iSoul In the beginning is reality

Science in the center

There are many different musical temperaments that have been used to tune musical instruments over the centuries. They all have their advantages and disadvantages. But there is one musical temperament that is optimally acceptable: the equal temperament method in which the frequency interval between every pair of adjacent notes has the same ratio. This produces a temperament that is a compromise between what is possible and what is agreeable to hear.

Science faces many situations such as the challenge of musical temperament. Conventions and methods need to be adopted and there are multiple options, each with their advantages and disadvantages. There are those who promote one method and those who promote another method, often the opposite method. Should science pick one and force everyone to conform? Or should science find a compromise of some sort?

There is a way in the middle that is a compromise between extremes and alternatives. It is a conscious attempt to avoid extremes and biases, and seek a solution that is the most acceptable to all. This is science in the center, a science that minimizes bias. Although it might be called “objective,” that obscures the fact that it is a conscious choice.

I previously wrote about the need for a convention on the one-way speed of light. Science of the center would avoid the bias toward one direction of light and choose a one-way speed that is in the middle between all the possible speed conventions. This is the Einstein convention, which is part of his synchronization method.

Science in the center includes not biasing classifications either toward “lumping” or “splitting.” Nor should explanations of behavior be biased toward “nature” or “nurture.” The particulars of each case should determine the outcome, not a preference for one side or the other. If there’s any default answer, it’s in the center between such extremes.

Occam’s razor is understood to prescribe qualitative parsimony but allow quantitative excess. This is as biased as its opposite would be: to prescribe quantitative parsimony but allow qualitative excess. Science in the center would avoid the bias that each of these has by prescribing a compromise: there should be a balance between the qualitative and the quantitative. Neither should be made more parsimonious than the other. All explanatory resources should be treated alike; none should be more abundant or parsimonious than any other. I’ve called this the New Occam’s Razor, and it is an example of science in the center.

Circular orbits

With circular motion there is a radius and circumference that may be measured as distance or duration. Call the spatial circumference S, and the temporal circumference T, which is known as the period. Distinguish the spatial and temporal versions of the radius, R, by using Rs and Rt. Then S = 2πRs and T = 2πRt. Also, Rs = Rtv, and Rt = Rsu, with velocity, v, and celerity, u.

Circular orbits are a convenient entry into Kepler’s and Newton’s laws of planetary motion. Copernicus thought the orbits were circular, and most planetary orbits are in fact nearly circular. We have then three propositions:

  1. Each planet orbit the Sun in a spatially circular path.
  2. The Sun is at the center of mass of each planet’s orbit.
  3. The speed of each planet is constant.

Let’s follow the exposition given in Elements of Newtonian Mechanics by J.M. Knudsen and P.G. Hjorth (Springer, 1995), starting on page 6. Because the speed is constant, the acceleration follows the equation for uniform circular motion derived previously:

a = / Rs,

in which v is the speed and a is the acceleration. We have from the definition of speed:

v = S / T = 2πRs / T.

Elimination of v from these equations leads to

aRs / , or

a ∝ S / .

Kepler’s third law states that the orbital period, T, and the semi-major axis, A, are related as . For circular orbits this becomes

Rs³, or


Combining this with the equation for acceleration yields

a ∝ 1 / Rs², or

a ∝ 1 / , or

a ∝ 1 / T4/3, or

a ∝ 1 / Rt4/3.

Inserting the first acceleration into Newton’s second law leads to:

Fm / Rs²,

with force, F, and mass, m. The force is directed toward the Sun, with a magnitude inversely proportional to the square of the distance from the Sun.

Because of Newton’s third law, there is an equal and opposite force from each planet toward the Sun. Which is to say:

FM / Rs².

for mass, M. The combined law of gravitation is thus:

F = GmM/Rs²,

for some constant G.

Now consider gravity at the surface of the Earth. Newton showed that the gravitational force of a body would be the same if the mass were concentrated at its center, so the above equation can be used with the radius of the Earth, ρ²:

F = GmM/ρ² = mg,

with g as the acceleration of gravity on Earth. Then

g = GM/ρ².

If the known values of G, M, and ρ are inserted into this equation, the result is g = 9.8 m/s².

Let us now modify this derivation so that distance is the independent variable instead of time. This is better called a law of levitation since it is naturally directed toward the smaller mass. We have then three propositions:

  1. The Sun orbits each planet in a temporally circular path.
  2. The Sun is at the center of vass of its orbit.
  3. The pace of the Sun is constant.

Because the pace is constant, the prestination follows the equation for uniform circular motion derived previously:

b = / Rt,

in which u is the pace and b is the prestination. Again, distinguish the spatial and temporal versions of the radius, R, by using Rs and Rt. Then S = 2πRs and T = 2πRt. Also, Rs = Rtv, and Rt = Rsu, with velocity, v, and celerity, u.

We have from the definition of pace:

u = T / S = 2πRt / S.

Elimination of u from these equations leads to

bRt / , or

bT / , or

bT / Rs².

Kepler’s third law states that the orbital period, T, and the semi-major axis, A, are related as . For circular orbits this becomes

Rs³, or

, or


Combining this with the equation for prestination yields

b ∝ 1 / Rt1/3, or

b ∝ 1 / T1/3, or

b ∝ 1 / S1/2, or

b ∝ 1 / Rs1/2.

Inserting the latter prestination into Newton’s second law in the form of gorce, Γ, gives:

Γ ∝ n / Rs1/2,

with gorce, Γ, and vass, n. The gorce is directed away from the Sun, with a magnitude inversely proportional to the square root of the distance from the Sun.

Because of Newton’s third law, there is an equal and opposite gorce toward each planet from the Sun. Which is to say:

Γ ∝ N / Rs1/2,

for vass, N. The combined law of levitation is thus:

Γ = HnN/Rs1/2,

for some constant H. Since this is a function of the inverse square root instead of the inverse square, it is more sensitive to changes in Rs, as with a body in radial motion.

Consider levity at the surface of the Earth, using the above equation with the radius of the Earth, ρ:

Γ = HnN/ρ1/2 = nh,

with h as the prestination of levity on Earth. Then

q = HN/ρ1/2.

The values for H, N, and ρ may be inserted into this equation to determine the value of h.

Foundations of mechanics for time-space

The first edition of New Foundations for Classical Mechanics (1986) by David Hestenes included “Foundations of Mechanics” as Chapter 9. This was removed for lack of space in the second edition, but is available online as a pdf here. This space-time foundation may serve as a guide for the foundation of mechanics for time-space. To do so requires terms and correspondences in addition to switching space and time:

particle → eventicle (point event), body → event, instant → spot, clock → odologe, reference frame → reference timeframe.

Let’s focus on section 2 “The Zeroth Law of Physics” and start with the second paragraph on page 8, revising it for time-space:

To begin with, we recognize two kinds of entities, eventicles and events which are composed of eventicles. Given an event R called a reference timeframe, each eventicle has a geometrical property called its time position with respect to R. We characterize this property indirectly by introducing the concept of Temporal Space, or Relative Time, if you prefer. For each reference timeframe R, a temporal space T is defined by the following postulates:

  1. T is a 3-dimensional Euclidean space.
  2. The time position (with respect to R) of any eventicle can be represented as a point in T.

The first postulate specifies the mathematical structure of a temporal space while the second postulate supplies it with a physical interpretation. Thus, the postulates define a physical law, for the mathematical structure implies geometrical relations among the time positions of distinct eventicles. Let us call it the Law of Temporal Order.

Notice that this law asserts that every eventicle has a property called time position and it specifies properties of this property. But it does not tell us how to measure time position. Measurement is a separate matter, since it entails correspondence rules as well as laws. In actual practice the reference timeframe is often fictitious, though it is related indirectly to a physical event. Our discussion is simplified by feigning that the reference timeframe is always a real event.

We turn now to the problem of formulating the scientific concept of space. We begin with the idea that space is a measure of motion, and motion is a change of time position with respect to a given reference timeframe. The concept of space embraces two distinct relations: spatial order and temporally remote coincidence. To keep this clear we introduce each relation with a separate postulate.

First we formulate the Law of Spatial Order:

The motion of any eventicle with respect to a given reference timeframe can be represented as an orbit in temporal space.

This postulate has a semantic component as well as a mathematical one. It presumes that each eventicle has a property called motion and attributes a mathematical structure to that property by associating it with an orbit in temporal space. Recall that an orbit is a continuous, oriented curve. Thus, an eventicle’s orbit in temporal space represents an ordered sequence of spatial positions. We call this order a spatial order, so we have attributed a distinct spatial order to the motion of each eventicle.

To define a physical space scale as a measure of motion, we select a moving eventicle which we call an eventicle odologe. We refer to each successive time position of this eventicle as a spot. We define the distance interval Δs between two spots by

Δs = cΔt,

where c is a positive numerical constant and Δt is the arc time of the odologe’s orbit between the two spots. Our measure of space is thus related to the measure of duration in temporal space.

To use this distance scale as a measure for the motions of other eventicles, we need to relate the motions of eventicles at different times. The necessary relation can be introduced by postulating the

Law of Coincidence:

At every spot, each eventicle has a unique time position.

This postulate determines a correspondence between the points on the orbit of any eventicle and points on the orbit of an odologe. Therefore, every eventicle orbit can be parametrized by a space parameter defined on the orbit of an eventicle odologe.

Note that this postulate does not tell us how to determine the time position of a given eventicle at any spot. That is a problem for the theory of measurement.

So far our laws permit orbits which are nondifferentiable at isolated points or even at every point. These possibilities will be eliminated by Newton’s laws which require differentiable orbits. We include in the class of allowable orbits, orbits which consist of a single time point through some distance interval. An eventicle with such an orbit is said to be fixed with respect to the given reference timeframe through that distance interval. Of course, we require that the eventicles composing the reference timeframe itself be fixed with respect to each other, so the reference timeframe can be regarded as a fixed event.

Note that the pace of an eventicle is just a comparison of the eventicle’s distimement to the distimement of an eventicle odologe. The pace of the eventicle odologe has the constant value 1/c = Δt/Δs, so the odologe moves uniformly by definition. In principle, we can use any moving eventicle as an odologe, but the dynamical laws we introduce later suggest a preferred choice. It is sometimes asserted that a periodic process is needed to define an odologe. But any moving eventicle automatically defines a periodic process, because it moves successively over time intervals of equal length. It should be evident that any real odologe can be accurately modeled as an eventicle odologe. By regarding the eventicle odologe as the fundamental kind of odologe, we make clear in the foundations of physics that the scientific concept of space is based on an objective comparison of motions.

We now have definite formulations of time and space, so we can define a reference system as a representation x for the possible time position of any eventicle at each distance r in some distance interval. Each reference system presumes the selection of a particular reference timeframe and eventicle odologe, so x is to be interpreted as a point in the temporal space of that timeframe. Also, a reference system presumes the selection of a particular destination for time and space and particular choices for the units of distance and duration, so each position and time is assigned a definite numerical value. The term “reference system” is sometimes construed as a system of procedures for constructing a numerical representation of time and space.

After we have formulated our dynamical laws, it will be clear that certain reference systems called alacritous systems have a special status. Then it will be necessary to supplement our Law of Coincidence with a postulate that relates coincident bodies in different alacritous systems. That is the critical postulate that distinguishes classical mechanics from special relativity, but we defer discussion of it until we are prepared to handle it completely. It is mentioned now, because our formulation of time and space will not be complete until such a postulate is made.

It is convenient to summarize and generalize our postulates with a single law statement, the Zeroth (or Temporospatial) Law of Physics:

Every real event has a continuous history in time and space.

Distance, duration, and angles

Let’s follow the orbit of a particle or the route of a vehicle as a curvilinear function with associated directions at every point. Measurement produces travel distance r, travel time t, with directions θ and φ. The directions may be considered as functions of either travel distance or travel time: θr, φr, θt, or φt. There are accordingly four possibilities:

(r, t, θr, φr), (r, t, θt, φt), or (t, r, θr, φt), or (t, r, θt, φr).

The latter two may be made equal by a change of convention for measuring the angle. These may be represented rectilinearly as:

(t, rx, ry, rz), (r, tx, ty, tz), (rw, rx, ty, tz), or (tw, tx, ry, rz).

The latter two may be made equal by a change of convention for the axes.

Three possibilities remain: (3D space + 1D time), (1D space + 3D time), or (2D space + 2D time).

An example of the third possibility would be a traveler who measured their horizontal angle relative to magnetic north and their vertical angle relative to the sun. Since magnetic north is (approximately) fixed, it serves to measure the horizontal angle spatially. Since the sun’s position continually changes, it serves to measure the vertical angle temporally. The result is (2+2) with (r, θr) and (t, φt).

Or one could do the opposite and measure the horizontal angle temporally, as with a sundial, and the vertical angle spatially, as with a theodolite. The result is (2+2) with (t, θt) and (r, φr).

If both angles are measured relative to a fixed point, then the result is (3+1) or (t, r, θr, φr). If both angles are measured relative to a moving point, then the result is (r, t, θt, φt). The moving point should be moving at a constant rate, or at least a constant acceleration.

If three coordinates are measured relative to a fixed axis, then the result is (1+3) or (t, rx, ry, rz). If three coordinates are measured relative to a rotating axis, then the result is (r, tx, ty, tz). The moving axis should be moving at a constant rate, or at least a constant acceleration.

The potential reality of (r, t, θr, φr, θt, φt) collapses to one of the possibilities above in the act of measurement. The potential reality of (rx, ry, rz, tx, ty, tz) collapses to one of the rectilinear possibilities above in the act of measurement.

Alphabetical glossary

Here is an updated glossary for time-space (1D space + 3D time). See also the Time-Space Glossary in the menu above.

alacrity is the nonresistance of an event to a change in its state of motion.

celentum (ce∙len′∙tum) is the vass times the celerity. From Latin cele(r), swift, + (mome)ntum. Units of s kg‑1 m‑1. Symbolized by q = nu.

angular celentum for a point particle is the cross product of the particle’s time vector, t (relative to some origin), and its celentum vector, q = nu.

celerity (ce∙ler′∙i∙ty) is the ratio of the distimement to the magnitude of displacement of an event. It is a vector with temporal direction and magnitude, which is a pace. From literary term celerity, swiftness of movement. Units such as s/m or min/km. Symbolized by or u. Mean celerity is Δtr. Instantaneous celerity is dt/dr.

center of vass is the harmonic mean point in time of an event’s points, weighted by their vasses.

distimement (dis∙time′∙ment) is the vector from an initial point in time to a final point in time. From dis + time + ment. Units such as s, hr. Symbolized by t.

effort is a constant gorce, Γ, that moves an event the distimement t: V = Γt.

event is a collection of points in time that can change position over a distance of space.

exertion is the space rate of making effort: V/s = Γu.

Galilei (or Galilean) time-space is a context in which the measurement of space is the same for all observers (i.e., absolute space), whereas the measurement of time is relative to the motion of each observer. The Galilean time-space transformation is: = t – ur, and for all other coordinates the primed and unprimed values are equal.

gorce is the prestination times vass or divided by mass. Units of s kg-1 m-2. Symbolized by Γ = nb.

Lorentz time-space is the relativistic 1D space + 3D time. It includes a factor, γ, along with the modal pace, þ: r′ = γ (rt/u) and t′ = γ (trþ²/u), with γ = (1 – þ²/)–1/2. This applies only if |u| > þ. The superluminal case is: t′ = γ (t – ur) and r′ = γ (rut/þ²) with γ = (1 – /þ²)–1/2, which applies only if |u| < þ.

odologe (o′∙do∙loge) is a device that measures 1D distance continuously. From Greek odo(s), way/path/road + (horo)loge, clock.

pace is the ratio of the magnitude of distimement to the magnitude of displacement of an event. It is a scalar quantity with magnitude but no direction, but it may be a component of a vector. The magnitude of a celerity is a pace. From Latin passus, a step or stride, from the unit of length in the denominator. Units of s/m, min/km, etc. Mean pace is Δtr. Instantaneous pace is dt/dr.

pace of light is the pace of light in a vacuum, which equals 3.335641 ns/m. Symbolized by þ (thorn).

plime is a particular portion of time, whether of definite or indefinite extent. From pl(ace) + (t)ime.

prestination (pres∙ti∙na′∙tion) is the rate of change of celerity with respect to the magnitude of displacement. Negative prestination is deprestination. Verb is prestinate. From Italian, presto, quickly. Units of s/m². Symbolized by b.

rotational alacrity (moment of alacrity) is the second moment of vass with respect to time from an axis, t: J = ∫ t2 dn.

strophence is the rate of change of angular celentum of an event, σ = Iβ.

synmacronize means to calibrate measuring rods for 3D time + 1D space. From syn + macron + ize (“to occur at the same length”).

temporal direction is direction in a 1D + 3D time geometry, which is direction toward an event. Examples: toward the sunrise, toward magnetic north (which moves), toward the final stop (on a transit schedule).

tempus is a location in time, or the set of all points in time whose location is determined by specified conditions. From the Latin word for time.

time is either (1) the duration of a trajectory synchronously compared with a reference trajectory; (2) distimement; (3) position within a standard of continual change, i.e., a clock. Units of seconds, hours, etc. Symbolized by t (with or without subscripts).

time-space is a four-dimensional continuum with one dimension of space and three dimensions of time.

travel distance is the 1D distance measured along the world line of a body. Also called proper distance.

vass is the nonresistance of an event to a change in its state of motion when a net gorce is applied; it is the inverse of mass. Units of kg-1. Related to vast, as in sparse mass. Symbolized by n.

Center of vass

The vass is to time (duration) as the mass is to space (distance). As noted before here, the vass can be measured by a similar procedure as the mass. The mass and vass are inverses with opposite uses.

center of mass

The center of mass is the point that two or more particles (point masses) are balanced (or one large mass is balanced). For two particles, m1 and m2 that are located at points x1 and x2, respectively:

Center of Mass (CM) = (m1x1 + m2x2)/(m1 + m2),

which is the weighted arithmetic mean with the masses as the weights. This is similar to the momentum, in which the velocity is weighted by the mass: mv.

The center of vass is the point in time that two or more particles are balanced. For two particles, m1 and m2 that are located at points in time t1 and t2, respectively:

Center of Vass (CV) = ((n1/t1 + n2/t2)/(n1 + n2))–1

= ((1/(m1t1) + 1/(m2t2))/(1/m1 + 1/m2))–1,

which is the weighted harmonic mean with the vasses as the weights. This is similar to the celentum, in which the celerity is weighted by the vass: nu.

In order to generalize this, let’s use the derivation of the center of mass, as in Knudsen and Hjorth’s Elements of Newtonian Mechanics, chapter 9. Start with the time position vector, T, to find the center of vass for a system of eventicles:

N Tcv = Σ ni ti,

where ni and ti are the vass and the time position vector of the ith eventicle, and N = Σ ni is the total vass of the system. Then differentiate with respect to space (distance) to get

N Tcv´ = Σ ni ti´= Σ qi := Q,

where the total linear celentum of the system is denoted Q. In other words, the total linear celentum Q of a system of eventicles is the same as that of an eventicle with vass N moving the the celerity of the center of vass. This is also stated as

Q = N ucv,

where ucv is the celerity of the center of vass.

Equations of motion in space-time and time-space

First, here is a derivation of the space-time equations of motion, in which acceleration is constant. Let time (distimement) = t, position = r, initial position = r(t0) = r0, velocity = v, initial velocity = v(t0) = v0, v = |v| = speed, and acceleration = a.

First equation of motion

v = ∫ a dt = v0 + at

Second equation of motion

r = ∫ (v0 + at) dt = r0 + v0t + ½at²

Third equation of motion

From v² = vv = (v0 + at) ∙ (v0 + at) = v0² + 2t(av0) + a²t², and

(2a) ∙ (rr0) = (2a) ∙ (v0t + ½at²) = 2t(av0) + a²t² = v² ‒ v0², it follows that

v² = v0² + 2(a ∙ (rr0)).

Next, here is a derivation of the time-space equations of motion, in which prestination is constant. Let position (displacement) = r, time = t, initial time = t(r0) = t0, celerity = u, initial celerity = u(r0) = u0, u = |u| = pace, and prestination = b.

First equation of motion

u = ∫ b dr = u0 + bt

Second equation of motion

t = ∫ (u0 + br) dr = t0 + u0r + ½br²

Third equation of motion

From u² = uu = (u0 + br) ∙ (u0 + br) = u0² + 2r(bu0) + b²r², and

(2b) ∙ (tt0) = (2b) ∙ (u0r + ½br²) = 2r(bu0) + b²r² = u² ‒ u0², it follows that

u² = u0² + 2(b ∙ (tt0)).

Christianity and science

A good summary of the myth of a long-running conflict between Christianity and science is in Timothy Larsen’s “War is Over, If You Want It” (September 2008). This warfare myth was invented in the 19th century by people such as TH Huxley who either should have known better or were purposely stirring up animosity. It is composed of individual myths that “support” it, such as the myth that Christians thought the earth was flat in the Middle Ages or the myth that Christians opposed the use of anesthesia during childbirth in the 19th century.

Larsen references Frank M. Turner’s “Contesting Cultural Authority” (Cambridge, 1993), as someone who “persuasively argued that the notion of a conflict between theology and science was generated as part of a campaign of professionalization by would-be scientists.” (p.150) It’s almost forgotten today, but the profession of a scientist didn’t exist until the late 19th century. Before that, science was developed by amateurs (including clerics) who had the leisure and interest. TH Huxley and others fought against such people because they stood in the way of a new class of professional scientists.

Although the warfare meme is vastly exaggerated, there are enough misunderstandings that the opposite idea of integration isn’t realistic. For example, it is said that many Christians quickly accepted Darwin’s theory of evolution in the 19th century and later. But what is overlooked is the fact that Christians misunderstood Darwin and substituted their own ideas of evolution by law or miracle.  Theistic evolution is common among Christians who either insert a law-bound version for Darwin’s undirected version or else invent undetectable miracles that make it God-directed.

Many have noted that modern science developed in a Christian matrix. If science jettisons its Christian roots, it loses a reason to expect an ordered universe that can be understood by human beings. It may either adopt a multiverse that just happens to have order in one universe or drift toward non-causal explanations in a chaotic universe.

Some scientists want to deepen the Christian roots of science rather than cut them off. They are mostly creationists or intelligent design proponents. Those who follow TH Huxley will have nothing of it. But some are willing to entertain new proposals. As the modern era comes to a close, we can expect that modern science will change into something else.

Derivation of Newton’s second law

It is often said that Newton’s laws are laws of nature, which can only be determined by observation. That’s true in the sense that the definitions required are based on inductive reasoning. However, once these definitions are in hand, it should be a deductive science. This is classical science, in the sense of Euclid, Archimedes, Plato, and Aristotle.

Here is a derivation of Newton’s second law in space-time, with distance, s, time, t, and mass, m:

Velocity, v := ds/dt.

Acceleration, a := dv/dt.

Mass flow rate, := dm/dt.

Weighted distance, ș := ms.

Momentum, p := dș/dt = d(ms)/dt = (mds + sdm)/dt = m(ds/dt) + s(dm/dt) = mv + sṁ.

If mass is constant, then p = mv.

Force, F  := dp/dt = d(mv + sṁ)/dt = d(mv)/dt + d(sṁ)/dt = (mdv/dt + vdm/dt) + (sd/dt + ds/dt) = ma + vṁ + sṃ + ṁvma + 2ṁv + sṃ, where = d/dt.

If mass is constant, then F = ma.

Here is a derivation of Newton’s second law in time-space, with vass, n = 1/m:

Celerity, u := dt/ds.

Prestination, b := du/ds.

Vass flow rate,  := dn/ds.

Weighted durationț := nt.

Celentum, q := dț/ds = d(nt)/ds = (ndt + tdn)/ds = n(dt/ds) + t(dn/ds) = nu + tṅ.

If vass is constant, then q = nu.

GorceΓ := dq/ds = d(nu + tṅ)/ds = d(nu)/ds + d(tṅ)/ds = (ndu/ds + udn/ds) + (td/ds + dt/ds) = nb + uṅ + tṇ + ṅu = nb + 2ṅu + tṇ, where  = d/ds.

If vass is constant, then Γ = nb.

Transformations with two speeds of light

I’ve discussed before how there may be two different one-way speeds of light since they are matters of convention. See here and here. This post is concerned with the transformation between two observers in such a case. I continue to work with the standard configuration, see here.

The standard transformation of reference frames begins with two frames in uniform relative motion along one axis (usually called x). Here we take the spatial axis to be the r-axis, which parallels the spatial axis of motion. Similarly, the temporal axis is taken to be the t-axis, which parallels the temporal axis of motion.

The notation here is indifferent as to the existence of other dimensions. If they exist, they are orthogonal to the direction of motion, whether spatial or temporal, and their corresponding values are the same for both frames. One can generalize the results here to other directions by rotation.

The two frames are differentiated by primed and unprimed letters. They coincide at time t = 0 and their relative speed is v. The one-way speed of light is c1 in one direction and c2 in the opposite direction such that the round-trip speed equals c, the standard speed of light in a vacuum. That is, the harmonic mean of c1 and c2 equals c.

The trajectory of a reference particle or wave that travels at the speed of light follows these equations in both frames:

r =  c1t or r/c1 = t and r′ = c2t′ or r′/c2 = t′.

Consider a point event such as a flash of light that is observed from each reference frame. How are its coordinates in each frame related?

The basic relations are: r′ = rtv = r (1 – v/c1) and t′ = t (1 – v/c1). Next a factor, γ, is included in the transformation equations:

r′ = γ (rvt) = γr (1 – v/c1), and

t′ = γt (1 – v/c1),

with equal values for the other corresponding primed and unprimed coordinates. The inverse transformations use the other speed of light:

r = γ (r′ + vt′) = γr′ (1 + v/c2), and

t = γt′ (1 + v/c2).

Multiply each corresponding pair together to get:

rr′ = γ²rr′ (1 – v/c1)(1 + v/c2).

Dividing out rr′ yields:

1 = γ2 (1 – v/c1)(1 + v/c2).

Solving for γ leads to:

γ = [(1 – v/c1)(1 + v/c2)]–1/2, which applies if |v| < |c|.

This is the Lorentz transformation with two one-way speeds of light.