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Tag Archives: Transportation

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).

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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.

Adding them together gives

rx′ + rx = rx + rx′ − vtx + vtx′,

which easily leads to

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.

Adding them together leads to

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.

Time at Mach 1

In a sense every speed is a conversion speed, that is, a way to convert time into space and vice versa because multiplying a time interval (duration) times the speed of an object leads to the corresponding space interval (length). In some contexts, i.e., a transportation mode or physical medium, there is a particular speed, the conversion speed, that reflects the context in general and does not depend on the speed of any particular object in that context. This conversion speed applies to all objects in its context, except for the known speed of an object or signal.

In the context of high-speed jet travel, the speed of sound may be the typical speed of travel. Also, in the context of sound waves in air a standard speed reflecting ideal conditions may be the reference speed. In both these cases the conversion speed is the speed of sound, also known as Mach 1 because in that case the Mach number equals 1.

T. S. Shankara takes a related approach in his article, Tachyons via Supersonics (Foundations of Physics, Vol. 4., No. 1, 1974, p. 94-104). He draws a parallel between acoustic and electromagnetic waves, derives the Lorentz transformations in this way, and shows that the signal velocity is unrelated to its maximality. His goal is to suggest the possibility of tachyons — of which there is a large literature. This is consistent with what we have shown, too.

Converting space and time

To convert a length of space into a corresponding length of time requires a conversion factor. For physical reality that conversion factor is the speed of light: r = ct, where r is a spatial displacement, t is a temporal displacement, and c is the conversion factor. For a mode of transportation the conversion factor between space and time is a typical or conventional speed. One reason for such a convention is to communicate the length of time expected in order to traverse a corresponding length of space.

That this conversion factor is also a speed is extraneous to its status as a conversion factor. Whether or not anything travels at such a speed does not matter to the conversion of space and time. Whether or not one measures something going at such a speed does not matter either. All that matters is that the convention is accepted. The science community has agreed to define the speed of light in a vacuum as exactly 299,792,458 metres per second. A particular map may use a single conversion between space and time; everyone following that map has the same conversion factor.

Does that make the conversion of space and time subjective? Not necessarily, because a convention may be justified by an argument about objective reality. But it could be subjective if someone adopts their own conversion factor which no one else is using. That may reflect their driving style or preference for under or over estimation. The purpose of the conversion is relevant.

A conversion factor applies to a real or virtual phenomena. If people with frame of reference S have the same conversion factor as those with frame S′, then the conversion applies to both. So r = ct in S and r′ = ct′ in S′ because of the status of c as the conversion factor for both S and S′. This is how the Lorentz transformation can apply both to particle physics and everyday transportation.

Actual and potential time and space

I’m made the point that if we’re asked how far away a place is, we can answer in units of space or time. If we’re talking about stable places such as on land, the distance will not change but the length of time depends on a characteristic speed. In that case the length of time is a potential time it would take assuming the characteristic speed. The actual speed and thus the actual time may be different.

Here’s a scenario for example: Person X asks, How far is it downtown? Person Y responds, About half an hour, which assumes a typical speed. Person X starts driving downtown and after half an hour they haven’t arrived yet. How much longer will it take? Based on this simple scenario, the time potentially remaining is the distance remaining divided by the typical speed. The actual time is not known yet.

If there is a characteristic travel speed as for a guideline or a map, then space and time are convertible into one another as potentials. The conversion is actual either after the fact when the actual speeds are known or if the conversion is fixed, as with a vehicle that only goes one speed or light which has a constant speed.

Is there a potential length of space as well as this potential length of time? Yes, if the travel time is predetermined but the distance traveled is not. For example, if someone agrees to go for a walk but only has a certain amount of time, the potential length of the walk is the allotted time divided by the typical pace (e.g., in minutes per mile).