[FOM] Re: On foundations of special relativity (Twin Paradox)

[Istvan Nemeti]

On February 06 2004 Jose Felix Costa wrote

>consider the following thought experience:
>
>subject moving inside a ship at 0.99 c from earth to alpha-centauri;
when >arriving to alpha-centauri then
>      subject jumps instantaneously from that ship
>      to another ship in the opposite direction
>      moving at 0.99 c from alpha-centauri towards earth.
>
>the instantaneous character of the jump (infinite instantaneous
>acceleration at alpha-centaury) leeds to results on the twin paradox
>(within special
>relativity) coinciding to those given by general relativity (with
finite >negative / positive accelaration near destination).
>
>my thought experience is not physical! but it is mathematicall.
>
>can you carry out this calculation on the Twins inside your system?

Thank you for the question. Smile.

In short: the answer is "yes".

In more detail: 

Yes: There are answers on two levels, one is "mathematical", the second
is "physical". 

To fix our language, let there be two twins, one is inertial  B  while
the other twin  A  accelerates. So  B  remains stationary while  A
makes a round-trip. We want to compare the clocks of  A  and  B  before
and after the trip. We would like to conclude that less time elapsed on
the clock of accelerating twin  A  than on the clock of inertial twin
B.

(1)

First answer: In the simple framework of our posting of February 5 we
can do this by approximating the life-curve of twin  A  with a sequence
of life-lines of inertial observers. So  A  is approximated by, say,  A1
and  A2  and  when  A1  and  A2  meet, they synchronize their clocks. A1
and A2 are inertial.

We formalized this approximation of the twin paradox in our FOL language
in [3], p.140 (see also pp.139-150). This is basically the same
arrangement that Felix described. (FOL abbreviates first-order logic.)

So the period of act of infinite acceleration can be replaced by a
synchronization of clocks. This eliminates the non-physical element from
the thought experiment, but one might still feel that this is not
perfectly the same as the original twin paradox (with finite
acceleration). We address this below:

(2)

Second, more ambitious answer: Let  Specrel  denote the FOL-theory
Axioms 1-5  in our FOM posting of February 5. We generalized our
FOL-theory  Specrel  of special relativity to a similar FOL-theory
Accrel  allowing accelerated observers, too.  Accrel  is presented in
section 3 (pp.15-19) of our paper [1]. Accrel  is almost the same as
Specrel,  the only difference is that in  Accrel  we permit accelerated
observers, too. 

Now, in formulating  Twin Paradox  we can choose twin  A  to be an
appropriately accelerated observer and twin  B  to be inertial. Now, the
acceleration of  A  remains finite (differentiable etc) all the time. So
we can formulate the conjecture that less time passes on the accelerated
clock (that of  A ) than on that of  B.

This is formulated as the FOL-formula  Twinp  on p.17 of  [1]. Then, it
is stated as a theorem of  Accrel  on p.18 (Thm.3.1).

Further discussion of the twin paradox and of  Accrel  with references
is on pp.18-19 of the above paper [1]. See also the introductions of [4]
and [7].

A further generalization towards general relativity of our FOL-based
approach is making the FOL-theory  Accrel  local. (Here, local is
understood in the sense of both general relativity and in topology.) So
we obtain a local FOL-theory  Loc(Accrel). This is outlined in [1,
section 4] and in Madarasz-Nemeti-Toke [7].

Thank you again:

Istvan and Hajnal

[1] Andreka-Madarasz-Nemeti: Logical axiomatizations of space-time.
Samples from the literature.
http://www.math-inst.hu/pub/algebraic-logic/lstsamples.pdf

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

[4] Andreka-Madarasz-Nemeti: Logical analysis of relativity theories.
http://www.math-inst.hu/pub/algebraic-logic/foundrel03nov.pdf

[7] Madarasz-Nemeti-Toke: Generalizing the logic-approach to space-time
towards general relativity: first steps (draft)
http://www.math-inst.hu/pub/algebraic-logic/loc-mnt04.pdf