[FOM] 215:Special Relativity Corrections
friedman at math.ohio-state.edu
Tue Feb 24 20:13:36 EST 2004
We make some corrections to the last two postings on the foundations of
special relativistic kinematics - numbers 5,6. The need for these
corrections was pointed out to us by Hajnal Andreka and Istvan Nemeti, who
also made valuable comments.
There is just one small correction to #214:Foundations of special
relativistic kinematics 6, 2/14/04, 9:43AM.
There I wrote
5c. AGREEMENT axiom. There exist two distinct observers and two distinct
events such that the observers agree on the time interval and the distance
between the two events.
This should read
5c'. For any two observers there are two distinct events such that the
observers agree on the time interval and the distance between the two
There are more corrections to #213:Foundations of special relativistic
kinematics 5, 2/8/04, 9:33PM. We give a restatement of a tail of that
We now wish to define the fundamental equivalence relation
*on pairs of events*
that motivates the c-Minkowski inner product.
Let E1,E2,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
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
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;
Furthermore, these 6 cases are mutually exclusive. The choice of t and d
is unique in cases i),ii),v). In cases iii),iv), t can be any positive real
number. Thus for given c,M,E1,E2, we cannot have one observer place
(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),ii), the quantity d in v), and the quantity 0 in cases
iii),iv),vi), 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 more than one case.
We now relate these unique t's and d's to the so called Minkowski distance
between two 4-vectors.
Let (x,y,z,w) and (x',y',z',w') be given. The c-Minkowski form from
(x,y,z,w) to (x',y',z',w') is the quantity
c^2(x-x')^2 - (y-y')^2 - (z-z')^2 - (w-w')^2.
The c-Minkowski distance from (x,y,z,w) to (x',y',z',w') is the square root
of the absolute value of the c-Minkowski form from (x,y,z,w) to
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),ii),v), the indicated positive quantity is the
c-Minkowski distance from (x,y,z,w) to (x',y',z',w'). In cases iii),iv),vi),
the c-Minkowski distance from (x,y,z,w) to (x',y',z',w') is 0.
COROLLARY 5. Let c > 0. In M, let (E1,E2) be a pair of events. The
c-Minkowski distance and c-Minkowski form from E1 to E2 is independent of
the choice of observer.
THEOREM 6. Let c > 0. In M, let E1,E2,E3,E4 be given, where E1,E2 are
distinct, and E3,E4 are distinct. Then the following are
i) (E1,E2),(E3,E4) are equivalent or (E1,E2),(E4,E3) are equivalent;
ii) the c-Minkowski form from E1 to E2, computed by any observer, is
equaled to the c-Minkowski form from E3 to E4, computed by any other
iii) there is an observer which thinks that the c-Minkowski form from E1 to
E2 is the same as the c-Minkowski form from E3 to E4.
Theorem 4 best illustrates the fundamental importance of the c-Minkowski
distance. It DEFINES the c-Minkowski distance in fundamental relativistic
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;
vi) "zero" or "degenerate".
as in Theorem 3 above.
If we just wanted to understand the division into the six cases i) - vi) 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
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 Euclidean distance between the two events
is less than c times the time interval from the first event to the second
ii) According to any observer, the Euclidean distance between the two events
is less than c times the time interval from the second event to the first
iii) According to any observer, the Euclidean distance between the two
events is c times the time interval from the first event to the second
iv) According to any observer, the Euclidean distance between the two
events is c times the time interval from the second event to the first
v) According to any observer, the Euclidean distance between the two events
is greater than c times the magnitude of the time interval between the two
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 form of a 4-vector is the c-Minkowski form from the origin
to that 4-vector. There is a unique inner product, the c-Minkowski inner
product, such that
x dot x = the c-Minkowski form of x.
This is because of the inner product formula
(x + y) dot (x + y) = (x dot x) + (y dot y) + 2(x dot y).
However, it would be more satisfying to start with the c-Minkowski distance
and "derive" the c-Minkowski inner product. This is because we defined the
c-Minkowski distance, in Theorem 4, in fundamental relativistic terms.
Given the above equation, we would like to
*start with the c-Minkowski norm of 4-vectors and "derive" the c-Minkowski
form of 4-vectors*.
THEOREM 8. The c-Minkowski form function on 4-vectors is the only continuous
function from R^4 into R whose magnitude is always the square of the
c-Minkowski norm function, and which is positive at (1,0,0,0),(-1,0,0,0),
and negative at (0,0,0,1).
Looking at the explicit way in which the c-Minkowski inner product is
defined explicitly from the c-Minkowski form, we immediately see the
THEOREM 9. 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.
I use http://www.mathpreprints.com/math/Preprint/show/ for manuscripts with
proofs. Type Harvey Friedman in the window.
This is the 215th 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
213:On foundations of special relativistic kinematics 5 2/8/04, 9:33PM
214:On foundations of special relativistic kinematics 6 2/14/04, 9:43AM
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