[FOM] 213:On foundations of special relativistic kinematics 5
Harvey Friedman
friedman at math.ohio-state.edu
Sun Feb 8 21:33:17 EST 2004
NOTE: From what I can tell, the researchers whose perspective is closest to
mine concerning foundations of physics are Andreki, Nemeti, and associates,
in Budapest. See http://www.math-inst.hu/pub/algebraic-logic/Contents.html
Obviously there is a large overlap between what I have been saying and what
they have done. I have just started to think about such things mainly as an
$$$INITIAL EXPERIMENT (by me) in (my style of) foundational thinking outside
mathematics.$$$
In contrast, they have a draft (at least) of an over 1300 page book on
foundations of special relativity, and lots of weighty papers. I am
beginning to learn from these materials.
Our respective preferred ways of talking about these things differ somewhat,
and I expect that they may diverge later in more substantive ways.
#################################
We begin with some clarifying remarks about the axiomatization SRK(math)
given in the previous posting 212:On foundations of special relativistic
kinematics 4.
Recall that we very deliberately pared down the primitives, not including
photons, and not including any concept of positions of an observer. We now
indicate how these concepts fit into our pared down framework.
1. Given an observer O and two events E,E', there is the derived concept of
"the velocity vector from E to E'" and "the signed speed from E to E'".
These concepts are only defined if, according to O, E and E' are NOT
simultaneous (we don't want to divide by zero). The former is the vector
obtained by subtracting the space coordinates of E, according to O, from the
space coordinates of E', according to O, and dividing by the time interval
from E to E' according to O. The latter is the Euclidean distance from E to
E' (as used in the axioms) divided by the time interval from E to E'.
2. We can relate the velocity vector from E to E' according to O, to "moving
particles" as follows. It is the "net velocity vector" of any particle
taking up position E in event space and taking up position E' in event
space, according to O. We can relate the signed speed from E to E' according
to O, as follows. It is the "net signed speed" of any particle taking up
position E in event space and taking up position E' in event space (and not
considering other positions it takes up in event space). Note that this
depends only on events E,E' and observer O. It has nothing to do with the
nature or motion of particles.
3. Fix an observer O. If the particle is "moving uniformly" then we can take
ANY distinct events E,E' that the particle "appears at" and get the exact
same velocity vector and signed speed defined above in 2.
4. In particular, if the particle happens to be a photon, then the signed
speed computation is always c, NO MATTER WHAT THE OBSERVER IS, NO MATTER
WHAT PHOTON IT IS, NO MATTER HOW THE PHOTON IS "MOVING", ETC. This follows
A. Einstein. (We are discussing special relativity).
5. Our distillation of 1-4 is of course our LIGHT PATH axiom:
LIGHT PATH axiom. Let two events be given. If some observer thinks that
the Euclidean distance between the two events is c times the time interval
from the first event to the second event, then every observer thinks so.
6. Note again that we do not use photons, paths of photons, velocity
vectors, signed speeds, etc., in formulating the LIGHT PATH axiom.
We can rephrase the LIGHT PATH axiom as follows, given our formal definition
of signed speed:
LIGHT PATH axiom. Let two events be given. If some observer thinks that
the signed speed from the first event to the second event is c, then every
observer thinks so.
7. Every observer O can be viewed as occupying positions in event space. In
particular, we associate to O, the following set of events: the events whose
coordinates according to O, are of the form (t,0,0,0), where t is a real
number.
8. Let O' be another observer (possibly O again). We can consider the
opinion of O' with regard to 7 above. I.e., we can look at the coordinates
(quadruples) according to O', of the events of the form: (t,0,0,0) according
to O. These quadruples form a line in R x R^3 = R4, and tell us what O'
thinks about O's "motion". Note that O thinks that O is always at rest;
i.e., with signed speed 0, and velocity vector (0,0,0,0).
9. We can continue this development and carefully discuss relative
velocities of observers; i.e., one observer's opinion of any observers
velocity (velocity vectors, signed speeds). We won't continue with this now,
but it should be clear what the connection is between this kind of
completely transparent treatment and various talk of physicists and other
very strange people.
#########################
Note that the axiomatization we presented in the previous posting #212, On
foundations of special relativistic kinematics 4 (and the earlier obsolete
versions) DELIBERATELY AVOIDS any mention of Minkowski space.
Of course, the Minkowski space is absolutely central to the NORMAL
mathematical presentations of special relativity.
We of course have already used some of Minkowski space in order to define
the Lorentz and Poincare transformations.
But that is after the fact. That involves only some mathematics hidden
behind the curtain. Our presentation begins from a more fundamental
standpoint.
So from our axiomatic point of view, where does Minkowski space fit in?
What we need is a fundamental way to define the Minkowski distance and
Minkowski inner product directly out of the unique models of our axioms
(unique up to isomorphism for a given c > 0 using the reals). These
definitions should make the central importance of the Minkowski space
apparent.
Recall our definition of these "unique" models SRK(MATH,c), for c > 0.
SRK(MATH,c) has a representation where
i) the "observers" are the c-Poincare transformations in R x R^3 = R^4;
ii) the "events" are the elements of R x R^3 = R^4;
iii) the "reals" are the usual ordered field of real numbers.
iv) the four coordinate functions, TC, XC, YC, ZC, are just the 4 coordinate
functions of function application.
We now wish the define the fundamental equivalence relation
*on pairs of events*
that motivates the c-Minkowski inner product.
Let E1,E1,E3,E4 be events in (any isomorphic copy of) SRK(MATH,c). We say
that (E1,E2) and (E3,E4) are SRK(MATH,c) equivalent if and only if there
exist observers O and O' such that
*the 4 coordinates of E1 and the 4 coordinates of E2, according to O - 8
real numbers in all - are respectively identical to the 4 coordinates of E3
and the 4 coordinates of E4, according to O' - 8 real numbers in all.*
THEOREM 1. Let c > 0. (E1,E2) and (E3,E4) are equivalent in SRK(MATH,c) if
and only if there is an automorphism of SRK(MATH,c) that sends E1 to E3 and
E2 to E4.
Note that so far, we have given such conceptually basic definitions (two of
them for the same concept) that we have not even used coordinate
subtraction!
THEOREM 2. Let c > 0. There are continuumly many equivalence classes.
As soon as one sees continuumly many here, there is the obvious suggestion
that there should be a way to represent each equivalence class by a real
number.
As is standard in such situations, one looks for a preferred representative
of each equivalence class, with the plan of assigning real numbers in some
obvious way to each of the preferred representatives.
THEROEM 3. Let c > 0. In M, for every (E1,E2) there is an observer such that
the coordinates of (E1,E2) according to that observer is
i) ((0,0,0,0),(t,0,0,0)) for some t > 0; or
ii) ((0,0,0,0),(-t,0,0,0)) for some t > 0; or
iii) ((0,0,0,0),(t,0,0,ct)) for some t > 0; or
iv) ((0,0,0,0),(-t,0,0,ct)) for some t > 0; or
v) ((0,0,0,0),(0,0,0,d)) for some d > 0;
vi) ((0,0,0,0),(0,0,0,0)).
Furthermore, these 6 cases are mutually exclusive, and the choice of t and d
is unique. Thus for given c,M,E1,E2, we cannot have one observer places
(E1,E2) in one of the six categories, whereas another observer places
(E1,E2) in a different category.
In light of Theorem 3, it is of course most natural to simply assign the
quantity t in i) - iv), and the quantity d in v), with an indication of
which case one is in. We will not QUITE have assigned a real number to each
equivalence class, in a one-one fashion, because the same real number may be
used in one than one case.
We now relate these unique t's and d's to the so called Minkowski distance -
actually, the c-Minkowski distance. It is not really a metric by normal
mathematical standards (e.g., the usual triangle inequality fails, and there
is a reverse triangle inequality). We define it after we present the
following crucial result.
THEOREM 4. Let c > 0. In M, let (E1,E2) be a pair of events such that,
according to some observer, the coordinates are ((x,y,z,w),(x',y',z',w')).
In cases i) - vi), the indicated positive quantity (0 in case vi)) is the
c-Minkowski distance between these two 4-tuples.
COROLLARY 5. Let c > 0. In M, let (E1,E2) be a pair of events. The
c-Minkowski distance between E1 and E2 is independent of the choice of
observer.
THEOREM 6. Let c > 0. In M, let E1,E2,E3,E4 be given. Then the following are
equivalent.
i) (E1,E2),(E3,E4) are equivalent or (E1,E2),(E4,E3) are equivalent;
ii) the c-Minkowski distance between E1 and E2, computed by any observer, is
equaled to the c-Minkowski distance between E3 and E4, computed by any other
observer.
In particular,
Theorem 4 best illustrates the fundamental importance of the c-Minkowski
distance. It DEFINES the c-Minkowski distance in fundamental relativistic
terms.
Here is the official mathematical definition of c-Minkowski distance. Let
(a,b,c,d) and (e,f,g,h) be given. Their c-Minkowski distance is the square
root of
|c^2(a-e)^2 - (b-f)^2 - (c-g)^2 - (e-h)^2|.
In the literature on special relativity, it is said that:
the separation from E1 to E2 is
i) future timelike;
ii) past timelike;
iii) future lightlike;
iv) past lightlike;
v) spacelike;
vi) "zero" or "degenerate".
as in Theorem 3.
If we just wanted to understand the division into the six cases i) - v) of
Theorem 3, then we would not need the c-Minkowski distance. Put another way,
consideration of the division into cases i) - v) DOES NOT motivate the
Minkowski distance.
This is because of the following.
THEOREM 7. Let c > 0. In M, let a pair of events be given. Then the cases in
Theorem 3 correspond to the following conditions:
i) According to any observer, the distance between the two events is less
than c times the time interval from the first event to the second event;
ii) According to any observer, the distance between the two events is less
than c times the time interval from the second event to the first event;
iii) According to any observer, the distance between the two events is c
times the time interval from the first event to the second event;
iv) According to any observer, the distance between the two events is c
times the time interval from the second event to the first event;
v) According to any observer, the distance between the two events is greater
than c times the magnitude of the time interval between the two events;
vi) The events are identical.
How does the c-Minkowski inner product enter?
We have seen just how the c-Minkowski distance pops out of consideration of
the crucial equivalence relation on pairs of events (see Theorem 1 and just
before Theorem 1).
The c-Minkowski norm is the c-Minkowski distance to the origin. There is a
unique inner product, the c-Minkowski inner product, such that
x dot x = ||x||^2
where || || is the c-Minkowski norm, in exact analogy to the case of the
Euclidean distance, Euclidean norm, and Euclidean inner product.
Looking at the obvious way in which the c-Minkowski inner product is defined
explicitly from ||x|| = d(x,0), where d is the c-Minkowski distance, we
immediately see the following.
THEOREM 8. Let c > 0. In M, let four events E1,E2,E3,E4, be given. The
c-Minkowski inner product of E2 - E1 and E4 - E3, when computed by any
observer, is the same.
*********************************************
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This is the 213th in a series of self contained numbered postings to
FOM covering a wide range of topics in f.o.m. The list of previous
numbered postings #1-149 can be found at
http://www.cs.nyu.edu/pipermail/fom/2003-May/006563.html in the FOM
archives, 5/8/03 8:46AM. Previous ones counting from #150 are:
150:Finite obstruction/statistics 8:55AM 6/1/02
151:Finite forms by bounding 4:35AM 6/5/02
152:sin 10:35PM 6/8/02
153:Large cardinals as general algebra 1:21PM 6/17/02
154:Orderings on theories 5:28AM 6/25/02
155:A way out 8/13/02 6:56PM
156:Societies 8/13/02 6:56PM
157:Finite Societies 8/13/02 6:56PM
158:Sentential Reflection 3/31/03 12:17AM
159.Elemental Sentential Reflection 3/31/03 12:17AM
160.Similar Subclasses 3/31/03 12:17AM
161:Restrictions and Extensions 3/31/03 12:18AM
162:Two Quantifier Blocks 3/31/03 12:28PM
163:Ouch! 4/20/03 3:08AM
164:Foundations with (almost) no axioms, 4/22/0 5:31PM
165:Incompleteness Reformulated 4/29/03 1:42PM
166:Clean Godel Incompleteness 5/6/03 11:06AM
167:Incompleteness Reformulated/More 5/6/03 11:57AM
168:Incompleteness Reformulated/Again 5/8/03 12:30PM
169:New PA Independence 5:11PM 8:35PM
170:New Borel Independence 5/18/03 11:53PM
171:Coordinate Free Borel Statements 5/22/03 2:27PM
172:Ordered Fields/Countable DST/PD/Large Cardinals 5/34/03 1:55AM
173:Borel/DST/PD 5/25/03 2:11AM
174:Directly Honest Second Incompleteness 6/3/03 1:39PM
175:Maximal Principle/Hilbert's Program 6/8/03 11:59PM
176:Count Arithmetic 6/10/03 8:54AM
177:Strict Reverse Mathematics 1 6/10/03 8:27PM
178:Diophantine Shift Sequences 6/14/03 6:34PM
179:Polynomial Shift Sequences/Correction 6/15/03 2:24PM
180:Provable Functions of PA 6/16/03 12:42AM
181:Strict Reverse Mathematics 2:06/19/03 2:06AM
182:Ideas in Proof Checking 1 6/21/03 10:50PM
183:Ideas in Proof Checking 2 6/22/03 5:48PM
184:Ideas in Proof Checking 3 6/23/03 5:58PM
185:Ideas in Proof Checking 4 6/25/03 3:25AM
186:Grand Unification 1 7/2/03 10:39AM
187:Grand Unification 2 - saving human lives 7/2/03 10:39AM
188:Applications of Hilbert's 10-th 7/6/03 4:43AM
189:Some Model theoretic Pi-0-1 statements 9/25/03 11:04AM
190:Diagrammatic BRT 10/6/03 8:36PM
191:Boolean Roots 10/7/03 11:03 AM
192:Order Invariant Statement 10/27/03 10:05AM
193:Piecewise Linear Statement 11/2/03 4:42PM
194:PL Statement/clarification 11/2/03 8:10PM
195:The axiom of choice 11/3/03 1:11PM
196:Quantifier complexity in set theory 11/6/03 3:18AM
197:PL and primes 11/12/03 7:46AM
198:Strong Thematic Propositions 12/18/03 10:54AM
199:Radical Polynomial Behavior Theorems
200:Advances in Sentential Reflection 12/22/03 11:17PM
201:Algebraic Treatment of First Order Notions 1/11/04 11:26PM
202:Proof(?) of Church's Thesis 1/12/04 2:41PM
203:Proof(?) of Church's Thesis - Restatement 1/13/04 12:23AM
204:Finite Extrapolation 1/18/04 8:18AM
205:First Order Extremal Clauses 1/18/04 2:25PM
206:On foundations of special relativistic kinematics 1 1/21/04 5:50PM
207:On foundations of special relativistic kinematics 2 1/26/04 12:18AM
208:On foundations of special relativistic kinematics 3 1/26/04 12:19AAM
209:Faithful Representation in Set Theory with Atoms 1/31/04 7:18AM
210:Coding in Reverse Mathematics 1 2/2/04 12:47AM
211:Coding in Reverse Mathematics 2 2/4/04 10:52AM
212:On foundations of special relativistic kinematics 4 2/7/04 6:28PM
Harvey Friedman
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