[FOM] Ants, Elephants, and Special Principle of Relativity

[Istvan Nemeti]

Recall that our, as well as Harvey's, symmetry axiom (Axiom 5) for SRK
states that if two events  e1  and  e2  are simultaneous for both of two
observers, then the two observers measure the same Euclidean distance
between  e1  and  e2.

Why do we call this a symmetry axiom?

Because it says that two observers "see" something the same way. So in
this respect, these observers are "equivalent", which is a kind of
symmetry principle.

Is then Axiom 5  related to Einstein's Special Principle of Relativity
(SPR)? Einstein's SPR states that the same laws of nature are true for
all observers.

Interestingly, Tim Chow in his FOM posting of Jan 24 12:54:14 mentioned
that many relativity texts "derive" predictions of relativity from 2
postulates only,

(i)  SPR  and

(ii) finiteness of the speed of light.

As Tim pointed out, these derivations "secretely" use lots of tacit
assumptions not mentioned explicitly in the "axiom system" {(i), (ii)}.

This is indeed so, cf. e.g. Rindler's relativity book Springer 1977 or
Landau & Lifsic [LL].

A careful formalization of  SPR  was published in Madarasz [MD] p. 84
(section 2.8.3). There it is also investigated what follows from the
official system of postulates {(i), (ii)}.  E.g. Axiom 5 does not
follow.

Axiom 5 was systematically investigated e.g. in the dissertation [MD]
beginning with p.83 under the name Ax(eqspace).
It is shown in [MD] and in [AMNBook] that Axiom 5 implies the following
principle of the Ant and the Elephant.

The life-line of an observer is the set of events he/she takes part in. 

(Ant-elephant) Any two observers  o1, o2  sharing the same life-line use
the same "units of measurement" for time. That is, for any pair of
events  e,e'  on their joint life-line they measure the same
time-interval between  e  and  e'. (This will imply that they agree on
spatial distances, too.)

So it cannot happen that  o1  is the ant,  o2  is the elephant and (by
their sheer sizes) they disagree on the length of some fixed (,say,
temporal) distance.

Axioms 1 - 5  were introduced in our FOM-posting of Feb 5 17:13:12.

CLAIM 1.  Axioms 1 - 5  imply   (Ant-elephant).

But strangeley enough,

CLAIM 2. {Axioms 1 - 4, SPR} does not imply  (Ant-elephant),  and also
         {Axioms 1 - 4, (i), (ii)}  does not imply (Ant-elephant).


The reason for this is that Axioms 1 - 4  do not exclude dilations in
the world-veiw transformations  M[o1, o2] (by e.g. [AMNBook]p.154),
Axioms 1 - 4 imply (ii), and that SPR does not exclude dilations either,
e.g. by Madarasz[MD] section 2.8.3.

Moreover, we conjecture that if we omit one from Axioms 1 - 4, then the
rest together with (i),(ii) do not prove the omitted axiom.  E.g. we
conjecture that {Axiom 1, 3 - 4, (i), (ii)} does not imply Axiom 2.

All this imply how right Tim Chow was with his remark. Also, this
implies how important the goals formulated in Harvey's postings are.

More information on the above, on the logical form of SPR, on the
logical analysis of SPR, its connections with the symmetry axiom Axiom 5
etc can be found in [MD] section 2.8.3 and in [AMNBook] sections 2.8,
3.9.

References:

[LL] Landau, L. D. and Lifsic, E. M., Classical Theory of Fields,
Pergamon Press, Oxford, 1975. 

[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


Hajnal Andreka and Istvan Nemeti