On Saturday, September 2, 2023 at 1:32:10=E2=80=AFAM UTC-7, Pat Dolan wrote=
:
Consider a distant observer traveling at .867 c ( =F0=9D=9B=BE=3D2 ) rela=
tive to the solar system...
In his inertial frame of reference the earth's orbital velocity is only h=
alf the velocity
necessary to keep the earth in stable orbit...
Not true, the earth follows a helical geodesic trajectory through spacetime=
, and=20
this helical geodesic is not intrinsically altered by being described in te= rms of a=20
different system of coordinates (such as the asymptotically flat inertial c= oordinates
in which the distant observer is at rest). By the way, the *extrinsic* cur= vature of the=20
earth's trajectory is the same for those two coordinate systems, which may = be=20
surprising to you if you aren't taking the time component of the trajectory=
into=20
account. (Misner, Thorne, Wheeler illustrates this with a bullet and baseb= all.)
Invariant spacetime curvature...
Be careful... as noted above, the *extrinsic* curvature of the trajectory i=
s invariant=20
under Lorentz transformation (which is essentially what you're applying by = switching
to the asymptotically flat background inertial coordinates in which the dis= tant high=20
speed object is at rest, superimposed on the mildly curved spacetime surrou= nding the=20
sun), but the components of the *intrinsic* curvature of spacetime are not = invariant
under coordinate transformations, they change along with the components of = the=20
metric as expressed in terms of the different coordinate systems. These th= ings
are all coordinated so that the invariant intervals are, well, invariant.
Will the earth spiral into the sun?
No, describing the phenomena in terms of a different coordinate system does= n't change
the intrinsic phenomena. For example, if you draw two chalk grids on a put= ting green,=20
and describe the trajectory of a putt going into the hole in terms of one c= oordinate=20
system, it will also go into the hole when described in terms of the other = coordinate=20
system. Yes, the ball has different coordinates at the end, but the cup als=
o has different=20
coordinates, so the ball still goes into the cup.=20
The idea that changing the coordinate system used to describe the phenomena=
can=20
somehow change the phenomena is wrong. And no, this does not imply that loc= al=20
Lorentz invariance has no physical dynamical effects. The dynamical equatio=
ns of=20
physics are locally Lorentz invariant, which is the physical content of spe= cial relativity.
Ridiculous! See Einstein's First vs. Kepler's Third, ibid.
I'll assume that by "Einstein's First" you are referring to the principle o=
f special
relativity, i.e., that the equations of physics take the same simple homoge= neous=20
and isotropic form in terms of every standard system of inertial coordinate=
s, and=20
that by "Kepler's Third" you are referring to Kepler's proposition that the=
squares of=20
the orbital periods of the planets are directly proportional to the cubes o=
f the=20
semi-major axes of their orbits. =20
There's no conflict here, and nothing that makes the above explanation=20 "ridiculous". The principle of relativity is contained in local Lorentz in= variance,=20
which is clearly satisfied in this situation. It also happens that Kepler'= s=20
proposition remains satisfied (to the same approximation that it ever was),= =20
since the angular periods of the helical paths of the planets remain in the= =20
same proportion to each other in terms of the asymptotic inertial coordinat=
es
in which your distant observer is at rest.
You may be getting confused by trying to apply the Newtonian concepts of=20 instantaneous gravity and Galilean invariance of physical laws, etc., (even though you we not invoking Newtonian concepts), leading to the quantitative= =20
Newtonian extrapolation of Kelper's law, relating Newtonian mass to force=
=20
and orbital periods, etc., and pointing out that if all those things were t= rue, then=20
special relativity would be false. That is correct, but it essentially amo= unts to=20
saying if special relativity was false then special relativity would be fal= se. It's=20
a true statement, but it has no meaningful cognitive content.
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