Band 31a Zeitschrift für Naturforschung Heft 11 Meaningless Questions in Cosmology 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 science 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 initial singularity of the universe?" or "What is the fate of matter in gravitational collapse inside the event horizon?" turn out to be empirically meaningless. We also show that a "wormhole" 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 black 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 black hole 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. General relativity 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 light 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 photons, 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 "photon", "observer" etc. are used (for mationally symmetric with itself. Minkowski flat brevity) instead of "history (or world line) 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 proper time 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 light cone 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.