iSoul In the beginning is reality.

# Three arguments for 3D time

There are three main arguments for duration to have three dimensions:

(1) The speed of light is a conversion factor between length space (distance) and duration (distime). Transportation conversion factors include the maximum, minimum, or typical speeds associated with different travel modes. Since length space is three dimensional, its conversion into duration space is also three-dimensional. The scale of maps may be in units either of length or of duration using a standard conversion speed.

(2) Observation follows the movement of light, which is three dimensional. In astronomy it is often said that observation of the sky is a way of looking into the past. Observation is a form of communication and any type of signal (sound, mail, etc.) will suffice though not as exact as light. As the observation of length space is three-dimensional, so the observation of duration space is also three-dimensional. Maps of duration space are similar to maps of length space: they show observations (signals) in different directions.

(3) Movement in orthogonal directions entails moving in three dimensions. Each dimension of movement has an average speed from the change in length divided by the change in duration . As movement is in three dimensions, so the duration aspect of movement is in three dimensions. Travel shows the same result. Maps of movement may show either stantial positions or temporal positions or both.

Each argument may be illustrated by a map. The conversion factor argument shows how any map may be scaled in units of length or duration. The observation argument shows that maps of observations have units or length or duration. The movement argument shows that maps of movement may show length space or duration space or both (as in a isochrone map).

Note that every argument has an analogue in ordinary travel and so is not unknown to nonspecialists. The arguments are however exact in the case of physics.

# Conversion of space and time

If there exists a constant, characteristic speed, then one may speak of the characteristic conversion of space and time. For example, the speed of light in a vacuum is a defined constant in the SI system of units. So in physical science and its applications one may speak of the characteristic conversion of space into time and vice versa. This means that even if in some sense light curves (as by gravity), then the path of light is a geodesic, that is, equivalent to a straight line.

In other contexts, there may be no such characteristic speed but still there may be a constant speed within a specified context, which serves as a contextual conversion of space and time. This allows a map with a consistent scale, for example this map of the London Tube:

http://www.oskarlin.com/images/timetravel_no_zones_old_colours.pdf

Informally, this is done quite often. When asked how far away something is, we answer with the travel time by car or other mode.

Now the surprising thing is that the Lorentz transformation arises just because there exists such a conversion between space and time. It shows how to transform particular velocities in the context of a conversion speed between space and time. See the previous posts on the Lorentz transformation.

# Homogeneity and isotropy

A circle or sphere are omnidirectional in two or three dimensions, respectively. This is equivalent to isotropy, uniformity in all directions. A straight line is unidirectional but multiple straight lines may require multiple dimensions. This is equivalent to rectilinear homogeneity.

Pure space or average space is homogeneous and isotropic. Then space may be modeled by one dimension, although since the word dimension usually has to do with degrees of freedom or potential directionality, we say it has three dimensions.

It’s the same with time. Pure time or average time is homogeneous and isotropic, and may be modeled by one dimension, though it has three degrees of freedom and so we say it has three dimensions. If time is isotropic, only one dimension is needed to model it. If time is anisotropic, then three dimensions are needed to model it.

This is like the duality of wave and particle in quantum mechanics. Space and time have one or three dimensions depending on the aspect modeled.

Universal simultaneity requires homogeneity: “the transport of an ideal clock without distortion of time-intervals, requires a homogeneous space” (*).

In surface transportation a distinction can be drawn between congestion-type and current-type hindrances to travel. Radial congestion, such as a simple model of a city with a central business district, is isotropic. Travel across or in a river current could be modeled as rectilinearly homogeneous.

The conclusion is that homogeneity and isotropy come with a pure or average conception of space or time and require only one dimension to model. But the particulars of many situations do not exhibit either homogeneity or isotropy and so require three dimensions to model.

# Multiple dimensions of time in transportation

There are many examples of two-dimensional time maps in transportation, although the authors do not acknowledge the multiple dimensions of time in their maps. Let’s start with a map of New York travel times on commuter rail which shows the travel time in minutes from Manhattan to commuter rail stations during the evening rush period. The distances on the map match travel times rather than travel distances.

A second map shows travel times for trains from Paris (stunden means hours). Another gives travel times in Paris under three modes of transportation (in minutes). Another shows isochronous travel from Paris, between 2 hours to 15 hours long.

This website shows time-space maps with railway times in Europe and a series of maps showing a day’s journey through history. Here are two travel time contour maps of Atlanta (15 minute contours).

All these maps show two dimensions of time. Can you see it, too?

# Optimizing travel time routes

It is not unusual to seek the route in space that minimizes travel time, for example, a drive from point A to point B may go out of the way to include a high-speed facility that reduces travel time even if it increases distance traveled.

But what about routes in time? Does it ever make sense to minimize the distance traveled? Yes, for example, when a resource cost is related to the distance traveled, as with some taxi fares, or the wear on tires, or for railroad track access. In other cases, minimizing time and distance go together, as with the great circle routes of ships or aircraft.

A race could be delimited by an amount of time rather than a distance. The goal would be to maximize the distance traveled in a fixed time period, rather than to minimize the travel time over a fixed distance. For example, walk-a-thon participants may seek pledges of support for every mile they travel within a specified time period.

An indirect example would be those sports that take place over a fixed time period, such as basketball, football, and hockey: the goal is to score the most points, which usually involves moving the ball or puck the greatest distance (though there are strategies to control the ball and run out the clock).

Commuters seek to minimize the travel time rather than the distance traveled, so a map with distances is not as important as a map with travel times during rush hour. There are apps that show (or speak) the route with the shortest time. Restaurants near businesses need to take the fixed lunch hours of their potential customers into account; short travel time routes may lead through walkways, highways, or public transit stops.

In all these cases, the route through time is more important than the route through space.

# Modes of travel

Travel, that is, the movement of something, includes transporting and signalling. To transport means to take something (e.g., people or goods) from one place to another by means of a vehicle or the like (e.g., a car). To signal means to transmit information or instructions from one place to another through a medium or the like (e.g., sound).

A mode of travel is a means, technology, or technique for moving something. Travel modes may be distinguished by whether they are on or through a solid (e.g., land), on or through a liquid (e.g., water), through a gas (e.g., air), or in a vacuum (e.g., outer space).

A mode of travel has a free-flow speed, which is the speed attained in which there are no impediments to travel in that mode. This is the highest normal speed in the mode but may not be the highest possible speed. If local conditions (e.g., topography) do not exert significant influence, the free-flow speed serves as a reference speed for the mode because it is homogeneous and isotropic.

There are two basic perspectives on travel and the measurement of travel: (1) the most common perspective looks from a state of rest and observes something traveling relative to it; (2) the second perspective looks from a state of travel and observes something that is not moving (e.g., a landscape). The basic measure for perspective (1) is velocity, the change in distance traveled per unit of travel time. The basic measure for perspective (2) is the inverse of velocity, the change in travel time per unit of distance traveled, which could be called invelocity from the words inverse and velocity.

The first perspective (1) is the spatial perspective because it is from a state of rest, which is associated with space, with something that seems to be there apart from time. The second perspective (2) is the temporal perspective because it is from a state of travel, which is associated with the passage of time, with something that takes time. The Galilean and Lorentz transformations apply to the spatial perspective but as we have seen there are similar transformations that apply to the temporal perspective.

# Lorentz for space and time

Consider again the now-classic scenario in which observer K is at rest and observer K′ is moving in the positive direction of the x axis with constant velocity, v. This time there is a characteristic constant speed, c. The basic problem is that if they both observe a point event E, how should one convert the coordinates of E from one reference frame to the other?

We return first to the Galilean transformation and include a factor, γ, in the transformation equation for the positive direction of the x axis:

rx′ = γ (rx − vtx)

where rx is the spatial coordinate and tx is the temporal coordinate. Only the coordinates of the x axis are affected; the other coordinates do not change.

The inverse spatial transformation is then:

rx = γ (rx′ + vtx).

# Galileo for space and time

Consider the now-classic scenario in which observer K is at rest and observer K′ is moving in the positive direction of the x axis with constant velocity v. The basic problem is that if they both observe a point event E, how should one convert the coordinates of E from one reference frame to the other?

First assume time is absolute and space is relative with no characteristic speed. Only the spatial coordinates in the positive direction of the x axis are affected. The other coordinates do not change. The transformation equation for the positive direction of the x axis is

rx′ = rx − vtx

where rx is the spatial coordinate and tx is the temporal coordinate in the positive direction of the x axis. The inverse transformation is

rx = rx′ + vtx.

rx′ + rx = rx + rx′ − vtx + vtx′,

tx = tx.

This is called the Galilean transformation.

Now consider the case in which space is absolute and time is relative with no characteristic speed. Only the temporal coordinates in the positive direction of the x axis are affected. The other coordinates do not change. The transformation equation for the positive direction of the x axis is

tx′ = tx − rx/v

and the inverse transformation is

tx = tx′ + rx′/v.

rx = rx.

This could be called the dual Galilean transformation since only temporal coordinates change.

# The speed of spacetime

For each mode of travel there are four speeds to consider: the minimum speed, the typical speed, the maximum speed, and the speed of particular objects. The more that impediments to travel are removed (e.g., other objects, the topography, the network), the more that speed reaches free flow.

In transportation, the free flow speed is slower than the maximum speed. For example, the maximum speed for a highway vehicle might be determined by the fastest speed of a vehicle on the Bonneville Salt Flats. Or by the fastest speed at a NASCAR stock car race. These speeds would be much faster than what is safe on a highway. In physics, the free flow speed and the maximum speed are the same because of the principle of least action.

If there exists a speed that is constant for all particles or vehicles, then there is a way to relate the space and time coordinates of every event. Depending on whether this special speed is the minimum, the maximum, or the typical speed, there will be a kind of Lorentz transformation of the coordinates.

A coordinate system is a map for representing objects in space and/or time. The origin point of a coordinate system represents the reference point for the other points represented.

Maps may be static and represent a rest frame or they may be dynamic and change with their location as with GPS devices. For example, the origin point of a dynamic map may represent the location of a moving vehicle.

A spacetime coordinate system needs a way to relate spatial and temporal coordinates. The relation may be very particular, localized, and complex or it may be general, universal, and simple. The simplest relation between space and time is a constant. Such a constant represents a speed.

All maps have a scale, e.g., one centimetre represents 500 metres. A map with a reference speed also has a time scale, e.g., one centimetre represents one minute. To represent an arbitrary point in spacetime requires two maps: one for the space coordinates and the other for the time coordinates. These may be combined if different axes have different units. Or time coordinates may be put on a space map (and vice versa) as isoline plots, i.e., isochrones or isodistances.

The relation between the scale of space and the scale of time on two related maps is the speed that ties them together. That is the speed of spacetime.

# Actual and default speeds

The actual speed of a particle or vehicle is the local conversion of duration and distance, that is, local time and space. The potential speed of a particle or vehicle is a characteristic speed for similar particles or vehicles. This characteristic speed is a default speed, to be used if the actual speed is not known.

A characteristic speed may be a local default speed, that is, relative to a specific particle or vehicle and not necessarily applicable elsewhere. A characteristic speed may be relative to someone’s viewpoint, perhaps the way they drive or an estimate of the amount of congestion. A characteristic speed may be a universal speed, an absolute for all particles and vehicles, and so for time and space in general. In any case, an actual speed over-rides any default speed.

A signal speed used in determining the location or velocity of a remote particle or vehicle is similar to a characteristic speed. It is as if a generic particle or vehicle were used to bring the news of what actually happened. This speed might be known but if a generic service is used, then the characteristic speed would be the signal speed.

A characteristic speed may be the typical speed, which may be subject to change, or a maximum speed, which may never change. There are advantages and disadvantages in either case. If the characteristic speed is the typical speed, then it errors are minimized if it is substituted for an unknown actual speed. If the characteristic speed is the maximum or optimum speed, then it shows the extreme case, which may bound the problem.

The speed of light in a vacuum is a maximum or perhaps optimum speed. A typical speed may be the average speed from some speed data or a nominal speed in round units. The specific medium may set the default speed, for example, if the medium is air and sound is used for signaling, the speed of sound may be the logical characteristic speed. For a specific particle or vehicle type, their typical or optimal characteristics may be decisive for the characteristic speed.