Re: Is Curved Space An Improvement Over The Use of the Concept of Forces?

On 11/16/2024 11:50 PM, ProkaryoticCaspaseHomolog wrote:

On Sat, 16 Nov 2024 23:48:09 +0000, LaurenceClarkCrossen wrote:
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Is Curved Space An Improvement Over The Use of the Concept of Forces?

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[SNIP]
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The following text has been edited very little from the version that
I added to Wikipedia in April 2018.
https://en.wikipedia.org/wiki/Spacetime#Is_spacetime_really_curved?
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Is spacetime really curved?
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In Poincaré's conventionalist views, the essential criteria according
to which one should select a Euclidean versus non-Euclidean geometry
would be economy and simplicity. A realist would say that Einstein
discovered spacetime to be non-Euclidean. A conventionalist would say
that Einstein merely found it more convenient to use non-Euclidean
geometry. The conventionalist would maintain that Einstein's analysis
said nothing about what the geometry of spacetime really is.
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Such being said,
  1) Is it possible to represent general relativity in terms of flat
     spacetime?
  2) Are there any situations where a flat spacetime interpretation
     of general relativity may be more convenient than the usual
     curved spacetime interpretation?
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In response to the first question, a number of authors including
Deser, Grishchuk, Rosen, Weinberg, etc. have provided various
formulations of gravitation as a field in a flat manifold. Those
theories are variously called "bimetric gravity", the "field-
theoretical approach to general relativity", and so forth. Kip
Thorne has provided a popular review of these theories.
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The flat spacetime paradigm posits that matter creates a gravitational
field that causes rulers to shrink when they are turned from
circumferential orientation to radial, and that causes the ticking
rates of clocks to dilate. The flat spacetime paradigm is fully
equivalent to the curved spacetime paradigm in that they both
represent the same physical phenomena. However, their mathematical
formulations are entirely different. Working physicists routinely
switch between using curved and flat spacetime techniques depending on
the requirements of the problem. The flat spacetime paradigm is
convenient when performing approximate calculations in weak fields.
Hence, flat spacetime techniques tend be used when solving
gravitational wave problems, while curved spacetime techniques tend be
used in the analysis of black holes.

The space-time curvature is a mere mental model
to effect to reflect a milieu of "gravity" in a theory
with massy bodies and a universal law of gravitation.
So, all it is, is, a "geodesy", that has all these level or
plane curves according to integral analysis, that
accordingly, a "test-mass", which is an arbitrarily
small amount of mass, that the test-mass in this
field of potentials, follows in the geodesy, its
"world-line", that is merely the gradient descent,
that the test-mass follows its world-line.
Then, the theory has a "mass-energy equivalency"
combined with an L-principle, and indeed two L
principles, that "relativistic mass" reflects inertially
resistance to acceleration, yet as well, tendency
to fall faster in the world-line.
So, there's an L-principle that light's speed is a
constant, and that light, though mass-less,
behaves as a test-mass in the geodesy, that's
what it's said to do. Then also there's that
the energy according to arbitrarily referenced
velocity, has that this "c" is effectively infinity
in this theory, that to reach it requires infinite
energy because a given massy body accumulates
or radiates any, "relativistic mass".
Then, in this "potential well", of gravity, this geodesy,
it's always evaluated instantaneously with regards
to all massy-bodies everywhere all the time constantly.
This is usually left out yet otherwise has that nothing
would ever happen or move.
So, most usually it's considered with regards to
the gravity well of Earth the terrestrial, that
humans are so small as to be like test-masses,
and any "curvature" of space-time, is no different
than Newton's universal law of gravitation,
according to Earth's mass and Galileo's what
results the constancy there, of gravity, is yet
only because the 2-body system has it so negligeable,
what's otherwise the mutually proportional product,
inverse the distance square.
Then it results that the gradient descent the
world-line, is merely "down", that it's "down"
in the gravity well, that of course there's an
implicit tendency of massy bodies to accelerate
according to gradient descent and a tautochrone,
that's also usually left out as it was already given
back to the universal law of gravitation, and
Galileo's simplifications of what's otherwise
inter-actions.
"Down, Einstein?" "Yeah, straight down."
So, the frames are the containers as it were
of mass, and relativistic mass, and live in a space.
Thusly you can notice that a variety of things
are left under-defined, for example how a
photon can be mass-less and travel at c,
versus being a test-mass and follow a world-line.
Similarly there's that a given frame _has space in it_,
that a frame has spaces in it and a space has frames
in it, space-frames and frame-spaces, that usually
the theory is nested frames, and space is just a
great outside local "flat" section, whatever is
the nearest center gravitationally.
It doesn't have to be that way, and in the same theory,
that there are implicits and "un-stated" assumptions
in the theory, that are always givens, while yet it's
always world-lines the geodesy and a flat locality
where things go straight down, as down the ramp,
the ramp, of the gradient descent, of "space-time",
keeping up the tauto-chronous like that, as after
a most sort of machine in operation, the inclined
plane, or ramp.
Then, in a sense, it's always _of_ and "flat space-time",
what's down, and always in a "flat space-time",
what's around.
Then, as with regards to what defines the ramp
the gradient descent, it's classical universal law
of gravitation what makes a geodesy, considering
relativistic mass, and, classical motion down a ramp
in classical linear downward gravitation, about
all of classical mechanics.
So, these are as well under-defined, and also,
classical mechanics is even under-defined, as
with regards to whatever's more than a
space-time where everything orbits or falls,
apart from each other, and nothing ever meets.
That's all given to "classical mechanics".
So, anyways, "curved space-time", is a mental model
about the universal law of gravitation and
a classical model of classical mechanics, of it,
and then also "and photons follow world-lines
like test-masses".
That's all there is to it, yet, there's another
theory, that has all the same actions, where
space is a frame and a frame is also space,
and objects in motion and massy-bodies are,
..., mostly space, and their space goes along
with them. This then is in a space, a great
altogether flat, as empty, with potential
about orbits the geodesy, that the model
of space-time as, "curved", is, dispensable.
Then there's the cosmological constant
"vanishing, yet, non-zero: an infinitesimal"
that effects to reflect a gradient, of gravity
at all, explained as a fall gravity its mechanism,
for, a mechanical reduction of mechanics.
So, mass-energy equivalency, for relativistic mass,
and, a cosmological constant, for, the universal
law of gravitation to lean into effect, and,
three L-principles "light's speed is a constant"
and "light's mass-less" and "light follows world-lines
like a test-mass", make for most of the things.
Then here as that's all merely to satisfy
the Lorentzian invariant then there are
other ways to go about it and whatever
may be said to keep the sum of partials
the Lorentzian, the Laplacian,
from being incomplete suffices.
Here it's "fall-gravity", a "FitzGeraldian"
for the linear and then that the rotational
and linear are different, with regards to
the "space-time wheel" and such notions,
yet of course for a deconstructive account
of _mechanics_, since, classical _mechanics_
itself is under-defined, and usually applied
senselessly to the mental model, since it's "givens".
So, it's agreeable that, "curved space-time",
is _not_ satisfying, while it simply is what it is,
a simple model in classical machines in classical gravity
of classical mechanics. The sky survey since some
decades has established "the cosmological constant,
if positive and non-zero, is vanishing", so, "flat".
While "curving", ....