On Sep 8, 3:20 pm, "Dan Drake" dandrake.com> wrote:

The Church condemned Galileo because he was wrong. But what they
neglected to mention is that they were wrong too.

Both sides were still attached to the premise of a fixed reference
frame and that (in modern language) there were only 7 degrees of
symmetry governing space and time, rather than 10 governing spacetime:
6 of the symmetries pertained to space alone; 1 to time alone. The
remaining 3, which involve both space and time were not recognized by
either Galileo (or other heliocentrists), nor by the Church.

On the larger issue, the Church had a point: Galileo and his
contemporaries were doing nothing more, in effect, than replacing one
sacrosanct ground by another -- the sun.

It was ONLY during his confinement that Galileo recognized the error
of his ways -- and the error of everyone else's ways. What's not
widely advertised and what's widely forgotten is that (1) Galileo was
the one who first formulated the Principle of Relativity (i.e. the
remaining 3 degrees of symmetry), and (2) he did this while in
confinement.

Very likely, being forced to admit he was wrong before a committee had
the effect of planting the seeds of doubt in his own mind. It's always
a healthy exercise to actually do this at some point (even if you
don't believe you're wrong). It forces you to think out of the box.

It was only during confinement that he finally realised what
landlubbers seem to be completely blind to; and what any sailor or
navigator for the past 5000 or 10000 years or more could have told
you: motion is relative!

At the time (and even at the present time) people were so used ot
thinking in terms of the the ground as being the "Cosmic Floor" it
didn't fully dawn on them (even the most "progressive" thinkers of the
time) just what the consequence of the ground NOT being the ultimate
ground of the cosmos really was.

Only a few thought that far ahead. Kepler, for instance -- though not
widely known -- was one of the first to write a Science Fiction story
about going to the Moon. I haven't seen the story, but clearly he must
have realised that out there in space there are no landmarks and that,
like the sailor in the middle of the ocean on a downwind heading where
everything seems perfectly still, no matter how fast you move in any
direction you'll seem like you're standing absolutely still. The
vacuum has no reference speed attached to it.

If that seems obvious to you, in retrospect, that I'd argue that even
YOU don't believe it! You just think you do! For, one of the
consequences of that fact drives a stake deep into concepts that you
and almost everyone still adhere to -- the Absoluteness of
Genidentity. If motion is relative, then a consequence is that the
"sameness" of two points at different times is relative too. You can
no longer say that "New York" in 2008 is the "same place" or even the
"different place" as New York in 2001.

Therefore, it makes no more sense to have a memorial there, than it
does to have it somewhere else at the same latitude on Earth (where
the Earth would have been oriented in 2001 relative to some fixed
axes), or out in the middle of space where the Earth "used to be in
2001".

The property whereby two things at different times are considered the
"same thing" is called Genidentity. Almost everyone holds to
Genidentity of points as a fundamental assumption and they act on that
belief. Even if they say they don't, they believe it (even if they
BELIEVE they don't believe if, they do). Actions count, not words.

Nor is it totally obvious from the point of view of a typical freshman
Physics course. A 200 level course generally only treats rotational
and translation symmetries (and, correspondingly, the conservation of
angular momentum and linear monentum); and time translation (and
consequently the conservation of energy). Few that I know of
explicitly account for boost-invariance (much less the relativity of
genidentity), much less the corresponding conserved quantity (moving
mass moment). Not too surprisingly the only place where you see this
conserved quantity in prominent use is by those involved with
navigation (moments are important for aircraft, ships and rockets) --
again, the landlubbers are one step behind.

Boost invariance, along with its full ramifications (relativity of
genidentity, explicit treatment of the moving mass moment) is the odd
one out, from a typical 200 level course.

Nor was boost-invariance obvious to those who were around even as late
as the 19th century. Even Maxwell, who was famous for articulating the
vivid description of 'space having no reference speed or landmarks"
nonetheless attached a reference speed to the vacuum in his equations
of electromagnetism -- the G vector. So, his equation for the E field,
in terms of the potentials, read E = -grad phi - dA/dt + G x B; and he
wrote the D field as D = K E. In modern notation, we'd simply define E
= -grad phi - dA/dt, so that Maxwell's original equations (in modern
form) read: D = epsilon (E + G x B).

Notice there's no G vector in Maxwell's list: A, B, C (= J + dD/dt),
D, E, F (= rho E + J x B), H, I (= magnetization), J? Lorentz, in
effect, took it out by raising to a postulate the empirical fact that
G = 0 in every frame of reference. Yet, even he tried to explain away
the newly (re)found boost invariance in terms of something similar to
Maxwell's boost non-invariant theory. Only Einstein (re-)raised
Galileo's axiom of boost invariance back to a postulate ... albeit at
the cost of replacing the (infinite) absolute speed by a (finite)
absolute speed. This is why, despite the occurrence of an absolute
speed in the foundation, the theory is designated "Relativity". The
name is meant to contrast it to the boost symmetry-breaking formalisms
that immediately preceded it.

Even Newton had a major issue with it and tried to argue against it --
he implicitly recognized that with boost invariance, and the
corresponding loss of genidentity, there could no longer be a cohesive
notion of spatial geometry -- since the very notion of "point" is
premised on genidentity. With relativity, your only out is to delve
one level deeper and replace the concept of a point by the even more
basic concept of a point-at-an-instant -- what we now call an "event".

Thus, it's not Minkowski's geometry which forced space and time into a
union, but Galileo's axiom of relativity! Minkowski merely consummated
the union that was already in place; the boost symmetries already mix
space and time (in one direction); the Poincare' version of boost mix
space and time in the reciprocal direction, but did not innovate the
mixing of space and time.

Newton was not in a position to handle that, and specifically set out
to formulate his physics on SPATIAL geometry. So, this forced his
hand: he had no choice but to reject the relativity of motion and
reject the notion of boost-invariance. His arguments about
acceleration being absolute were specious (all that proves is that
velocities are elements of an affine geometry, not that they too are
absolute!)

They were all wrong, and only the later (post-confinement) Galileo had
it right. But the (pre-confiment) Galileo was just as wrong as the
Church.