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Locally-defined parameters chosen to be minimally frame-variant can be useful for describing motion in accelerated frames and in curved spacetimes. In particular the metric-equation's synchrony-free ``traveler-point parameters'', namely proper-time, proper-velocity, and proper-acceleration, are useful in curved spacetime because extended arrays of synchronized clocks (e.g. for local measurement of the denominator in Δx/Δt) may be hard to find. These same parameters can better prepare intro-physics students for their everyday world, as well as for the technological world e.g. of GPS systems where differential aging must be considered explicitly. The net frame-invariant proper-force, which is simply rest-mass times proper-acceleration, is not generally a rate of momentum-change, but is the quantity reported by accelerometers moving along with ``the traveler''.
For more on this topic, see this working draft pdf archived at [1], and this note on the scientific observation cycle.
This is partly an issue with the author's style. Further suggestions for making it better in this context are invited.
Classic conversions:
Classic conversions:
To give you a taste of the traveler-point variable notation in classic terms, imagine a traveler with book-keeper coordinates x[t] seen from the vantage point of a "free-float" or inertial frame in flat spacetime with no geometric forces, so that the equation of motion DUλ/dτ-ΓλμνUμUν = dUλ/dτ in the form proper + geometric = observed predicts the net proper-force ΣFo alone (e.g. a Lorentz force like eFλβUβ) will equal mα, where m is frame-invariant rest-mass and α is "frame-invariant" proper-acceleration (with 3-vector direction defined by the traveler). Then proper-velocity w ≡ dx/dτ = γv where coordinate-velocity v ≡ dx/dt, differential-aging factor γ ≡ dt/dτ = √1+(w/c)², kinetic energy K = (γ-1)mc2, momentum p = mw, rate of energy change dK/dτ = mα•w, and the rate of momentum change is dp/dτ=mα+(γ-1)mα||w. As Tony French suggested in his classic text, coordinate-acceleration a≡d2x/dt2 is simply related to neither of these latter two dynamical quantities, and so mainly approximates the frame-invariant proper-acceleration (e.g. measured by your phone) at low speeds.
The table below shows that proper acceleration α (in the absence of geometric forces i.e. from the vantage point of an inertial frame in flat spacetime) is directly connected to rates of energy change dE/dt thru the geodesic equation in the "second-law" form ΣFo =mα. Proper-acceleration is furthermore indirectly connected, through net frame-variant force, to rates of momentum exchange dp/dt and hence the action-reaction forces of the "third-law".
One reason that the foregoing remains a working draft is perhaps clarified by this very helpful 2017 referee report, provided to the author by the editor of the American Journal of Physics:
"In general, while I get the impression that the author has a clear argument and compelling reasons for this approach in his/her head, he/she simply did not get it across to me. The article reads too much like someone who has a lot to say but is saying it too quickly and urgently to fill in all the steps. If the author would slow down and think more empathically about the needs of likely readers of this article (teachers in introductory-level classes with little familiarity with general relativity and maybe even an incomplete understanding of special relativity), this could be a really interesting and useful article. But I am sorry to say that I don't think it is yet acceptable in its current form."
Do the flat-space "inertial-frame" entries in this table look right?
Scaling relations:
Scaling relations:
We use the term scaling relations, following Messerschmidt, to describe the γ ≡ dt/dτ dependence of a target vector, on the components of a given vector that are parallel and perpendicular to a reference vector, which in this case is proper-velocity w ≡ dx/dτ (same direction as coordinate-velocity v ≡ dx/dt). Hence the entries in these tables might be used to "find" the components of the target vector, from the components of the "given" vector, using the relation:
.
Here the × symbols simple denote scalar multiplication.
The first table lists parameter a, which describes the γ dependence of the component parallel to proper velocity w, for conversion between several quantities (namely coordinate-acceleration, proper-acceleration, and frame-variant force per unit mass in units of momentum-change per unit of coordinate-time or proper-time) which all have units of acceleration.
This second table then describes parameter b, the relative (perpendicular to parallel ratio) γ factor for the perpendicular component, which of course can be ignored in (1+1)D spacetime.
These notes were inspired by Messerschmidt's draft paper, whose work I think focused on components with respect to the "heading" or velocity. The same questions of course might be asked about components with respect to the proper-acceleration, the frame-variant force, or the coordinate-acceleration. In other words, for dynamically moving objects there may be at least 4 different directions of interest.
Footnotes
P. Fraundorf (2016/2017) Traveler-point dynamics, hal-01503971 working draft.
P. Fraundorf (2016) "The proper-force 3-vector", HAL-01344268 current pdf.
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