Visual physics and mathematics/What is a space-time reference frame?

(currently under reflexion)

A space-time reference frame without massive particles

Let us consider a one-dimensional space.

Let A and B be two photon trains moving in opposite directions. We assume that the photons within a given train are evenly spaced, like the crests of a periodic wave.

Let A(i) be the i-th photon of train A and B(i) be the i-th photon of train B. We assume that the photons of A and B are numbered in the direction opposite to their motion: A(i+1) follows A(i), and B(i+1) follows B(i).

The encounter between A(i) and B(i) constitutes an event e(i) in space-time. All events e(i) lie along the same time-like trajectory. The interval between e(i) and e(i+1) is constant and can serve as a unit of time measurement.

More generally, the encounter between A(i-j) and B(i+j) constitutes an event e(i,j) in spacetime. For a given j, all events e(i,j) lie on the same timelike trajectory T(j). All trajectories T(j) are parallel to one another, in the sense that they never intersect. They can therefore be identified as the trajectories of objects at rest relative to each other. The interval between e(i,j) and e(i,j+1) depends on neither i nor j; it is spacelike and can serve as a unit of distance measurement.

The events e(i,j) can therefore be regarded as a space-time reference frame, as they allow for the measurement of distances and durations.

In three-dimensional space, six photon trains traveling in opposite directions along three spatial axes are required to establish a space-time reference frame.

Consider two monochromatic plane waves that do not travel in exactly the same direction. There always exists a reference frame—indeed, an infinite number of them—in which these two monochromatic plane waves travel in opposite directions and have the same wavelength. In such a frame, the superposition of the two plane waves forms a standing wave. A standing wave simultaneously defines a unit of time (its period) and a unit of length (its wavelength). Two monochromatic plane waves traveling in different directions always define reference frames in which their superposition is stationary. Six monochromatic plane waves traveling in suitably chosen directions determine a unique rest frame: the frame in which their superposition is stationary.

Conclusion: restless particles, which always travel at the speed of light, such as photons, suffice to determine the geometry of spacetime. They define rest frames from which all geometric relations can be determined. Massive particles are not necessary to determine the geometry of spacetime. Restless particles therefore appear to be more fundamental than massive particles.

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