• Acceleration compared to which reference system?

    From Luigi Fortunati@21:1/5 to All on Mon Jul 27 15:26:06 2020
    The second principle (F=ma) states that, if we apply a force F to the
    body of mass <m>, it accelerates by a proportional quantity <a>.

    What acceleration is it?

    Is it an acceleration with respect to anyone or with respect to a
    specific reference system?

    --
    - Luigi Fortunati


    [[Mod. note -- I think you're looking for the concept of an inertial
    reference frame,
    https://en.wikipedia.org/wiki/Inertial_reference_frame
    -- jt]]

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  • From J. J. Lodder@21:1/5 to Luigi Fortunati on Tue Jul 28 12:31:51 2020
    Luigi Fortunati <fortunati.luigi@gmail.com> wrote:

    The second principle (F=ma) states that, if we apply a force F to the
    body of mass <m>, it accelerates by a proportional quantity <a>.

    What acceleration is it?

    Is it an acceleration with respect to anyone or with respect to a
    specific reference system?

    The proper relativistic generalisation of Newton's F=ma is the 4-vector equation F=ma, with F the 4-vector force, m the scalar mass, aka the
    rest mass, and a the 4-vector acceleration.

    Since it is a 4-vector equation it is coordinate independent,

    Jan

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  • From is sad@21:1/5 to All on Thu Jul 30 10:13:22 2020
    (F=ma) is an external force
    with respect to every object in every coordinate system

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  • From Rock Brentwood@21:1/5 to Luigi Fortunati on Fri Aug 14 05:56:51 2020
    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.

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