the same in all directions, so is time. For example, the duration measured by a clock is the same whether it is facing north, south, east, west, up, or down. And the duration is the same whether it is oriented horizontally, longitudinally, or transversely.

It is said that in classical mechanics time is reversible. This is a confused statement. What can be shown is that if a classical particle moves in one direction, its movement in the opposite (“reverse”) direction is also classical. Since both space and time are directional, that would equally well be true of space as of time but no-one says that space is reversible. It is best to leave questions of (ir)reversibility to thermodynamics, causality, etc.

Noether’s theorem shows that the homogeneity of space leads to the conservation of momentum, the homogeneity of time leads to the conservation of energy, and the isotropy of space leads to the conservation of angular momentum. I haven’t checked it yet but it is natural to expect that the isotropy of time leads to the conservation of rotational energy.