[[Mod. note -- I have rewrapped overly-long lines. -- jt]]
On Saturday, May 7, 2022 at 5:38:42 AM UTC-5, Luigi Fortunati wrote:
A free-falling brick is an inertial frame?
If it's not rotating. Otherwise, the free-falling object is in a
rotating frame, if the frame is attached to and associated with the
object, itself. If it's not rotating, then it is locally inertial.
The distinction between the two rests on Newton's bucket thought
experiment.
Are a pair of free-falling half-bricks an inertial frame or are they
two distinct inertial frame?
The curvature of space-time is precisely the warping of a field of
locally inertial free-fall frames whereby those that are initially
at rest with respect to one another start to accelerate with respect
to each other. If you display them as worldlines in a 4-dimensional
graph (or in a 3-dimensional graph, where one of the spatial
dimensions is suppressed, for the benefit of the unlucky few who
are visually impaired to see in 4 dimensions) then you'll see the
worldlines for locally inertial free-fall frames - initially parallel
in a time-like direction - starting to curve into one another -
hence the "falling" action associated with gravity. In this sense,
the gravity one feels and experiences is actually a warping in time,
first and foremost, rather than a warping in space. The actual
contraction is quantified and accounted for in the Raychaudhuri
equation, which is closely related to the "geodesic deviation
equation".
In a flat space-time spatially separated locally inertial frames,
initially at rest with respect to one another, remain at rest; and
so can be said to comprise the different locations of a global
inertial frame.
All of the foregoing applies independent of paradigm - to *both*
relativistic *and* non-relativistic theory; so it is neither a
construct nor innovation of "general relativity", but rather one
which first fully emerged at the onset of general relativity and
so has been (falsely) associated with it as a characteristic feature
of it. It is a general feature of any theory of gravity that respects
the Equivalence Principle.
In fact, both Newtonian gravity and Einsteinian gravity (specifically:
the Schwarzschild solution) can be unified as a one-parameter family
of geometries, that are warped versions of the 5-dimensional Bargmann
Geometry, via the line element + constraint:
dx^2 + dy^2 + dz^2 + 2 dt du + a du^2 - 2 V dt^2 + 2aV/(1 + 2aV) dr^2 = 0
where r = root(x^2 + y^2 + z^2), dr = (x dx + y dy + z dz)/r and
V = -GM/r is the potential of a gravitating body of mass M located
at r = 0.
The extra, u, coordinate is the non-relativistic limit of (s - t)
c^2, as c goes to infinity, where s is proper time. This has meaning
... which also (by the way) shows that such things as "time dilation"
and "twin paradox" are *also* rooted in non-relativistic theory in
disguised form as u, and are not features specific to Relativity!
The u coordinate shows up, physically, as negative the action per
unit mass for an inertial particle.
When a = 0, this is Newtonian gravity, and it can be generalized
by having V be the total potential for all gravitating bodies,
rather than just for one. The geodesics for this geometry are the
orbits of Newtonian gravity. Since V is a function of the coordinates
and velocities of individual bodies, rather than a bona fide field
quantity, it is very tempting to try and quantize this geometry
directly in quantum *mechanics*. But for the fact that you still
have the self-energy and self-force problems to deal with (in starker
form, in fact) you'd almost have a full-fledged *geometric* quantum
theory for Newtonian gravity - one in which space-time itself is
quantized. But the whole "quantizing field theory as mechanics"
strategy has these same issues roadblocks, here, as did Feynmann
and Wheeler's attempt to do the same with electromagnetic theory
in the 1940's.
The case where a = 0 and V = 0 is the Bargmann geometry, which is
the natural geometric arena for non-relativistic theory.
When a > 0, this is the Schwarzschild solution in which the proper
time is given as s = t + a u, and in which c = root(1/a) is an
invariant speed (i.e. "light speed").
The term "dx^2 + dy^2 + dz^2" is the legacy of Euclidean geometry;
while "2 dt du" is the legacy of *Galileo*'s principle of Relativity,
which is where space and time *actually* became unified into the chrono-geometry of spacetime. The secret eloping of the two, however,
went largely unnoticed until it was fully consummated by the addition
of the Poincare' term "a du^2", which turns this into a geometry
for Minkowski space. The warping of time associated with Newtonian
gravity is in the "-2V dt^2" term, while the warping of space,
itself, associated with General Relativity is limited to the
substantially smaller "2aV/(1 + 2aV) dr^2" term.
The only effects of paradigm, here, are those limited to the "Special Relativity" term a du^2 and the "General Relativity" term
2aV/(1 + 2aV) dr^2 (so that whatever "relativistic corrections" there
are, which are made to trajectories have to come from these terms).
The main thrust of gravity - and the essential background behind your
query - resides with the "Newtonian Gravity" term -2 V dt^2.
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