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.