[FOM] Time in relativity.

[Istvan Nemeti]

One of the most generally noticed striking features of relativity is
that it predicts that our common sense intuition about time is
hopelessly wrong. In some sense this was the main motivation for Kurt
Godel in his breakthrough work in general relativity, which lead to
rotating black holes later, and eventually e.g. to the Etesi-Nemeti
paper [EN] on black-hole driven approach to Church Thesis.

Let us look at the simplest incarnation of "misbehaviour" of
relativistic time. Let us make it real simple: "What time is it? (on
your watch)"

Let us contemplate the question what the various observers "think" about
each other's clocks. Does the other observer's clock run slow, or does
it show the right time, or does it run fast?

Let  o  and  o1  be observers. Let's look at the intuitive statement:

"in o's worldview it is time  t  when  o1's clock shows time  t1".

This can be formalized in our FOL-language for SRK as follows:

The event  E  which in  o1 's worldview is at position <t1,0,0,0,>, in
o 's worldview is at position with time-coordinate  t.

In order to appreciate our next axiom, we include here the statement
expressing the "principle of absolute time":

PRINCIPLE OF ABSOLUTE TIME. Assume that  o  and  o1  synchronize their
clocks  to 0 when they meet (i.e. assume that  o  and  o1  meet in event
E, and the time-coordinate of the position of  E  in  o 's worldview is
0, and the time-coordinate of the position of  E  in  o1 's worldview is
also 0). Then  o  and  o1  always agree on time,  i.e. for all  t, in  o
's worldview it is time  t  when  o1 's clock shows time  t.

Our next axiom will turn out to be equivalent with the SIMULTANEOUS
DISTANCE axiom. In our works we usually denote it by AxSym or by AxSyt,
abbreviating "Axiom of Symmetry of Time".

AXIOM OF SYMMETRY OF TIME. Let  o  and  o1  be observers who synchronize
their clocks to  0  when they meet. Then for any  t  and  t1, if in o 's
worldview it is time  t  when  o1 's clock shows  t1, then in  o1 's
worldview, too, it is time  t  when  o 's clock shows t1.

The above axiom talks about time only, and in this it differs from
Harvey Friedman's similar axioms (Axioms 5, 5a - 5d in his posting of
Feb 14 [FoSRK6]) which talk both about time and spatial distance.

The above axiom of SYMMETRY OF TIME implies that whenever  o thinks that
o1 's clock runs fast,  o1  will also think that  o 's clock runs fast,
as opposed to thinking that  o 's clock then runs slow (as in normal
everyday thinking we would infer). How can this happen at all? Doesn't
the axiom of SYMMETRY OF TIME above then imply the PRINCIPLE OF ABSOLUTE
TIME? 

The answer is that, assuming the axiom of SYMMETRY OF TIME, it is *not*
possible that  o  thinks that  o1 's clock runs FAST, but it *is*
possible that  o  thinks that  o1 's clock runs SLOW. How is this latter
possible? The solution is that  o  and  o1  will "observe" different
events as simultaneous. A *visual explanation* (a drawing) of the axiom
of SYMMETRY OF TIME can be found e.g. in [FOL75] Figure 12 on p.22 or in
[J50] Figures 17, 18 on pp.21,22.

Before stating the equivalence of the axiom of SYMMETRY OF TIME and the
SIMULTANEOUS DISTANCE axiom, we introduce the notion of TIME-SPHERE of
an observer, because this notion will be useful in characterizing the
content of the axiom of SYMMETRY OF TIME.

Definition. The TIME-SPHERE of an observer  o  is the set of all those
coordinate points  in  o 's world-view where some other observer's clock
shows time 1 and this other observer synchronized his clock with  o  at
time  0. The UPPER HYPERBOLOID is defined as the the set of those
coordinate points  <t,x,y,z> for which  t>0  and  t^2 - x^2 - y^2 - z^2
= 1.


THEOREM 1.  Let  M  be any model satisfying Axioms 1 - 4 in our FOM
posting of Feb 6 together with Axiom 6 (THOUGHT EXPERIMENT axiom) of our
posting of Feb 14 (or, equivalently, satisfying axioms 1,4,6 of Harvey's
posting of Feb 14). Statements (i) - (vi) below are equivalent.

(i) the axiom of SYMMETRY OF TIME holds in  M.

(ii) the SIMULTANEOUS DISTANCE axiom holds in  M.

(iii) there are two observers  o, o1  not at rest relative to each other
which have the same TIME-SPHERE.

(iv) all observers have the same TIME-SPHERE.

(v) there is an observer whose TIME-SPHERE is the UPPER HYPERBOLOID.

(vi) the TIME-SPHERE of all observers is the UPPER HYPERBOLOID.


Pictures representing possible time-spheres (under Axioms 1-4, 6) are
e.g. in [AMNBook] on pp.88, 441.

References

[EN] Etesi, G. and Nemeti, I.: Non-Turing computability via
Malament-Hogarth spacetimes. Internat. Journ. Theoretical Physics 41,2
(2002), pp.341-370.

[FOL75] Andreka-Madarasz-Nemeti: Logical analysis of relativity
theories. In: First-order Logic Revisited (Proc. FOL75 Berlin), Kluwer,
to appear.

[J50] Andreka-Madarasz-Nemeti: Logical Analysis of Special Relativity
Theory (23 pp).
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