Rates of motion are almost always expressed as a ratio with respect to time. For example, the average speed of a body is the travel distance of the body divided by the travel time. This makes the independent variable time and distance the dependent variable.

However, there is no physical dependency of motion on time rather than distance. One could just as well express the average rate of motion as the travel time of the body divided by the travel distance. The ratios are equally valid.

This is a general result. There is a binary symmetry between space and time. Travel distance and travel time are interchangeable as far as the equations of physics are concerned. J. H. Field has expressed this as a postulate for space-time exchange (STE):

(I) The equations describing the laws of physics are invariant with respect to the exchange of space and time coordinates, or, more generally, to the exchange of the spatial and temporal components of four vectors. (A four-vector has three components of length and one of distime.)

He avoids the question of 3D time by limiting the STE to the direction of inertial motion. Here we generalize the STE postulate to include 3D time:

(II) The equations describing the laws of physics are invariant with respect to the exchange of space and time coordinates, or, more generally, to the exchange of the spatial and temporal components of six-vectors. (A six-vector has three components of length and three of time.)

Field found the STE to violate Galilean symmetry, but this is incorrect because time is three dimensional, and there is a co-Galilean transformation symmetric to the Galilean transformation.

The STE postulate affirms the complete symmetry of space and time, which is built on the symmetry of length and duration. As distance is the metric of space, a kind of length, so distime is the metric of time, a kind of duration. The metric of space or time may be used to organize events linearly, with equivalence classes defined for events at the same position in the order.

In a paper titled Nothing but Relativity (Eur. J. Phys. 24 (2003) 315-319) Palash B. Pal derived a formula for transformations between observers that is based on the relativity postulate but not a speed of light postulate. In a paper titled Nothing but Relativity, Redux (Eur. J. Phys. 28 (2007) 1145-1150) Joel W. Gannett presented an alternate derivation with fewer implicit assumptions. Here we’ll use Pal’s approach to derive the time-space version.

Consider two inertial timeframes S and S´, where the second one moves with legerity u, along the t-axis, with respect to the first one. There are two other time axes. The coordinates and radial distance in the S-timeframe will be denoted by t and x, and in the timeframe S´ will be denoted with a prime. The time-space transformation equations have the form:

t´ = T(t, x, u) and x´ = X(t, x, u),

and out task is to determine these functions. A few properties of these functions can readily be observed. First, the principle of relativity tells us that if we invert the legerity in these equations, we must obtain the same functional forms:

t = T(t´, x´, –u) and x = X(t´, x´, –u).

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*** Revised from June 2017 ***

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, for space, and Q, for time. Then S = 2πR and T = 2πQ. Also, R = Qv, and Q = Ru, with speed, v, and pace, 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 from the perspective of the Sun toward each orbiting planet:

1. Each planet orbits the Sun in a circular path with radius R in 3D space.
2. The Sun is at the center of mass of each planet’s orbit.
3. The speed of each planet is a constant, v.

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The Galilean transformation is typically presented for motion in direction of the x-axis, with the other axes unchanged:

x´ = x – vt, y´ = y, z´ = z, and t´ = t,

where v is the relative velocity of the observers. This is incompatible with the Lorentz transformation, but more than that, it is inconsistent with the two-way (round-trip) speed of light in a vacuum.

The Lorentz transformation can be made compatible with the round-trip speed of light if light is considered to travel instantaneously to its observer, which is usually the final leg. The speed of light for the other part of the round trip can be inferred so that their harmonic mean equals c, which is the most that is known (see One-way speed of light).

That is, if the speed of light in the Lorentz transformation is allowed to approach infinity, then the transformation will approach the Galilean transformation. Here the Galilean transformation arises as a limit of the Lorentz transformation by the speed of light approaching infinity, rather than usual the relative velocity approaching zero.

The Lorentz transformation for motion in the direction of the x-axis is:

x´ = γ (x − vt), y´ = y, z´ = z, and t´ = γ (t – vx/c²), with γ = (1 – v²/c²)^–1/2,

where γ (gamma) is the Lorentz factor. As c → ∞, γ → 1 and t´ → t. This can only be the case for one part of the light trip, which we’re taking as the last part of the trip.

Why the last part? Because that’s what is observed, directly or reflected in a mirror. And in everyday conversation the place where something is observed to be is spoken of as where it is now. Even with the convention of a constant speed of light, one has to be very pedantic to keep correcting others and oneself by saying that where something is seen to be is in fact where it was in the past.

For a round trip, the speed of light for the part not directly observed can be inferred from the empirical result that the round trip speed equals the constant, c:

x´ = γ (x − vt), y´ = y, z´ = z, and t´ = γ (t – 4vx/c²), with γ = (1 – 4v²/c²)^–1/2.

The speed of light for the unobserved part is inferred from the necessity that the harmonic mean equals c:

(1/c₁ + 1/c₂) = 2/c,

where c₁ → c/2 as c₂ → ∞. This harmonic mean of speeds is the arithmetic mean of paces. What is actually measured is the pace of light from the independent length traversed in the dependent time.

This post continues the parallelism between length and time, and includes some new terms.

Length and time both have base units in SI metric: the meter (or better: metre to distinguish it from a device) and the second. They can both be associated with direction. Length in a direction is from or toward an event place. Time in a direction is from or toward an event time.

Multiple dimensions of length are called space. However, space can mean merely the space between two points. To designate 3D space, let’s use the Latin spatium (space). Analogously, let’s use the term tempium to designate 3D time (cf. Latin tempus, time).

Events ordered by time are in time order. Events ordered by length are in length order. Events ordered by importance could be said to be in magna order.

Things are persistent events. Things have length. Things have three dimensions of length. Events have duration. Events have three dimensions of duration. The extent of space between things is called distance. The extent of tempium between events could be called temstance.

Relative space is divided into here and there; “here I am, there I was, there I will be.” The present tense of space is here. The past or future tense of space is there. Here I am. Some places were traversed in the past. Some places will be traversed in the future.

Relative tempium is divided into now and then; “now I am, then I was, then I will be.” The present tense of tempium is now. The past or future tense of tempium is then. Now I am. Some times were traversed in the past. Some times will be traversed in the future.

Matter is a spatial substance. Figure is a temporal substance. Matter has mass, solidity. Figure has vass, lightness. Many sports move matter, such as a baseball pitcher throwing the ball. A figure skater traces out figures in space and time, that is, spatium and tempium.

Observers detect objects and events with objects. These objects are essentially passive; they must be made to do things by force and work.

Participants are subjects among subjects, actively engaging in events and making them happen. Subjects participate and participants are subjects.

The perspectives of an observer and a participant are inverses of each other. They are different attitudes. An observer has an attitude of standing apart from the world. A participant has an attitude of being part of the world. The world is placed in different contexts because of the attitude of the contact.

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