Natural Science Forum / Physics / Research / December 2007

This is an expansion of the reply issued back in April or May in
sci.math.research to:
This Week's Finds: Week 250
2007 April 26
http://groups.google.com/group/sci.physics/msg/68b6191b0c963d6d?dmode=source

A more complete version may be found under:
         http://federation.g3z.com/Physics/index.htm#TWF250

There are a lot of issue here, including a clarification of the roles
Maxwell and Lorentz in the emergence of Special Relativity (and a
clearing up of a major misconception of what Maxwell actually did and
said). There's also a novelty posed here: linking the notion of
relativity to affine and projective spaces, and linking the genesis of
the space-time concept to none other than the principle of relativity,
itself!

  "Well, even though I can't allow you to move the sun through the
   skies, what I can do is to stop the sky."
   -- Untold Story of Phaeton (what *really* happened)
      http://federation.g3z.com/FedSeries/FourthWave/Phaeton%20Myth.htm

"Right now I'm in a country estate called Les Treilles in southern
France, at a conference organized by Alexei Grinbaum and Michel
Bitbol:
1) Philosophical and Formal Foundations of Modern Physics,
http://www-drecam.cea.fr/Phocea/Vie_des_labos/Ast/ast_visu.php?id_ast=762"

"Okay - now for our Tale.  I want to explain double cosets as spans of
groupoids... but it's best if I start with some special relativity."

[...]

"Though Newton to have believed in some form of "absolute space",  the
idea that motion is relative predates Einstein by a long time..."

(Foray into the concept of classical Galilean Relativity and the
Poincare' Relativity of special Relativity)

These following are my comments relating to the original, a few parts
kept intact.

Though Newton is said to have believed in some form of absolute space,
the idea that motion is relative predates Einstein by a long time. In
1632, in his Dialogue Concerning the Two Chief World Systems, Galileo
wrote:
(To paraphrase, since the original comment was in Baez' article):
"In a deckhold of a large steady-going vessel, it looks and feels like
you're standing still, the same as when you're ashore."

(Of course, Galileo didn't discover that. Any sailor who's ever gone
on a downwind heading already knows all about that.)

As a result, the coordinate transformation we use in Newtonian
mechanics to switch from one reference frame to another moving at a
constant velocity relative to the first is called a Galilei
transformation. For example:
    (t, x, y, y) --> (t, x + vt, y, z).

The problem Newton had with the concept can be more clearly understood
if put in modern perspective. When we say motion is relative, what
we're actually saying is that there is no distinguished 0 for
velocity. In a typical Physics course, we are taught to regard
velocity as a vector. But the absence of a distinguished origin is
precisely what differentiates a vector space from an affine geometry!
Thus, what we are actually saying, when we say that motion is
relative, is that velocities are not vectors at all, but are points in
an affine space [and that the present-day Earth-bound civilization is
not yet advanced enough to fully understand and fully accommodate the
notion of relativity.]

One can go further, in fact, and treat velocities as points not just
in an affine space, but projective space, if we also include the
infinite velocities. Though worth mentioning, we won't actually do
this here, and will only make brief reference to it below when
explaining the difference between Galilean relativity and special
relativity.

In either case, the problem that Newton had emerges here. If
velocities are relative, then what are positions? Even more relative?

Since affine points are relative, then the vector operations
  (r,v) |-> rv, (v,w) |-> v+w
now have to be replaced by affine operations
  (r,o,v) |-> (1-r)o + rv, (o,v,w) |-> v-o+w
with the vector operations recast so as to make the origin explicit,
rv = (1- r)0 + rv and v + w = v - 0 + w . In a way we already
acknowledge this: the so-called parallelogram rule for vector addition
actually pertains to the 4 points, v,0,w, v - 0 + w , not just 3. So,
it's actually the operative rule for affine addition, not vector
addition!

The difference of affine points, however, can still be considered a
vector. Indeed, that is the very genesis of the concept. Vector, in
Latin, means to carry or transport. So the arrow depicting a vector
has a head h and tail t corresponding to the 2 affine points
respectively situated, and the vector itself is their difference h -
t . It is visually represented by drawing an arrow from point t to
point h .

It's at this point, we see Newton's dilemma. Accelerations are
differentials of velocities, so they must be vectors. In particular, a
distinguished 0 has meaning for acceleration. But, if vector
accelerations are the differentials of affine velocities, then affine
velocities are differentials of what? That is, how does one complete
the analogy,
  (acceleration):(velocity) = (velocity):(position);
that is,
  (vector):(affine) = (affine):(???).

Classically, positions are treated as points in a Euclidean geometry
-- that is as points in an affine geometry. But if positions are
affine points, then their differences would be vectors, and velocities
would not be affine at all, but vectors!

This was the dilemma Newton faced. There was simply nothing in the
vocabulary of the time to complete the analogy above. That is, the
question
     The differential of (???) is an affine point
could not be resolved.

This is a very important point that almost always passes by
unnoticed.  Though it is rarely discussed explicitly, the most
important ramification of the relativity of motion cuts deep into the
very core of how we think about space, itself, and conduct our lives.

The relativity of motion means an even more relative relativity of
position. In particular, if it is relative whether one is standing
still "at a point" or moving, then the very question of whether that
point at an earlier time is the "same place" or "different place" as
that at a later time is, itself, relative. For instance, the very
question of whether the World Trade Center in 2001 in New York is the
"same place" as the hallowed ground commemorated in succeeding
Septembers, itself, becomes relative! After all, the Earth is moving
around the Sun and is rotating around its axis. The Sun is moving
around the center of the Milky Way galaxy and the galaxy is moving
toward the Great Attractor. So, is the "same place" even on Earth at
all, or is it somewhere in the middle of outer space where the "Earth
was"? The answer is: it's relative.

Thus, the very underpinning of such hallowed ceremonies as memorials
-- one of the major foundations of everyday life -- is undercut at its
very root. So, though people may "know" that motion is relative, their
lives have not actually accommodated that principle. And even
unbeknownst to themselves, they actually do not know that motion is
relative!

One should not, however, generalize too much. There actually are
languages where the essential unity of the concepts of motion and
stasis are explicitly recognized. For instance, in Japanese, the same
root is used for all three concepts, "to be at", "to come" and "to
go".

The underlying problem, of course, is that humans are still currently
an Earth-bound race and so are caught in the warp of an "absolute
space" frame of mind fixed on the Earth, as if the surface of the
planet were a Cosmic Floor. In the near future, as this civilization
evolves from a 2-dimensional geography into the 3-dimensional
cosmography of a spacefaring civilization, the relativity of the
sameness of position will bear more directly on peoples' experiences.

The more adventurous sorts will refer back home to the "Earthlubbers
who still think places are fixed, are stuck in their Real World and
are out of touch with the Real Cosmos".

Just from asking the right question here, one sees that the
ramification of the relativity of motion is that points in space
cannot form an affine geometry

For whereas the question of motion vs. rest is relative, the mere
question of whether a point at one time is the same as a point at
another time now becomes relative! The very cohesiveness of the
concept of point, when meant in the sense of a location or place
enduring in time, is lost.

In place of "point", one needs to adopt a new primitive that runs
deeper: a "place at a given time" or, what is known in the parlance of
modern Physics, an "event". This is actually what makes the concept of
space-time necessary, in place of space and time.

Thus, we see here that the true genesis of the space-time concept is
the *Galilean* principle of relativity, not Minkowski, and the
necessity of the concept goes all the way back to Galileo, not
Einstein, Lorentz or Minkowski.

So, what is the anti-differential of an affine space? A more general
type of geometry whose points are "events" and where the space of
velocities at each point is an affine space. In modern language, this
is known as an Affine Bundle. It is the concept that had been absent
in the time of Newton and is what made necessary the expedient Newton
took with his concept of "absolute space" in order to make it possible
to properly establish a geometric foundation for his physics.

Now, skipping ahead a couple centuries past Galileo and Newton...

By the time Maxwell came up with his equations describing light, the
idea of relativity of motion was well established.

In 1876, he wrote:
(to paraphrase, since the original comments are in the Baez article)
"In space, they can't hear you scream. There are no landmarks. Not for
position, not for time, nor even for your motion."

However, the terrestrial frame of mind still pervaded Maxwell's
thinking and the concept underlying these remarks (notwithstanding the
intellectual awareness of it) had not been fully assimilated. In fact,
the theory he posed in his treatise even undercut his own remarks
here! One of the more significant deficits of Maxwell's treatise is
the absence of a comprehensive analysis of the effect of a change in
the frame of reference on the equations defining the theory he laid
out. There was a strong tendency, in particular, to refer everything
to a fixed ground frame.

This actually made it harder to fully appreciate and understand one of
the more important, but little-remembered, features of Maxwell's
theory: his equations were still Galilean invariant. This invariance
was achieved, by breaking the relativity of motion, hypothesizing the
existence of a universal velocity G . The frame where G = 0 was then
the frame where light propagation would take place in a sphere on a
fixed center. The breaking of relativity was brought in by, what in
modern terms, would be relations of the following forms
  D = epsilon_0 (E + GxB), B = mu_0 (H - GxD)
between the two sets of fields. In actuality, Maxwel's E is equivalent
to what we would now call E + GxB , since he posed the following
equation between the field and potential:
  E = -del phi - dA/dt + GxB
whereas, today, we just write
  E = -del phi - dA/dt.

Because Maxwell never carried out a complete analysis, he never fully
discerned the necessity of the second relation
  B = mu_0 (H - GxD),
though the - GxD term had been mentioned by Heaviside in a posthumous
footnote.

It is an unfortunate circumstance in timing that Maxwell's treatise
came out only in time to quote figures for light speed of accuracy of
100,000 km/hour. For, this put the quotes just outside the threshold
where one could begin to discern G from the Earth's motion.

And because he was still fixed in the terrestrial "Earthlubbing" frame
of mind, and because Maxwell had in fact not sufficiently incorporated
the principle of relativity into his own thought, he never even raised
the question of what effect the motion of the Earth would have on the
measurement of light speed!

Needless to say, it eventually reached the point where the uncertainty
of the figures quoted for light speed fell under the magic threshold
of the Earth's orbital speed. So, by the 1880's it gradually became
apparent that no motion could be discerned at all, and that it was as
if every frame of reference had G = 0 !

Eventually, this led to the modification of the two constitutive laws
by Lorentz to the form
  D = epsilon_0 E, B = mu_0 H.
Consequently, the theory that presently goes under the name of
"Maxwell's Theory" and the equations under the name "Maxwell's
Equations" are actually a hybrid creation of Maxwell and Lorentz and,
in earlier times, had been known as Maxwell-Lorentz theory or the
Maxwell-Lorentz equations to make this distinction explicit.

This is the true genesis of what came to be known as Special
Relativity, though it was not properly appreciated at the time as
such. What's significant about this turn of events is that it
repudiates Maxwell's own repudiation of relativity! It reasserts the
relativity of motion -- but now in a difference guise.

The contrast between the two forms of relativity is not too difficult
to explain: whereas relativity under Galileo had entailed that
infinity is an absolute speed, relativity under Maxwell-Lorentz theory
entails that a finite speed -- the speed at which light travels -- is
absolute.

That is, under changes of reference in the physics of Galileo and
Newton, an infinite velocity remains an infinite velocity. This is
actually what justifies treating velocities there as merely points in
an affine space, rather than points in a projective space. We can
distinguish an invariant "plane at infinity" in the geometry
comprising velocities in Newtonian/Galilean physics.

In contrast, in a change in frame of reference under a form of
relativity consistent with the Maxwell-Lorentz theory, light speed
remains the same. The roles are reversed. While light speed is now
absolute, it's now infinity that is a relative speed. Under a change
in frame of reference, an infinite speed transforms into a faster-
than- light speed.

As a consequence of the revised principle of relativity, it is as if G
= 0 were true in every frame of reference, so the G is superfluous. It
is for this reason that the alphabet soup that Maxwell had devised: A
(magnetic potential), B (magnetic induction), C (total current), D
(electric displacement), E (electric intensity), F (force density), H
(magnetic intensity), I (magnetization) and J (current) now has a hole
in it where the letter G used to be.

So, the big deal about special relativity is not that motion is
relative. The principle of relativity goes much deeper back to Galileo
and had already been present. Rather, it's that it is possible to
assert the relativity principle while keeping the speed of light the
same for everyone - as the modern form of Maxwell's equations insist,
and as we indeed see! In place of Galilean transformations, which
leave infinite velocities invariant, we have Lorentz transformations,
which leave motions at light speed invariant.

Making infinite speeds relative means the same thing as making
simultaneity relative. Think about this for a while: two events are
simultaneous, precisely one would need to go at infinite speed to get
from one event to the other. Since infinite speed is absolute in
Newtonian physics, then so is the simultaneity of events. Thus, it is
possible to attach a clock time to every single event in such a way
that two events have the same time precisely when they are
simultaneous. This labeling of events by time is, then, independent of
reference frame.

In popularizations this feature is couched by the unnecessarily vague
and misleading term "absolute time".

In contrast, under the relativity of Maxwell-Lorentz theory, infinity
is a relative speed. Consequently, two events deemed simultaneous in
one frame of reference are not so under a change in frame of
reference! This is what forced people to replace Galilei
transformations by "Lorentz transformations", whose most important
distinguishing feature is that two coordinate systems moving relative
to each other will disagree not just on where things are, but when
they are.

As Einstein wrote in 1905:
(To paraphrase)
"We can take G = 0 in all frames AND consistently assert a (new form)
of relativity for motion; thus allowing us to take, unchanged,
Maxwell's theory in a 'stationary frame' to one for arbitrary moving
frames."

Einstein, himself, was not fully cognizant of the distinction between
Maxwell's older theory and the then more recent Maxwell-Lorentz
theory, and so thought he was correcting an internal discrepancy
latent in Maxwell's theory, itself.

But once the fog of war from that time period is lifted, we see that
what he was actually doing was removing the discrepancy between
Maxwell and Lorentz from the Maxwell-Lorentz theory. What he was
referring to is that one can, indeed, remove Maxwell's hypothesis of a
universal velocity G , and assert the constitutive law
  D = epsilon_0 E, B = mu_0 H
in every frame of reference, as if G = 0 were true in each one,
precisely as Lorentz had said could be done. Only he didn't try to
force-fit it into Galilean relativity, as Lorentz had been trying to
do up to then.

So, what really changed with the advent of special relativity? First,
our understanding of precisely which transformations count as
symmetries of spacetime. These transformations form a group. Before
special relativity, it seemed the relevant group was a 10-dimensional
gadget consisting of:
* 3 dimensions of spatial translations
* 1 dimension of time translations
* 3 dimensions of rotations
* 3 dimensions of Galilei transformations
Nowadays this is called the Galilei group.

With special relativity, the relevant group became the Poincare'
group:
* 3 dimensions of spatial translations
* 1 dimension of time translations
* 3 dimensions of rotations
* 3 dimensions of Lorentz transformations
It's still 10-dimensional, not any bigger. [Actually: smaller. See
following note] But, it acts differently as transformations of the
spacetime coordinates
  (t, x, y, z) .

(Note: Actually, it's smaller. A deeper analysis of the Galilei group
uncovers the need for an 11th degree of symmetry. The passage from
Galilei to Poincare' involves mixing this extra degree of symmetry
with the Galilean notion of time translation to give us the Poincare'
notion of time translation. In fact, to define a consistent limiting
relation for Poincare' -> Galilei, one needs to add an 11th parameter
to Poincare'!)

Another thing that changed was our appreciation of the importance of
symmetry! Before the 20th century, group theory was not in the toolkit
of most theoretical physicists. Now it is.

Okay. Now suppose you're the only thing in the universe, floating in
empty space, not rotating. To make your stay in this thought
experiment a pleasant one, I'll give you a space suit. And for
simplicity, suppose special relativity holds true exactly, with no
gravitational fields to warp the geometry of spacetime.

Would the universe be any different if you were moving at constant
velocity? Or translated 2 feet to the left or right? Or turned
around?  Or if it were one day later?

No! Not in any observable way, at least! It would seem exactly the
same.

Unlike the Earth, the vacuum of outer space does not have, in and of
itself, fixed landmarks, just as Maxwell had described. Where but for
the presence of stars, planets and other celestial bodies, your
sojourn into space would be the very definition of Lost. Like the
stagnantly-unchanging homogeneous residential district in the working
class neighborhood of a Communist-era nation, it would look so much
the same everywhere, with the landscape filled with buildings and
apartments that can barely be distinguished from one another and look
the same from year to year, that you wouldn't be able to find your way
around and wouldn't even know what year it was!

So in this situation, it doesn't really make much sense to say "where
you are", or "which way you're facing", or "what time it is".

Spacetime is actually more deadening, still. You could run away from
the Communist state (or try), and it would look like it's receding
from you. You can distinguish being stuck in it from getting out of
it. In contrast, the vacuum has no reference velocity (notwithstanding
Maxwell or Newton's assertions to the contrary). Therefore, you can't
even tell if you're moving anywhere or standing still.

The remainder of the discussion can be found both in the Baez article
and the discussion whose link is posted at the top of the article.