On Monday, July 27, 2020 at 5:26:08 PM UTC-5, Luigi Fortunati wrote:
Is it an acceleration with respect to anyone or with respect to a=20
specific reference system?=20
Newton, in the background to the formulation of his laws, said that all mot= ion was to be referred to a specific frame - putatively one in which the st= ars are "fixed". The laws were formulated in such a way that they hold equa= lly well when taken in reference to any other frame that is in uniform moti=
on with respect to this one ... so that one could not determine which was t=
he actual stationary one and which ones were moving.
Galileo (it is NOT well-known here) was Newton's great-grand-math-father in=
the Mathematicians' Genealogy. Galileo (eventually) held to the propositio=
n that each of these frames stands on equal ground. Newton rejected it by w= ord, but implemented it by deed. In other words,. he tried to have it both = ways, saying one thing and doing another. But it gets the job done: of dist= inguishing a family of frames (out of all those that are possible) as the o= nes that are inertial. They each move at a constant speed in a constant dir= ection with respect to one another.
The closest modern equivalent of Newton's assertion (and one which revokes = his doctrine of Unknowability on the matter of which frame is the Stationar=
y one) is the co-moving frame that is almost literally tied to "fixed stars=
" - namely the one given by the CMB: the one which makes it maximally isotr= opic, minimizing all its Doppler shifts.
The reason Newton had to take this route (though not clearly stated or even=
understood by him) is Genidentity. Everything is formulated in the languag=
e of spatial geometry. The fundamental object of spatial geometry is the Po= int. A Point defines a location. The concept, however, has no meaning unles=
s and until you can say what's to count as the "same" location at two diffe= rent times. Is New York in 2001 the "same place" as New York in 2020? Or is=
that "location" somewhere else on the Earth at the same latitude (because = the Earth rotates) or different latitude (because the rotation wobbles) or = different altitude (because the crust fluctuates) or different part of the = Earth's orbit (because the Earth goes around the sun) or different part of = the galaxy (because the solar system orbits the galaxy) or out in intergala= ctic space (because the galaxies move mostly away from each other)? What co= unts as the "same place" at a different time? That's the property of Genide= ntity. And, as you can see, Genidentity is just a back-door way of saying w= hat is Stationary and what is not.
Without this, you have no Genidentity. Without Genidentity, you have no con= cept of a Location that endures in time or of a Point. Without Point, you h= ave no foundation for Spatial Geometry. Newton's treatise is cast in the la= nguage of spatial geometry, so he needs Point, Location, Genidentity and St= ationarity. Therefore, he had no choice but to refute Galileo's principle, = even if he still tried to have it through the back door by making his laws = invariant under Galilean boosts.
To fully implement Galileo's principle requires delving deeper than the con= cept of a Point, and deconstructing it into even more fundamental constitue= nts - as a sequence of Point-Instants. The geometry required for this is no=
t a geometry at all, but a chrono-geometry: one whose fundamental objects a=
re point-instants. That is made necessary by Galileo's principle of Relativ= ity - or by any other principle of Relativity that supersedes it.
So, it is also the case that the true origin of chrono-geometry (that is: t=
he concept of spacetime) lies rooted in Galileo but that it just happens to=
also be the case that neither he nor anyone else realized this or that thi=
s had to be so until after Relativity was changed from Galilean to Lorentzi= an. In other words, Minkowski didn't marry space and time, nor did Einstein=
or Poincare'. They merely ordained the eloping of the two, which took plac=
e nearly 300 years before that.
A chain of Point-instants can be any one-dimensional subspace of this chron= o-geometry. Those are worldlines. Of all the possible ones, a distinguished=
subset of them have the property of possessing zero acceleration at each p= oint-instant - one for each direction in space at each speed. So, ultimatel=
y, the answer to the question is that an additional structure is imposed on=
the chrono-geometry which singles out which of the worldlines are inertial=
; such that at each point-instant, in each direction at each speed, passe=
s through exactly one such worldline. This structure is embodied by what we=
now call an affine connection.
In Newtonian Physics, all such worldlines represent either motions that are=
at a constant speed and direction relative to one another, or an instantan= eous line of points in a snapshot of 3D space at a specific time (a spatial=
geodesic). In the relativistic world, in place of the spatial geodesics ar=
e the light-like and space-like geodesics.
Accelerations are taken relative to geodesics. So, for a given motion passi=
ng a given point-instant in a given direction and speed, you compare it to = the geodesic possessing those same attributes.
--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)