Mark's Elements, III

[Mark Hopkins]

Mark's Elements, III

This time, we're going to provide a much deeper analysis in which the
the POINT and DISTANCE concepts become defined in terms of more basic
primitives. The resulting structure will be a Minkowski space, familiar
to those who study Relativity theory. This treatment follows closely that
given in [1], but cleans up the notation, concepts and proofs and reveals a
structure similar in nature to a differential manifold -- and relates it to
the material presented in Mark's Elements I and II. In many ways, the
treatment also follows closely Einstein's original formulation of Relativity
theory, rather than the coordinate-based approach of Minkowski. In fact, one
can draw a rough parallel between the axioms imposed here and his as follows:

Equivalence <--> Clock Axiom + Definition of Inertial Points
Invariant Speed <--> Signal Axiom
Existence <--> Axiom of Inertia

>From our Axiom of Inertia, the whole pseudo-Euclidean structure of the
Minkowski space will be derived!

This presentation, much more than the others, is highly technical so that
it will have to be studied and reviewed carefully.

Contents:
(12) Basic Concepts
(13) Derivation of Metric Space Axioms
(14) The Differential Structure
(15) The Fixed Point Lemmas
(16) Properties of Inertial Motion
(17) The Connectivity Axiom and the Euclidean Structure
(18) The Axiom of Inertia

Time is the basic concept here, not space. At the most fundamental level
distances are not measured, but rather only time differences are. This is
a unique feature of the geometry underlying Relativity theory: Space is a
redundant concept, and was a discovery initially made by A.A. Robb in 1914
[2], [3], [4].

The following notation will be used, partly in order to adapt this
presentation to ASCII:

f.g: the composition of functions -- f.g(x) = f(g(x)).
f': the inverse of a function.
f': the derivative of a function (in Sections 16 and 17, the
context will make it clear which meaning is intended).
inf A: The least upper bound of the set A.
x in A: x lies in the set A.
C^n: the class of functions with contiuous derivatives up to the
nth order.
x != y: x is not equal to y.
R: the metric space closure of the set of fractions
(i.e., the real number system).
*word*: italics
WORD: boldface
sqrt(x): the square root of x

(12) Basic Concepts
The POINT is analysed here as a continuing succession of INSTANTS, which
occur "at the same place". Not too suprisingly this also captures the
concept of MOTION, provided one remembers that "same place" and "different
place" are relative (e.g., since the Earth is moving about the Sun, are
you at the "same place" as you were a moment ago?).

There are languages, in fact, which treat all the concepts (to BE (AT),
to COME/GO), using a common morpheme -- notably Japanese. Even in English,
one will often mix terms in this way, often using "go" to denote continuation
in time, as if the continuation of something at a point were a form of *in
situ* motion. We also use phrases such as "how goes it?" = "how are you",
and "... going to ..." to express continuation into the future, as well as
"still going" = "still alive".

The main difficulty philosophers have had in the past in trying to
capture the essence of the logical relations in space and in time
(particularly the attempt by Leibnitz), was exactly their inability to
analyse the concept of point further down to the more basic, and logically
prior, concept of instant; to unify the concepts of point and motion;
to eliminate the prior standing of the concept of space, in favor of
time; and to recognize that the ordering relation provided by "before" and
"after" can only have been inferred to be a *partial ordering*, since (a)
that's all that our perceptions actually imply and (b) there is simply no
direct way to infer any such thing as a "there and now" in the first place,
except by assumptions which, in virtue of (a), have no direct connection to
anything we perceive.

DEFINITION 1: Let I denote the set of all instants.

THE CLOCK AXIOM: There is a partition, P, of I such that for each A in P, a
one-to-one map T_A: A -> R exists. The map T_A is "A's
internal clock".

DEFINITION 2: Each element of P is called a POINT. Thus, a point is a
continual succession of instants.

A natural ordering structure is provided on the instants that comprise a
point with the following notation:

For every A in P; e, f in A:
e <= f <==> T_A(e) <= T_A(f)
e >= f, e < f, e > f defined similarly

The next axiom allows us to extend this ordering structure to instants
which lie on different points, by use of a "signalling function":

THE SIGNAL AXIOM: For every A, B in P, there is a function
s_AB: B -> A
such that
(1) s_AB is monotonically increasing:
For all e, f in B, if T_B(e) < T_B(f) then
T_A(s_AB(e)) < T_A(s_AB(f)).
(2) The Identity Property -- s_AA(a) = a
(3) The Causality Property -- s_AB(s_BC(c)) >= s_AC(c).

DEFINITION 3: Let a, b be instants respectively on points A and B:
a <= b <==> s_BA(a) <= b
a == b <==> a <= b and b <= a
a >= b <==> b <= a
a < b <==> a <= b, but not b <= a
b < a <==> b <= a, but not a <= b

These define the concepts of "before/at", "at", "after/at", "before" and
"after" respectively. The == relation allows us to establish coincidence
relations between points.

DEFINITION 4: a is at B <==> a == b for some b in B
A meets B at c <==> c is at A and at B
A meets B <==> A meets B at some instant c.

Thus, an ordering relation is established among points, modulo the
coincidence relation:

THEOREM 1: <= is a pre-order and == an equivalence relation.
Proof:
The second property follows directly from the first. To prove the
first, we need to show that <= is reflexive and transitive. Reflexivity
comes directly from the Identity Axiom:

s_AA(a) = a, therefore a <= a

Transitivity ultimately comes from the Causality Axiom. Suppose a, b and c
are instants respectively on points A, B and C with a <= b <= c. Then

s_CA(a) <= s_CB(s_BA(a)) by the Causality Axiom
<= s_CB(b) by monotonicity of s_AB and a <= b
<= c since b <= c

Therefore, a <= c.

THEOREM 2: a == b <==> s_AB(b) = a and s_BA(a) = b
If a is an instant on point A, then a is at A.
Proof:
The second property is an immediate consequence of the first since
s_AA(a) = a. So assume that a and b are instants respectively on points
A and B with a == b. Then

s_AB(b) <= a, s_BA(a) <= b.
Therefore,
a = s_AA(a) <= s_AB(s_BA(a)) <= s_AB(b) <= a
b = s_BB(b) <= s_BA(s_AB(b)) <= s_BA(a) <= b

The Signal Axiom also makes it possible to define distances and to extend
each point's internal clock to all of I.

DEFINITION 5: Let A, B be points and b any instant on B. Then let
t_A(b) = 1/2 (T_A(s_AB(b)) + T_A(s_BA'(b)))
d_A(b) = 1/2 (T_A(s_AB(b)) - T_A(s_BA'(b)))
and
T_AB = t_A.t_B', R_AB = d_A.T_B'

The function t_A is an extension of T_A, and the expected properties with
respect to the coincidence relation are satisfied:

THEOREM 3: T_A(a) = t_A(a) for all instants a on point A
d_A(b) = 0 <==> b is at point A.
Proof:
Let a be an instant on A. Then

t_A(a) = 1/2 (T_A.s_AA(a) + T_A.s_AA'(a))
= 1/2 (T_A(a) + T_A(a))
= T_A(a)

Therefore, we will use the notation T_A for t_A from here on.
If b is an instant on point B that is at A, then let a be an instant on A
such that s_AB(b) = a and s_BA(a) = b. Then

d_A(b) = 1/2 (T_A.s_AB(b) - T_A.s_BA'(b))
= 1/2 (T_A(a) - T_A(a))
= 0

Conversely, if d_A(b) = 0, then

T_A.s_AB(b) = T_A.s_BA'(b)
s_AB(b) = s_BA'(b), by the monotonicity of T_A

Letting a = s_AB(b), we get a = s_BA'(b), or s_BA(a) = b. Therefore a == b.

THEOREM 4: Let b and c be instants respectively on points B and C, with
b == c. Then the following identities holds for all points A:
(a) s_AC(c) = s_AB(b)
(b) s_CA'(c) = s_BA'(b)
(c) d_A(c) = d_A(b)
(d) t_A(c) = t_A(b)
Proof:
If b and c are as indicated, then
s_BC(c) = b and s_CB(b) = c.
Therefore
s_AB(b) = s_AB.s_BC(c) >= s_AC(c)
= s_AC.s_CB(b) >= s_AB(b)

both by the Causality Axiom.
To prove part (b), let a = s_BA'(b). Then

a >= s_CA'(c) by the monotonicity of s_BA'.
= s_CA'.s_CB(b)
= s_CA'.s_CB.s_BA(a) >= s_CA'.s_CA(a) = a
b = s_BC(c) = s_BC.s_CA.s_CA'(c) >= s_BA.s_CA'(c)

where the last inequality in each sequence is by the Causality Axiom
(and the monotonicity of s_CA', in the sequence for a).
Parts (c) and (d) are immediate since

1/2 (s_AC(c) +/- s_CA'(c)) = 1/2 (s_AB(b) +/- s_BA'(b))

Based on these results, we can factor out the relation ==, which is a
congruence with respect to all of the operations defined. This leads to
the following refinements:

DEFINITION 6: An EVENT is an congruence class of instants, modulo ==.

This captures the idea that an event somehow should represent the meeting of
two or more moving points at an instant. We could also define a WORLDLINE to
be a point modulo the relation ==, though preference will be given to the
term "point" below -- and also to the term "instant" below.

For what follows, it will be easier to just skip the extra pedantry and
use the set P of points to refer to P/==, and I to I/==, and therefore
understand two instants to be identical whenever they coincide. With
respect to this refinement, points can now have intersecting sets of
instants and THEOREM 2 will become exact:

THEOREM 5: Let A, B be points and b an instant on point B.
(a) d_A(b) = 0 <==> b is an instant in point A.
(b) A = { a in I: d_A(a) = 0 }
= { a in I: a is at A }
(c) A meets B <==> A and B intersect.
Proof:
If d_A(b) = 0, then as argued above b == a for some instant a in A. By
our convention, a and b are then considered identical. Part (c) is an
immediate consequence of (b), which immediately follows from (a).

(13) Derivation of Metric Space Axioms
An REST FRAME is thought of as a collection of clocks scattered all
throughout space, each remaining at a constant distance from each other and
all synchronized by the continual exchange of signals. In the original
treatment of Relativity, the concept of "rigid ruler" was also employed as
a device to enforce the constancy of distances between the clocks. This
concept, as seen above, is superfluous, and the desired condition can be
directly captured by requiring a constancy of "echo time" between the
clocks. From this, a Metric Space will emerge for free!

DEFINITION 7: A REST FRAME, F, is a set of points such that:
(1) F contains at least two distinct points,
(2) For all points A, B in F, and all b in B
* d_A(b) = constant = AB
* T_A(b) = T_B(b),
(3) Every instant is at some point in F:
union F = I = union P.

Using a rest frame, one can independently establish the Identity Axiom,
if one interprets the use of the relation "c is at A" in part (3) of the
definition above in the sense d_C(a) = 0

COROLLARY 3: s_AA(a) = a
Proof:
Let a be any instant on point A. Let C be any point in the rest frame
F which meets A at a: 0 = d_C(a). Then

0 = d_C(a) = 1/2 (s_CA(a) - s_AC'(a))
==> s_CA(a) = s_AC'(a)
==> s_AC.s_CA(a) = a
or
s_AC(c) = a, where s_CA(a) = c

Therefore a == c.

Since THEOREM 4 was established indepdendently of the Identity Axiom,
we may then apply part (a) of the theorem to obtain a = s_AC(c) = s_AA(a).

A rest frame is a Metric Space which has duration in time. This
conception provides a better fit of the way we perceive space than the
classical treatment given in Euclidean Geometry and is the main reason
I prefer to call successions of instants "points" rather than "worldlines".
In our actual perception, we don't actually see "worldlines" as lines,
but as points which endure in time, possibly engaging in a state of
motion.

THEOREM 6: If A, B are points in a rest frame then for all instants b on B:
T_A.s_AB(b) = T_B(b) + AB
T_A.s_BA'(b) = T_B(b) - AB
Proof:
T_A.s_AB(b) = T_A(b) + d_A(b) + T_B(b) + AB
T_A.s_BA'(b) = T_A(b) - d_A(b) = T_B(b) - AB

THEOREM 7: Every rest frame is a metric space with respect to its distance
function:
* AB >= 0, with equality if and only if A = B
* Symmetry: AB = BA
* Triangle Inequality: AB <= AC + CB
Proof:
By monotonicity, s_BA'(b) <= b <= s_AB(b), for any instance b in B.
Therefore, d_A(b) >= 0. This property holds for arbitrary points A and B.
Therefore for any points A and B in a rest frame: AB = d_A(b) >= 0.
The Symmetry property comes from THEOREM 6, which can also be written in
the form:
T_A(a) = T_B.s_AB'(a) + AB
T_A(a) = T_B.s_AB(a) - AB

Reversing the roles of A and B and comparing the result with the two
equations above leads directly to the desired result.
The Triangle Inequality comes from Causality and THEOREM 6. Let A, B
and C be points in a rest frame with b and instant on B. Then

T_B(b) + AB = T_A.s_AB(b)
<= T_A.s_AC.s_CB(b) by the Causality Axiom
= T_C.s_CB(b) + AC
= T_B(b) + CB + AC
Therefore,
AB <= AC + CB

In virtue of our prior treatment on Euclidean Spaces and Metric Spaces,
all we need to do in order to prove that this metric makes a rest frame
a Euclidean Space is to verify the properties:

(1) For all points A, B and numbers r, there is a point C such that
AC = |r| AB, BC = |1 - r| AB
(2) If AB + BC = AC, then
AB CD^2 - AC BD^2 + BC AD^2 = AB AC BC

The uniqueness of the point described in (1) is actually a consequence of
(2), and we will define it as C = [A, r, B] as before. The operations of
the underlying Inner Product space will have the following compatibility
relations then:

In each space V_O, for each point O of a rest frame:
0 = O, rX = [O, r, X], X + Y = 2 [X, 1/2, Y]
<X, Y> = (OX^2 + OY^2 - XY^2)/2
|X| = OX

To prove (1) and (2), we will need another axiom.

(14) The Differential Structure
The ultimate intent behind the formalism, as stated at the very end of
reference [1], was to come up with a concept analogous to a manifold which
involved a network of "geodesic motions". The function

f_AB = T_A s_AB T_B'

plays the analogous role of the function that links coordinate patches in a
manifold and we have the commutative diagram, expressing the f's as
pullbacks of the s's, whose analytic structure is then inherited from
those of the f's:
s_AB
B -----------> A
| |
T_B | | T_A
v f_AB v
R = = = = = => R

In terms of the functions f_AB, both R_AB and T_AB (see DEFINITION 5) can
be written as:

T_AB(s) = T_A.T_B'(s)
= 1/2 (T_A.s_AB.T_B'(s) + T_A.s_BA'.T_B'(s))
= 1/2 (f_AB(s) + f_BA'(s)).
R_AB(s) = d_A.T_B'(s)
= 1/2 (T_A.s_AB.T_B'(s) - T_A.s_BA'.T_B'(s))
= 1/2 (f_AB(s) - f_BA'(s))

These are the forms originally presented in [1] in lieu of T_A and d_A.

The following then yields the analogue of a C^n manifold:

DEFINITION 8: A C^n TIME NET is a set of instants satisfying the
Clock and Signal axioms (taken modulo the coincidence relation
==), in which f_AB(s) are C^n for all s where R_AB(s) > 0,
and continuous when R_AB(s) = 0.

EXAMPLE 1:
The reason for the restriction can be made clear by this fairly mundane
example in 2-dimensional Minkowski Geometry. Let

A = { (0, t): t in R }
and B = { (gvt, gt): t in R }

where v is between -1 and 1 and g = 1/sqrt(1 - v^2) with

T_A(0, t) = t = T_B(gvt, gt).

The function s_AB(gvt, gt) is defined as follows:

s_AB(gvt, gt) = (0, g(t + v |t|))
Then:
f_AB(t) = T_A.s_AB.T_B'(t)
= T_A.s_AB(gvt, gt)
= T_A(0, g (t + v |t|))
= g (t + v |t|)

As it turns out, this is also the value of f_BA(t), and note that the
inverse of both functions has nearly the same form: g(t - v|t|). As will be
seen, these symmetries will be characteristic of points in mutually inertial
motion. From this, T_AB and R_AB can be calculated:

T_AB(t) = 1/2 (T_A.s_AB.T_B'(t) + T_A.s_BA'.T_B'(t))
= 1/2 (g(t + v|t|) + g(t - v|t|))
= gt

R_AB(t) = 1/2 (T_A.s_AB.T_B'(t) - T_A.s_BA'.T_B'(t))
= 1/2 (g(t + v|t|) - g(t - v|t|))
= gv |t|

The function T_AB(t) remains continuous to all orders, but R_AB(t) undergoes
a sharp turn corresponding to the instant when points A and B meet. This is
because the distance function R_AB(t) must remain non-negative, and for all
t != 0:
(d/dt T_AB)^2 - (d/dt R_AB)^2 = g^2 - (+/-gv)^2
= g^2 (1 - v^2) = 1

Both of these properties will hold in general.

(15) The Fixed Point Lemmas
Owing to the nature of the functional equations that will occur, a few
fixed-point properties will have to be established.

LEMMA 8: If f: R -> R is monotonically increasing with f(f(A)) = A,
then f(A) = A.
Proof:
If f(A) > A, then by monotonicity f(f(A)) > f(A) > A, and by a similar
argument if f(A) < A, then f(f(A)) < f(A) < A. The only possibility left,
therefore, is f(A) = A.

LEMMA 9: If (a_0, a_1, a_2, ...) is a sequence with:
A = a_0 <= B
a_{n+1} = F(a_n), for all n >= 0
F is continuous and monotonic with x <= F(x) and F(B) = B
then
a_n -> inf { x in [A, B]: F(x) = x }
as n -> infinity
If there are no other fixed points to F(x) between A and B, then
a_n -> B.
Proof:
First, it's easy to see that no fixed point of F can lie between x and
F(x) unless x = F(x). If there were one, call it B', then we'd have:

x <= B', with F(B') = B' <= F(x)

But F is monotonic, therefore F(x) <= F(B') = B', and x = B'. So B' cannot
lie between x and F(x).
Let
S = { x in [A, B]: F(x) = x }

Then the considerations above show that the sequence (a_n) is bounded
above by the set S. Therefore a_n <= inf S for all n >= 0. But the
sequence is strictly increasing, therefore it must have a limit, a, for
which: A <= a <= inf S.
This limit must be a fixed point at least as large as A, therefore

a is in S, therefore inf S <= a.

But a <= inf S since a_n <= inf S, therefore the limit of a_n is a = inf S.
This also shows that any set of fixed points of F bounded below has a
greatest lower bound.

Further, if B is the least fixed point of F(x) at least as large as A,
then a_n -> inf S = B.

(16) Properties of Inertial Motion
Based on the example, we first define inertial points

DEFINITION 9: Let F be a rest frame and C any point. Then C is inertial
with respect to F if for each point A in F there is a
constant L such that:

f_AC(s - L) = f_CA(s) + L

By redefining T_C(s) to T1_C(s) = T_C(s) + L, we can write

f1_CA(s) = T1_C.s_CA.T_A'(s)
= T_C.s_CA.T_A'(s) - L = f_CA(s) + L
and
f1_AC(s) = T.A.s_AC.T1_C'(s)
= T_A.s_AC.T1_C(s - L) = f_AC(s - L)

so that f1_CA = f1_AC. The calibration of T_C does not affect any of the
axioms or properties stated above, therefore we may always adjust T_C so
that f_AC = f_CA, for a specific point A in the rest frame.

Given a rest frame, a set of points that are inertial with respect to it
is defined. Due to the symmetry relation above, these points will turn
out to exhibit the time dilatation relations of Minkowski Geometry. In the
following theorems, we will be working over a C^2 Time Net.

In EXAMPLE 1, we noted that even though R_AB had a sharp break at the
instant points A and B met, T_AB was still continuous to all orders. This
result applies generally to points in a rest frame since all the points in a
rest frame share a common clock:

THEOREM 10: Let A be any point in a rest frame F and B be an arbitrary point.
The function T = T_AB is C^2 everywhere.
Proof:
Let b any instant along point B, and s = T_B(b). If d_A(b) > 0, then
T_AB is C^2 at s. Otherwise if d_A(b) = 0, we can choose a distinct point
C in the rest frame F, for which:

d_C(b) = d_A(b) + AC = AC > 0

Therefore, T_AB = T_A.T_B' = T_C.T_B' = T_CB has a continuous second order
derivative at s.

In EXAMPLE 1, we also noted that R encountered a sharp break when the
two points met and noted the relation T'(s0)^2 - R'(s0)^2 = 1 as the reason.
This result can also be generalized to a limited extent:

THEOREM 11: Let A be a point in a rest frame F, and B be any point
inertial with respect to F. Then at the instant B meets A:

T'(s0)^2 - R'(s0)^2 = 1
where R = R_AB, T = T_AB,
R' = dR/ds, T' = dT/ds

where the derivatives of R are one-sided derivatives as s -> s0
above or below, and s0 = T_B(b) where b is the abovementioned
instant.
Proof:
Let A, B be as given above, and b be an instant on B which is at A.
Then defining s0 = T_B(b), we have

R_AB(s0) = d_A.T_B'(s0) = d_A(b) = 0
or
f_AB(s0) - f_BA'(s0) = 0
Therefore
s0 = f_BA.f_AB(s0)

There is a constant L such that

f_AB(s - L) = f_BA(s) + L
or shifting s:
f_AB(s) = f_BA(s + L) + L
Therefore
s0 = f_BA.f_AB(s) = f_BA(f_BA(s0 + L) + L)

Define a function f(s) = f_BA(s + L). Then f inherits monotonicity
and continuity from f_BA, so that LEMMA 8 applies, to yield:

f(s0) = s0
or
f_BA(s0 + L) = s0
and f_AB(s0) = s0 + L

Taking linear combinations, f_AB(s0) +/- f_BA'(s0), we get: R(s0) = 0
and T(s0) = s0 + L, where R = R_AB and T = T_AB.

So now if we take the symmetry relation:

f_AB(s) = f_BA(s + L) + L
and invert it
s = f_BA'(f_AB(s) - L) - L

we can then also invert the relations

2 R = f_AB - f_BA' and 2 T = f_AB + f_BA'
and write
f_AB = T + R, f_BA' = T - R.
Therefore
s = T(s1) - R(s1) - L, where s1 = T(s) + R(s) - L

Taking derivatives with respect to s, we get:

1 = (T'(s1) - R'(s1)) (T'(s) + R'(s))

As s -> s0 from above or below, then s1 -> T(s0) + R(s0) - L = s0. Therefore

1 = T'(s0)^2 - R'(s0)^2

where the derivatives are taken in the limit as s -> s0 from above or below.

This result can be immediately generalized because even where an inertial
point B does not meet a point A in the rest frame, we can always find another
point C that it does meet at that instant and then use some relations between
A and C to arrive at the following inequality:

THEOREM 12: Let A be a point in a rest frame F and B any point inertial
with respect to F. Then

T'(s0)^2 - R'(s0)^2 >= 1
where R = R_AB, T = T_AB,
R' = dR/ds, T' = dT/ds
limits of R', T' taken as s -> s0 +/-.
Proof:
Given the assumptions above, let C be any point in F which meets B at
s0, so that T_CB'(s0)^2 - R_CB'(s0)^2 = 1. For an arbitrary s, we may
use the Triangle Inequality to write:

AC + R_CB(s) = AC + CD >= AD = R_AB(s)
and
R_AB(s) = AD >= AC - CD = AC - R_CB(s)

where D is a point in F which meets point B at instant s. Therefore:

AC + R_CB(s) >= R_AB(s) >= AC - R_CB(s)
or
|R(s) - AC| <= R_CB(s)

Since |R(s0) - AC| = R_CB(s0) = 0, the derivatives must then satisfy a
similar set of inequalities:

+R'(s0) <= lim (+R_CB'(s): s -> s0+) = +R_CB'(s0+)
-R'(s0) <= lim (-R_CB'(s): s -> s0-) = -R_CB'(s0-)

If in our definition of a C^n Time Net we had required all functions to
be C^n everywhere, we'd end up deriving R'(s0) = R_CB'(s0) = 0. But the
example cited illustrate how R_CB' may indeed switch sign abruptly at s0
where a point C in a rest frame F meets another point B. Since R_CB'
need not have one-sided limits at s0 (and generally does not), then the
more refined argument is necessary.

Noting that both +R_CB'(s0+) and -R_CB'(s0-) are both positive, we may
write:
R'(s0 +/-)^2 <= R_CB'(s0 +/-)^2
= T_CB'(s0)^2 - 1 by THEOREMS 10 and 11.
= T_AB'(s0)^2 - 1

since T_A = T_C everywhere. This establishes the result.

Together these results imply the linearity of T_AB:

THEOREM 13: Let A be any point in a rest frame F and B any point inertial
with respect to F. Then T_AB''(s0) = 0.
Proof:
Since T = T_AB is independent of the point A chosen from the rest frame, we
can always choose A so that R(s0) = R_AB(s0) = 0 and apply THEOREM 11 to get:

T'(s0)^2 - R'(s0 +/-)^2 = 1

The inequality in THEOREM 12 shows that T'(s)^2 - R'(s)^2 approaches 1 from
above as s approaches s0 from either side. Therefore the following
inqualities are satisfied upon differentiation:

+T'(s0) T''(s0) >= +R'(s0+) R''(s0+) >= 0
and -T'(s0) T''(s0) >= -R'(s0-) R''(s0-) >= 0.

which implies that T'(s0) T''(s0) = 0. The case T'(s0) = 0 is ruled out
by the relation T'(s0)^2 = 1 + R'(s0 +/-)^2 >= 1. Therefore T''(s0) = 0.

This implies that T_AB(s) = as + b, where a and b are constants of
integration. Noting, as above, that T_B may be calibrated so as to
make f_AB = f_BA and noting that none of this affects the linearity
of T_AB, we may write:

as + b = T_AB(s) = (f(s) + f'(s))/2

[where f' denotes the inverse of f, not its derivative]. Substituting
f(s) for s, we get:
f(f(s)) + s = 2(a f(s) + b)

By THEOREM 11, we also note that a = T'(s) >= 1, and in particular

(d/ds f(s) - a)^2 = (d/ds R(s))^2 <= a^2 - 1

by THEOREM 12.

In the following theorem, we are essentially duplicating the arguments
involving fixed point functional equations posed in the original Relativity
paper by Einstein to arrive at the time diilatation and length contraction
effects.

THEOREM 14: Let A be a point in a rest frame F, and B a point inertial with
respect to F, which meets A. Then there is a constant a >= 1
such that:
R_AB(s) = sqrt(a^2 - 1) |s - S|
T_AB(s) = S + a(s - S)
where S corresponds to the instant where A and B meet.
Proof:
With the assumptions above, after following the previous discussion, and
after calibrating T_B appropriately so that f_AB = f = f_BA, we may write:

f(f(s)) + s = 2(a f(s) + b) (1)

An instant where A and B meet corresponds to where R_AB(s) = f(s) - f'(s)
becomes 0. This occurs where f(f(s)) = s, or (by LEMMA 8) where f(s) = s.
Any fixed point, S, of f, must satisfy (1), thus implying:

f(f(S)) + S = 2(a f(S) + b)
==> 2 S = 2(a S + b)
==> S = -b/(a - 1)

So this is the one and only fixed point to f(s).
Since R(s) = R_AB(s) >= 0, then R(s) approaches 0 from above as s -> S.
This leads to the inequality in the first order differentials:

0 <= R'(S+) - R'(S-) = (f(S+) - a) - (f(S-) - a)
or
f(S+) >= f(S-)

Differentiating (1) yields:

f'(f(s)) f'(s) + 1 = 2a f'(s)
or
f'(f(s)) = 2a - 1/f'(s) (2)

Since the derivative f' is continuous away from S then taking the limits
as s -> S+ or S-, results in

f'(S+) = 2a - 1/f'(S+)
and f'(S-) = 2a - 1/f'(S-)

with f'(S+) - f'(S-) = R'(S+) - R'(S-) > 0 in virtue of the fact that
R(s) > 0 when s moves away from S. This implies:

f'(S +/-) = a +/- sqrt(a^2 - 1)

Alternatively, we may take the sequence a_n given by:

(f'(s0), f'.f(s0), f'.f.f(s0), ...) s0 < S

which can be represented, using equation (2), by:

a_0 = f'(s0)
a_{n+1} = 2a - 1/a_n

Noting that a - sqrt(a^2 - 1) <= a_0 <= a_n <= a_{n+1} <= a + sqrt(a^2 - 1),
with a +/- sqrt(a^2 - 1) being the only two fixed points of F = 2a - 1/F,
we may use LEMMA 9 to conclude that either

a_n = a - sqrt(a^2 - 1) for all n
or a_n -> a + sqrt(a^2 - 1)

Since s0 < S, then (s0, f(s0), f.f(s0), ...) -> S from below, therefore

a_n -> f'(S-) = a - sqrt(a^2 - 1)

which rules out the second case. Therefore

f'(s0) = a_0 = a - sqrt(a^2 - 1).

A similar argument (with the inverse function, f', of f) shows that f'(s0)
must be a + sqrt(a^2 - 1) for all s_0 > S. Therefore:

d/ds f(s) = a + sqrt(a^2 - 1) sgn(s - S)

and upon integration

f(s) = S + a(s - S) + sqrt(a^2 - 1) |s - S|

Therefore T(s) must be the linear part and R(s) the remainder:

T(s) = S + a(s - S)
R(s) = sqrt(a^2 - 1) |s - S|

COROLLARY 15: Let A be any point in a rest frame F, B any point
inertial with respect to F. Then either R_AB(s) is
constant or goes to infinity, as s -> +/- infinity.
Proof:
Let A and B be given as above, and let C be a point in the rest
frame which meets B at T_B(S). Then by the previous theorem:

R_CB(s) = sqrt(a^2 - 1) |s - S|

When a = 1, then R_CB(s) = 0 and R_AB = R_AC = AC. Otherwise a > 1 and we
have, using the Triangle Inequality:

R_CB.T_B(b) = d_C(b) >= |d_A(b) - AC| = |R_AB.T_B(b) - AC|

Therefore
R_CB(s) >= |R_AB(s) - AC| = |sqrt(a^2 - 1) |s - S| - AC|

which goes to infinity as s -> +/- infinity.

COROLLARY 16: Let A be any point in a rest frame F, and B any point
inertial with respect to F. Then R_AB(s) assumes its
minimum value for some s = S.
Proof:
Since R_AB(s) >= 0, it must have an least upper bound at least as
large as 0. But it approaches infinity as s -> +/- infinity, therefore
this least upper bound must be that of some finite range:

inf { R_AB(s) } = inf { R_AB(s): s0 < s < s1 } = R_0

for some s0 and s1. Since R_AB is continuous then R_AB(s) must
actually take on the value R_0 for some S in the range s0 < S < s1.

(17) The Connectivity Axiom and the Euclidean Structure
The previous result establishes time dilatation for an inertial motion that
coincides at some moment with a given point in a rest frame. However, it does
not tell us how the inertial points remotely situated synchronize with respect
to the point. This, in turn, is what is necessary to construct a Euclidean
Metric Space in a rest frame. The missing element is the ability to
synchronize remote points with an agent that moves arbitrarily slowly from one
point to the other (thus approximating a resting state). This is covered by
the following:

DEFINITION 10: A rest frame is CONNECTED if every two points can be
connected by an arbitrarily slow inertial motion. In
particular, for every two points A, B in a rest frame F
and for all v > 0, there is a point, C, inertial with respect
to F such that
(a) C meets A at instant a
(b) C meets B at instant b, with a < b
(c) d/ds R_AC(s) < v as s -> T_C(a) from above.

Condition (c) is stated weaker than otherwise anticipated since we've
already shown that R_AC' is piecewise constant in the previous theorem.

We can now prove the two properties which define a Euclidean Metric.

THEOREM 17: Let F be a connected rest frame and A, B points in F. Then for all
r there is a point C in F such that

AC = |r| AB, and BC = |1 - r| AB
Proof:
Without loss of generality, we can take r > 0, since if r <= 0, we could
first establish the existence of a point C such that

BC = |1-r| BA, and AC = |1 - (1-r)| BA

and then note that this point, C, will also have the property stated in the
theorem.
Let D be any point inertial with respect to F that meets point A at instant
a, and B at instant b > a. Assume that T_D(a) = s1 and T_D(b) = s2, and note
that s1 < s2. Let c be the instant defined by T_D(c) = r s2 + (1 - r) s1 > s1
and let C be a point in the rest frame that meets D at c. Then

AB = d_A(b) = R_AD(s2)
= sqrt(a^2 - 1) (s2 - s1)

AC = d_A(c) = R_AD(r s2 + (1 - r) s1)
= sqrt(a^2 - 1) (r s2 + (1 - r) s1 - s1)
= r sqrt(a^2 - 1) (s2 - s1)
= r AB
= |r| AB
and
BC = d_B(c) = R_BD(r s2 + (1 - r) s1)
= sqrt(a^2 - 1) |r s2 + (1 - r) s1 - s2|
= |1 - r| sqrt(a^2 - 1) (s2 - s1)
= |1 - r| AB

Based on these considerations define the following:

DEFINITION 11: Given a pair of points A, B in a rest frame F and a point D
inertial with respect to F, let [D; A, r, B] be the point C
in F that meets D at instant c, where

T_D(c) = (1 - r) T_D(a) + r T_D(b)

In the process of proving the second property that defines a Euclidean metric,
we will show that [D; A, r, B] is independent of D.
The following properties are readily apparent:

[D; A, 0, B] = A A [D; A, r, B] = |r| AB
[D; A, 1, B] = B B [D; A, r, B] = |1 - r| AB

To prove the second property, we will need to generalise THEOREM 14 to cases
where points A and B do not meet.

THEOREM 18: Let A be a point in a rest frame F, and B a point inertial with
respect to F, for which R_AB(s) takes on the minimum R_0 at
some s = S. Then
* |R(s) - X| <= N R_0 (a^2 - 1)^{1/4}
* where
X^2 = R_0^2 + U^2
U = (s - S) sqrt(a^2 - 1)
N = N(R_0) is a constant of O(1) as a -> 1.
Proof:
Applying the constructions of the previous theorem, we may once again
write equation (1)

f.f(s) + s = 2(a f(s) + b)

The constant b can be expressed in terms of S since:

as + b = T_AB(s) = T_CB(s) = S + a(s - S)

where C is a point in the rest frame that meets point B when s = S.
Therefore b = S(1 - a). So the sequence:

(s_n: n >= 0) = (s0, f(s0), f.f(s0), ...)

will satisfy the recursion relations:

s_{n+2} = 2 a s_{n+1} - s_n + 2 S (1 - a)
or
(f(s_n) - a s_n - b)^2 - (a^2 - 1) (s_n - S)^2 = constant = C(s0)

Since f is strictly increasing and has no fixed points, then f maps the
successive intervals:

... [f'(S), S) --> [S, f(S)) --> [f(S), f.f(S)) --> ...

which cover the real line so that every number s maps to or from some number
s0 = F(s) in the range [S, f(S)) by successive applications of f, or its
inverse f'. Therefore, we may write:

(f(s) - a s - b)^2 = (a^2 - 1) (s - S)^2 + C(s0)
where s0 = F(s), with S <= s0 < f(S)

The function f(s) takes the form R(s) + S + a(s - S), we get:

R(s)^2 = (a^2 - 1) (s - S)^2 + C(s0)
where s0 = F(s), with S <= s0 < S + R_0

>From THEOREM 12: T'(s)^2 - R'(s)^2 >= 1, or |R'(s)| <= sqrt(a^2 - 1).
This implies that:
|R(s0) - R_0| <= sqrt(a^2 - 1) |s0 - S|
Therefore,
R(s0) = R_0 + e sqrt(a^2 - 1) f R_0
where s0 = S + f R_0, |e| < 1 and 0 <= f < 1.

Using this we can write C(s0) in terms of R_0:

C(s0) = R(s0)^2 - (a^2 - 1) (s0 - S)^2
= R_0^2 (1 + 2ef sqrt(a^2 - 1) + (e^2 - 1) f^2 (a^2 - 1))
and with
U = (s - S) sqrt(a^2 - 1)
X^2 = R_0^2 + U^2
we get
|R(s) - X| = |sqrt(U^2 + C(s0)) - sqrt(U^2 + R_0^2)|
<= sqrt(|C(s0) - R_0^2|)
= R_0 sqrt(|2ef sqrt(a^2 - 1) + (e^2 - 1) f^2 (a^2 - 1)|)
= N R_0 (a^2 - 1)^{1/4}
where N^2 = |2ef + (e^2 - 1) f^2 sqrt(a^2 - 1)|

The multiplier N is independent of s and S. It cannot take on a larger value
than where e -> 1 and f = 0, in which case N would approach the value
2 + sqrt(a^2 - 1), which approaches 2 as a -> 1.

Using the connectivity property, we can find a sequence of inertial points
for which a -> 1. Then it will be possible to make the previous result exact.
But it will first be necessary to prove that [D; A, r, B] is independent of D.

THEOREM 19: For points A, B in a connected rest frame F, [D; A, r, B] is
uniquely defined.
Proof:
For a given D0 and r0, set C = [D0; A, r0, B]. Let D be another inertial
point that meets A and B, and define C(r) = [D; A, r, B]. We then have:

C(0) C = AC
C(1) C = BC
|r AB - AC| <= C(r) C <= |r| AB + AC

so that C(r) C is asymptotically linear in r.
From the previous theorem, we can write:

R_CD(s) = sqrt(R_0^2 + U^2) + e N R_0 (a^2 - 1)^{1/4}
where
U = (s - S) sqrt(a^2 - 1)
N = N(R_0) is a constant of O(1) as a -> 1.
|e| <= 1
with
s = (1 - r) sa + r sb
D meets A at sa, and meets B at sb.

The asymptotic condition then implies that

AB = (sb - sa) sqrt(a^2 - 1)
or
U = AB (s - S)/(sb - sa)

and the other two conditions become:

AC = R_CD(sa); BC = R_CD(sb)

If a were equal to 1, we would get:

R_0 = 0, S = (1 - r0) sa + r0 sb

and then C(r0) C = R_0 = 0. More generally, however, R_0 will be a smoothly
varying function of a (except where R_0 = 0!). When the rest frame is
connected, we can choose v = T_AC'(s) = sqrt(a^2 - 1) as close to 0 as we want
and let a -> 1.

So then, taking a sequence of inertial points (C0, C1, C2, ...), whose
corresponding parameters a_n -> 1 as n -> infinity, we get a convergence
of point Cn(r0) to point C. Since the limit of the convergence is unique
(ultimately due to the positive definiteness of the distance function!),
then the point C, itself, must be uniquely defined.

From this, we will be able to strengthen THEOREM 18 to an exact result:

THEOREM 20: With the same assumption as THEOREM 18:
R(s)^2 = R_0^2 + (s - S)^2 (a^2 - 1)
with a = d/ds T(s)
Proof:
Taking a family of inertial points (B0, B1, B2, ...) which trace out the
same path as B, whose parameters a_n -> 1 as n -> infinity, we note that:

|R_n(s) - X_n| -> 0 uniformly, as n -> infinity
where X_n^2 = R_0^2 + U_n^2
U_n = (s - S_n) sqrt(a_n^2 - 1)
R_n(s) = R_ABn(s), minimized at R_0 when s = S_n.

Choosing point C as the point where the distance is minimized, and choosing an
arbitrary point D which meets B at instant sd, and Bn at instant sd_n, we have:

CD = sqrt(a^2 - 1) (sd - S) = sqrt(a_n^2 - 1) (sd_n - S_n)
and
R(s) = A [B; A, r, C] = A [Bn; A, r, C] = R_n(s')
where
r = (s - S)/(sd - S) = (s - S)/CD sqrt(a^2 - 1)
s' = S_n + r (sd_n - S_n)
= S_n + r CD / sqrt(a_n^2 - 1)
= S_n + (s - S) sqrt( (a^2 - 1)/(a_n^2 - 1) )

But
R_n(s') -> sqrt(R_0^2 + (s' - S_n)^2 (a_n^2 - 1))
= sqrt(R_0^2 + (s - S) (a^2 - 1))

since R_n(s) -> X_n uniformly. Therefore R(s) must be the limit of this
sequence, and we have:

R(s)^2 = R_0^2 + (s - S)^2 (a^2 - 1)

The quadratic dependence in this function is exactly what the second
defining property of the Euclidean metric requires. Therefore, as a direct
consequence, we find that:

COROLLARY 21: The distance function of a connected rest frame is Euclidean.
Proof:
Let F be a rest frame, and A, B and C be points in F with AB + BC = AC.
If F is connected then there is an inertial motion, E, that passes through A
and C. It must also pass through B, with B = [E; A, r, C] where AB = |r| AC.

According to the previous result, we may write the following distance
function for E:
R_DE(s)^2 = R^2 + (a^2 - 1) (s - S)^2

where R represents the distance of closest approach to D, and S the time of
closest approach. From this, we get:

AD^2 = R^2 + (a^2 - 1) (sA - S)^2
BD^2 = R^2 + (a^2 - 1) (sB - S)^2
CD^2 = R^2 + (a^2 - 1) (sC - S)^2
AB = sqrt(a^2 - 1) |sB - sA|
BC = sqrt(a^2 - 1) |sC - sB|

which leads to the following results:

AD^2 - AB^2 - BD^2 = (a^2 - 1) (sA - sB) (sB - S)
CD^2 - CB^2 - CD^2 = (a^2 - 1) (sC - sB) (sB - S)
or to:
BC (AD^2 - AB^2 - BD^2) = - (a^2 - 1)^{2/2} (sB - sA)^2 (sB - S)
AB (CD^2 - CB^2 - BD^2) = + (a^2 - 1)^{2/2} (sB - SA)^2 (sB - S)

So these two quantities add up to 0:

BC (AD^2 - AB^2 - BD^2) + AB (CD^2 - CB^2 - BD^2) = 0

and with a little manipulation:

BC AD^2 - (AB + BC) BD^2 + AB CD^2 = AB (AB + BC) BC
or

BC AD^2 - AC BD^2 + AB CD^2 = AB AC BC


(18) The Axiom of Inertia
The previous result is conditional on the existence of a connected
rest frame. From it, the Euclidean structure of the rest frame's
distance function will be deriveable, which leads to a Minkowski
Geometry for the set I of instants. The existence of such a rest
frame, the main premise behind the "Special" in "Special Relativity"
will therefore be granted by postulate:

THE AXIOM OF INERTIA: There exists a C^2 Time Net containing a connected
rest frame.

THEOREM 20 and COROLLARY 21 establish that the set, I, of instants has
the structure of a Minkowski Geometry. Once we have this, we can then
prove that all rest frames must be connected, that each point of each
rest frame must be inertial in all other rest frames, and that the
Lorentz Transformations apply between mutually inertial rest frames. But
that's standard material, so it won't be repeated here [besides which, I've
developed a set of introductory articles on Relativity Theory which may get
posted from time to time to sci.physics].

So at last we come full circle. The one last item to handle is to impose
a restriction on the dimension of the rest frame's Euclidean space, e.g.,
by requiring that there exists a rest frame containing 4 equally spaced
points in it, but that no rest frame has 5 equally spaced points in it.

References:
[1] P. Suppes SPACE, TIME AND GEOMETRY, 1973
A compilation of approaches related to ours. Includes:

R. Hudgin "Coordinate-Free Relativity"

which provides a complete axiomitization of Minkowski geometry in
terms of a "signalling time" primitive and 5 axioms.
[2] A.A. Robb A THEORY OF TIME AND SPACE, Cambridge 1914, 2nd ed.
Robb's axiomitization uses *only* the before/after relation, and
is therefore much larger and more complex.
[3] A.A. Robb THE ABSOLUTE RELATIONS OF TIME & SPACE, Cambridge 1921
[4] A.A. Robb GEOMETRY OF TIME AND SPACE, Cambridge 1936