iSoul In the beginning is reality

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.

Transportation and physics

Theoretical physics has been applied to a variety of disciplines such as economics and traffic flow theory. Here we are returning the favor by considering transportation as a model for physics; in other words, physics is like a transportation system.

Consider the space-time continuum as an infinitely dense transportation network. The spatial extent of the network is the union of the lengths traveled by all light rays. The temporal extent of the network is the union of the durations taken by all light rays.

We begin with the minimal assumptions that space is 3-dimensional, homogeneous, isotropic, and locally Euclidean. The space-time network is self-contained; there is no external space or time in which it exists. So measures of space and time are measures of travel on the space-time network.

Network regularities, called laws, are the same for every trip in which the speed is constant (the principle of relativity). We will focus for now on the special case in which each trip has a constant speed (special relativity).

There exists a speed that is constant for all travelers (also called observers). For transportation this is usually an average or typical speed. For physics this speed is the speed of light in a vacuum, c (the principle of invariant light speed).

The speed that is constant for all travelers enables a conversion between space (travel length) and time (travel duration). In physics this means that for any length, x, the corresponding duration is x/c and for any duration, t, the corresponding length is ct, i.e., x = ct. Since space is linearly related to time (and vice versa), time possesses all the properties of space as well: time is 3-dimensional, homogeneous, isotropic, and locally Euclidean.

As is well known, from these assumptions the Lorentz transformation may be derived. If time is reduced to its magnitude only, spacetime may be represented by the 4-dimensional Minkowski space. If space is reduced to its magnitude only, spacetime may be represented by a 4-dimensional space that is isomorphic to Minkowski’s.

From this we know that Δr2 = Δx2 + Δy2 + Δz2 and Δt2 = Δx12+ Δx22+ Δx32 with the convention that Δx2 = (Δx)2. Then Δr2 = c2Δt2 or equivalently c2 Δt2 – Δr2 = 0 so that is an invariant of spacetime. This means that there is a kind of conservation of spacetime; the network remains the same no matter what people do.

Average spacetime conversion

It may seem an exaggeration at this point to speak of “spacetime” while focusing on examples from everyday life rather than the physics laboratory. Yet after all we live in a physical world so physics should include that, too. But we’ve put off considerations such as the constant speed of light until we have a better handle on what the symmetry of space and time means.

Consider two vehicles on a highway, traveling in the same direction on different lanes. Vehicle S is traveling at speed b and vehicle S’ at speed b’ which is greater than b so at some point vehicle S’ overtakes vehicle S: call this point in spacetime the origin. The relative speed of vehicle S from vehicle S’ is b’ – b. The relative speed of vehicle S’ from vehicle S is b – b’.

Now consider that each vehicle has an odometer and a speedometer; we will ignore any clock or watch since they are not integrated with the frame. From the odometer readings, say r and r’ for vehicle S and S’ respectively, and the speedometer readings, which are b and b’, we can determine the travel times: r/b and r’/b’. The problem now is that we want to use b or b’ as conversion factors between space and time, that is, length and duration. It would be confusing or perhaps contradictory to use both b and b’ for this purpose, so it is natural to use their average:

b* = (b + b’)/2.

Then b* is the spacetime conversion factor in this case. It should not be surprising that an average would be used for the conversion factor; after all, the average or typical travel time between two points is what an isochrone map or travel distance and duration map would show. We’re not saying that the whole fabric of spacetime depends on these two vehicles — that would be ridiculous — but that the spacetime geometry of a transportation system would be represented in this way.

We may then follow the logic of the post on Galileo revised to determine that the transformation from one frame of reference to the other is the same as that given except for time:

t’ = t – vx / (b*)².

The average speed, b*, could be a typical speed rather than a calculated average. There might be reason to take it as the posted speed limit instead. When we open it to the speed of any object, there is good reason to take it as the absolute speed limit, c, the speed of light.

Education in a democracy

We interrupt this series of blog posts to address education in a democracy, especially in America where the public education system is deplorable — beyond reform and beyond revival — so it’s ready for replacement. Another reason for replacement is the right of the people to control their own education, rather than the state and the public sector unions. In order to give some flesh to how this could actually happen, let me sketch out a proposal.

There is considerable investment in state-controlled infrastructure for education; this should not be simply dismantled or sold. Let the state continue to own and control its operation — a contractor may be better than state employees, but that is a secondary matter. Let the classrooms be leased to independent (aka private) schools. The state then would control facility maintenance and operation of the common areas and common activities such as school lunch programs and extracurricular activities (e.g., team sports).

There is always the objection that all the independent schools in an area will not be acceptable to all the students (and their parents) or vice versa (the schools will have requirements or limitations so they don’t accept all the students. In that case — and in that case only — the state might control the schooling. For example, special needs children who require expensive assistance could be schooled by the state — but it would be better if the state provided financial assistance for independent schools to include special needs students. The other example is children who are not wanted by independent schools because they cause discipline problems. Or there are no schools acceptable to students because of their religion. Again, the state could be the educator of last resort in such cases.

In order to pay for this kind of education system, the parents should be given tax credits, vouchers, or state assistance. That way, the parents have the lead in selecting which school their children will attend, which assumes that the parents are the best ones to make that decision — they have the best interests of their children in mind. That is a democratic assumption, that the people are to be trusted with decisions that directly affect them.

The state can and should support education but the state need not operate school systems, not only because the state isn’t particularly suited to that but because parents are particularly suited to know what’s best for their children and because competition between independent schools is the best way to obtain excellence in education.

Galileo revised

Galileo was the first to see clearly that someone traveling in uniform motion would not be able to discern any difference from being at rest (without looking out the window). He imagined someone on a ship eating peas, and if a few dropped off their fork, there would be no difference from what would happen on land. This is called Galilean relativity, to distinguish it from the later Lorentzian or Einsteinian relativity, in which the speed of light is a constant.

Let’s consider a standard situation with two observers and their respective frames of reference, one moving with a constant velocity relative to the other, as in this illustration from this article. So there are two observers/frames, S and S’, with S’ moving at a velocity v such that observer S uses coordinates x (length) and t (time) and observer S’ uses x’ and t’. The coordinates are arranged so that they are coincident at time t = t’ = 0. Then, as is well known, x’ = x – vt.

What about the time coordinate? (Here only the time coordinate in the x direction is considered.) If, as Newton assumed, the measurement of time is the same for all observers, then t’ = t. However, this implicitly assumes a third frame which provides the independent time measurement. With the relativity of both time and space, we cannot use clocks as if they were from a reference frame independent of all others (see Movement and measurement).

Instead it is best to measure length and duration together. Let us do that by having (or imagining) measuring wheels moving along each axis (in both directions) in each frame of reference. To keep things simple, let there be two measuring wheels moving at constant speed, b, in opposite directions as a point event is jointly observed. Then b is a conversion factor between length and duration: x = bt and x’ = bt’. Combine these with the transformation above for x’ and the result is: t’ = t – vx/b2.

So the revised Galilean transformation for frames moving with the x axis is:

x’ = x – vt, y’ = y, z’ = z, and t’ = t – vx/b2.

Note that if the conversion factor is the speed of light, c, the time transformation becomes t’ = t – vx/c2. This is correct apart from considering the constancy of the speed of light, which leads to the Lorentz factor.

Addendum: The problem with this transformation is that it does not form a transformation group. The transformations for space and time need to be similar. This revised transformation will do:

x’ = (1 – v/b) x, y’ = y, z’ = z, and t’ = (1 – v/b) t.

Movement and measurement

If an object or event is in one position so that it can be measured at leisure, then time is not an explicit factor in its measurement. However, length units are defined in terms of time: “The meter is the length of the path travelled by light in vacuum during a time interval of 1/299 792 458 of a second.” –NIST

Also, it takes time to make two measurements or to position both ends of an object next to a measuring rod and read the result. Measuring lengths by using signals traveling at the speed of light explicitly includes time and leads to the Lorentz contraction of relativity.

When it is movement itself that is being measured, the role of time is critical. The result is that we need to be careful about what it means to measure time.

Clocks are commonly used to measure time. This means that there is a movement external to whatever is under observation that is used as a standard of time. Is this necessary? No, it is entirely possible to calibrate a clock-like apparatus that is within the frame of reference, not independent of it. Moreover, it is misleading to use clocks as if time were an absolute in its own independent frame.

It is best to measure length and duration together. One way to do this is to have (or imagine) measuring wheels traveling along each axis (both positive and negative directions) to measure both lengths and times together. When a measuring wheel comes to an object or event in its direction, the value is recorded in units of length and time. Another method would be to use pulses of light.

Distance without time

“You can have time without moving but you can’t move without any time.” Actually, no, that is not correct. I introduced this topic here but let me go into more detail in this post.

The previous post on measurement sets the background: we need to be very careful what it is we’re measuring and how. I’ll use the illustration of two trains again but will analyze it further. We won’t get into relativistic issues like clock synchronization here.

Let’s start with a standard scenario: there are two frames of reference, one attached to train A and the other attached to train A’. Both trains (or frames) are moving with constant speed in the same direction. Train A’ is traveling with a speed V relative to train A, that is, the speed of train A’ = V1 + V where V1 is the speed of train A.

It is important to note that both train A and train A’ are moving relative to the ground. The ground is an important aspect because, like an electrical ground, it provides the context for measurement of train movements. So there is an implicit frame G that represents the external perspective from the ground.

In order to make the parallels between space and time clearer, let’s dispense with ordinary clocks and rods and use two methods that work for measuring both distance and time. For movement relative to the ground we’ll use the click-clack of the train wheels as a measure of time (say, one second) and distance traveled (say, 25 metres). For movement relative to each train we’ll use a measuring wheel traveling down the aisle of each train at a constant speed.

Here’s where the independence of the denominator becomes important. With velocity, the elapsed time is the independent variable but with the co-velocity, the distance traveled is the independent variable. These independent variables are chosen first, then the dependent variable is measured. The independent variable must be chosen to be non-zero in order to avoid zero in the denominator.

The two trains are traveling along side one another and gradually train A’ pulls ahead of train A since it’s traveling faster. Their relative position is measured by the measuring wheel going down the aisle. Now say that when the two trains are side-by-side that train A’ slows down and goes at the same velocity as train A so they are at rest relative to one another.

The click-clack of the trains acts like a clock, showing that time flows ever onward. But it also shows that the distance traveled flows ever onward, too. None of this business about time being unique or unidirectional.

The measuring wheel now shows the two trains with a relative distance traveled of zero: the measuring wheel just begins and it’s done measuring their relative positions. But the measuring wheel also measures a relative elapsed time of zero; it takes no time to go between the beginning and ending of the relative positions of the trains.

Let’s put this together. When we measure velocity, we pick an independent time interval and then measure the distance traveled during this interval. Say we pick 10 click-clacks as the time interval (10 seconds). What distance does the measuring wheel measure during this time? Zero, so the relative velocity is zero.

When we measure the co-velocity, we pick an independent distance interval and then measure the travel time during this interval. Say we pick 10 click-clacks as the distance traveled (250 metres). What elapsed time does the measuring wheel measure during this distance traveled? Zero, for the same reason that the relative velocity is zero. So there is a distance traveled with zero elapsed time.

The key is that the denominator comes from an external movement, whereas the numerator comes from an internal movement. Is this correct? It’s exactly what is done with clocks: they are external to the movement observed, keeping time without regard to the phenomena under observation. In order to switch space and time, we have to completely switch them: so we use travel distances instead of clocks and relative elapsed times instead of distances traveled.

We have again shown that travel time and distance traveled may be interchanged, that is, space and time are symmetric.

Measurement

Measurement is the act of comparing something, X – an object, an event, a phenomenon, anything that can be compared – with an independent standard unit and its multiples, and then assigning the corresponding quantity of units to X as the measure of that aspect (characteristic, property) of X.

I want to focus on the independence of the measurement standard. This is easy to see in the case of clocks. Every clock is independent of events that happen at points of time measured by the clocks. But so is every ruler or measuring wheel independent of the objects they measure.

In order to measure movement we need measurements with the same standard at two or more points. To measure rates of movement requires (1) measurements with the same standard at two points in time or space along with (2) measurements of a different property at the same two points in time or space. The first two measurements (1) are chosen independently of the second two measurements (2). The second two measurements (2) use an independent standard of measurement but are measured at the times or places corresponding to the first two measurements (1); that is, the second two measurements (2) are dependent on the first two measurements.

Measurement is only possible because of the homogeneity and isotropy of space and time. Because of that, measuring devices can be moved to an object, event, phenomenon, etc. in order to measure it by contact in space and time. Or signals may be used to measure non-contact objects and non-simultaneous events, but relativistic considerations will apply.

Velocity puzzle

A number of word problems involve vehicle or aircraft speeds over two distances or two time periods and ask what the average speed is. The student is expected to understand the difference between the space-mean speed and the time-mean speed (though these terms are not typically used).

What about the “average velocity”? Since velocity is a vector, is the average velocity the velocity of the resultant motion (displacement), with its magnitude (speed) and direction? Or does it mean the total distance traveled divided by the total travel time — but for what direction?

Say a vehicle travels east for 10 miles in 20 minutes, then travels north for 17 miles in 15 minutes. What is the average velocity? Here are two answers:

(1a) The total distance traveled is 10 + 17 miles, or 27 miles. The total travel time is 20 + 15 minutes, or 35 minutes. The average speed would be 27/35 miles per minute, or about 46 mph.

(1b) Then what is the direction? Take the direction of the displacement velocity. The vehicle travels east 10 miles in 20 minutes, or 30 mph. Then it travels north 17 miles in 15 minutes, or 68 mph. If we follow the triangle formed by these two velocities, the resultant vector is the hypotenuse of a triangle with sides of 30 and 68 mph, which is at an angle equal to the arctangent of 68/30 or about 66 degrees.

(2a) If we follow the triangle formed by the distances traveled, the resultant vector (displacement) is the hypotenuse of a triangle with sides of 10 and 17 miles, or about 20 miles. The direction is the arctangent of 17/10, or about 60 degrees.

(2b) If we follow the triangle formed by the travel times, the resultant vector (displacement) is the hypotenuse of a triangle with sides of 20 and 15 minutes, which is about 28 minutes. The direction is the arctangent of 20/15, or about 53 degrees.

(2c) Then the displacement velocity is 20 miles in 28 minutes or about 43 mph. For the direction, we would have to pick either the one from (2a) or (2b).

What is the answer? And why? Is it mere convention? If so, then we’re dealing with a symmetry.

Personality types of science

Aristotle described the four “causes” (really “becauses” or explanatory factors) that are required for a full explanation. These are called the material cause, the efficient cause (or mechanism), the formal cause (or design), and the final cause (or purpose).

It seems as though trying to cover all four causes at once is either too much to expect or that investigators have preferences as if there’s a kind of scientific personality. Compare the Myers-Briggs-Jung psychological typology, which is based on people’s preferences for two of four functions, characteristic of their psychological type. So, for example, an individual may prefer “intuitive feeling” to sensing and thinking, as these words are defined in the typology. All four functions are used in some respects but people have personal preferences for two of the four, a dominant function and an auxiliary function.

Aristotle’s approach to science focused on the material and formal causes, and so is called “hylomorphic” (a combination of matter and form). Modern science modifies this aspect of Aristotle and focuses on the efficient cause (the mechanism) and the material cause (perhaps it should be called “hylodynamic”). Some modern scientists even have a difficult time acknowledging the existence of other causes.

Intelligent design advocates are focusing on formal causes (or designs) and efficient causes (or mechanisms). They see a strong role for information such that the material cause is played down. This portends the rise of a new kind of science, one with a “scientific personality” preferring formal and efficient causes (perhaps it should be called “dynamorphic”).

It seems that creationists are increasingly focusing on formal and final causes. Without denying the existence of efficient and material causes, they emphasize the importance of God’s purpose and design over all (perhaps this should be called “telomorphic”). It may be that some will focus on final and efficient causes. These are other scientific personality types.

Can all these scientific personalities get along? There may be conflicts but it is to be hoped that over time as with human relations, scientific personality types will be able to at least understand one another and perhaps to cooperate.