The Large Numbers Hypothesis: Outline of a self-similar quantum-cosmological Model H. GENREITH ∗ Institut f¨ur Geophysik und Meteorologie der Universit¨at zu K¨oln Abstract leads to a scaled quantum of action 3 In 1919 A. Einstein suspected first that gravita- H =Λ h (2) · tional fields could play an essential role in the in order to handle the Universe like an elementary structure of elementary particles. In 1937, P.A.M. particle. Hence the Universe should have an angu- Dirac found a miraculous link between the prop- lar momentum of about H/2π and it can be sus- erties of the visible Universe and elementary par- pected to be a huge rotating black-hole close to its ticles. Both conjectures stayed alive through the Kerr limit, or close to G¨odel’s spin [1, 2, 35, 10], following decades but still no final theory could be which is the value for the rotating cosmological so- derived to this issues. The herein suggested frac- lution to Einstein’s equations of General Relativity. tal model of the Universe gives a consistent expla- The recent discussion regarding the origin of and nation to Dirac’s Large Numbers Hypothesis and destiny of the Universe raises, for example, the combines the conjectures of Einstein and Dirac. question of where the Universe comes from. The main explanation for this is that time and space 1 Introduction were created with the big-bang and therefore the question of where the Universe is embedded or what 1.1 The Large Numbers Hypothesis happened before the creation, is not admissible. This takes the assumption that there was no rea- son for the creation and that the Universe was cre- arXiv:gr-qc/9909009v1 2 Sep 1999 Dirac[4, 5] first mentioned in 1937 a presumable connection between the properties of elementary ated from ’nothing’ by a type of quantum fluctua- particles and the Universe by multiplying elemen- tion and should have an overall-energy of zero. Al- tary properties with certain large numbers 1. Since though this explanation sounds plausible, it does then this conjecture is called the Large Numbers not solve the real problem that if the Universe was Hypothesis. It states, that the cosmological quan- created from ’nothing’, why should there not be tities M,R,T can be related to the particle quan- countless other Universes, and what relationship tities m,r,t through the scale relation[1] could there be between them? Numerous scientists have tried to give an explanation to this question (e.g. A. Linde [17, 18]). Whatever the correct an- T R M 38 41 = = =Λ 10 − (1) t r m ∼ swer is, the explanation must be a kind of infinite r regress, as otherwise the embedding problem would which may easily be verified for example for the recur. properties of the Universe and the proton. Mass, An additional question that is also unsatisfactory time and length is enough to construct a complete answered is, what matter, such as elementary parti- system of units. A simple dimensional analysis cles, really is. To answer this question, one tries to ∗ find the most elementary particles by experimenta- Email:[email protected] 1Nota bene: The Large Number Λ has nothing in com- tion and theory, all matter should be composed of mon with the cosmological constant Λ. these smallest ’grains’ of matter. But this explana- 1 tion always raises the question that these ’grains’ the Oppenheimer/Snyder-solution may be taken could be composed of something that could be de- for a speculation of a Friedmann-Universe inside scribed by an even more elementary description. a black hole, shrinking down to a singularity and This is where the so-called string-theory comes from there expanding back to its maximum expan- in; this gives a topological explanation, with the sion again. strings as the origin of matter. But what are these If someone does not like such an interpretation, strings composed of, they should be pure topology, there is the problem to explain how the expand- as space-time curvatures are [6, 12]. ing Universe let behind its own event-horizon. As the standard Big-Bang Universe starts to expand at 1.2 Can the Universe be described vanishing spatial dimensions, this leads to the fol- as a Black Hole ? lowing seemingly paradoxical context: Either the Schwarzschild-radius is greater-equal to the world The critical mass density of the Universe is the radius 2 of the Universe, as large estimations of amount of mass which is needed to bring the uni- the Universe mass assume, then the Universe may versal expansion to a halt in the future and to a be called a black hole 3. Or, as small estimates collapse in a final big crunch. This mass is, depend- of the mass of the Universe assume, the Schwarz- ing on the world model, about 1053 kg. The visible schild-radius is about some 10 percent of the today mass of the Universe is sufficient for only about world radius of the Universe. But then the Universe some percent of the critical mass density. On the should have expanded beyond 4 its Schwarzschild- other hand the mass density of the baryons should radius in former times when it crossed a radius of be 10 to 12 percent of the critical mass density [3] as at least 1.5 billion light-years, which is an event can be derived from the theory of nucleosynthesis that is supposed to be not in casual harmony with and the measured photon density. Therefore the so- general relativity. By this one could guess that our called dark matter should be about 10 times larger Universe should be indeed a black hole. than the visible bright mass. This matter shows The theory of inflation [18] tries to avoid such itself in the extinction of light on its way through a paradox by a kind of extremely fast inflationary the Universe as well as by its gravitational force expansion. But finally the standard inflationary on galaxies and galaxy clusters. Including this, the model also demands a critical mass density for the mass density of baryonic matter seems to be not Universe. Recent models prefer cosmologies with large enough to close the Universe. Besides the a cosmological constant Λ = 0 and small densities, classical baryonic matter therefore also exotic mat- which are also in agreement6 with the interpretation ter is taken into consideration as far as the overall of the luminosities of far away Type-Ia-Supernovae mass of the Universe is concerned. which seem to be about 0.3m darker than expected. 2GM The Schwarzschild-radius Rss = c2 of a given These more complex Λ = 0-models open a large mass M is derived from the Einstein field equations 6 2 considering a point-like mass. The Schwarzschild Although the dimension of the world radius and the Schwarzschild-radius may be equal, there is a considerable solution is a static, homogeneous and isotropic so- difference: The world radius is the ’visible’ dimension which lution for the region outside the Schwarzschild ra- is defined by an infinite red shift. On the other hand, the dius. However, the inside solution may look other. Schwarzschild-radius can never be seen. Even when an ob- A very interesting solution is the solution of Op- server is placed close to this border (as seen from ’outside’): In this case the line of sight would be curved when seen from penheimer and Snyder [31, 26] which shows the as- outside but an inside observer would observe it as free of any tonishing result that the inside solution must be forces and straight. Because of this any number of world ra- a Friedmann-Universe [9]. This result is an out- dius’ can be placed into the area of the Schwarzschild- radius come from the fitting of the outside to the in- even if both have the same dimension. 3A black hole, if watched from inside, may be called a side solution of a collapsing star: after the burn- white hole because it is guessed that from its inside singu- ing out of the star every pressure vanishes when larity everything can come into existence [13, 9]. A black the star shrinks to 9/8-th of its Schwarzschild- hole actually needs an outer space which it is defined on, radius. When the collapsing star reaches the event- but the existence or non-existence of an outer space is first of all a question of belief. horizon one has to set p = 0 and one gets a 4A ’pushing in front’ of the event- horizon is easily pos- Friedmann-Universe at its maximum expansion. So sible due to the increasing scale-factor R(t) of the Universe. 2 variety of possible large scale structures of the Uni- is a solution of RR˙ 2 = Const., which gives verse. It shall be emphasized, that especial the 2/3 non-critical, so-called open (hyperbolic) Universes, te R(te)= R0 ( ) . (3) must not have an infinite volume in any case[27]. · t0 The correspondance between an open and an in- Therein t is the time of emission of a signal and finite Universe is only true for the very special e t = 0 is the time of the big bang and t is the case of a zero cosmological constant in a simple- e 0 present time. With the Hubble equation dR = connected topology of the Universe [21]. Also there dt H(t)R(t) follow the relations for the age and size exist competing theories to explain the luminosi- of the Universe: ties of the Type-Ia-Supernovas, such as taking into account the nonhomogenity[20] of the Universe on 2 2c tU = and RU = medium scales and others.