Mister T said:
You reveal the answer in your next line.
The definition of physical units of measurement is subject to our consensus. Gravitational waves either stretch the arms of the LIGO detector or they don't. That was my take-away on the goal of the post.
I have seen discussions of gravitational waves use both approaches. Some approaches stress the physical distance, defined ultimately by the SI definition. This changes because the round-trip light time changes, and that's how the SI system defines distance. Other approaches use a convenient set of coordinates defined by a set of free-fall observers. In this approach, the number associated with an observer on each mirror does not change, by definition.
Both approaches can provide insight and are commonly used.
I suspect many people do not use the modern SI definition of distance, and are confused about what happens according to their own views, which in some cases may not be fully thought out and hence not fully explainable. One can't hit a moving target such as an unspecified definition, but one can attempt to address the issue of what happens from a pre-relativistic definition of distance, which was based on rigid rods, physically implemented out of steel bars.
I haven't seen any discussion of this in textbooks, so the answer will necessarily be my own, not a textbook answer. My answer to this is that the SI definition in this case best approximates what a steel bar would measure. Basically, according to my take, measuring the round-trip time of light is by the principle of the constancy of the local speed of light is simply a better implementation of a "rigid ruler". The fact that this is the SI definition of distance supports the view. If you regard the SI definition of distance as being a refinement of how to accurately measure distance, then one can accept that's the defintion want to use it as the "physical" distance, and one can regard other notions of distance (such as assigning constant coordinates to inertial objects) as "non-physical". Non-physical doesn't mean useless, some of these notions can be very useful in understanding. But one needs a connection between the math, and what one can actually measure.
The complicating factor here is that rigid objects don't actually exist in a general curved space-time :(. But that seemingly alarming discussion might best belong in another thread. My personal take is that it's not vital to understanding GW's, but to justify my take requires some rather advanced mathematics.
It doesn't take much math to demonstrate a simple example to demonstrate that "rigid" two dimensional surfaces don't exist on the two dimensional surface of a 3d manifold unless the object and it's surface manifold is highly symmetrical. This won't explain why I don't think this issue isn't vital to explaining GW's, but it can be interesting in its own right, though it's a bit of a digression. I'll take the risk of explaining further, with the suggestion that followup questions probably belong in a different thread.
Let's set some ground rules. To keep things simple, we have some 3 dimensional object, and we treat it's surface as a 2d manifold. Then we address the issue of whether we can construct "rigid" objects on this 2d manifold. What do we mean by a rigid object? We could use a Born's definition, but a good treatment of that gets rather mathematical (and seems hard to find elementary treatments, a lot of the original papers aren't in English as well). I'll suggest that the notion of congruent geometric figures (such as triangles) serves as a good notion of rigidity. Basically, if all triangles are congruent, then we can move a triangle from one spot to another without changing its shape. We can break up other shapes into triangles, and then argue that if all the triangles are congruent, so are the larger figures.
But when we look at the sum of the angles of a triangle on a curved surface (such as a sphere), we start to see the issues. The sum of the angles of a triangle on a plane is always 180 degrees, but this is not true in spherical geometry. It's rather well known that the sum of angles of a spherical triangle is greater than 180 degrees, and depends on the size of the triangle. (I regard this as not being "much math", though i suppose readers unfamiliar with spherical trig might disagree. But it's certainly a lot simpler math than a full understanding of General Relativity and Riemannian geometry, though it can be a useful motivational tool for why we need Riemannian geometry).
If we only demand that triangles of the same size be congruent, we can get around this issue on a sphere. We go from "all triangles are congruent" to "all triangles of the same area are congruent". Unfortunately, when we consider more general cases, such as a flat geometry with a "bump", we see that we just can't make all triangles congruent. If we ask that all triangles be congruent, we can't have the sum of the angles be 180 degrees at one locaction, and some different number at a different location. But if we move a triangle with three equal sides from a location "on the bump" to a location "off the bump", the sum of it's angles changes (as long as the triangle has a finite area). So, we are lead to the notion that saying that "all triangles" are congruent is simply wrong, which then suggests that there are severe difficulties with "rigid objects" as a definition of distance.
Sorry, this got rather long. But I'll try to summarize my point. I believe that pre-relativistic notions of distance relied on "rigid rulers", and the lack of rigid objects on curved manifolds leads to a lot of confusion about what distance really is. Unfortunately I can't point to the literature as to what the "real" definition of distance is, because as near as I can tell there isn't much agreement on how best to formulate it. As a consequence, students wind up with little guidance on this point. I can point out that relying on "rigid objects" to define it is going to cause some issues. And I think it's fair to say that the previous incarnation of the definition of distance used "rigid rulers", so I don't think I'm too far off in suggesting that this is the source of a lot of confusion about the nature of distance.
I should stop here, but I want to add one more thing. I believe it can be fruitful to go from the notion that "all traingles are congruent" to "all sufficiently small triangles are congruent". But I haven't seen any text actually take this approach. I believe, though, that it could be used to motivate Reimannian geometry.
