This Quantum World/Appendix/Relativity

The ABCs of relativity

See also the Wikibook Special relativity that contains an in-depth text on this subject.

The principle of relativity

If we use an inertial system (a.k.a. inertial coordinate system, inertial frame of reference, or inertial reference frame), then the components $x,y,z$ of the position of any freely moving classical object ("point mass") change by equal amounts $\Delta x,\Delta y,\Delta z$ in equal time intervals $\Delta t.$ Evidently, if ${\mathcal {F}}_{1}$ is an inertial frame then so is a reference frame ${\mathcal {F}}_{2}$ that is, relative to ${\mathcal {F}}_{1},$

1. shifted ("translated") in space by any distance and/or in any direction,
2. translated in time by any interval,
3. rotated by any angle about any axis, and/or
4. moving with any constant velocity.

The principle of relativity states that all inertial systems are "created equal": the laws of physics are the same as long as they are formulated with respect to an inertial frame — no matter which. (Describing the same physical event or state of affairs using different inertial systems is like saying the same thing in different languages.) The first three items tell us that one inertial frame is as good as any other frame as long as the other frame differs by a shift of the coordinate origin in space and/or time and/or by a rotation of the spatial coordinate axes. What matters in physics are relative positions (the positions of objects relative to each other), relative times (the times of events relative to each other), and relative orientations (the orientations of objects relative to each other), inasmuch as these are unaffected by translations in space and/or time and by rotations of the spatial axes. In the physical world, there are no absolute positions, absolute times, or absolute orientations.

The fourth item tells us, in addition, that one inertial frame is as good as any other frame as long as the two frames move with a constant velocity relative to each other. What matters are relative velocities (the velocities of objects relative to each other), inasmuch as these are unaffected by a coordinate boost — the switch from an inertial frame ${\mathcal {F}}$ to a frame moving with a constant velocity relative to ${\mathcal {F}}.$ In the physical world, there are no absolute velocities and, in particular, there is no absolute rest.

It stands to reason. For one thing, positions are properties of objects, not things that exist even when they are not "occupied" or possessed. For another, the positions of objects are defined relative to the positions of other objects. In a universe containing a single object, there is no position that one could attribute to that object. By the same token, all physically meaningful times are the times of physical events, and they too are relatively defined, as the times between events. In a universe containing a single event, there is not time that one could attribute to that event. But if positions and times are relatively defined, then so are velocities.

That there is no such thing as absolute rest has not always been as obvious as it should have been. Two ideas were responsible for the erroneous notion that there is a special class of inertial frames defining "rest" in an absolute sense: the idea that electromagnetic effects are transmitted by waves, and the idea that these waves require a physical medium (dubbed "ether") for their propagation. If there were such a medium, one could define absolute rest as equivalent to being at rest with respect to it.