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Tag Archives: Space & Time

Matters relating to length and duration in physics and transportation

Multiple dimensions of time

This post is another in a series on the duality of space and time. I have emphasized that the basis for space is length and the basis for time is duration. What, then, about direction? Does direction apply to both space and time? Yes, and in the same manner.

If someone says, “The hotel is 10 minutes away by car” how is that different from saying “The hotel is 5 miles away by car”? One provides a duration and the other a distance. Neither provides a direction. Both require movement to measure. They are exactly parallel.

If someone says, “The hotel is 10 minutes north by car” how is that different from saying “The hotel is 5 miles north by car”? One provides a duration and the other a length, each with a direction. Both require movement to measure. They are again exactly parallel.

Is the direction “north” part of space in one case but not in the other case? Then what does “north” mean in the sentence “The hotel is 10 minutes north by car”? It means that the direction “north” and the duration “10 minutes” are combined, just as we combined the direction “north” and the length “5 miles”. It would be arbitrary to say that direction applies to space (length) and not to time (duration).

So what is direction? It is something independent of length and duration, that is, it is independent of space and time but can be applied to either space or time. Direction is what makes the scalar “lengths” into a vector of directed lengths, often called a displacement. In the same way, direction is what makes the scalar “duration” into a vector of directed duration, which could be called a temporal displacement.

Is the concept angle only related to space? Look at the hand of a clock. Is it measuring an angle of space or of time? Both. We read a clock directly as time, a duration measured by revolving hands. But we recognize the spatial angles, too, and can use clock numbers to indicate space, as in “10 o’clock high” for a direction in space.

But if there are three dimensions for direction in space, does that mean there are three dimensions for direction in time? Yes, and they are the same three dimensions. For example, an isochrone map shows contour lines (isolines) for durations in two dimensions. It is like an isodistance map which shows travel lengths in two dimensions. The only difference between these maps is whether durations or lengths are shown; the two dimensions are the same.

So when we say that looking into outer space is looking back in time, that includes the three dimensions we see.

Space, time, and arrows

This post is a continuation on the duality of space and time. The basis of space is distance (or length) and the basis of time is duration. It must be emphasized that both distance and duration are scalars, i.e., they have magnitude but no direction. They are not one-dimensional because that would entail direction, represented by a positive and negative quantity. So scalars are non-negative real numbers (zero is a degenerate case).

Consider two sentences: “The Arcade building was a block long.” “They were stuck in traffic on Lake Shore Drive for 12 hours.” The first sentence expresses a distance and the second expresses a duration. Note that both sentences use the past tense. In the present tense the Arcade building in Chicago’s Pullman district doesn’t exist because it was demolished in 1926, and the mammoth traffic jam on Chicago’s Lake Shore Drive in February 2011 is over. This parallel shows that anyone who wants to say the past time is in the opposite direction from the present time could equally well say that past space is, too.

The problem is Arthur Eddington’s “arrow of time” which says time is one-way or asymmetric. But time in this sense has to do with tense, not duration, and has no application to space and time. Note that more recent work on “arrows of time” has focused on thermodynamics, causality, etc., and not on space and time.

What, then, is the meaning of a time line that goes from negative to zero to positive? If “now” is at the zero point, isn’t the negative part in the past? In fact, this is no different from a “space line” that goes from negative to zero to positive with “here” at the zero point. The location of a point in the past would be negative, but that does not lead us to say that space is one-way or asymmetric.

Putting “-t” into an equation of physics does not change the tense or make the present precede the past. It simply reverses the direction of the duration. If “+t” is to the right, then “-t” is to the left.

Arrow of tense

The arrow of time is a concept developed by Arthur Eddington in 1927. It is an arrow that points from the past through the present into the future. One problem with this concept is that multiple futures are possible; it would have to be a many-headed arrow. Another problem is that it could just as well be pointing from the future through the present to the past. The choice is arbitrary and may simply reflect a progressive bias.

One could as well speak of an arrow of place that points from there (where one was) to here (where one is) to there (where one is going). So both space and time have their arrows.

A deeper problem with the concept is that it’s really about tense in language. Different languages have different ways of indicating the time when an action or event occurs, or when a state or process holds. The past, present, and future tenses are one means of doing this. But there are other tenses such as the still sense, indicating that that a state is still the case. And some languages such as Chinese are tenseless.

So the arrow of time would be better called the arrow of tense and understood as a property of language. If the arrow of time is used at all, it should be paired with the arrow of place.

Duality of space and time

Several dualities of space and time are known, but there are thought to be exceptions for the dimensions of space and the arrow of time. It turns out these are not exceptions; space and time are fully dual. To understand this first note that movement is required for the measurement of time and space, and then compare the various meanings of the words time with the parallel meanings of space (or place):

(1) Time as duration, a period of time, is a length of time, analogous to a length of space. Duration and length are both scalar quantities.

(2) Time as points in time, instants of time, associated with specific actions or events. This is analogous to points in space, locations, which may also be associated with actions or events. Duration is a difference between two points in time as length is a difference between two points in space.

(3) Time as tense, a grammatical sense which expresses how an action or event relates to the present time, usually relative to the moment of speaking. This corresponds to language which expresses how an action or event is oriented toward the present location, usually relative to the place of speaking.

(4) Adverbs of time are relative to the speaker and include now, yesterday, tomorrow, later, etc. These correspond to adverbs of place, which are relative to the speaker and include here, there, down here, over there, etc.

(5) Time as the arrow of time is the forward flow from past times to the present time to future times. There is a corresponding flow from past places to the present place to future places which could be called the arrow of place. These are one-dimensional views of time and space which could be reversed by looking backwards.

(6) The speed of an object is a scalar measure of its rate of movement, expressed either as the travel length divided by the time taken (average speed) or the rate of change of position with respect to time at a particular point (instantaneous speed). To examine the relation of speed with time and space, consider highway traffic flow measurement which distinguishes two types of average speed:

The time-mean speed is the arithmetic mean of the vehicle speeds measured at one roadside location. The space-mean speed is the harmonic mean of speeds measured by the travel times collected between roadside locations (or on probe vehicles between two locations). Why the harmonic mean? Because the units are in the numerator, so it is a kind of inverse speed (the inverse of the duration of travel divided by the unit of travel length).

From this we may define the speed as the travel length divided by the unit of travel duration. If measured at a point or “spot” it is called a spot speed, which is the instantaneous speed of a vehicle at a specified location. Another speed may be defined as the reciprocal of the duration of travel divided by the reciprocal unit of travel length. For constant speeds, these values are equal but they are conceptually different.

The ordinary speed has spatial direction but no time direction because the temporal denominator is a scalar. For the alternate speed the measurement of duration has temporal direction but no spatial direction because the spatial denominator is a scalar.

Rectilinear motion is along a straight line, with the distance from a point in that line varying with the time. Angular motion is the rotation of an object about a fixed point or fixed axis in a given time period.

Velocity is the rate at which an object changes its position. A velocity may be defined as the vector of travel through space divided by the scalar unit of travel duration. An alternate velocity may be defined as the inverse of the vector of travel through time divided by the scalar unit of travel length.

(7) Direction is a vector of orientation or movement whose magnitude may be a length or a duration. A movement from here to there is also a movement from now to then which may be expressed as a vector. We tend to think of this in spatial terms but it may equally well be thought of in temporal terms.

There will be more to come on this topic but the bottom line is that length and duration are both scalars that may become dimensioned or tensed in an appropriate context. Space and time are dual.