Zeno Marx said:
I'm very much simpatico with least action physics richard feynman being my major scientific hero my point was really - ok maybe the thought experiment i posted didn;t do it properly but the idea wa to ask how you define velocity at all if you eliminate totally the concept of inertial observers because no observer over the course of his/her/its observership is ever completely inertial
If Newton's laws hold well enough that you can use them, then you have a de-facto inertial observer. This is most of the time. You can then describe things in the famliar manner of inertial frames, and inertial observers, etc, just by going to any of the various coordinate systems in which Newtons laws work well, or well enough that you can't tell the difference.
Occasionally, one finds situations (such as considering the universe on a cosmological scale) where one can't approximate physics via Newtonian laws.
If curvature is present (and in a truly general situation I would expect it to be - no curavature is a special case) I would say that there isn't any good , truly general, coordinate independent definition of velocity except for two observers who are close to each other. See for instance Baez's remarks in "The Meaning of Einsteins' Equations."
http://math.ucr.edu/home/baez/einstein/node2.html
In general relativity, we cannot even talk about relative velocities, except for two particles at the same point of spacetime -- that is, at the same place at the same instant. The reason is that in general relativity, we take very seriously the notion that a vector is a little arrow sitting at a particular point in spacetime. To compare vectors at different points of spacetime, we must carry one over to the other. The process of carrying a vector along a path without turning or stretching it is called `parallel transport'. When spacetime is curved, the result of parallel transport from one point to another depends on the path taken! In fact, this is the very definition of what it means for spacetime to be curved. Thus it is ambiguous to ask whether two particles have the same velocity vector unless they are at the same point of spacetime.
There might be some ways to avoid some of the impact of what Baez says in specific cases - for instance, taking advantage of particular symmetries, such as the "Hubble Flow" in the case of the universe at large.
My experience is that a lot of physicists do want to define relative velocity, and they use various tricks to try to avoid the issues Baez mentions, with various degrees of success. So the idea of velocity can be useful, but one needs to pay careful attention to how it's being defined in any specific case - it's usually based on some trick or feature of the particular problem at hand or the use of some particular coordinates.
Semi-philosophically, though, there isn't any need for relative velocity - and in fact, we don't even really need observers, either.
Consider Misner's remarks in
http://arxiv.org/abs/gr-qc/9508043 "A Precis of General Relativity". It was written in response to some remarks by Neil Ashby,.
A method for making sure that the relativity effects are specified correctly (according to Einstein’s General Relativity) can be described rather briefly. It agrees with Ashby’s approach but omits all discussion of h ow, historically or logically, this viewpoint was developed. It also omits all the detailed calculations. It is merely a statement of principles.One first banishes the idea of an “observer”. This idea aided Einstein in building special relativity but it is confusing and ambiguous in general relativity. Instead one divides the theoretical landscape
into two categories. One category is the mathematical/conceptual model of whatever is happening that merits our attention. The other category is measuring instruments
and the data tables they provide.
For GPS the measuring instruments can be taken to be either ideal SI atomic clocks in trajectories determined by known forces, or else electromagnetic signals describing the state of the clock that radiates the signal.
<...snip...>
What is the conceptual model? It is built from Einstein’s Gen
eral Relativity which asserts that spacetime is curved. This means that there is no
precise intuitive significance for time and position. [Think of a Caesarian
general hoping to locate an outpost. Would he understand that 600 miles
North of Rome and 600 miles West could be a different spot depending on
whether one measured North before West or visa versa?] But one can draw
a spacetime map and give unambiguous interpretations.
Misner goes on to explain that the "space-time map" is just a metric - a mathematical construct which allows one to calculate geodesics ("straight lines" - which can include the motion of force-free bodies as disucssed above) and their lengths (which for moving bodies, are time-like, the proper times we've been discussing).
So really, the only reason we need to talk about 'observers' is to try to allow people to leverage as much as they can off their Newtonian intuition. They aren't actually needed for anything, and frequently there is a lot of confusion generated by trying to pound the square peg of the mathematics of General Relativity into the round hole of Newtonian-based preconceptions.
To expand on this viewpoint a bit, in General Relativity, we can assign labels to events in space-time in any manner that we find convenient, and from any such assignment, we can create a metric, a "map" of space-time, which assigns particular labels (which we call coordinates) to specific events.
These labes are purely human inventions - though typically, some will be easier to work with than others, when they respect fundamental physical symmetries, for instance. But the point is that the coordinates are not "physical", they're human inventions. A very simple point, but it seems difficult to get people to appreciate it :-(.
A useful replacement for "an observer" using this semi-philosophical approach is to define some particularly useful coordinates called Fermi normal coordinates. Given any particular worldline, such coordinates can be constructed, and will behave in a manner that is, in some sense, as close to Newtonian as one can manage.
I've seen a few people on PF who seem to prefer a different set of coordinates, which are based on a particular point in space-time, rather than a particular worldine. These can be useful, they don't to me have the same feature of being associated with an "observer", because they are associated with only one point in space-time, not a worldline that moves "through" it.
I'm not sure it'd be relevant to get into more details of Fermi normal coordinates, and this has gone on more than long enough already, so I'll cut it here.
To recap, though - we don't really NEED to define velocities - or even observers - to do physics. We can make all the physical predictions we need from the metric.