We have recently exchanged with Richard a few posts
dealing with the mental experiment of Rotating Disk.
This discussion exceeded my expectations and preparing
new posts I could refine and improve my presentation of
the issue. Due to its general interest I decided to post
it to several lists from which I may expect helpful
feedback.
CONTEXT
Galilean Relativity postulated that all laws of mechanics
are covariant among all moving at constant speed or non
accelerating "inertial" aka "galilean" referentials, by
the Galilean Transformation (speed cumulation). Galilean
absolute 4D SPACE consisted of 3D Euclidean space and of
affine with respect to it 1D time.
Special Relativity extended the Galilean over Electro-
Magnetism, always for galilean referentials. Time
dimension was replaced by "lighttime" CT which has the
measure of distance, commensurable with space dimensions.
Thus the 4D space/lighttime SPACE is metric and admits
the pseudo-rotational Lorentz Transformation, by virtue
of which relatively moving bodies appear shorter and
clocks - slower.
However, galilean referentials are only an approximation
and no strictly galilean referential has ever been
observed. The cosmos appears to be entirely non-galilean.
Consequently, Einstein considered extension of SR over
non-galilean referentials. He saw it not as developing
a new model from the scratch, but as continuing the
extending attitude: SR extended the Galilean Relativity
for subluminal (close to C) speeds, but for speeds
negligible with respect to C SR regresses to Galilean
Relativity which, embodied by the Newton model,
coordinates till today all slow speed mechanics
including the space travel.
Similarly, SR is a sufficiently precise approximation
for subluminal speeds of particle physics and its
extension to non-galilean referentials will concern
only few exceptional phenomena exceeding the scope
of SR. In other terms we have to look for a non-galilean
extension retrogressive to SR for particular conditions,
which we still have to determine.
We shall do it with help of Einstein's mental experiment
od Rotating Disk.
UNDERLYING PRINCIPLES
1.Local Euclidean approximation.
For every curved geometrical construct like line L, or
surface S an "infinitesimal" differential dL, dS may be
approximated as Euclidean. In other terms, any curved
construct may be locally approximated as flat.
2.Einstein's Covering Principle
Einstein's intuitive foundation of space by rigid bodies
is embodied in his Covering Principle which requires
abstract, geometrical distance to be measured, (also in
mental experiments), by covering it with (imaginary)
physical rods complying with physical rules, such as the
Lorentz Contraction. The Covering Principle underlies
directly or indirectly the entire Extended Relativity.
For the first time in the history of science an ontological
principle is directly incorporated in defining of physical
theorems.
3.Equivalence Postulate
Newtonian model applied the term "mass" as homonym of
two distinct concepts: "inertial mass (IM)" and
"gravitational mass (GM)". Since Galileo's experiment
demonstrating that all objects have equal gravity
acceleration, IM was known to be equal to GM, but this
was considered as a coincidence. Several physicists hoped
that more precise measurements may show some slight
difference. In spite of extremely fine precision, the
equality persisted, but was always considered as
accidental.
By asserting that there are no distinct IM and GM, but
only one mass M Einstein extended Relativity over
gravitation. Mass being a mathematical coefficient void
of phenomenal sense, the postulate's physical meaning is
that inertial and gravitational fields are equivalent,
i.e. indistinguishable within an experiment.
SETUP AND TERMINOLOGY
Note: The term "inertial referential" denoting a non-
accelerating referential may be misleading due to the
historical mix up where "inertial referential" does NOT
experience inertial forces and the non-inertial DOES
experience them. We'll replace "inertial-" with "galilean
referential".
Let's imagine two disc-formed referentials superposed like
gramophone plates, one galilean, stationary "DS" and the
other, rotating "DR", with observers, respectively "ObS"
and "ObR".
Let "R" and "r" radii identical in DS and DR, R>>r, attached
to both centers; R long enough for its extremity having in
DR subluminal tangential speed; r short enough for its
extremity having in DR tangential speed negligible with
respect to C.
Let "SSR", "SSr" Circumferences of circles drawn in DS
respectively at R and r and "SRR", "SRr" Circumferences
of circles drawn in DR respectively at R and r.
MEASUREMENTS OF CIRCUMFERENCES PERFORMED BY ObS
ObS measures first the circumferences SSR and SSr at his
own referential DS covering them with "infinitesimal" dS
rods approximated as rectilineal. He gets SSR=2piR and
SSr=2pir.
Then he procedes to measure SRR and SRr at the rotating
DR by dropping on them his dS rods which start rotating.
Tangential speed of SRR is subluminal, which may suggest
that SRR will be affected by Lorentz contraction and
shrink. However, on the one hand Lorentz transformation
applies only to galilean referentials and not to the
centripetally accelerating DR and, on the other hand,
it applies to physical bodies and not to abstract
geometrical constructs like a circumference.
But it applies to the covering rods which have been chosen
as approximately rectilineal and may be considered as
galilean mini-referentials embodying the local regression
to SR. Thus, the rods will be Lorentz contracted and, in
order to get covered, SRR will need more of them than did
the stationary SSR, i.e. the measurement will give SRR>SSR.
On the other hand, R - perpendicular to the tangential
speed - stays unaffected. We had SSR=2piR, thus SRR>SSR
gives SRR>2piR.
Now circumference > 2piR defines hyperbolic, Lobatchevskian
SPACE and we arrive at the first definition of the General
Relativity:
[D1] CENTRIPETAL ACCELERATION DETERMINES CURVED
HYPERBOLIC SPACE
Let's note that the tangential speed of the SRr being
negligible with respect to C, the rods covering it stay
unaffected and SRr=2pir. We have thus four measurements:
SSr=2pir
SSR=2piR
SRr=2pir
SRR>2piR
With D1 we have essentially derived the General Relativity.
In next posts we shall use the Rotating Disk to define
qualitatively several GR's features such as event horizon,
black hole, precession of the periphelion of elliptic
orbits and Einstein's lens (curved trajectories of light).
All of them will be derived from D1.
Georges.
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