[FOM] On a natural definition of relativistic distance of events in SRK . (Ants, elephants, and Minkowski-distance.)

[Istvan Nemeti]

This is in connection with Harvey Friedman's posting #213 (Feb 8
21:33:17, Fo SRK 5). 

All what we do below goes through in Harvey's axiom system  SRK(math)
in place of ours. We work in ours to illustrate flexibility (and also to
show how things work in an autonomous system).

Consider a new axiom called THOUGHT EXPERIMENT axiom as follows.

Axiom 6 (Thought Experiment Axiom) Everywhere, in every direction, it is
possible to travel with any speed less than 1 (the speed of light).
I.e.: For any observer  o1  for every line  L (as subset of  F^4)  with
slope less than  1  there is an observer  o2  such that the life-line of
o2  as seen by  o1  is  L .

Axioms 1 - 5  were presented in our posting of Feb 5 17:13:12.  Below we
will use Axioms 1 - 6.

We note that Axiom 6 is a weakened version of Harvey's axiom of
maximality (axiom 6 in his posting #212, Feb 8).


SLOGAN: We want to define an observer-independent mathematical structure
on the set  EVE  of events. (Preferably, this extra structure should
make  EVE  into a kind of familiar mathematical structure.)


DEFINITION 1. (relativistic distance)  Let  E, E1  be two events. The
relativistic distance  d(E,E1)  is  defined to be  r  if there is an
observer  o  who either sees both  E  and  E1  on his time axis and
measures the time-difference between them as -r (note that in this case
r  is negative), or else  o  sees  E  and  E1  as simultaneous and
measures their Euclidean distance as  r.  If no observer sees  E, E1
this way (i.e. either both on the time axis or simulteneous), then
d(E,E1)=0  by definition.

THEOREM 1.  Axioms 1 - 6  imply that the above definition is correct,
i.e.  d(E,E1)  exists and is unique. Further,  d(E,E1)=0  iff  E  and
E1  are connected by a photon.

Distance is symmetric, i.e.  d(E,E1)=d(E1,E). If both  E  and  E1
happened on the life-line of some observer  o  then their distance is
negative or zero. If they are simultaneous for some  o  then their
distance is non-negative.

TERMINOLOGY: Assume  E  and  E1  are distinct.  They are called timelike
separated if  d(E,E1)  is negative. They are called spacelike separated
if  d(E,E1)  is is positive. They are called light-like separated if
d(E,E1)=0.

THEOREM 2.  Assume Axioms 1 - 6. Then the relativistic distance of  E,
E1  as defined above coincides with the official Minkowski distance of
their coordinates (in  F^4) according to any observer. In more detail,
if  o  "sees"  E  at coordinate-point  p  and  E1  at coordinate-point
q, then  d(E,E1) = "official Minkowski-distance of  p  and  q ".

REMARK 1. Let  i  denote the square root of -1. We note that many
relativity books use  (i  times  r)  where we used the simpler  -r .
I.e. for them  d(E,E1)  is an imaginary number if  E,E1  are timelike
separated. This is only a notational matter and serves only to code the
claim that the events in question are timelike separated and their
distance is  |r|  (the absolute value of  r ). Let  D(E,E1) :=
|d(E,E1)|. Then Harvey's distance on events (in his posting #213 Feb 8)
is the same as  D  defined herein.

REMARK 2.  The definition of distance  d  in Definition 1  above (which
is basically the same as Harvey's definition in his posting of #213, Feb
8) is an explicit definition in the sense of first-order logic (FOL).
I.e. it is of the form

d(E,E1) = r   iff   phi(E,E1,r)

where  phi  is a FOL-formula (in the language of SRK).

An advantage of this logical formula  phi  over the official
computational definition of Minkowski distance is that  phi  can be used
meaningfully in modified or weaker versions of  Axioms 1 - 6  or of
SRK(math)  while the computational, official definition loses its
relevance if context is even slightly modified.   

Theorem 1 stating that our definition of the relativistic distance  d
is correct used Axioms 1 - 6. Actually, the same idea for defining  d
works under more general assumptions.
   
E.g.: if we replace  Axiom 5  (the anti-dilation axiom) with the weaker
ANT AND ELEPHANT axiom in our posting of Feb 12, then the above
definition of  d  still works but the official Minkowskian definition
does not (uniqueness of Minkowski-distance between events fails, hint:
cf. Figure 29 on p.88 in [AMNBook]). 

This shows the advantage of defining the relativistic distance between
events in the present logic-oriented style as opposed to the official
style of Minkowski distance. (This of course shows the advantage of
Harvey's definition over the official one, too, since Harvey's
definition is of the same spirit as ours.)

The Ant-elephant axiom can be further generalized (weakened) by saying
that "There is a smallest animal". I.e., there may be ants, there may be
elephants, and they may use their own units of measurement [e.g. feet],
but [among co-moving observers] there are smallest animals such that no
observer uses shorter units of measurement than these smallest animals. 

The logic oriented definition of  d  admits a natural refinement such
that it works under this weaker "smallest animal condition", too (in
place of Axiom 5). Namely, we define  d(E,E1)  to be the supremum of the
distances measured between  E  and  E1  by those observers who can
measure it as either purely temporal or purely spatial. These ideas are
explored e.g. in [MD] chapter 6, see especially pages 51-53, 145,
343-345; and in [AMNBook] chapter 4, especially see pp. 88, 89, 441,
796.

If we apply Definition 1 to the Newtonian case, then we obtain that the
distance  d of two simultaneous events is their Euclidean distance while
if they are not simultaneous then their distance  d  is the (negative
of) time which elapsed between them. At the same time, it is not clear
at all how one could use the official Minkowski computation here. 

Our point is that, as Harvey indicated in his January essays, logical
minded definitions can be made flexible, easily adaptable to change,
while "orthodox", ad-hoc, "quantitative"  definitions tend to be rigid.
We think that Harvey's posting of Jan 24 05:14:58 (Knowledge -
quantity/quality) is relevant here.

In connection with intuition/philosophy related to the ant-elephant
axiom and the smallest animal axiom cf. e.g. the Incredible Shrinking
Man in [Scifi] pp.194-5.

[MD] Madarasz, J. X., Logic and Relativity (in the light of definability
theory), PhD Dissertation, ELTE-University, Budapest, 2002.
http://www.math-inst.hu/pub/algebraic-logic/Contents.html

[AMNbook] Andreka-Madarasz-Nemeti: On the logical structure of
relativity theories.
http://www.math-inst.hu/pub/algebraic-logic/Contents.html

[Scifi] Nicholls, P. (ed.): The science in science fiction. Crescent
Books, New York 1982.



Hajnal Andreka and Istvan Nemeti