Dainton, Teller, and the construction of neo-Newtonian spacetime

I told the class last week that I would look again at Dainton’s discussion of the order in which neo-Newtonian spacetime is constructed and the charge of emptiness against that construct.  I forgot to mention it in class this week, so here are my thoughts.

I still think that it is not literally the order in which the elements or components of neo-Newtonian spacetime are constructed that is really at issue here.  Paul Teller argues that the inertial frames and trajectories of neo-Newtonian spacetime are distinguished only by their lack of inertial effects (effects like the tension on the cord between Newton’s globes, the tendency of the water in Newton’s bucket to rise up the sides of the bucket, and the feelings of pressure, weight, and perhaps queasiness that we feel in a rapidly rising airliner).  Since it is (supposedly) only on the basis of lacking such effects that we pick out the inertial trajectories, Teller claims that we are not really explaining the lack of such effects (or their presence in other cases) by pointing out that those in which the effects do not occur are inertial.  The suggestion seems to be that we are in effect saying nothing more than ‘there are no inertial effects associated with these trajectories because there are no inertial effects associated with them’.   

Perhaps this can be understood as a matter of order:  We first observe that certain trajectories or frames are not associated with inertial effects, and we then label these the inertial frames/trajectories.  But what really matters in Teller’s argument is not the order but the basis, i.e., the reason for distinguishing the trajectories in question.  In principle, we might have called them inertial first, and then justified this later in terms of the lack of inertial effects.

In any case, I think Teller and Dainton have both overlooked something important here.  It is not as if just any old trajectory counts as inertial just because of an observed lack of inertial effects.  In fact, the inertial trajectories all have a special relation to one another:  They all exhibit uniform relative velocity to each other.  I think this suggests that when we say that these trajectories are not associated with inertial effects because they are inertial trajectories, we are not just blurting out an empty tautology, but identifying something special about a particular class of trajectories, something that is plausibly due to the particular structure of the spacetime they inhabit and their relation to that structure.

Announcement:  I am about to post a corrected version of the differential equations "handout".  The arguments in Equation (2) were reversed.  Thanks to Somayeh for catching that.