- Email: [email protected]
Hubble expansion in static spacetime
Chaos, Solitons and Fractals 33 (2007) 770–775 www.elsevier.com/locate/chaos
Hubble expansion in static spacetime Otto E. Rossler a
a,*
, Dieter Fro¨hlich a, Ramis Movassagh b, Anthony Moore
c
Division of Theoretical Chemistry, University of Tubingen, Auf der Morgenstelle 8, 72076 Tu¨bingen, FRG b ETH and Collegium Helveticum, Switzerland c Academy for Media Arts, Cologne, FRG Accepted 14 June 2006
Communicated by Prof. L. Marek-Crnjac
Abstract A recently proposed mechanism for light-path expansion in a static spacetime is based on the moving-lenses paradigm. Since the latter is valid independently of whether space expands or not, a static universe can be used to better see the implications. The moving-lenses paradigm is related to the paradigm of dynamical friction. If this is correct, a Hubble-like law is implicit. It is described quantitatively. A bent in the Hubble-like line is predictably implied. The main underlying assumption is Price’s Principle (PI3). If the theory is sound, the greatest remaining problem in cosmology becomes the origin of hydrogen. Since Blandford’s jet production mechanism for quasars is too weak, a generalized Hawking radiation hidden in the walls of cosmic voids is invoked. A second prediction is empirical: slow pattern changes in the cosmic microwave background. A third is ultra-high redshifts for Giacconi quasars. Bruno’s eternal universe in the spirit of Augustine becomes a bit less outlandish. 2006 Published by Elsevier Ltd.
1. Introduction Recently, hopes were raised for a new cosmology based on the moving-lenses paradigm [1]. Rather than having the whole spacetime expand, it may suffice to expand the light-rays passing through a stationary spacetime. The frequencyindependent light-stretching effect would justify Hubble’s lifelong belief in a first-principles ‘‘tired-light’’ explanation of his distance-proportional redshift law. The well-known falsification of all tired-light effects, implicit in the supernova project’s finding that astrophysical events of a duration of weeks are still proportionally redshifted (elongated in time) [2], turns out to be inapplicable since the light-ray expansion caused by moving Einstein lenses does [3–6] extend to ultralow frequencies. In the following, the proposal is put into quantitative terms and a new implication (a bent in the Hubble line) is predicted. Finally, the approach is given the benefit of the doubt by placing it into a wider empirical and historical context.
*
Corresponding author. Tel.: +49 7071 66323.
0960-0779/$ - see front matter 2006 Published by Elsevier Ltd. doi:10.1016/j.chaos.2006.06.046
O.E. Rossler et al. / Chaos, Solitons and Fractals 33 (2007) 770–775
771
2. Dynamical friction of the second kind The idea belongs into the context of statistical mechanics which until now has lacked a role in cosmology. There exists a not very well known example in statistical mechanics called ‘‘dynamical friction’’ which resembles entropy because it exhibits dissipative behavior in both time directions. Nevertheless, it lives in ordinary space rather than phase space. Classical dynamical friction applies when a heavy-slow particle is injected into a dilute gas of light-fast particles endowed with long-range repulsion. Even though there is no direct contact (as in ordinary friction – hence the name), the intruding particle loses energy to the many particles. But light passing through the cosmos is not an ultra-heavy particle. Indeed the proposed new sub-variety of dynamical friction starts out not with a heavy-slow, but with a light-fast particle injected into a dilute gas, not of light-fast but of heavy-slow particles which are endowed, not with long-range repulsion but with long-range attraction. Despite these three differences, the injected particle can again be predicted to lose energy to the many uncorrelated particles [6]. The mechanism of this effect (a simulation is still outstanding) goes like this: When the little fast particle enters the next ‘‘region of dominating influence’’ (potential dip) of a moving heavy slow particle, this entry in general occurs in a more or less tangential fashion. The funnel-shaped dip then, with the same probability, either approaches or veers away from the path of the intruded fast particle. Hence the closest passing distance from the center, R, is not the same in the two cases: The approaching path comes closer, leading deeper down the throat of the lily-shaped funnel to thereby generate a stronger damping effect, than does the path leading out of the throat into a shallower terrain, generating a weaker accelerating effect there [3–6]. The same asymmetry was implicitly noticed by Wucknitz and Sperhake [7]. Let us be a bit more explicit. The fast particle may be a photon and the soup through which it passes a cloud of randomly moving gravitating masses that act as weak Einstein lenses. Birkinshaw’s law for moving gravitational lenses [8] then applies at each encounter: z ¼ 2ðv=cÞðRs =RÞ;
ð1Þ
with the approximate factor 2 included for specificity. Eq. (2) describes the frequency shift z caused in a light-ray by a transversely moving gravitational lens [8]: When the lens is approaching (v positive), z is positive (redshift), and vice versa if v is negative (blueshift). The effect thus is perfectly time-symmetric if the passing distance R is the same in the two cases. The two remaining constants are Rs, the Schwarzschild radius (mass) of the attraction center and c, the speed of light. The point is that the two distances of closest encounter R (call them R+ and R) are not the same in the situation of our present interest. They deviate (to first approximation) by an equal amount d from their common mean R. For, owing to the assumed lack of correlation between the passing photon and the moving gravitating mass, the ‘‘decision’’ of whether the newly entered funnel is approaching or receding from the fast particle’s path is in effect only made at the moment of definitive engagement. At that moment, when the region of dominating influence of one funnel is left and that of the next is entered, the actual and the opposite direction of motion of the freshly entered funnel have exactly the same probability of being applicable as far as the uncorrelated light particle is concerned. Hence the difference between R+ and R.
3. Quantitative formulation The dependency of z on R in Eq. (1) is nonlinear since R is in the denominator. Therefore, a positive net difference (redshift) applies between two equiprobable passages characterized by an equal numerical value of v except for sign. For the same ‘‘distance of engagement’’, D (half mean distance between equal funnels) leads to two unequal R’s (R+ and R) that deviate by d from their joint mean R as we saw. Hence after many encounters (or pairs of encounters, respectively), a cumulative ‘‘net redshift’’ applies in proportion to the number (1/2 Æ n) of equiprobable pairs of encounters: znet ¼ ð1=2Þ n2ðv=cÞRs ½1=ðR dÞ 1=ðR þ dÞ ¼ nðv=cÞRs 2d R2 d2 Þ 2nðv=cÞRs d=R2 :
ð2Þ
Putting in d 1/2(v/c)D where D is the effective funnel radius (half mean distance between equal funnels), as an estimate for the mean differential change d of the passing distance R after the funnel has been entered, one obtains znet ð1=2Þ 2nðv=cÞ2 Rs D=R2 :
ð3Þ
If the mean distance of passage, D/2, is inserted for R, this becomes znet 4nðv=cÞ2 Rs =D:
ð4Þ
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This equation means (since all terms on the right hand side are positive and n is proportional to the distance traveled) that a distance-proportional Hubble-like redshift law is implicit in the assumption made of uncorrelated initial conditions between an injected light particle and a soup of uncorrelated randomly moving gravitational masses.
4. Evaluation of the formula In order to obtain a quantitative estimate, it is first necessary to insert empirical parameters. The latter are the mean mass of a galaxy (Rs), the mean transverse velocity of galaxies (v), the mean distance between galaxies (2D), the visible diameter of the cosmos (d) divided by the third root of the visible number of galaxies (N), jointly yielding the mean number (n) of galaxies passed per visible diameter. By inserting empirically plausible values for these parameters into Eq. (4) (namely: Rs = 1 light year, d = 1010 light years, N = 1012, n = N1/3 = 104 encounters per visible cosmic diameter d, v/c = 3 · 103, and 2D = d/n = 1010/104 = 106 light years between encounters), one obtains an appallingly small value for znet. Specifically: znet = 4 · 104 · 105 · 2 · 106 106. This would be a rather ridiculous explanation of the Hubble law if instead of a value of the order of unity, a value 6 orders of magnitude too small were offered! However, the statistics was not yet done correctly. Instead of one’s inserting mean values only, the underlying distributions need to be taken into account. In this case, several ‘‘gain factors’’ come into play. A first gain factor is obtained if we revert from Eq. (4) to Eq. (3) in which R is not fixed at D/2 but rather is allowed to vary about this mean value in equidistribution between zero and D. Then the 1% innermost passages (at R = D/100) jointly generate a 502 times larger effect, since R is present in squared form in the denominator while simultaneously n only decreases by a factor of 100. Hence an effect larger at least by a net factor of 502/100 = 25 is obtained compared to the mean value taken alone. An analogous smaller second gain factor can be expected if v is allowed to vary about its own mean. So we have by now reached an order of magnitude of znet 104. A third favorable enhancement mechanism has not only quantitative but also qualitative implications (for it will turn out that the linear ‘‘Hubble-like’’ law of Eq. (3) is linear only if the randomly moving gravitation centers are fully equidistributed so that they do not ‘‘lump’’). First of all though, if we allow the galaxies to form moving clusters and superclusters as realistic, a gain factor is obtained once again. Specifically, if the mass per gravitation center Rs is increased a thousandfold in the transition from the level of galaxies to that of galaxy clusters, as realistic, the number of encounters per line of sight (n) decreases only tenfold (third root of 1000) and the mean funnel distance (2D) increases only twofold (third root of 10). Hence a net gain factor of 1000/(10 · 2) = 50 results. A similar gain factor applies once more in the transition to the next hierarchical level (from clusters to superclusters). Taken together, the individual gain factors suffice to catapult Eq. (3) into a realistic domain.
5. A qualitative prediction and a cautioning remark The gain factors are not the whole story, however, as already mentioned. The funnel of a cluster is not only by a factor of 1000 more massive, but also gets ‘‘blunted’’ more effectively at its downward tip, since the center of a cluster is even ‘‘more empty’’ relatively speaking than that of a galaxy. Owing to this progressive effective flattening of the bottom of the lily as one climbs up the hierarchy, the redshift at some point ceases to profit from the transition toward the next higher level. Eventually even a decrease in the overall redshift factor is bound to occur for very large distances. As a consequence, not only is a linear Hubble-like law predictable [3–6] but so is an eventual downward bent in the slope of the Hubble-like line. As it is well known, the empirical Hubble line does exhibit a surprise decrease of its own slope beyond redshifts of about unity – a fact which gave rise to the reluctant introduction of a ‘‘dark energy term’’ into the canonical big-bang equation [2]. Now, it would be tempting to instead predict that both ‘‘space expansion’’ and ‘‘dark energy’’ can be explained-away independently by the same mechanism. Since this prediction sounds almost too good to be true, a point of cautioning is on line. The ultimate basis of the proposal is Boltzmann’s ‘‘hypothesis of molecular chaos’’ (cf. [9]). Price [9] prefers to instead speak of the ‘‘Principle of Independence of Incoming Influences’’ (PI3). This principle is recognizably implicit in the above assumption of ‘‘uncorrelatedness’’ between the intruding particle‘s motion and the random motions of the many slow attraction centers that it is diving into. What is unfamiliar is only the fact that the time-asymmetric implication for once lives not in phase space (as entropy and its cousins do) but rather in ordinary space. This conceptually foreign fact (the parallelism to dynamical friction) could be the reason why the proposal has so far escaped criticism.
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6. Comparison with observation Since theory has reached a critical stage a side glance at empirical reality could be helpful to see whether the latter does not automatically exclude any isolated new interpretation of the Hubble law. For if this turned out to be the case, an independent empirical falsification of the present approach would be on hand. All of the other major ‘‘explananda’’ of cosmology beside the Hubble redshift itself are currently ‘‘explained-along’’ by the big-bang postulate in one way or another, as is well known. Hence in the hypothesized absence of this postulate, they suddenly need an individual explanation of their own each. A brief evaluation of the two most burning questions is on line. The hardest case is clearly the outside the big bang and its steady-creation analog mysterious, origin of the lightest elements in the cosmos – those out of which all others are ‘‘bred’’ in stars by the familiar nucleosynthetic mechanisms. The first problem therefore is: Where does the cosmic hydrogen come from if not from the primordial fire ball or steady creation? A form of nuclear fission (with generation of electrons and protons), in the jet formation by the accretion disk around a black hole, is about the only candidate that spontaneously comes to mind. It was proposed [10] as an explanation of the huge clouds of hydrogen that are injected into the surrounding universe by the two symmetric ‘‘jets’’ of an active galaxy’s nucleus that stretch out so glaringly over millions of light years. Analogously, the (uptil now unexplained) copious hydrogen ejections which on a smaller scale extend many thousands of light years away from the plane of an ordinary galaxy [11] would be a product of the many micro-quasars present in the galaxy and hence be based on the same mechanism. However, in spite of its obvious theoretical importance, Blandford’s mechanism is clearly insufficient for the present purpose since only a fraction of the mass of a rotating black hole and its accreted matter can be recirculated in principle. An independent second mechanism evaporating dead mass is therefore needed in the present context. What immediately comes to mind is Hawking evaporation [12]. The latter is known to be very slow for large and mediumsized black hole masses [12]. Moreover, it was recently proposed that every locally stationary particle deep down in a gravitational well and close to the horizon suffers a mass decrease proportional to the local gravitational redshift factor [13]. If this were correct, a further mechanism beside Hawking evaporation would be needed in the present context. An unfamiliar phenomenon, Newton–Einstein particle-acceleration [6,14], comes to mind at this point. Hereby, a ‘‘Newtonian void’’ is focused on for once. A hollow sphere of constant wall thickness is well known to be internally free of any gravitation, as Newton first demonstrated with the aid of an ingenious geometric argument. Einstein later recognized that special relativity is bound to modify Newton’s famous result [15]. For some reason though, the simplest case – constant relative speed of an internal particle – remained unaddressed (only a ‘‘sudden jolt’’ applied to the particle was focused on [15]). In the absence of a cue to Einstein’s self-restraint, it cannot be ruled out for the time being that any initially fast-moving particle inside a Newtonian void suffers a forward acceleration. For special relativity changes the energy- (and hence mass-) distribution across the surrounding sphere for the fast-moving internal particle, so that Newton’s condition of ‘‘equal wall thickness’’ is no longer fulfilled for such a particle – so that an acceleration proportional to the relativistic longitudinal blueshift would apply [14,6]. The energy going into the acceleration would then have to stem from the surrounding inertial mass (in accordance with Einstein’s mass-energy law) which hence would get ‘‘drained’’ in a Hawking-analogous fashion. This prediction would be consistent with general relativity since its two ingredients are, (i) general relativity reproducing Newton’s zero gravity result and (ii) special relativity. Unfortunately, this simple conclusion is surprisingly hard to formalize though. If it can be confirmed, a second re-circulation mechanism ‘‘drying out’’ black holes (and other mass accretions) in the walls of cosmic voids would exist [6]. Since this is speculation, the ‘‘re-circulation problem’’ remains largely unsolved. Hence a cosmology without ‘‘primordial hydrogen creation’’ appears almost inconceivable for the time being. This fact throws an independent cautioning light on the above approach. The second maximally hard problem in any cosmology without space expansion is, of course, the origin of the famous microwave background radiation (CBR). The latter can no longer be explained as a cosmic background-bound relic of the big bang in the absence of the latter. The oldest explanation – and indeed prediction – of the phenomenon, as ‘‘mean bolometric cosmic temperature’’ [16], is surprisingly little known. The observed radiation can in the absence of space expansion no longer come from very far-away (as the accepted name ‘‘cosmic background radiation’’ would suggest). Rather, owing to the persistent distance-proportional light-ray expansion described, the incoming radiation is then bound to become a ‘‘cluster-background radiation’’ (CBR). In other words, a ‘‘local’’ origin of the radiation needs to be sought. The observed excessive thermal homogeneity of the radiation then would require a ‘‘local equilibration mechanism’’ too. A type of classical cold dark matter that would be up to the task in our own supercluster is not
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currently known. Only the fact that large quantities of some type of dark matter of baryonic type do exist in our own supercluster (as well as most others) is well-established for decades.
7. Two empirical predictions Fortunately, there exists an empirical question which ‘‘overrides’’ the theoretical impasse just mentioned: Do there exist any pattern changes in the observed microwave background radiation? Note that no matter whether it is streaming plasmas that are assumed present in the local supercluster [17] or moving dust clouds [18] or anything remotely similar to these as required in the present context, in each case changes of pattern on a scale of years or decades will be unavoidable. Such changes in the microwave background still remain to be ruled out empirically for two reasons. First, barely enough time has accrued in three years of accurate observation. Second, no pressing need for excluding slow pattern changes was felt up till now in the context of the ruling space-expansion paradigm. This is somewhat surprising since only the demonstrated absence of such changes can falsify the competing Guillaume–Assis theory. Conversely will any change of pattern in the CBR automatically falsify the cosmological origin of the radiation. Falsification of that negative prediction is however crucial for the survival of the ruling paradigm itself. A second empirical question of overriding importance involves high-tech space-bound instruments once more. In a recent talk [19], nobelist Riccardo Giacconi drew attention to the unfamiliar fact that X-ray astronomy is already a thousand times more sensitive than optical astronomy – despite the latter’s 1000 times longer history. More specifically, X-ray point sources 1000 times dimmer than any optical source can in principle be detected to date even if but a single photon were to arrive per hour [19]. Unexpectedly, such point sources do exist in reality in large quantity in apparent equidistribution across the sky [19]. The familiar population of equidistributed maximally distant optical quasars (that is, very far-away active galactic nuclei that are still visible because their beacons happen to point in the right direction) would thereby acquire a natural extension. However, if ultra-distant X-ray quasars are the source of the Giacconi phenomenon (and almost nothing else comes to mind), their redshifts ought to go up to about 30 (square-root of 1000). Any such measured record redshift would falsify the space-expansion paradigm directly. The pertinent and by their very nature extremely time-consuming redshift measurements are in progress since 2003 [19]. The whole world is waiting for the result.
8. Discussion The paradigm of ‘‘dynamic weak-lensing’’ – moving attracting masses that bend light the Einstein way – was taken up and brought in tentative contact with other elements of modern cosmology. The direct precursor of the same paradigm – ‘‘static weak-lensing’’ – enjoyed a number of successes of its own, all of which are very well known. For example, it can explain the glaring discrepancy that exists between the lumpiness (high lacunarity) in the distribution of low and medium-distance quasars on the one hand, and the equidistribution (near-zero lacunarity) of the large number of quasars found at very large distances on the otherhand. The equidistribution could be a weak-lensing induced optical artifact. The well-known observed increase of the fractal dimensionality in the distribution of galaxies at very large distances could then be a natural corollary. Almost a quarter century ago, Benoit Mandelbrot demonstrated with the aid of an explicit example that a ‘‘reduced lacunarity’’ (his own term) in the absence of any accompanying change of fractal dimensionality is mathematically possible [20]. The observed sharp increase in cosmic fractal dimensionality with distance (from 1.2 to 3) at very large distances is still awaiting re-calculation from the raw data in light of this possibility. The decades-long delay strikes one as a surprise since falsification of Mandelbrot’s prediction is a precondition for the survival of the space-expansion paradigm itself. Static weak-lensing thus is of great theoretical importance, not only with respect to a more or less Euclidean cosmos as just discussed, but also with respect to alternative cosmic topologies like those recently proposed by Luminet [21] and Iovane [22]. Now, at last the non-static (‘‘dynamic’’) version of the same ‘‘weak-lensing paradigm’’ is called upon to contribute its own share to cosmology. The details presented above show that the implied postulate-free explanation of the Hubble phenomenon is still a conjecture. Secondly, it turned out that no convincing answer can be given as yet for the origin of the cosmic hydrogen in the absence of space-expansion, and only a preliminary one exists at best for the origin of the cosmic microwave background. Thirdly, a conceptual surprise was in store along the way which may be worth emphasizing at last. Hawking’s 30-year-old evaporation proposal taken at face value possesses a previously unappreciated aspect: it amounts to a partial return to Anaxagoras’ two-and-a-half millennia old perpetual recycling paradigm. The remarkable empirical success of the fractal paradigm with its transfinite spirit acquires a dynamical sibling in this fashion. The E-infinity paradigm of El Naschie [23] with its string connection toward the smallest and the largest [22] stands
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in the middle. The specter of a ‘‘created eternity’’ in the spirit of Augustine would return if it were true that lack of visible miracles is the greatest miracle of them all [24]. Presently, the least miraculous principle (PI3) almost seems to be the most miraculous. To conclude, Hubble’s unique discovery of cosmological redshift can possibly be explained as a statistical-mechanical phenomenon living in ordinary space (rather than phase space) as Haversion of dynamical friction. The conjecture is open for four years. This undecided state gives one the unusual opportunity to speculate constructively about the origin of hydrogen and the cosmic background radiation. A new synthesis is hovering on the horizon.
Acknowledgements We thank Martine Moore, Peter Weibel, Werner Ebeling, Siegfried Zielinski, Rene´ Marburger, Tassilo Ku¨pper, Andreas Scheider, Jakob Nill, Wolfgang Wettlaufer, Norbert Pailer, Michael Baecker, Jonathan Kemp, Michael Langer and Tim O’Riley for discussions. For J.O.R.
References [1] Rossler OE, Lasker GE, Andonian G, Movassagh R. A possibly consistent cosmology based on the dynamic weak-lensing paradigm. In: Advances in education, vol. VI, in press. [2] Kirshner RP. The extravagant universe: exploring stars, dark energy and the accelerating cosmos. Princeton: Princeton University Press; 2002. [3] Rossler OE, Fro¨hlich D, Kleiner N. A time-symmetric Hubble-like law: light-rays grazing randomly moving galaxies show distance-proportional redshift. Z Naturforsch 2003;58a:807–9. [4] Rossler OE. Cosmic shear’s temporal fluctuations generate a distance-proportional redshift in both time directions: minibangtheory. Chaos, Solitons & Fractals 2004;19:1335–8. [5] Rossler OE, Movassagh R. Bitemporal dynamic Sinai divergence: an energetic analog to Boltzmann’s entropy? Int J Nonlinear Sci Simul 2005;6:337–8. [6] Rossler OE. Static cosmology from chaos-borne Hubble law. Nonlinear Phenom Complex Syst 2006;9:53–60. [7] Wucknitz O, Sperhake U. Deflection of light and particles by moving gravitational lenses. Phys Rev D 2004;69:063001. 1–13, p. 8. [8] Birkinshaw M. Moving gravitational lenses. Lect Notes Phys 1988;330:59–64. [9] Price H. Time’s arrow and Archimedes’ point. Cambridge: Cambridge University Press; 1996. [10] Blandford RD, Begelman MC, Rees MJ. Cosmic jets. Sci Amer 1982;246(5):84–94. [11] Rand, Kulkarni, Hester. Astrophys J 1990;352:L1–4. [12] Hawking SW. Black-hole evaporations? Nature 1974;248:30–2. [13] Rossler OE, Kuypers H. The scale change of Einstein’s equivalence principle. Chaos, Solitons & Fractals 2005;25:897–9. [14] Rossler OE. A morphogenetic instability in gravitation. Physica D (invited paper submitted in honor of Yoshiki Kuramoto 2004). [15] Einstein A. Does there exist a gravitational effect analogous to electrodynamic induction? (in German). Vierteljahresschrift fu¨r Gerichtsmedizin o¨ffentliches Sanita¨tswesen 1912;14:37–40. Reprinted in: Collected papers of Albert Einstein, English Translation edition, vol. 4. Princeton: Princeton University Press; 1996. p. 126–9. [16] Guillaume CE. La tempe´rature de l’espace (The temperature of outer space). La Nature 1896;24(2):210–1. 234; Assis AKT, Neves CD. History of the 2.7 K temperature prior to Penzias and Wilson. Apeiron 1995;2:79–84. [17] Peratt T, Peter W, Alfve´n H. quoted in: Lerner EJ. The big bang never happened. Discover 1998;9:70–9. [18] Hoyle F. Home is where the wind blows. Sausalito: University Science Books; 1994. [19] Giacconi R. Kepler Lecture held at the University of Tu¨bingen; July 2003. [20] Mandelbrot BB. The fractal geometry of nature. San Francisco: Freeman; 1982 [Table 334]. [21] Luminet JP. A cosmic hall of mirrors. Physics World September 2005:23–8. [22] Iovane G. Mohamed El Naschie’s e(1) Cantorian space–time and its consequences in cosmology. Chaos, Solitons & Fractals 2005;25:775–9. [23] El Naschie MS. From Hilbert space to the number of Higgs particles via the quantum two-slit experiment. In: Weibel P, Ord G, Rossler OE, Diebner HH, editors. Space time physics and fractality, Festschrift in honor of Mohamed El Naschie on the occasion of his 60th birthday. Vienna, New York: Springer-Verlag; Wien; 2005. p. 231–2. [24] Rossler R, Rossler OE, editors. Jonas’ World – the thinking of a child (in German). p. 10. ‘‘The dear Lord is ashamed of (working) visible miracles (age 2.3 years)’’.
Hubble expansion in static spacetime Otto E. Rossler a
a,*
, Dieter Fro¨hlich a, Ramis Movassagh b, Anthony Moore
c
Division of Theoretical Chemistry, University of Tubingen, Auf der Morgenstelle 8, 72076 Tu¨bingen, FRG b ETH and Collegium Helveticum, Switzerland c Academy for Media Arts, Cologne, FRG Accepted 14 June 2006
Communicated by Prof. L. Marek-Crnjac
Abstract A recently proposed mechanism for light-path expansion in a static spacetime is based on the moving-lenses paradigm. Since the latter is valid independently of whether space expands or not, a static universe can be used to better see the implications. The moving-lenses paradigm is related to the paradigm of dynamical friction. If this is correct, a Hubble-like law is implicit. It is described quantitatively. A bent in the Hubble-like line is predictably implied. The main underlying assumption is Price’s Principle (PI3). If the theory is sound, the greatest remaining problem in cosmology becomes the origin of hydrogen. Since Blandford’s jet production mechanism for quasars is too weak, a generalized Hawking radiation hidden in the walls of cosmic voids is invoked. A second prediction is empirical: slow pattern changes in the cosmic microwave background. A third is ultra-high redshifts for Giacconi quasars. Bruno’s eternal universe in the spirit of Augustine becomes a bit less outlandish. 2006 Published by Elsevier Ltd.
1. Introduction Recently, hopes were raised for a new cosmology based on the moving-lenses paradigm [1]. Rather than having the whole spacetime expand, it may suffice to expand the light-rays passing through a stationary spacetime. The frequencyindependent light-stretching effect would justify Hubble’s lifelong belief in a first-principles ‘‘tired-light’’ explanation of his distance-proportional redshift law. The well-known falsification of all tired-light effects, implicit in the supernova project’s finding that astrophysical events of a duration of weeks are still proportionally redshifted (elongated in time) [2], turns out to be inapplicable since the light-ray expansion caused by moving Einstein lenses does [3–6] extend to ultralow frequencies. In the following, the proposal is put into quantitative terms and a new implication (a bent in the Hubble line) is predicted. Finally, the approach is given the benefit of the doubt by placing it into a wider empirical and historical context.
*
Corresponding author. Tel.: +49 7071 66323.
0960-0779/$ - see front matter 2006 Published by Elsevier Ltd. doi:10.1016/j.chaos.2006.06.046
O.E. Rossler et al. / Chaos, Solitons and Fractals 33 (2007) 770–775
771
2. Dynamical friction of the second kind The idea belongs into the context of statistical mechanics which until now has lacked a role in cosmology. There exists a not very well known example in statistical mechanics called ‘‘dynamical friction’’ which resembles entropy because it exhibits dissipative behavior in both time directions. Nevertheless, it lives in ordinary space rather than phase space. Classical dynamical friction applies when a heavy-slow particle is injected into a dilute gas of light-fast particles endowed with long-range repulsion. Even though there is no direct contact (as in ordinary friction – hence the name), the intruding particle loses energy to the many particles. But light passing through the cosmos is not an ultra-heavy particle. Indeed the proposed new sub-variety of dynamical friction starts out not with a heavy-slow, but with a light-fast particle injected into a dilute gas, not of light-fast but of heavy-slow particles which are endowed, not with long-range repulsion but with long-range attraction. Despite these three differences, the injected particle can again be predicted to lose energy to the many uncorrelated particles [6]. The mechanism of this effect (a simulation is still outstanding) goes like this: When the little fast particle enters the next ‘‘region of dominating influence’’ (potential dip) of a moving heavy slow particle, this entry in general occurs in a more or less tangential fashion. The funnel-shaped dip then, with the same probability, either approaches or veers away from the path of the intruded fast particle. Hence the closest passing distance from the center, R, is not the same in the two cases: The approaching path comes closer, leading deeper down the throat of the lily-shaped funnel to thereby generate a stronger damping effect, than does the path leading out of the throat into a shallower terrain, generating a weaker accelerating effect there [3–6]. The same asymmetry was implicitly noticed by Wucknitz and Sperhake [7]. Let us be a bit more explicit. The fast particle may be a photon and the soup through which it passes a cloud of randomly moving gravitating masses that act as weak Einstein lenses. Birkinshaw’s law for moving gravitational lenses [8] then applies at each encounter: z ¼ 2ðv=cÞðRs =RÞ;
ð1Þ
with the approximate factor 2 included for specificity. Eq. (2) describes the frequency shift z caused in a light-ray by a transversely moving gravitational lens [8]: When the lens is approaching (v positive), z is positive (redshift), and vice versa if v is negative (blueshift). The effect thus is perfectly time-symmetric if the passing distance R is the same in the two cases. The two remaining constants are Rs, the Schwarzschild radius (mass) of the attraction center and c, the speed of light. The point is that the two distances of closest encounter R (call them R+ and R) are not the same in the situation of our present interest. They deviate (to first approximation) by an equal amount d from their common mean R. For, owing to the assumed lack of correlation between the passing photon and the moving gravitating mass, the ‘‘decision’’ of whether the newly entered funnel is approaching or receding from the fast particle’s path is in effect only made at the moment of definitive engagement. At that moment, when the region of dominating influence of one funnel is left and that of the next is entered, the actual and the opposite direction of motion of the freshly entered funnel have exactly the same probability of being applicable as far as the uncorrelated light particle is concerned. Hence the difference between R+ and R.
3. Quantitative formulation The dependency of z on R in Eq. (1) is nonlinear since R is in the denominator. Therefore, a positive net difference (redshift) applies between two equiprobable passages characterized by an equal numerical value of v except for sign. For the same ‘‘distance of engagement’’, D (half mean distance between equal funnels) leads to two unequal R’s (R+ and R) that deviate by d from their joint mean R as we saw. Hence after many encounters (or pairs of encounters, respectively), a cumulative ‘‘net redshift’’ applies in proportion to the number (1/2 Æ n) of equiprobable pairs of encounters: znet ¼ ð1=2Þ n2ðv=cÞRs ½1=ðR dÞ 1=ðR þ dÞ ¼ nðv=cÞRs 2d R2 d2 Þ 2nðv=cÞRs d=R2 :
ð2Þ
Putting in d 1/2(v/c)D where D is the effective funnel radius (half mean distance between equal funnels), as an estimate for the mean differential change d of the passing distance R after the funnel has been entered, one obtains znet ð1=2Þ 2nðv=cÞ2 Rs D=R2 :
ð3Þ
If the mean distance of passage, D/2, is inserted for R, this becomes znet 4nðv=cÞ2 Rs =D:
ð4Þ
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This equation means (since all terms on the right hand side are positive and n is proportional to the distance traveled) that a distance-proportional Hubble-like redshift law is implicit in the assumption made of uncorrelated initial conditions between an injected light particle and a soup of uncorrelated randomly moving gravitational masses.
4. Evaluation of the formula In order to obtain a quantitative estimate, it is first necessary to insert empirical parameters. The latter are the mean mass of a galaxy (Rs), the mean transverse velocity of galaxies (v), the mean distance between galaxies (2D), the visible diameter of the cosmos (d) divided by the third root of the visible number of galaxies (N), jointly yielding the mean number (n) of galaxies passed per visible diameter. By inserting empirically plausible values for these parameters into Eq. (4) (namely: Rs = 1 light year, d = 1010 light years, N = 1012, n = N1/3 = 104 encounters per visible cosmic diameter d, v/c = 3 · 103, and 2D = d/n = 1010/104 = 106 light years between encounters), one obtains an appallingly small value for znet. Specifically: znet = 4 · 104 · 105 · 2 · 106 106. This would be a rather ridiculous explanation of the Hubble law if instead of a value of the order of unity, a value 6 orders of magnitude too small were offered! However, the statistics was not yet done correctly. Instead of one’s inserting mean values only, the underlying distributions need to be taken into account. In this case, several ‘‘gain factors’’ come into play. A first gain factor is obtained if we revert from Eq. (4) to Eq. (3) in which R is not fixed at D/2 but rather is allowed to vary about this mean value in equidistribution between zero and D. Then the 1% innermost passages (at R = D/100) jointly generate a 502 times larger effect, since R is present in squared form in the denominator while simultaneously n only decreases by a factor of 100. Hence an effect larger at least by a net factor of 502/100 = 25 is obtained compared to the mean value taken alone. An analogous smaller second gain factor can be expected if v is allowed to vary about its own mean. So we have by now reached an order of magnitude of znet 104. A third favorable enhancement mechanism has not only quantitative but also qualitative implications (for it will turn out that the linear ‘‘Hubble-like’’ law of Eq. (3) is linear only if the randomly moving gravitation centers are fully equidistributed so that they do not ‘‘lump’’). First of all though, if we allow the galaxies to form moving clusters and superclusters as realistic, a gain factor is obtained once again. Specifically, if the mass per gravitation center Rs is increased a thousandfold in the transition from the level of galaxies to that of galaxy clusters, as realistic, the number of encounters per line of sight (n) decreases only tenfold (third root of 1000) and the mean funnel distance (2D) increases only twofold (third root of 10). Hence a net gain factor of 1000/(10 · 2) = 50 results. A similar gain factor applies once more in the transition to the next hierarchical level (from clusters to superclusters). Taken together, the individual gain factors suffice to catapult Eq. (3) into a realistic domain.
5. A qualitative prediction and a cautioning remark The gain factors are not the whole story, however, as already mentioned. The funnel of a cluster is not only by a factor of 1000 more massive, but also gets ‘‘blunted’’ more effectively at its downward tip, since the center of a cluster is even ‘‘more empty’’ relatively speaking than that of a galaxy. Owing to this progressive effective flattening of the bottom of the lily as one climbs up the hierarchy, the redshift at some point ceases to profit from the transition toward the next higher level. Eventually even a decrease in the overall redshift factor is bound to occur for very large distances. As a consequence, not only is a linear Hubble-like law predictable [3–6] but so is an eventual downward bent in the slope of the Hubble-like line. As it is well known, the empirical Hubble line does exhibit a surprise decrease of its own slope beyond redshifts of about unity – a fact which gave rise to the reluctant introduction of a ‘‘dark energy term’’ into the canonical big-bang equation [2]. Now, it would be tempting to instead predict that both ‘‘space expansion’’ and ‘‘dark energy’’ can be explained-away independently by the same mechanism. Since this prediction sounds almost too good to be true, a point of cautioning is on line. The ultimate basis of the proposal is Boltzmann’s ‘‘hypothesis of molecular chaos’’ (cf. [9]). Price [9] prefers to instead speak of the ‘‘Principle of Independence of Incoming Influences’’ (PI3). This principle is recognizably implicit in the above assumption of ‘‘uncorrelatedness’’ between the intruding particle‘s motion and the random motions of the many slow attraction centers that it is diving into. What is unfamiliar is only the fact that the time-asymmetric implication for once lives not in phase space (as entropy and its cousins do) but rather in ordinary space. This conceptually foreign fact (the parallelism to dynamical friction) could be the reason why the proposal has so far escaped criticism.
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6. Comparison with observation Since theory has reached a critical stage a side glance at empirical reality could be helpful to see whether the latter does not automatically exclude any isolated new interpretation of the Hubble law. For if this turned out to be the case, an independent empirical falsification of the present approach would be on hand. All of the other major ‘‘explananda’’ of cosmology beside the Hubble redshift itself are currently ‘‘explained-along’’ by the big-bang postulate in one way or another, as is well known. Hence in the hypothesized absence of this postulate, they suddenly need an individual explanation of their own each. A brief evaluation of the two most burning questions is on line. The hardest case is clearly the outside the big bang and its steady-creation analog mysterious, origin of the lightest elements in the cosmos – those out of which all others are ‘‘bred’’ in stars by the familiar nucleosynthetic mechanisms. The first problem therefore is: Where does the cosmic hydrogen come from if not from the primordial fire ball or steady creation? A form of nuclear fission (with generation of electrons and protons), in the jet formation by the accretion disk around a black hole, is about the only candidate that spontaneously comes to mind. It was proposed [10] as an explanation of the huge clouds of hydrogen that are injected into the surrounding universe by the two symmetric ‘‘jets’’ of an active galaxy’s nucleus that stretch out so glaringly over millions of light years. Analogously, the (uptil now unexplained) copious hydrogen ejections which on a smaller scale extend many thousands of light years away from the plane of an ordinary galaxy [11] would be a product of the many micro-quasars present in the galaxy and hence be based on the same mechanism. However, in spite of its obvious theoretical importance, Blandford’s mechanism is clearly insufficient for the present purpose since only a fraction of the mass of a rotating black hole and its accreted matter can be recirculated in principle. An independent second mechanism evaporating dead mass is therefore needed in the present context. What immediately comes to mind is Hawking evaporation [12]. The latter is known to be very slow for large and mediumsized black hole masses [12]. Moreover, it was recently proposed that every locally stationary particle deep down in a gravitational well and close to the horizon suffers a mass decrease proportional to the local gravitational redshift factor [13]. If this were correct, a further mechanism beside Hawking evaporation would be needed in the present context. An unfamiliar phenomenon, Newton–Einstein particle-acceleration [6,14], comes to mind at this point. Hereby, a ‘‘Newtonian void’’ is focused on for once. A hollow sphere of constant wall thickness is well known to be internally free of any gravitation, as Newton first demonstrated with the aid of an ingenious geometric argument. Einstein later recognized that special relativity is bound to modify Newton’s famous result [15]. For some reason though, the simplest case – constant relative speed of an internal particle – remained unaddressed (only a ‘‘sudden jolt’’ applied to the particle was focused on [15]). In the absence of a cue to Einstein’s self-restraint, it cannot be ruled out for the time being that any initially fast-moving particle inside a Newtonian void suffers a forward acceleration. For special relativity changes the energy- (and hence mass-) distribution across the surrounding sphere for the fast-moving internal particle, so that Newton’s condition of ‘‘equal wall thickness’’ is no longer fulfilled for such a particle – so that an acceleration proportional to the relativistic longitudinal blueshift would apply [14,6]. The energy going into the acceleration would then have to stem from the surrounding inertial mass (in accordance with Einstein’s mass-energy law) which hence would get ‘‘drained’’ in a Hawking-analogous fashion. This prediction would be consistent with general relativity since its two ingredients are, (i) general relativity reproducing Newton’s zero gravity result and (ii) special relativity. Unfortunately, this simple conclusion is surprisingly hard to formalize though. If it can be confirmed, a second re-circulation mechanism ‘‘drying out’’ black holes (and other mass accretions) in the walls of cosmic voids would exist [6]. Since this is speculation, the ‘‘re-circulation problem’’ remains largely unsolved. Hence a cosmology without ‘‘primordial hydrogen creation’’ appears almost inconceivable for the time being. This fact throws an independent cautioning light on the above approach. The second maximally hard problem in any cosmology without space expansion is, of course, the origin of the famous microwave background radiation (CBR). The latter can no longer be explained as a cosmic background-bound relic of the big bang in the absence of the latter. The oldest explanation – and indeed prediction – of the phenomenon, as ‘‘mean bolometric cosmic temperature’’ [16], is surprisingly little known. The observed radiation can in the absence of space expansion no longer come from very far-away (as the accepted name ‘‘cosmic background radiation’’ would suggest). Rather, owing to the persistent distance-proportional light-ray expansion described, the incoming radiation is then bound to become a ‘‘cluster-background radiation’’ (CBR). In other words, a ‘‘local’’ origin of the radiation needs to be sought. The observed excessive thermal homogeneity of the radiation then would require a ‘‘local equilibration mechanism’’ too. A type of classical cold dark matter that would be up to the task in our own supercluster is not
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currently known. Only the fact that large quantities of some type of dark matter of baryonic type do exist in our own supercluster (as well as most others) is well-established for decades.
7. Two empirical predictions Fortunately, there exists an empirical question which ‘‘overrides’’ the theoretical impasse just mentioned: Do there exist any pattern changes in the observed microwave background radiation? Note that no matter whether it is streaming plasmas that are assumed present in the local supercluster [17] or moving dust clouds [18] or anything remotely similar to these as required in the present context, in each case changes of pattern on a scale of years or decades will be unavoidable. Such changes in the microwave background still remain to be ruled out empirically for two reasons. First, barely enough time has accrued in three years of accurate observation. Second, no pressing need for excluding slow pattern changes was felt up till now in the context of the ruling space-expansion paradigm. This is somewhat surprising since only the demonstrated absence of such changes can falsify the competing Guillaume–Assis theory. Conversely will any change of pattern in the CBR automatically falsify the cosmological origin of the radiation. Falsification of that negative prediction is however crucial for the survival of the ruling paradigm itself. A second empirical question of overriding importance involves high-tech space-bound instruments once more. In a recent talk [19], nobelist Riccardo Giacconi drew attention to the unfamiliar fact that X-ray astronomy is already a thousand times more sensitive than optical astronomy – despite the latter’s 1000 times longer history. More specifically, X-ray point sources 1000 times dimmer than any optical source can in principle be detected to date even if but a single photon were to arrive per hour [19]. Unexpectedly, such point sources do exist in reality in large quantity in apparent equidistribution across the sky [19]. The familiar population of equidistributed maximally distant optical quasars (that is, very far-away active galactic nuclei that are still visible because their beacons happen to point in the right direction) would thereby acquire a natural extension. However, if ultra-distant X-ray quasars are the source of the Giacconi phenomenon (and almost nothing else comes to mind), their redshifts ought to go up to about 30 (square-root of 1000). Any such measured record redshift would falsify the space-expansion paradigm directly. The pertinent and by their very nature extremely time-consuming redshift measurements are in progress since 2003 [19]. The whole world is waiting for the result.
8. Discussion The paradigm of ‘‘dynamic weak-lensing’’ – moving attracting masses that bend light the Einstein way – was taken up and brought in tentative contact with other elements of modern cosmology. The direct precursor of the same paradigm – ‘‘static weak-lensing’’ – enjoyed a number of successes of its own, all of which are very well known. For example, it can explain the glaring discrepancy that exists between the lumpiness (high lacunarity) in the distribution of low and medium-distance quasars on the one hand, and the equidistribution (near-zero lacunarity) of the large number of quasars found at very large distances on the otherhand. The equidistribution could be a weak-lensing induced optical artifact. The well-known observed increase of the fractal dimensionality in the distribution of galaxies at very large distances could then be a natural corollary. Almost a quarter century ago, Benoit Mandelbrot demonstrated with the aid of an explicit example that a ‘‘reduced lacunarity’’ (his own term) in the absence of any accompanying change of fractal dimensionality is mathematically possible [20]. The observed sharp increase in cosmic fractal dimensionality with distance (from 1.2 to 3) at very large distances is still awaiting re-calculation from the raw data in light of this possibility. The decades-long delay strikes one as a surprise since falsification of Mandelbrot’s prediction is a precondition for the survival of the space-expansion paradigm itself. Static weak-lensing thus is of great theoretical importance, not only with respect to a more or less Euclidean cosmos as just discussed, but also with respect to alternative cosmic topologies like those recently proposed by Luminet [21] and Iovane [22]. Now, at last the non-static (‘‘dynamic’’) version of the same ‘‘weak-lensing paradigm’’ is called upon to contribute its own share to cosmology. The details presented above show that the implied postulate-free explanation of the Hubble phenomenon is still a conjecture. Secondly, it turned out that no convincing answer can be given as yet for the origin of the cosmic hydrogen in the absence of space-expansion, and only a preliminary one exists at best for the origin of the cosmic microwave background. Thirdly, a conceptual surprise was in store along the way which may be worth emphasizing at last. Hawking’s 30-year-old evaporation proposal taken at face value possesses a previously unappreciated aspect: it amounts to a partial return to Anaxagoras’ two-and-a-half millennia old perpetual recycling paradigm. The remarkable empirical success of the fractal paradigm with its transfinite spirit acquires a dynamical sibling in this fashion. The E-infinity paradigm of El Naschie [23] with its string connection toward the smallest and the largest [22] stands
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in the middle. The specter of a ‘‘created eternity’’ in the spirit of Augustine would return if it were true that lack of visible miracles is the greatest miracle of them all [24]. Presently, the least miraculous principle (PI3) almost seems to be the most miraculous. To conclude, Hubble’s unique discovery of cosmological redshift can possibly be explained as a statistical-mechanical phenomenon living in ordinary space (rather than phase space) as Haversion of dynamical friction. The conjecture is open for four years. This undecided state gives one the unusual opportunity to speculate constructively about the origin of hydrogen and the cosmic background radiation. A new synthesis is hovering on the horizon.
Acknowledgements We thank Martine Moore, Peter Weibel, Werner Ebeling, Siegfried Zielinski, Rene´ Marburger, Tassilo Ku¨pper, Andreas Scheider, Jakob Nill, Wolfgang Wettlaufer, Norbert Pailer, Michael Baecker, Jonathan Kemp, Michael Langer and Tim O’Riley for discussions. For J.O.R.
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