iSoul In the beginning is reality

Coordinate lattices

Rindler’s Essential Relativity is a well-written monograph that we can use to explore time and relativity. He describes the coordinate lattice of a single inertial frame of reference (section 2.5). Let us consider it with an eye toward the corresponding temporal coordinate lattice.

Start with the observer at the origin of an inertial frame, with a clock, and measure the distance to any particle by bouncing a light signal off it and multiplying the elapsed time by c/2 (where c is the speed of light). Angle measurements are made with a theodolite to derive 3-dimensional coordinates of the particle. Test particles at rest are placed at all the lattice points — integer multiples of a given length — each with a clock synchronized with the master clock at the origin by a single light signal emitted at time t0; when this signal is received at any lattice clock, that clock is set to read t0 + r/c, where r is the predetermined distance from the origin.

This process is valid for a single inertial frame. With it the coordinates of all events can be read off locally, directly where the events occur, by suitable auxiliary observers. Rindler emphasizes that other types of signal could just as well have been used, for example cannon balls. I would add that probe vehicles on a street grid could also be used to set up such a lattice.

How does this read if we switch space and time? Let’s see.

Start with the observer at the spacetime origin of an inertial frame, with a ruler, and measure the duration to any particle by bouncing a light signal off it and dividing the distance traveled by 2c (where c is the speed of light). Angle measurements are made with a theodolite to derive 3-dimensional temporal coordinates of the particle. Test events at rest are placed at all the temporal lattice points — integer multiples of a given time — each with an event synchronized with the master event at the spacetime origin by a single light signal emitted at point p0; when this signal is received at any lattice event, that event is set to read p0 + ct, where t is the predetermined duration from the origin event.

The result is a lattice built out of durations, not distances. As these are related to the distances via the factor c, it is equivalent to a lattice of distances. But conceptually it is different and the result is that distance or length is the independent variable and time is the dependent variable.

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