iSoul In the beginning is reality.

Traffic flow in time and space

The following is based on Wilhelm Leutzbach’s Introduction to the Theory of Traffic Flow (Springer, 1988), which is an extended and totally revised English language version of the German original, 1972, starting with page 3 (with a few minor changes):

I.1 Kinematics of a Single Vehicle

I.1.1 Time-dependent Description

I.1.1.1 Motion as a Function of Time

Given any trajectory then, in the time-dependent case:

x(t) = distance: as a function of time [m];

f(t) = v(t) = dx/dt = speed: as a function of time [m/s];

a(t) = dv/dt = d²x/dt² = acceleration: as a function of time, the change of speed per unit time [m/s²];

If the initial conditions are denoted, respectively, by t0, x0, v0, a0, etc., the following equations of motion result, with integrals from t0 to t:

x(t) = x0 + ∫ v(t) dt   (I.1)

v(t) = v0 + ∫ a(t) dt   (I.2)

x(t) = x0 + ∫ v0 dt + ∫∫ a(t) dtdt   (I.3)


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Dual Galilean transformation

The Galilean transformation is based on the definition of velocity: v = dx/dt, which for constant velocity leads to

x = ∫ v dt = x0 + vt

So for two observers at constant velocity in relation to each other we have

x′ = x + vt

with their time coordinates unchanged: t′ = t if their origins coincide.

The dual Galilean transformation is based on the definition of lenticity: w = dt/dx, which for constant lenticity leads to

t = ∫ w dx = t0 + wx

So for two observers at constant lenticity in relation to each other we have

t′ = t + wx

with their length coordinates unchanged: x′ = x if their origins coincide.

These transformations reinforce the proposition that time is not necessarily the independent variable, and so is best understood as measured by a stopwatch rather than a clock.

Changing coordinates for the wave equation

The following is based on section 3.3.2 of Electricity and Magnetism for Mathematicians by Thomas A. Garrity (Cambridge UP, 2015). See also blog post Relative Motion and Waves by Conrad Schiff.

The classical wave equation is consistent with the Galilean transformation. Reflected electromagnetic waves are also consistent with classical physics using the dual Galilean transformation, in which linear location is the independent variable. The dual Galilean transformation for motion in one dimension is: x′ = x; t′ = twx = tx/v, where w = 1/v is the relative pace of the moving observer, which is equivalent to their inverse speed.

Suppose we again have two observers, A and B. Let observer B be moving at a constant pace w with respect to A, with A’s and B’s coordinate systems exactly matching up at location x = 0. Think of observer A as at rest, with coordinates x′ for location and t′ for time, and of observer B as moving to the right at pace w, with location coordinate x and time coordinate t. If the two coordinate systems line up at location x = x′ = 0, then the dual Galilean transformations are

x′ = x and t′ = t + wx,

or equivalently,

x = x′ and t = t′wx.

Suppose in the reference frame for B we have a wave y(x, t) satisfying the wave equation

\frac{\partial^2 y}{\partial x^2}-k^{2}\frac{\partial^2 y}{\partial t^2}=0.

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Michelson-Morley experiment

This post relates to a previous post here.

The Michelson-Morley experiment is a famous “null” result that has been understood as leading to the Lorentz transformation. However, an elementary error has persisted so that the null result is fully consistent with classical physics. Let us look at it in detail:

The Michelson-Morley paper of 1887 [Amer. Jour. Sci.-Third Series, Vol. XXXIV, No. 203.–Nov., 1887] explains it using the above figures:

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Temporo-spatial light clock

This post builds on the post about the Michelson-Morley experiment here.

One “Derivation of time dilation” (e.g., here) uses a light clock, pictured below:

The illustration on the left shows a light clock at rest, with a light beam reflecting back and forth between two mirrors. The distance of travel is set at the beginning with the separation of the mirrors. It is like a race: one lap is a round-trip between the two mirrors. So the independent variable is distance, and the dependent variable is time. This is key to properly understanding the experiment.

In the stationary frame the round trip longitudinal distance between the two mirrors is 2L, the pace is 1/c, and the time of one cycle is T = 2L/c.

The illustration on the right shows an observer moving with lenticity wt transversely to the light clock. In this case there are two components of motion: the longitudinal axis and the transverse axis. These components are independent of one another since they are in different dimensions. The transverse axis is the same as the stationary case above: the total distance is 2L and the total time is 2L/c. The mean pace is 2T/(2cT) = 1/c.

The longitudinal axis is simply the Galilean transform: t′ = t + wx since motion in different dimensions is independent. If the light clocks coincide at t = 0, this is t′ = wx, which is what was given.

Thus the distance, time, and mean speed are the same for both observers and independent of their relative velocity, which is what the Michelson-Morley experiment found, though didn’t expect.

Michelson-Morley re-examined

There are many expositions of the famous Michelson-Morley experiment (for example here) but they all assume the independent variable is time, which is not the case. As we shall see, distance is the independent variable, and so the experiment is temporo-spatial (1+3). Let us examine the original experiment as it should have been done:

The configuration diagrammed above is as follows: the apparatus is presumed to travel with pace w relative to the aether. In it a light source travels with pace k = 1/c to a beam splitter, whereupon part of it travels a distance L longitudinally and is reflected back, whereas another part travels a distance transversely and is reflected back. Part of the light is sent to an observer, who looks for an interference pattern.

Since the distance L is fixed, distance is the independent variable. In the stationary frame the round trip longitudinal distance between the beam splitter and the mirror is 2L. Let the time that light travels longitudinally from the beam splitter to the mirror be T1, and let the time for the return be T2. Then

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Principle of relativity

The relativity of uniform motion was stated by Galileo in the 17th century, though it was known to Buridan in the 14th century. Galileo’s statement of the principle of relativity is in terms of ships in uniform motion:

… so long as the motion is uniform and not fluctuating this way and that. You will discover not the least change in all the effects named, nor could you tell from any of them whether the ship was moving or standing still [Galileo Galilei, Dialogues Concerning the Two Chief World Systems (February 1632), Stillman Drake tr. (University of California Press, Berkeley, 1962, pp 186-8.]

This has been applied to constant speeds or zero accelerations, but it could just as well be applied to constant paces or zero retardations. In any case, if the speed is constant, so is its inverse, the pace. Let’s see how this operates.

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Reflected motion

This post was inspired by Chandru Iyer’s post here.

Consider a light ray sent a certain distance s that is immediately reflected back. According to Newtonian mechanics if a light ray travels at speed c, then for a body moving at speed v relative to the stationary frame, the light ray should travel at the speed c − v one way and at speed c + v the other way.

The total distance of the light ray is 2s. The total time of the light ray is

\frac{s}{c-v}+\frac{s}{c+v} = \frac{s(c+v)+s(c-v)}{c^2-v^2}=\frac{2sc}{c^2-v^2}=2s \gamma^2 /c

Then the mean speed is

\frac{2s}{\frac{s}{c-v}+\frac{s}{c+v}}=\frac{2s}{\frac{2sc}{c^2-v^2}}=\frac{c^2-v^2}{c}=c(1-(v^2/c^2))= c/\gamma^2

However, according to Einstein’s relativity theory, the mean speed of light is a constant, c. So the above speed needs to be multiplied by the gamma factor squared, γ². As Iyer notes, this is accomplished by contracting the moving rulers by the factor (1/γ) and dilating the moving clocks by the factor γ.

But that is not the correct approach.

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Space and time reciprocity

This post is related to others, such as here. Consider an analogue clock:

Analogue clock

The movement of the hand clockwise relative to the dial is equivalent to the movement of the dial couter-clockwise relative to the hand. That is, the motion of the hand relative to the dial corresponds to the opposite motion of the dial relative to the hand. This is the reciprocity principle of relative motion.

This means that motion for the space frame and the time frame are opposites. The displacement direction and the dischronment direction move in opposite directions. The velocity of A relative to B for space corresponds to the velocity of B relative to A for time, and vice versa.

The interchange of space and time needs to take this into account. Their vectors are in opposite directions.

Inverse units, inverse arithmetic

The use of space (stance) as an independent variable and time as a dependent variable leads to inverse ratios. There is pace instead of speed, that is, change in time per unit of length instead of change in length per unit of time. But a faster pace is a smaller number, which is counterintuitive and contrary to speed, for which faster speeds are larger numbers. There are two approaches to dealing with this:

  1. Use two kinds of arithmetic: the usual one, in which zero signifies the smallest quantity, and an arithmetic in which zero signifies the largest quantity (or infinity). Then they are isomorphic, with their extremes corresponding inversely. For example, speed and pace both measure how fast a body is going, but they use different arithmetics. A large number for speed corresponds to a small number for pace, and vice versa.

  2. Use the same arithmetic for both but invert one when making a comparison. For example, speed and pace are effectively inverses (apart from which is the independent and which the dependent variable). Given a speed and a pace for some body, to compare them requires inverting either the speed or the pace. An arithmetic means of one corresponds to the harmonic mean of the other.

The first approach inverts the arithmetic, whereas the second approach inverts the units. The second approach is preferable because we are so accustomed to ordinary arithmetic that introducing an alternative would be unnecessarily difficult. It’s much easier to change units than to change arithmetic.

The second approach is found in the comparison of time mean speed and space mean speed: the space mean speed is the arithmetic mean of speeds with a common time unit, and the time mean speed is the harmonic mean of speeds with a common length unit. The time mean speed is the mean pace inverted, which is a kind of parallel speed, which uses harmonic algebra.

Relativity uses a quotient, β, which is the speed of a body divided by the speed of light. Since the speed of light is the highest speed, β is always between zero and one, inclusive, in which zero signifies rest and one signifies the speed of light. What corresponds to β for pace? At first it seems to be the pace of a body divided by the pace of light. But such a quotient is the inverse of β, which would require an inverse arithmetic. The second approach is to take the inverse of this, which equals β. This is consistent with the inverse correspondence between space and time.