Reply to comment

Jim:

I've thought of a few more things. (Sorry, I've been on vacation, so I have plenty of time. It has been nice to be away from the office. :-) )

First, you state:

I’m inclined to believe there’s no place in the universe where curvature is actually zero, although it’s no doubt a legitimate construct for the purpose of illustrating a principle. I’ve used it myself to illustrate the nature of a “pure” inertial acceleration, “unadulterated” by gravitational curvature. But that relates to an issue you’ve brought into focus about the difference between empirical hypotheses or principles and mathematical models. I think it’s a mark of real progress that we’ve arrived at this point.

Unfortunately, I can think of at least two* interpretations of your statement concerning "an issue you’ve brought into focus about the difference between empirical hypotheses or principles and mathematical models." So I'm not quite sure whether this is a positive development. I'm hopeful, though.

You then go on with (starting with a quote of me):

You wrote “depending on other circumstances, like the distribution of mass-energy-momentum-stress surrounding [a uniform gravitational field], there may, indeed, be a definable magnitude” of the field.

While there may be a workable mathematical model of such a region, I don’t believe we can point to a place in the “real world” where it’s been found to exist. It may be worthwhile to hold the model in stock to help explain some new discovery, or to apply in a novel way to some existing unexplained phenomenon, but it has no standing against any empirical hypothesis that is otherwise coherent and useful.

To your last assertion ("it has no standing against any empirical hypothesis that is otherwise coherent and useful") I'll say that in the sense that such cannot be used as an experimental verification or refutation of such an "empirical hypothesis", you are correct. However, as a hypothetical situation (like any thought experiment) that may be used to distinguish between one "empirical hypothesis" that "refuses" to work with such and another (more complete) theory that naturally includes such within its framework (without any special accommodation), I would have to say that it may be quite relevant.

Secondly, I'm beginning to believe that your concept of the “real world” would have trouble with even nearby, actually realizable experiences.

For instance, consider a box in orbit around the Earth. Let's start by considering the box to be orbiting in such a way as to maintain one side always facing the Earth, so we have the "floor" being the side facing the Earth, the "ceiling" facing away from the Earth, the "front" facing the direction of "travel" of the orbiting box, in opposition we have the "back", with the "left" side being on the left and the "right" side on the right of an astronaut facing the "front". The question, then, is what will be the motions of free neutral test particles within this box? Especially, what will be the motions of the subset of such particles that don't hit any of the sides of the box?

Another instructive way of looking at essentially the same scenario may be to consider letting the "box" be a spherical shell, so symmetry allows us to ignore any issue of whether the "box" is "rotating" or not. (So any question of inertial "forces" may be avoided.) Then one may simply ask: What subset of geodesics (orbits) will always remain within the confines of the "box" without ever touching any of the walls?** Then one only needs to consider what such orbits will look like to an astronaut within the "box"? (This is to be considered because such is the nature of your other thought experiments.)

In both of the above scenarios (or two ways of looking at a single scenario, depending on how one looks at it) it is permissible to do it all within the framework of Newtonian mechanics and gravitation, since relativistic corrections are sufficiently small that the major features are all captured via the Newtonian approach.

If you attempt these exorcises at all I will think far greater of you for the effort. I expect you will find that these “real world” scenarios are far from being as trivial as your other thought experiments.

I wish you well.

David

* Perhaps this is a problem with my flexibility in being able to see multiple perspectives. :-)

** One may use the cylindrical symmetry of the system to reduce all the orbits to a class of orbits having their major axis and apogee all in the same direction, as seen from a vantage point along the axis of rotation. In addition, one has reflection symmetry above and below the plane of the "box's" orbit.