John Casti posed the following scenario
[7]
(for a
related discussion, see
[15]),
``[[Imagine some creatures-]]they may be carbon-based creatures just
like you and me. The difference is that they live in a world in which the
primary sensory inputs are not from the electromagnetic spectrum like
light, but rather come from sound waves. Note that this is not simply a
world of the blind; rather, it is a world in which there are no sensory
organs for perceiving any part of the electromagnetic spectrum. In this
case, then, ¼ such creatures would see the speed of sound
as a fundamental barrier to the velocity of any material object. Yet we
as creatures that do possess sensory organs for perceiving the
electromagnetic
spectrum see the sound barrier as no fundamental barrier at all. So, by
analogical extension, there may be creatures ``out there'' who regard the
speed of light as no more of a barrier than we regard the speed of
sound.''
Casti's scenario is one in which the observers are embedded in a universe which, to them, solely consists of sound waves. Such intrinsic observers develop a description [4,17,10,13,14,15,16] of the universe which may appear drastically different (cf. the emergence of complementarity [16]) from what some hypothetical ``super-observer'' perceives, who is peeking at the system from a ``God's eye,'' extrinsic position. One of the first researchers bothering about these issues has been the 18th century physicist Boskovich, ``¼ And we would have the same impressions if, under conservation of distances, all directions would be rotated by the same angle, ¼ And even if the distances themselves would be decreased, whereby the angles and the proportions would be conserved, ¼: even then we [[the observers]] would have no changes in our impressions. ¼ A movement, which is common to us [[the observers]] and to the Universe, cannot be observed by us; not even if everything would be stretched or shrinked by an arbitrary amount.''
Although generally not presented that way, relativity theory consists of two distinct parts: (i) it deals with conventions about how to operationalize certain concepts such as ``equal time at spatially separated points,'' in particularly synchronization procedures; and (ii) it states physical assumptions about the invariance of certain phenomena, such as the speed of light, and the laws of theoretical physics under changes of reference frames. Whereas the former conventions are mere working definitions, the latter statements are supposed to be God-given, eternal symmetries. Conventions can be changed at the price of complicating the theoretical formalism by means of a non-optimal representation of theory. Phenomena are a matter of physical fact. For instance, a preferred frame of reference, e.g., the one at rest with respect to cosmic background radiation, could be artificially introduced. To put it in the words of the late John Bell (cf. [2], p. 34): ``You can pretend that whatever inertial frame you have chosen is the ether of the 19th century physicists, and in that frame you can confidently apply the idea of the FitzGerald contraction [[of length of material bodies such as scales]], Larmor [[time]] dilation and Lorentz lag. It is a great pity that students dont understand this. ¼''
It was the radical operational feature in Einstein's theory of special relativity, which stimulated the physicist Bridgman. Bridgman demanded that the meaning of theoretical concepts should ultimately be based upon entities which are intrinsically representable and operational. That is, (cf. [6], p. V), ``the meaning of one's terms are to be found by an analysis of the operations which one performs in applying the term in concrete situations or in veryfying the truth of statements or in finding the answers to questions.'' More specifically (cf. [5], p. 103), ``¼ the meaning of length is to be sought in those operations by which the length of physical objects is determined, and the meaning of simultaneity is sought in those physical operations by which it is determined whether two physical events are simultaneous or not.''
Therefore, we are free to choose whatever physical concepts seem appropriate as long as they are operational. In particular, Casti's sound-perceiving creatures might choose sound, the author's water-fleas may use water waves, and human physicists may use light for the purpose of Einstein synchronization. The resulting frame of references will be different in all these cases. The space-time parameters by which events and phenomena are represented will be different, too. Using different types of formal representations of the same physical system-one may speak of layers or levels of description-might be an effective way to grasp different features of that system (associated, for instance, with different levels of complexity). This resembles Anderson's thesis of ``emerging laws'' ([1], p. 193; cf. Schweber [11]), ``The ability to reduce everything to simple fundamental laws does not imply the ability from these laws and reconstruct the universe. ¼ The constructionist hypothesis breaks down when confronted with the twin difficulties of scale and complexity. ¼ at each level of complexity, entirely new properties appear, and the understanding of the new behaviors requires research which I think is as fundamental in nature as any other.'' It is to be expected that in such an organization of physical concepts, the notion of causality need not be consistently defined for all descriptions combined: what may appear as causal connection (i.e., cause and effect) in one description needs not be causally connected in another description. Therefore it is of great relevance to make precise the limits and applicability of each of these descriptions.
Our primary concern will be the explicit construction of intrinsic space-time frames by adopting Einstein's synchronization conventions. Stated pointedly, we are interested primarily with ``virtual reality'' physics and physical epistemology. We do not attempt to reconstruct relativity theory one-to-one in the cellular automaton context. In particular, no Lorentz-invariant kinematic theory is introduced. Therefore, certain physical statements, in particular the relativity principle, stating that all laws of physics have an identical form in all inertial frames, needs not to be satisfied (cf. [3]).
However, it has been argued for quite some time [9] that, since all ``construable'' (in the sense of ``recursively enumerable'') universes are realisable within the CA framework, also the special theory of relativity can be implemented on such a structure.
Without loss of generality it is assumed that the maximal velocity, denoted by c, by which a body of information can move is one cell per cycle time. Flows of this kind will be called rays. In analogy to relativity theory, this velocity can be used to define clocks or synchronized events. We shall start by an explicit model of a ray clock. It consists of two mirrors, denoted by I, and a ray of velocity 1 cell per time cycle, denoted by > and < , which is constantly reflected back and forth between the two mirrors, and a one-place digital display right to the right mirror. In this model, after each backreflection of the light ray, the digit on the display increases by one modulus 10. The explicit transformation rules for a onedimensional CA with these properties are listed in appendix A. The time evolution of a ray clock is drawn in Fig. 1.
... __I>__I0__ ... ... __I>__I4__ ... ... __I>__I8__ ...
... __I_>_I0__ ... ... __I_>_I4__ ... ... __I_>_I8__ ...
... __I__>I0__ ... ... __I__>I4__ ... ... __I__>I8__ ...
... __I__<*0__ ... ... __I__<*4__ ... ... __I__<*8__ ...
... __I_<_I1__ ... ... __I_<_I5__ ... ... __I_<_I9__ ...
... __I<__I1__ ... ... __I<__I5__ ... ... __I<__I9__ ...
... __I>__I1__ ... ... __I>__I5__ ... ... __I>__I9__ ...
... __I_>_I1__ ... ... __I_>_I5__ ... ... __I_>_I9__ ...
... __I__>I1__ ... ... __I__>I5__ ... ... __I__>I9__ ...
... __I__<*1__ ... ... __I__<*5__ ... ... __I__<*9__ ...
... __I_<_I2__ ... ... __I_<_I6__ ... ... __I_<_I0__ ...
... __I<__I2__ ... ... __I<__I6__ ... ... __I<__I0__ ...
... __I>__I2__ ... ... __I>__I6__ ... ... __I>__I0__ ...
... __I_>_I2__ ... ... __I_>_I6__ ... ... __I_>_I0__ ...
... __I__>I2__ ... ... __I__>I6__ ... ... __I__>I0__ ...
... __I__<*2__ ... ... __I__<*6__ ... ... __I__<*0__ ...
... __I_<_I3__ ... ... __I_<_I7__ ... ... __I_<_I1__ ...
... __I<__I3__ ... ... __I<__I7__ ... ... __I<__I1__ ...
... __I>__I3__ ... ... __I>__I7__ ... .
... __I_>_I3__ ... ... __I_>_I7__ ... .
... __I__>I3__ ... ... __I__>I7__ ... .
... __I__<*3__ ... ... __I__<*7__ ...
... __I_<_I4__ ... ... __I_<_I8__ ...
... __I<__I4__ ... ... __I<__I8__ ...
A very similar configuration as for the light clock can be used as a
device for Einstein synchronization [8,16]:
Assume two clocks at two arbitrary points A and B which are ``of
similar
kind.''
At some arbitrary A-time tA a ray goes from A to B. At B
it is instantly (without delay)
reflected at B-time tB and reaches A again at
A-time tA¢. The clocks in A and B are synchronized
if
| (1) |
| (2) |
The ray velocity can then be defined to be identical for all frames, irrespective of whether they are moving with respect to the rest frame of the cellular space or not. Of course, this invariance of the ray speed with respect to changes of coordinate systems should ultimately be motivated by phenomenology and a proper choice of conventions. E.g., in relativity theory, the invariance of the Maxwell equations with respect to conformal, or angle preserving, coordinate transformations in four dimensions assures that for light-like vectors, ds2 = dx2-(c dt)2 = 0 and thus x = ct.
For synchronization, the same CA transformation rules as for the ray clock, which are listed in appendix 3 can be used. In Fig. 2, an example of synchronization between two clocks A and B is drawn.
clock1 A B clock2
... __I>__I0__I>______I__I>__I0__ ...
... __I_>_I0__I_>_____I__I_>_I0__ ...
... __I__>I0__I__>____I__I__>I0__ ...
... __I__<*0__I___>___I__I__<*0__ ...
... __I_<_I1__I____>__I__I_<_I1__ ...
... __I<__I1__I_____>_I__I<__I1__ ...
... __I>__I1__I______>I__I>__I1__ ...
... __I_>_I1__I______<*__I_>_I1__ ...
... __I__>I1__I_____<_I__I__>I1__ ...
... __I__<*1__I____<__I__I__<*1__ ...
... __I_<_I2__I___<___I__I_<_I2__ ...
... __I<__I2__I__<____I__I<__I2__ ...
... __I>__I2__I_<_____I__I>__I2__ ...
... __I_>_I2__I<______I__I_>_I2__ ...
... __I__>I2__I>______I__I__>I2__ ...
... __I__<*2__I_>_____I__I__<*2__ ...
... __I_<_I3__I__>____I__I_<_I3__ ...
... __I<__I3__I___>___I__I<__I3__ ...
... __I>__I3__I____>__I__I>__I3__ ...
... __I_>_I3__I_____>_I__I_>_I3__ ...
... __I__>I3__I______>I__I__>I3__ ...
.
.
.
This section deals with what happens with the intrinsic synchronization and the space-time coordinates when observers are considered which move with respect to the CA medium. For simplicity, assume constant motion of v automaton cells per time cycle. With these units, the ray speed is c = 1, and v £ 1.
There are numerous ways to simulate sub-ray motion on a CA. In what follows, the case v = 1/3 will be studied in such a way that every three CA time cycles the walls, symbolised by I, move one cell to the right. [Strictly speaking, there should be a periodic transformation of the wall such that I® a® b® I and the states a and b have the same reflection properties as I.]
Notice that two clocks which are synchronized in a reference frame which is at rest with respect to the CA medium are not synchronized in their own co-moving reference frame. Consider, as an example, the CA drawn in Fig. 3(a). (Strictly speaking, the CA rule here depends on a two-neighbor interaction.) By evaluating equation (1) for tA = 1, tB = 4, tA¢ = 5, and 4-1 ¹ 5-4. If the first clock is corrected to make up for the different time of ray flights as in Fig. f:3(b), tA = 2, tB = 4, tA¢ = 6, and 4-2 = 6-4. This correction, however, is the reason for asynchronicity of the two ray clocks with respect to the ``original'' CA medium.
__I>_I0__I___<___I__I>_I0________________ __I>_I1__I___<___I__I>_I0________________
__I_>I0__I__<____I__I_>I0________________ __I_>I1__I__<____I__I_>I0________________
__I_<*0__I_<_____I__I_<*0________________ __I_<*1__I_<_____I__I_<*0________________
___I>_I1__I>______I__I>_I1_______________ ___I>_I2__I>______I__I>_I1_______________
___I_>I1__I_>_____I__I_>I1_______________ ___I_>I2__I_>_____I__I_>I1_______________
___I_<*1__I__>____I__I_<*1_______________ ___I_<*2__I__>____I__I_<*1_______________
____I>_I2__I__>____I__I>_I2______________ ____I>_I3__I__>____I__I>_I2______________
____I_>I2__I___>___I__I_>I2______________ ____I_>I3__I___>___I__I_>I2______________
____I_<*2__I____>__I__I_<*2______________ ____I_<*3__I____>__I__I_<*2______________
_____I>_I3__I____>__I__I>_I3_____________ _____I>_I4__I____>__I__I>_I3_____________
_____I_>I3__I_____>_I__I_>I3_____________ _____I_>I4__I_____>_I__I_>I3_____________
_____I_<*3__I______>I__I_<*3_____________ _____I_<*4__I______>I__I_<*3_____________
______I>_I4__I______>I__I>_I4____________ ______I>_I5__I______>I__I>_I4____________
______I_>I4__I______<*__I_>I4____________ ______I_>I5__I______<*__I_>I4____________
______I_<*4__I_____<_I__I_<*4____________ ______I_<*5__I_____<_I__I_<*4____________
_______I>_I5__I___<___I__I>_I5___________ _______I>_I6__I___<___I__I>_I5___________
_______I_>I5__I__<____I__I_>I5___________ _______I_>I6__I__<____I__I_>I5___________
_______I_<*5__I_<_____I__I_<*5___________ _______I_<*6__I_<_____I__I_<*5___________
________I>_I6__I>______I__I>_I6__________ ________I>_I7__I>______I__I>_I6__________
________I_>I6__I_>_____I__I_>I6__________ ________I_>I7__I_>_____I__I_>I6__________
________I_<*6__I__>____I__I_<*6__________ ________I_<*7__I__>____I__I_<*6__________
_________I>_I7__I__>____I__I>_I7_________ _________I>_I8__I__>____I__I>_I7_________
_________I_>I7__I___>___I__I_>I7_________ _________I_>I8__I___>___I__I_>I7_________
_________I_<*7__I____>__I__I_<*7_________ _________I_<*8__I____>__I__I_<*7_________
__________I>_I8__I____>__I__I>_I8________ __________I>_I9__I____>__I__I>_I8________
__________I_>I8__I_____>_I__I_>I8________ __________I_>I9__I_____>_I__I_>I8________
__________I_<*8__I______>I__I_<*8________ __________I_<*9__I______>I__I_<*8________
___________I>_I9__I______>I__I>_I9_______ ___________I>_I0__I______>I__I>_I9_______
___________I_>I9__I______<*__I_>I9_______ ___________I_>I0__I______<*__I_>I9_______
___________I_<*9__I_____<_I__I_<*9_______ ___________I_<*0__I_____<_I__I_<*9_______
____________I>_I0__I___<___I__I>_I0______ ____________I>_I1__I___<___I__I>_I0______
____________I_>I0__I__<____I__I_>I0______ ____________I_>I1__I__<____I__I_>I0______
____________I_<*0__I_<_____I__I_<*0______ ____________I_<*1__I_<_____I__I_<*0______
_____________I>_I1__I>______I__I>_I1_____ _____________I>_I2__I>______I__I>_I1_____
_____________I_>I1__I_>_____I__I_>I1_____ _____________I_>I2__I_>_____I__I_>I1_____
_____________I_<*1__I__>____I__I_<*1_____ _____________I_<*2__I__>____I__I_<*1_____
______________I>_I2__I__>____I__I>_I2____ ______________I>_I3__I__>____I__I>_I2____
______________I_>I2__I___>___I__I_>I2____ ______________I_>I3__I___>___I__I_>I2____
______________I_<*2__I____>__I__I_<*2____ ______________I_<*3__I____>__I__I_<*2____
_______________I>_I3__I____>__I__I>_I3___ _______________I>_I4__I____>__I__I>_I3___
_______________I_>I3__I_____>_I__I_>I3___ _______________I_>I4__I_____>_I__I_>I3___
_______________I_<*3__I______>I__I_<*3___ _______________I_<*4__I______>I__I_<*3___
________________I>_I4__I______>I__I>_I4__ ________________I>_I5__I______>I__I>_I4__
________________I_>I4__I______<*__I_>I4__ ________________I_>I5__I______<*__I_>I4__
________________I_<*4__I_____<_I__I_<*4__ ________________I_<*5__I_____<_I__I_<*4__
. .
. .
. .
(a) (b)
Let us now explicitly construct the coordinate axes [`x] and [`t] of a system which moves with constant velocity with respect to the medium. For convenience, let us switch to continuous coordinates. That is, the ``grainyness'' of the CA medium is disregarded; i.e., coordinates will be represented by real numbers. In this construction, the convention of the constancy of the speed of rays for all reference frames will play an important role. The new space and time axes will both become rotated towards the ray coordinates.
Consider Fig. 4, which represents the process drawn in Fig. 3(b).
Assume that the two mirrors are a (arbitrary) units apart. For the system at rest with respect to the CA medium, a ray is emitted from the origin A = (0,0) and arrives at the mirror at B = (vtB+a,tB), where it is reflected and arrives at the original mirror at A¢ = (vtA¢, tA¢). Since B lies on the ray which comes from the origin, B = (tB,tB) = ([a/(1-v)],[a/(1-v)]). Since A¢ lies on the ray which comes from B, tA¢ = -xA¢+C and tB = -xB+C. By evaluating C, one obtains A¢ = ([2va/(1-v2)],[2a/(1-v2)]). In the coordinate system which is moving with respect to the CA medium, [`A] = (0,0). In order for the clocks to be synchronized, i.e., by equation (1), [`t]B = ([`t]A¢+[`t]A)/2. But this should also be the time coordinate [`t]1, since this is just half the time from A to A¢. Thus one obtains two points (events) 1 = ([(vtA¢)/2],[(tA¢)/2]) and B whose time coordinates in the moving reference frame is identical; i.e., [`t]1 = [`t]B. A short calculation shows that, with respect to the coordinate system which is at rest in the CA medium, the lines of equal time coordinates for a system wich is moving with constant velocity v, e.g., the [`x]-axis, have slope 1/v, whereas the lines of equal space coordinates, e.g., the [`t]-axis, have slope v.
These transformation of coordinate axes correspond to the
Lorentz transformations
| (3) |
So far, only rays propagating one CA cell per cycle have been considered. It is not entirely unreasonable to assume synchronization with signals which propagate slower (or faster) than these rays.
A typical example would be the use of a signal for synchronization with slower-than-ray speed, i.e., c¢ < c.
One consequence of sub-ray synchronization is the possibility of ``super-ray'' speeds v such that c ³ v > c¢, and of a ``time travel'' with respect to such space-time frames. That is, for certain observers moving with v > c¢, the time coordinate defined by c¢ would ``run backward.'' This ``time travel'' however, is nothing particularly mysterious, but the outcome of the specific synchronization convention chosen (cf. below).
Let us come back to Casti's sound-sensitive creatures-suppose that they call themselves smorons. Suppose further that they discover the possibility to generate and detect light; e.g., by sonoluminescence. It can be expected that this discovery will cause a major trauma for the smorons, because they will find out that they could communicate much faster than by sound waves, on which their space-time frames are based. If they have applied the Einstein conventions for defining space-time frames, they would observe light as a supersonic (i.e., faster-than-sound) signal, which propagates on space-like ((Dx)2-(Dt)2 > 0) world lines. Such a phenomenon which might allow forward in time signalling in one inertial frame could allow backward in time signalling in another inertial frame; there always exists some (orthochronous) Lorentz transformation L which transforms a space-like wordline with t > 0 into one with t < 0.
Given backward in time signalling, a classical time paradox can be formulated: Assume two observers A and B which can communicate backward in time; i.e., a signal traverses the distance xAB between them in time tAB such that tAB < 0. Then the observer A might emit a signal at time tA, which arrives the observer B in tB, where it is reflected and is back at the observer A at a time tA¢ < tA, i.e., before observer A has emitted the original signal. If one performs a ``diagonalization,'' i.e., if one assumes that observer A emits a signal at time tA if and only if no signal is absorbed at tA¢; observer A emits no signal at time tA if and only if a signal is absorbed at tA¢, one ends up with the simplest form of time paradox.
The syntactic structure of this paradox closely resembles Cantor's diagonalization method (based on the ancient liar paradox), which has been applied by Gödel, Turing and others for undecidability proofs in a recursion theoretic setup.
How could the smorons cope with such a time paradox?
One could argue that sound consists of elementary constituents (such as atoms or molecules or clusters thereof), whose motion is ultimately governed by electromagnetic forces. Therefore, the valid theory is electromagnetism, and any theory ``shell'' (``level'') such as the smoron theory of sound waves must ultimately be based upon (although not necessarily be totally derivable from) it. As has been pointed out earlier, such a ``shell'' is very similar to what Anderson [1] calls ``emerging law'' (cf. Schweber [11]). In this picture, the elementary constituents are a sort of ``hidden parameters'' for smoron physics; something they can neither observe nor control. Therefore, the smoron physics does not apply to configurations in which their physics ``shell'' is inappropriate. They are unable to control the events. This is why the theory is cryptodeterministic, and the smorons have no free will with respect to diagonalization-they will simply not be able to operationalize diagonalization purely in terms of sound waves.
One major goal of these considerations is the assertion that space and time are not God-given, metaphysical objects, but are subject to theoretical construction. The construction of space-time depends on conventions. Any inconsistency, for instance the possibility to construct time paradoxes, may be perceived as a problem of the improper, unfaithful construction of space and time rather than the impossibility to do certain tasks such as super-fast signalling. Any unfaithful construction of space-time may in turn be deeply rooted in the status quo of physical theory.
j( > ,_,X) = > , j(X,_, < ) = < , j(_,_,_) = _, j(X,_, > ) = _, j( < ,_,X) = _, j (_, > ,_) = _, j(_, < ,_) = _, j(_, > ,I) = < , j(I, < ,_) = > , j( > ,I,X) = *, j ( < ,*,X) = I, j(X, < ,*) = _, j(* ,1,X) = 2, j(*,2,X) = 3, j(*,3,X) = 4, j(*,4,X ) = 5, j(*,5,X) = 6, j(*,6,X) = 7, j(*,7,X) = 8, j( *,8,X) = 9, j(*,9,X) = 0, j(*,0,X) = 1, j(0,_,X ) = _, j(1,_,X) = _, j(2,_,X) = _, j(3,_,X) = _, j( 4,_,X) = _, j(5,_,X) = _, j(6,_,X) = _, j(7,_,X ) = _, j(8,_,X) = _, j(9,_,X) = _, j(X,*,0) = *, j(X ,*,1) = *, j(X,*,2) = *, j(X,*,3) = *, j(X,*,4 ) = *, j(X,*,5) = *, j(X,*,6) = *, j(X,*,7) = *, j(X ,*,8) = *, j(X,*,9) = *, j(X,1,X) = 1, j(X,2,X) = 2, j(X,3,X) = 3, j(X,4,X) = 4, j(X,5,X) = 5, j(X,6,X) = 6, j(X,7,X) = 7, j(X,8,X) = 8, j(X,9,X) = 9, j(X,0,X) = 0, j(X,I,0) = I, j(X,I,1) = I, j(X,I,2) = I, j(X,I,3) = I, j(X,I,4) = I, j(X,I,5) = I, j(X,I,6) = I, j(X,I,7) = I, j(X ,I,8) = I, j(X,I,9) = I, j(X,I,0) = I, j(X,I,X) = I, j(* ,_,X) = _, j(_,_,I) = _, j(I,_,_) = _, j(I, > ,_ ) = _, j(_, < ,I) = _ .
X stands for any state.