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Home , Event horizon, Through the Wormhole

Band 31a Zeitschrift für Naturforschung Heft 11

Meaningless Questions in and Relativistic Astrophysics M. Heller Obserwatorium Astronomiczne Uniwersytetu Jagiellonskiego, Krakow, Poland and M. Reinhardt Astronomisches Institut der Ruhr-Universität, Bochum, FRG

(Z. Naturforsch. 31a, 1271-1276 [1976]; received June 26, 1976)

After a general classification of meaningless questions in we concentrate on empirically meaningless questions. Introducing the concepts of informationally connected, semiconnected and disconnected observers, a formalism for the analysis of the informational structure of space-time is developed. We discuss some problems of epistemological nature in cosmology and blade hole phys- ics. A number of questions like "What was 'before' the of the universe?" or "What is the fate of in inside the horizon?" turn out to be empirically meaningless. We also show that a "" does not violate causality for the set of informationally connected observers who do not enter it.

I. Introduction going to define and discuss meaningless questions in relativistic astrophysics and cosmology. In gen- Questions of epistemological nature often occur eral we call a question meaningless, if there is in in cosmology and in relativistic astrophysics, in principle no answer to it. Since there are different particular after the boom in hole physics. kind of possible answers, we distinguish several Frequently these problems are either treated in a classes of meaningless questions. sloppy way, or their epistemological nature is not Meaningless questions of the first kind are of even recognized. In the present paper we attempt to tautological nature. These are meaningless questions sort out physically meaningless questions in this by definition. If, for example, we define the universe field. as "everything that exists", the question "what is In philosophy the term "meaningless" is histo- outside the universe?" is tautologically meaningless. rically burdened with the definition of "meaning" Meaningless questions of the second kind are by the positivistic and neopositivistic schools of theory dependent. These are questions to which one thought. The Vienna circle called a statement, even cannot get an answer within the framework of a a universal one, "meaningless", if it is not empiri- given theory. There are two subclasses: theoretically cally "verifiable" 1_4. They identified "non-empiri- and empirically meaningless questions. A question cal" and "non-verifiable" with "meaningless" and is theoretically meaningless in the framework of a "metaphysical". This definition was so narrow that given theory, if the theory does not provide any it excluded physical laws, which are always uni- tools to answer the question. E. g. questions about versal statements, from the realm of science. Pop- quantum effects are theoretically meaningless in a per 5 sharpened the "criterion of demarcation" be- classical field theory like Maxwell's theory. Ques- tween science and non-science by substituting the tions are empirically meaningless, if in principle term "verifiability" by "falsifiability" of a state- there cannot be an empirical (observational) an- ment. This was an improvement with respect to the swer to them within the framework of a given problem of empirical testability of scientific theo- theory. They relate to empirically untestable predic- ries. tions of a theory and do occur often in However even Popper's approach to the testabi- physics (see Section IV). lity of theories has been criticized as still too nar- In a given theory a theoretically meaningless 6 7 row ' . We will not deal here with the complicated question is also empirically meaningless, since the problem of the relations between theories and em- theory is unable to give any answer to the question, pirical data. Our scope is much simpler. We are which could be tested empirically. The reverse is not true. A question may be empirically meaningless Reprint requests to M. Reinhardt, Astronomisches Insti- and yet theoretically meaningful. This situation does tut der Ruhr-Universität, Postfach 102148, D-4630 Bo- chum, FRG. arise in relativistic astrophysics. 1272 M. Heller and M. Reinhardt • Meaningless Questions in Cosmology and Relativistic Astrophysics is quite capable to make predictions about regions time is measured by atomic, thermodynamical, or of space-time which will never be observable to us biological clocks, which can be non-metric and ir- in the framework of the theory (see Sects. Ill and regular. The only requirement is that these clocks IV). Although we will never be able to check these are monotonical, i. e. that informations arriving predictions of the theory empirically, their mathe- continuously to an observer form an ordered set8- matical treatment can help considerably towards a If a signal emitted by an observer Q is better understanding of the theory. In the following received by an observer P we call it a message8 we will concentrate on the problem of empirical from Q to P. If there is a message from Q to P we meaning. say that P is informationally semiconnected with Q, If one asks a question by setting up an obser- or more precisely: vational device, to which one cannot obtain, in prin- ciple, an answer in terms of measurable data, i. e. Definition 2 : any information, this question is empirically mean- P is informationally semiconnected with Q, with ingless. Even if the question is theoretically not respect to U cJll, P\\\ Q, iff for every pE P n U meaningless, it can never be used to test the theory. u Thus the central term of scientific epistemology is there is a q^Q such that q<^ p. Obviously we have: P |>| Q => P |\| Q . information. u u

Definition 3: II. Informational Structure of Space-Time If P|>|(?=>3 VeJA such that |P, we call We consider space-time, i. e. the set 7A = u v U and V informationally symmetric. {p, q, ..., x, y, ...} of elements called events, and two subsets of JA\ £. = L , ...}, the elements 2 Definition 4: of which are called light rays, light signals or If P |>| Q => Q |>| P, we call U informationally simply , and P = {P, Q, ..., X, Y, ...} the u u elements of which are called observers. Note that normal. Every informationally normal set is infor- the words "", "observer" etc. are used (for mationally symmetric with itself. Minkowski flat brevity) instead of "history (or ) of a space-time is informationally normal. photon or observer"8. In principle every timelike On on informationally normal set we have: curve may be considered as an observer, however in the following we shall consider only geodesic ob- P\>\QoQ\\\P AP\WQ servers, i. e. timelike geodesies. We define the set of informationally connected ob- If from an observer P a light signal is emitted servers to the observer P with respect to Uc'M: towards an observer Q, reflected at Q and received bade in a finite at P, we call it an Definition 5: echo 7 from Q to P and we write P |>| Q. SI C,(P) = {IeP:?Hl}. If there is an echo from Q to P we say that P is u informationally connected with Q. We express this A SICj' (P) is called an extension of SICJJ(P), if more precisely as follows. UcV and SIC[7(P)cSICF(P). A set of informa- tionally connected observers is called inextendible, Definition 1: if V V^U, VcWl:SlCu{P) =SICF(P). We define P is informationally connected with Q, with re- the set of informationally semiconnected observers spect to UcWl, P|>| Q, iff for every p^Pn U P u to the observer with respect to Ucy\{: there are a q^Q and a p2ePrs\ U such that Pi\q ! and q^po, with the condition that [ t (p2) — t (px) Definition 6: is finite. SlSu(P)={XeV:P\\\ X] . u Here ' t(p2) — f(pi)' denotes the proper time of the observer P elapsed between the events pi and p2. Inextendible sets of informationally semiconnected p ^ q is the so-called horizmodial relation; it should observers are defined as above with SIS instead of be read "q lies on the of p" 9. Proper SIC everywhere. 1273 M. Heller and M. Reinhardt • Meaningless Questions in Cosmology and Relativistic Astrophysics

Proposition 1 : One can construct as a "gedankenexperiment" Let N be an informationally normal region of multiply semiconnected sets of observers SICj (i = 1, ... ,n) space-time. Every observer in N belongs to exactly so that some SIC/,- can receive informa- one of pairwise disjoint sets of informationally tions from SIQ to SIQ + Wt' which are all mutually (semi) connected observers. informationally disconnected, but not vice versa, or so that SICo to SICn, all mutually informationally Proof. Within N, |>| and |\| are equivalence disconnected, can receive informations from SICX, relations. (Attention: relations |>| and |\| are de- but not vice versa (e. g. n — 1 black holes in the fined relatively to a subset U of y\il) universe SIC^. One can image also a hierarchy of For abstractions of these relations we may simply semiconnected sets, so that SICj can transmit in- write SICj, SIS; (i = 1,2, ...) without reference to formation to SIQ + i' but not the other way round. a single observer. Obviously by combination of these possibilities one Proposition 2: can construct very complicated informational struc- tures. We will not go into further details of this (P) . SIC[/(P)USISF(P) = SISj/u V informational topology. The main point is that, The set of observers may have a more complicated however complicated the structure for any given SIC, structure, for instance we may consider an infor- it is uniquely determined what empirically meaning- mationally semi-connected set of informationally ful and meaningless questions are. Thus the notion connected observers defined by: "empirically meaningless question" is relative, but (SIC)1|\|(SIC)2i. e. for every well defined for any given SIC, if the informational P e(SIC)i 3 Qe (SIC) 2 suchthat P|\| Q. structure is known. Definition 7 : An observer Q is informationally disconnected III. Meaningless Questions in Cosmology with P with respect to U, iff SISt/(P). For this we write P | | Q. Let us first define the universe. Definition 9: Definition 8: The universe is the maximum extension of the set An observer Q is informationally disconnected of informationally connected observers to which we with P, iff SISu(P), with SISV(P) inextendible, belong [SICj/(We) inextendible]. ^ II earth. "We" can be generalized to signify a suitable meaningless, if it relates to observers informa- informationally connected subset like our Galaxy, tionally disconnected with P. the Local Group, or even the Local Supercluster. If a question is empirically meaningless to P, it An alternative definition of the universe is is also empirically meaningless to all observers in u(P)- the inextendible SIC Definition 9': Let us consider for example two sets of informa- tionally connected observers SIQ and SIC2, such The universe is the set of all observers that can that SIC21\| SICj and SIQ | jSIC2 . This means that ever transmit information to us [SISj/(We), where U SIC2 can receive information from SICA. On the denotes the proper past]. other hand SICX can never receive any information It is worth to emphasize that even in an infor- U = y\i from any of the elements of SIC2. This implies that mationally normal space-time and even if it is empirically meaningful for an observer, who is the universe according to def. 9 is not the entire space-time [see proposition 1, and note in def. 1 an element of SIC2 , to ask questions about events the condition: \t{p2)—t(p1)\ is finite]. in SICX , but not vice versa. In other words SIQ and SIC2 are divided by an informationally semi- Note that the two definitions are not identical. permeable membrane. Examples of informationally Definition 9' is weaker than def. 9, for in principle semi-permeable membrans are the event horizons in we could be an element of SIC2 (SIC21\| SICX, the Schwarzschild and Kerr solutions of general SIQ || SIC2), or an element of an informationally relativity. more complicated structure. In this case we might 1274 M. Heller and M. Reinhardt • Meaningless Questions in Cosmology and Relativistic Astrophysics receive information from observers who are not started with a true singularity, no signals can have elements of the maximum extension of the SIC to reached us from the previous state *, if there ever which we belong. On the other hand def. 9 and 9' was one, nor will be any transmitted to the next would coincide if we were in SICX of a simply or state after the collapse of the universe to a second multiply informationally semiconnected set of ob- singularity, if the universe is closed (which seems servers — SlCi being inextendible and the infor- unlikely at present14). Thus if general relativity mational structure being such that no information does not break down at some finite , it is from other SIC can get to it. Note that this is true if empirically meaningless to ask about states of the general relativity holds. universe before the , or after a possible recollapse. On the other hand, if general relativity Let us consider the following questions: cannot be extrapolated to infinite , i. e. if (i) What is outside the universe? the universe did not start from a singularity, signals (ii) What was before the beginning of the from the state of the universe before the big bang universe? could have been transmitted to us. These difficulties (iii) What will be after the end of the universe? are apparently theory dependent. It is not impossible (and even desirable) that a future theory will over- On the first glance and intuitively they seem to come all of them. be meaningless questions of the first kind, i. e. In this case it would make sense to ask questions tautologically meaningless. In the light of chapter II about the state of the universe before the big bang things are more complicated. and after a possible recollapse to extreme densities. According to def. 9' these questions are empiri- But, of course, then the big bang would not be cally meaningless, since no observer, who is outside epistemologically the beginning of the universe. the universe, or who existed before the beginning, Question (ii) is closely related to the problem of or will exist after the end of the universe could the initial conditions of the universe. Zel'dovich and transmit information to us. This is trivial, for ac- Novikov12 have shown that in a bouncing closed cording to def. 9' an observer who can transmit in- Friedmann universe information will be transmitted formation to us must be part of the universe. On to the next state through the singularity in form of the other hand, according to def. 9, question (i) initial conditions, e. g. the per baryon. might not be empirically meaningless, if we were However, if the big bang started with a true sin- e. g. an element of SIC2 of an informationally semi- gularity, no signals, i. e. no direct informations, connected set of observers. In this case questions could have crossed it, and we can never check (ii) and (iii) would also be meaningful, if the in- whether the initial conditions are the consequence formationally semipermeable membrane between of previous cycles of a bouncing universe, or wheth- SI^ and SIC2 come into existence in the finite past er there were no previous cycles and the universe of the observers who are now elements of SIC2 . Of started with the initial conditions as they were, course these considerations are of a rather academic without any further "explanation". Thus, if general nature, since it is hardly possible to find out whether relativity holds strictly, all questions concerning two observers are informationally connected or semi- the origin of initial conditions are empirically connected, if the exchange of informations would meaningless, since we cannot receive signals from take several billion years of proper time. Question beyond the singularity. (ii) needs some discussion, since there might be some misunderstanding of the term "beginning". At There are attempts to connect the initial condi- present it is generally accepted that the universe tions of the universe with the laws of physics (see 15 was in a very dense and hot stage some lO10 a ago. e. g. ). Let us suppose that this program would be General relativity predicts a singularity as the start- successful, i. e. that one could prove a one to one ing point. The singularity would have been a state correspondence between physical laws and the ini- of infinite density (and infinite , see how- tial conditions of the universe. Certainly this would ever10'11). It has been repeatedly discussed what be a progress to some extent. However, it would preceded this singularity (see e. g.12'13). If general relativity holds up to infinite densities (which is * Information in the form of initial conditions can be transmitted through the singularity 12. See however the open to doubt), i. e. if the expansion of the universe discusion below. 1275 M. Heller and M. Reinhardt • Meaningless Questions in Cosmology and Relativistic Astrophysics only show that the laws of physics are as they are decided to commit suicide by plunging into a black because the initial conditions of the universe were hole will never disappear completely for his fellows as they were, or vice versa. It would not help a iota of SICX . For him it will be a quick and merciful towards the problem of the origin of the initial con- death. For them it will look like an infinitely pro- ditions of the universe. The question, what preceeded longed, but never completed suicide. the initial singularity would remain empirically Thus, jokes asides, it is empirically meaningless meaningless. to ask about the fate of matter once it crosses the of a singularity — which it never does for the SIQ outside — or about the physics inside IV. Meaningless Questions in Blade Hole Physics the event horizon, since no signal can ever cross it In the last section we have touched upon mean- from inside in gravitational collapse. ingless questions of the second kind. This section Now let us consider the "wormhole" hypothesis, deals wholly with theory dependent meaningless which was brought up by Wheeler17. In Kruskal questions under the assumption that general rela- coordinates the Schwarzsdiild solution consist of tivity is valid. There were repeated attempts to dis- two timelike and two spacelike regions. The idea of cuss the physics within the the "wormhole" is to identify the asymptotically (see e.g.13'16). We are going to show that it is flat timelike regions (R+ and R~), so that a gravita- empirically meaningless to ask questions about the tional collapse occuring in R+ would come out as a physics inside an event horizon for observers who reversed gravitational collapse in R~, i. e. in the are outside. We will also show that the existence of same space-time, but at another place and time. It "" can never be proven. has been claimed that "wormholes" of this kind Let us first deal with ordinary gravitational col- would violate causality 13. One can easily show that lapse. Suppose there is a black hole surrounded by for the observers who stay outside — and only they a SlCi. It does not matter whether the blade hole matter epistemologically — causality is not violated. was there from the beginning of the universe, or First of all, an observer falling into a black hole has whether it is in the process of formation. Suppose to cross another, "inner" timelike region before he further that an element of this SIC! (a sort of comes out through a in another, or the kamikaze pilot) descends into the black hole. Be- same universe 12. In this intermediate region all in- cause of the time dilatation he will never reach the formation from his previous life is erased12. But event horizon of the black hole within finite proper even if this were not so, all questions about "worm- time of any element of the SIC who has stayed holes" are empirically meaningless for the observers safely outside. Of course the kamikaze observer will outside (i. e. in R+ and R~, even if they are identi- cross the event horizon, which is in the simplest case fied) . Suppose there is a kamikaze observer, element the Schwarzschild radius, in a finite interval of his of a SICj, who tries to get into a "wormhole". In proper time. However, this does not matter to the his own proper time he will be able to cross the rest of SIQ outside. Since no signal can ever cross first event horizon, to pass through the "wormhole" the event horizon from inside in gravitational col- and come bade through the second event horizon. lapse (we will discuss the reversed gravitational Then he will see himself plunge into the blade hole collapse later) the observers inside the event horizon end of the wormhole. For him causality would be (SIC2) and the observers outside (SIQ) form a violated. However this does not matter. He will semiconnected set of observers. (In the language of never be able to tell anybody. The observers, who Section II: SIC^SIC^ SICjISICo.) are the elements of SICj which have stayed at a safe The kamikaze observer — assume for the moment distance to the black hole (or "wormhole"-nobody that he is not torn to pieces by tidal forces — will in SIQ will ever know) will never even see their know within finite proper time what physics is like suicidal fellow disappear, and thus of course not inside the event horizon, but he will never be able reappear at another point of space-time. Thus for to tell his former fellow observers outside. For the SICj outside the "wormhole" causality cannot them, he will never even reach the event horizon, he be violated by this procedure. If "white holes" would exist, i. e. if we could observe the reversed Avill always remain, though increasingly dimly, a gravitational collapse, questions about the physics member of the original SICX. The guy who has 1276 M. Heller and M. Reinhardt • Meaningless Questions in Cosmology and Relativistic Astrophysics inside the Schwarzschild radius would not be em- this questions is futile. If a question is theoretically pirically meaningless, since the SICj outside would meaningless, perhaps it would stimulate the develop- receive information from this region. However the ment of a new theory. If a question is experimentally existence of "white holes", i. e. of regions of space- meaningless, it might help us towards a better un- time which started to expand later than the rest of derstanding of the present theory. In this respect the universe12, is highly improbable because of our terminology is different from that commonly quantum effects in strong gravitational fields which accepted in the philosophy of science, which has re- limit the retardation to minute fractions of a second. served the term "meaningful" only for experimen- tally meaningful statements. We believe, however, that our understanding of the problem is not far from common practice in science and common under- V. Final Remarks standing in the philosophy of science. We strongly In science there is no place for tautologically stress the scientific role of empirically meaningful meaningless questions, they are completely empty questions. Even if a theory passes all tests, past and and useless. Questions like "What 'preceded' the future, in the maximum extension of a set of infor- initial singularity?" or "What happens to matter in mationally (semi) connected observers, the predic- gravitational collapse beyond the event horizon?" tions of this theory about observers (or events), are often considered as deep and fundamental issues who can never transmit information to any element in physics. We have shown that according to our of it, are untestable (unfalsiable) and thus experi- definition they are empirically meaningless because mentally meaningless. In other words, one may ask they relate to predictions of a theory on which in these questions (providing they are not tautologi- the framework of this theory no empirical data will cally meaningless) but they cannot be checked by ever be available. We do not say that research on observations or measurements.

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