I recently wrote about rest in space and time here. This post is about the opposite: immediate motion, arriving at a destination instantly.

Immediate motion means an infinite speed in space. An infinite speed results in an immediate change of place: something moves from one location to another in an instant. It’s here and there at the same time. The departure and arrival are simultaneous.

A body at infinite speed is at two places at the same time, but a speed ratio has a finite time interval. If it’s the same time, how can there be a finite time interval?

For speed the time interval is fixed as the length changes. If the speed approaches infinity, then the travel length in the numerator approaches infinity, so the time interval in the denominator becomes a smaller and smaller proportion and the ratio approaches infinity. The body is at two places sinultaneously.

Immediate motion also means a zero pace in time. A zero pace results in an immediate change of time: something moves from one time to another in an instant. It’s now and then at the same location. The departure time and arrival time are at the same location. I’m calling this simulocus.

But wait, two times at the same location seems like no motion at all. What gives?

For pace the length interval is fixed as the time changes. If the pace approaches zero, then the travel time in the numerator approaches zero, so the length interval in the denominator becomes a larger and larger proportion and the ratio approaches zero. The body is at two times simulocusly.

Does immediate motion exist? Not under the Lorentz transformation, in which there is a finite maximum speed. But the Galilean transformation implicitly uses an infinite speed of light. And the co-Galilean transformation implicitly uses a zero pace of light.

Two transformations of inertial reference frames are well-known: the Galilean and the Lorentz transformations. There is a third transformation as well, which will be called the co-Galilean transformation. Below is a derivation of all three transformations, closely following the paper Getting the Lorentz transformations without requiring an invariant speed by Andrea Pelissetto and Massimo Testa (American Journal of Physics 83 (2015), p.338-340). Their approach is based on the work of von Ignatowsky in the early 20th century.

We wish to characterize the transformations that relate two different inertial frames. Let us consider two inertial observers K and K′. Let r = (x, x₂, x₃) and w = (t,  t₂, t₃) be space and time coordinates for K and r´ = (x´, x₂´, x₃)´ and w´ = (t´, t₂´, t₃´) be the corresponding quantities for K′.

In order to simplify the argument, we will restrict our considerations to the subgroup of transformations involving x and t only, setting x₂´= x₂, x₃´ = x₃, t₂´ = t₂, and t₃´ = t₃. This is equivalent to choosing coordinates so that K and K′ are in relative motion along the x and t directions in K and the x′ and t´ directions in K´.

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Watches didn’t always exist. Neither did clocks that were transportable or manufactured in large quantities. I mention this because one way to determine the simultaneity of events is to have synchronized clocks transported to multiple locations – even an endless number of locations in theory.

How can an observer determine the simultaneous events from their frame of reference? Answer: simultaneous events are observed simultaneously by an observer. But how can this be reconciled with other observers who may observe the same events as non-simultaneous?

That is the point of relativity: applying transformations to coordinates from different frames of reference so that the equations of physics are the same in all reference frames. But relativity requires a convention of simultaneity (or a demonstration of what events are simultaneous events). Since I have defined time in terms of stopwatches rather than clocks, how can simultaneity be determined?

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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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