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Tag Archives: Transportation

Equality of space and time

“How far is it to X from Y?” That everyday question can be answered either by a distance or a duration with a mode of travel (e.g., walking, driving, flying). The interchangeability of a length of space and a length of time leads to two simple conclusions: (1) time has as many dimensions as space does, and (2) space and time are symmetric. In short, space and time are equals in an almost political way: we should not discriminate against one or in favor of the other if possible.

“What is the length of a coastline?” This question is used to point out that the coastline of a landmass does not have a well-defined length. If someone is walking the coastline or examining high-resolution aerial photographs, they will find a longer coastline than someone flying along the coastline or looking at low-resolution photographs. This reinforces that mode of travel or resolution are needed to specify a distance properly. It is similar for travel on land: flying between two cities is likely more direct than driving on a highway network. Which is the correct distance? It depends on the mode of travel.

In physics the “mode of travel” is a light wave — but there may be exceptions. For example, a sound studio or study of whale sounds would be more concerned with the distance that sound travels. Or a phase space might have its own kind of distance. In any case, the mode is there whether it is specified or implicitly understood.

Another aspect of distance and duration is the path that is taken. That may be clear from the mode or there may be alternate routes within a mode. For example, there are many ways to travel from X to Y on a highway network. One may be the shortest distance of travel and another the shortest travel time. So the question of “how far is it?” leads to the response, “with the shortest distance or travel time?”

There is a desire in physics to describe physical laws independently of any observer. If distance and duration aren’t convertible, e.g., by a conversion factor, then how can this be done? The answer is via knowledge of a field, which in transportation terms corresponds to a transportation network. Then physical laws can specify which path will be taken. The key way to do this is via the principle of least action (or stationary action).

Kinds of relativity

A simple way to look at the world is to assume that space and time are absolute: the locations, the distances, the durations, speeds, and so forth as measured by one person are the same for everyone. That is, if my automobile speedometer shows 50 mph (80 kph), then the police with a laser gun at the side of the road will show the same speed.

For many purposes of everyday life, that works just fine. But for those who think about it more or those who perform experiments, that breaks down. Galileo Galilei was the first express a principle of relativity in his 1632 work Dialogue Concerning the Two Chief World Systems using the example of a ship travelling at constant velocity on a smooth sea: any observer doing experiments below the deck would not be able to tell whether the ship was moving or stationary. He still accepted absolute time, however.

We can call Galilean relativity “spatial relativity” since it applies only to space. Since we have seen the symmetry between space and time, we could develop a similar “temporal relativity” in which time is relative but space is not. This may seem odd at first but it is as consistent (and limited) as spatial relativity. For reference, here are the transformations for spatial and temporal relativity, given two reference frames, S and S’, with an event having space and time displacements r and t (r’ and t’) respectively, with S’ moving at constant velocity v relative to S, then:

r’ = r – vt and t’ = t for spatial (Galilean) relativity, and

r’ = r and t’ = t – r/v for temporal relativity.

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Lorentz interpreted

The question is how to interpret the Lorentz transformation. In a previous post, Lorentz generalized, a modest generalization of the Lorentz transform was derived. Absolute reference speeds were combined with a relative actual speed.

Let’s step back and look at a map of space and time:

Interstate Drive Times & Distances Sample

Interstate Drive Times & Distances Sample

This map of nodes and links on the U.S. interstate highway system displays travel distances and driving times between cities. If you look closer, you can see that it is based on a standard travel speed of about 50 mph (with some local variations). So each point on the network represents a travel distance and a travel time: in other words, space and time are in sync.

Now compare this map with some actual travel experience, say, one traveler going at 40 mph and another at 60 mph. If they start together, after one hour of travel they will have gone 40 and 60 miles respectively, compared to the standard of 50 miles. After one hour, the standard “map” distance is 50 miles but the actual distances are 40 and 60, so space and time are not in sync with these travelers.

The problem is space and time can no longer be mapped together: either the distance traveled or the travel time can be mapped but not both. At most all the distances for one travel time or all the travel times for one distance can be mapped.

A physicist approaching this situation might ask, is there some function of space and time that can still be mapped? Is there a quantity that is invariant no matter what the travel speed is? Can an alternate map be constructed?

The answer is yes and the key is the Lorentz transformation. Note that this is for an alternate map: if travel speeds equal the standard speed, no new map is needed. So we’re looking at speeds u and u’ that differ from the standard speed, c.

The alternate map has one limitation: it’s from the point of view of one traveler. But an alternate map can be constructed for any traveler and the principles of its construction are the same for all travelers. That’s the best that can be done.



Lorentz generalized

In some ways transportation is more general than physics, which is surprising. In terms of extent, from the microscopic to the astronomical, from extremes of temperature, etc., physics is the more general subject. But because transportation includes people, there are some additional possibilities. Let’s look at one transportation situation in which this is the case. (Note: we are not talking about transport theory here.)

Consider transportation in terms of positions in space and time, directions and speeds plus the expectations people have for a trip — in particular, what they see as their typical or expected travel speeds. The point is that people use a particular speed for trip planning and forecasting purposes, which may reflect general travel conditions or their personal travel experience, or simply their driving style. Call this the reference speed to distinguish it from their actual speed(s).

Let there be observer-travelers going in the same direction but in different vehicles (or trains, boats, etc.). Distinguish them by their frame of reference, unprimed or primed. Call their frames S and S’, their positions in space r and r’, in time t and t’, the actual speed of the second frame relative to the first v, and their reference travel speeds b and c respectively. Allowing different reference speeds is more general than the Lorentz transformation.

To make it more general we could say they may begin at different positions or their units of measure are different, but we’ll leave these as an exercise for the reader. The actual speeds could also vary over time but we’ll consider them constant.

Consider only the path/trajectory followed, i.e., one dimension of space and time each. Then we have: r = bt and r’ = ct’ as time-space conversions for each frame. We will follow the derivation of the Lorentz transformation (wavefront approach). A general linear transformation between (r, t) and (r’, t’) can be written as: r’ = ex + ft and t’ = gr + ht where the constants e, f, g, and h depend only on b, c, and v. The derivation is an exercise in algebraic manipulation with the following result:

e = 1 / sqrt(1 – v2/ b2) = γb,

f = -v e = -v / sqrt(1 – v2/ b2) = – v γb,

g = – (v / (bc)) γc,

h = (b/c) / sqrt(1 – v2/ c2) = (b/c) γc,

where γc = 1 / sqrt(1 – v2/ c2).

So the general Lorentz transformation is:

r’ = γb (x – vt),

ct’ = γc (b t – vx / b).

If b = c, there is only one reference speed for both traveler-observers, which is the requirement of the Lorentz transformation.

r’ = γ (x – vt),

t’ = γ (t – vx / c2).

This is the case with the speed of light, which acts as a reference speed to which all speeds can be compared.