Brown's "inertial coordinate systems"

In his 1905 article "On the Electrodynamics of Moving Bodies", Einstein explained how clocks can be adjusted "in a coordinate system in which Newton's equations of mechanics hold" (p. 892).

There is a problem with this formulation since, if we have not already adjusted any distant clocks in a given frame of reference, we have no way of knowing whether or not Newton's laws hold in that frame, not to mention the fact that in special relativity Newton's laws do not precisely hold anyway.

So, what kind of coordinate system was Einstein really trying to identify? The answer given by Kevin Brown in his book Reflections on Relativity (2010) is that he was trying to identify "inertial coordinate systems", defined by Brown as systems "in terms of which inertia is homogeneous and isotropic, so that free objects move at constant speed in straight lines, and the force required to accelerate an object from rest to a given speed is the same in all directions" (p. 27).

This quote requires some unpicking.

First, what is "inertia"? According to Brown, the "principle of inertia" is the idea that "in the complete absence of external forces, an object would move uniformly in a straight line, and that, therefore, whenever we observe an object whose speed or direction of motion is changing, we can infer that an external force... is acting upon that object". Brown says this principle was first formulated in the 17th century and represents "the most successful principle ever proposed for organizing our knowledge of the natural world" (p. 22).

Of course, Brown's explanation of the principle of inertia introduces some new concepts that may be deemed to be in need of explanation, such as "uniform motion", "straight line" and "force". However, it seems to me that these can in turn be explained without resorting to any laws of mechanics and even without first adjusting distant clocks: "straight lines" can be equated with lines of sight in the absence of very massive bodies, such as stars; "uniform motion" can be established using signals emitted at regular intervals from a moving body; and instead of "in the absence of external forces" we could say "in the absence of specific sets of conditions, such as collisions, known to stop objects from moving uniformly and in straight lines".

So, a "spatial inertial coordinate system" could be defined as a series of markers which are arranged in straight lines at regular spatial intervals, established for example by means of standard rods, and relative to which objects that are not subjected to any force move uniformly in straight lines.

What is missing from this definition is time coordinates or, what amounts to the same thing, a procedure to adjust clocks in distant points.

As Brown points out, such a procedure to adjust distant clocks is implicit in Newton's "laws" of mechanics, which thus turn out to be not pure laws of nature but in part the result of a particular clock adjustment choice.

This choice was to use identical mechanisms to propel identical objects in opposite directions from the mid-point between two clocks at rest in a spatial inertial coordinate system and to set those clocks to the same time when the objects arrive there.

Once this has been done, the "speed" of such an object propelled by such a mechanism can be measured using those two clocks and the distance between them. Knowledge of that speed can then be used, together with knowledge of the distances between other clocks, to adjust any other clocks at rest in the same spatial inertial coordinate system and thus to establish a full inertial coordinate system. It is found empirically that the time coordinates in such a coordinate system do not depend on the kind of object or acceleration mechanism used to adjust distant clocks.

It is then clear that in such a coordinate system inertia is "isotropic", in other words if identical objects at rest in the system are accelerated by the same kind of mechanism in different directions, they reach the same speed. This is guaranteed by the way we have adjusted our clocks. And this clock adjustment convention, or choice, or decision, was written into Newton's second "law", which states that the "change of motion is proportional to the motive force impressed, and is made in the direction of the right line in which that force is impressed" (quoted in Brown, p. 23).

Newton's clock adjustment procedure may well seem perfectly plausible even to a modern mind: if we use identical mechanisms to propel identical objects in opposite directions along paths of equal length, then we may surely assume that they arrive at the end of those paths at the same time, by virtue of the very notion of simultaneity. So it seems that we are entitled to adjust clocks using Newton's procedure. Indeed, it would appear that we are obliged to do so if we want to synchronize our clocks, in other words if we want equal time coordinates to express a relationship of simultaneity.

But then Brown says on page 32 that Newton's procedure is empirically found to lead to the same clock adjustment as Einstein's light signal procedure, and I have previously stated that in general Einstein's procedure is not a synchronization procedure. Was I wrong? My next post will tell.