Journal of Modern Physics, 2013, 4, 1647-1654 Published Online December 2013 (http://www.scirp.org/journal/jmp) http://dx.doi.org/10.4236/jmp.2013.412205
The Large Numbers in a Quantized Universe
Yan Ryazantsev* 52, 43 Dimitrova Street, Zagoryansky, Shchelkovo District, Moscow, Russia Email: [email protected]
Received October 3, 2013; revised November 5, 2013; accepted November 29, 2013
Copyright © 2013 Yan Ryazantsev. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT The article relates to a decades-old problem of the mysterious coincidence between various large numbers of the mag- nitude ranging from 1040 to 10120 which sometimes appears in cosmology and quantum physics. Using well known clas- sical relations as well as the ideal Schwarzschild solution the exact relations of various large numbers, the fine structure constant and π were found. The new largest number law is claimed. The hypothetical approximations of the Hubble parameter—68.7457(82) km/s/Mpc, Hubble radius—14.2330(17) Gly, and some others were proposed. The exact formulae supporting P. Dirac’s large number hypothesis and H. Weyl’s proposition were found. It is shown that all major physical constants with the length dimension (from the Compton wave length of universe through the Planck and atomic scale up to the Hubble sphere radius) could be derived from each other, and the table of the specific conver- sion rules has been developed. The model shows that the Eddington-Weinberg relation can be transformed to precise identity. It is shown that both Bekenstein universal entropy bound and Bekenstein-Hawking Black Hole entropy bound are proportional to the largest number doubled.
Keywords: Large Numbers Hypothesis; Hubble Sphere; Eddington Number; Cosmological Constant
1. Introduction 1 q2 fe 2 (2) The problem of Large Numbers dates back decades. The 4π0 r first problem statement and attempts at resolving it can 2 me be found in the studies by H. Weyl [1-3] and Sir A. fGg 2 (3) Eddington [4,5], who drew attention to the incredibly r large numbers of the dimensionless physical constants The second is the ratio of the universe radius to the found in the cosmology, electrodynamics and quantum classical electron radius: mechanics. The magnitude of these constants is so big Rc 20 40 80 120 40 10 , 10 , 10 , 10 as compared with the con- NDR 4.63 10 (4) rHr ventional mathematical constants, like π 3.14 and ee
Euler’s constant e2.71 , that boggles imagination as where H is Hubble’s constant, re —classical electron P. Davies [6] noted. radius, R —Hubble sphere radius or radius of event
The first of the two most popular dimensionless large horizon, rg —gravitational electron radius, fe —electros- numbers is the classical ratio between the gravitational tatic force between two electrons at a distance r, 0 — force and the electromagnetic force in any given distance, permittivity of vacuum, q —electron charge, f g — called by H. Weyl [2] “even more mysterious than the gravitational force between two electrons at a distance r, fine structure constant ”: G —Newton gravitational constant, me —electron mass. r r The proximity of magnitude of e and g values free 42 R r NDF 4.16 10 (1) e frgg led H. Weyl to the idea that the incredible weakness of were: gravitational interaction may be due to the ratio of the electron and the universe radiuses or to the total quantity *Yan Ryazantsev, Independent researcher. of particles in the universe—the Eddington number [5].
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We refer everybody interested in the history of stu- On the other hand, we know the formula for the dying the problem of large numbers to the reviews by S. electron gravitational radius that includes more or less Ray, U. Mukhopadhyay, P. P. Ghosh [7] and K. A. precisely measured physical values: Tomilin [8]. Gm The hypothesis by P.A.M. Dirac is one of the best e rg 2 (9) known hypotheses put forward to explain the problem of c large numbers [9-11]. He supposed that the reason for Therefore, the formula for the classical electron radius appearance of great magnitudes of dimensionless values that uses G can be easily obtained from (8) and (9): is their reliance on the equally large value, so-called NGm “cosmological time”. This led P. Dirac to the hypothesis DFe re 2 (10) of dependence of the gravitational constant and the c universe mass on time: We will not discuss now if the value of the classical 1 electron radius has any real physical significance. It is G (5) T enough that it is one of the energy status representations of an electron, a particle with the minimum self-energy 2 M T (6) among all charged particles. where M is the mass of universe, T is cosmological For transition to energy values, we will use a large time. mass number introduced by H.Weyl: The idea of time-varying constants was developed, in M E particular, by E. A. Milne [12]. To establish the ratios N eU (11) M mE and laws between modern values of the fundamental eU e 2 2 constants, we suggest studying values of large numbers where EMcU —the universe self-energy, Emcee at the current point of time, without taking into account h —the electron self-energy, —Compton wave their time derivative. P. Dirac’s assumption that New- U Mc ton’s gravitational constant and the mass of universe are not true constants but change over time gave rise to an h length of the universe, e —Compton wave length array of scientific discussions, experimental and theore- mc tical studies devoted to verification of the fundamental of the electron. constants in subsequent decades. No reliable proofs of By dividing (8) by (10), one can get the following variability of the physical constants were found yet. ratio: In this article, we will use the Hubble time, the RM2 parameter inverse to the Hubble constant, as approxi- (12) rNm mation of the cosmological time: eDFe 1 Thus, we obtain the precise correlation among the T (7) three large numbers out of (4), (11) and (12): H 2NNN (13) We referred to the simplest mathematics in narrating M DF DR the article; however, the results we obtained provide Besides the above large numbers, we will use a large quite good approximation to the most precise and energy number NW representing the ratio of the generally accepted values of physical parameters, such as universe’s self-energy EU and “the minimum vacuum me and re , taking into account their uncertainties. In energy” E as proposed by J. Casado [15]: particular, CODATA 2010 [13] as well as the measure- W ments of Mission Planck 2013 [14] were used. EU NW (14) EW 2. Revealing the Large Numbers Ratios hc where EH —quantum of energy with wave To discover the correlation between large numbers in our W 2 R epoch, let’s begin with the well known vacuum solution length 2πR , h —Planck’s constant; to the Einstein’s equations for spherically symmetric and We will also need another large number equal to the static universe. According to Schwarzschild’s solution, cube of N value: the universe’s radius R that coincides with the Black DR 3 Hole radius with the mass of M is determined by a R 3 NNUDR3 (15) well-known formula (Schwarzschild radius): re 2GM Introduction of a new designation for a large number R 2 (8) c NU is quite justified, because, as we will see below, this
Open Access JMP Y. RYAZANTSEV 1649 number has a particular and substantial value. One of the Table 2. Calculated Large numbers values. simplest classical interpretations of this number: “the Parameter Value large number NU represents the total number of, say, N 4.16589 50 1042 ‘elementary clusters’ in the universe—the ratio between DF 40 volume VU of ball-like universe and the region Ve N DR 4.842 55 10 folded by sphere of radius r ”. Taking into account that 122 e N 1.135 39 10 this number has the greatest magnitude as compared with U 122 N 1.009 11 1081 other large numbers 10 , we suggest calling it “the M 121 Largest number”. NW 3.564 81 10
3. The Largest Number Law and Revealing the left-hand side, and the quantum ones in the right- of Dirac’s Proportionalities hand one the ratio (17) can be transformed into the Now let us use well known physical parameters with following relation: their uncertainties (see Table 1). The parameters listed in M 1 m n e (19) Table 1 enable us to calculate large numbers values (see x Rr22 e Table 2). To obtain NDF , we use (10): universe electron 1 rc2 The question is—what kind of a phisical law the last N e (16) DF equation represents? Gme In order to find out the answer we would like to We should pay an attention to the very close values of propose quite a simple classical model. We can apply the 121 NU and NW of the magnitude 10 . The ratio classical approach because we are dealing with constant between this two large numbers is macroscopic physical parameters and those ratios. Let us N consider a really large number of non-interacting quanta. n U 3.185(36) (17) x N All quanta are moving in all directions with a speed of W light, i.e. there are photons. Obviously, the set must be This is quite remarkable, taking into account the confined inside the Black Hole with a radius R If the extremely large magnitudes of the numbers involved. total energy of the whole quanta set is equal to EU . We Therefore, we can assume that this ratio is not just a have to conclude that absolutely every photon must be coincidence but some specific physical law. In order to reflected by the sphere’s bound in certain time. In other reveal the meaning of the ratio (17) one can rewrite it as words the inner side of the Hubble sphere plays a role of follows: an ideal diffusely reflecting surface. In a period of time TRc we will see that ultimately all quanta had 33Emcrm2 2π 2 RRWeee1 R experienced a reflection from the bound. It means that nx 3322 (18) rrMcRreeEMU 2π e the inner side of the Hubble sphere looks like a Lam- By simply grouping the cosmological parameters in bertian light emitter for an internal observer. Thus the constant radiant emittance from the inner side of our Table 1. Known physical parameters. Black Hole can be expressed by the formula: E 1 Parameter Known value Units WMHU 3 (20) U 4πRT2 4π km H 67.80 77 sMpc Now let us consider the observer-a spherical body in a H 2.197 25 1018 s1 vacuum with radius rR0 placed at the center of the Hubble sphere. The observer will find out a constant c R 1.364 15 1026 m isotropic quanta flow that is falling to an every surface H area S from a spatial hemisphere above the area. 0.0072973525698 24 - According to the Lambert’s cosine law a radiant energy 3 c 52 M 9.188 10 10 kg flux in through the area S (i.e. irradiance) will 2GH be: 15 re 2.8179403267 27 10 m π in 2π 1 17 2 T 4.551 52 10 s WU cos sin (21) H S 00 T 14.43 16 Gyr where is the angle between the beam and a line 2 11 Nm normal to the surface area S . G 6.67384 80 10 kg2 The total radiant flux in related to the entire surface
Open Access JMP 1650 Y. RYAZANTSEV of our spherical observer: The above Equations (13) and (29) readily yield the correlation of the best known large Weyl-Eddington-Di- in S (22) rac numbers, which include both the geometrical cons- in S S tant π and the fine structure constant : 2 Sr0 sin (23) πNNDFDR 2 (31) where and are spherical coordinates of the area This correlation is very notable. It enables one to S on observer’s surface. Thus: calculate the approximations for Hubble sphere radius
π and the Hubble parameter via the correlation of gra- 2ππ 2π 2 Wrcos sin2 sin vitational and electrostatic forces: in 0000 U 0 (24) 1 RNr π DFe (32) The last integral (24) can be simplified as follows: 2 2 2c 22 ErU 0 H (33) in4ππrW0 U (25) Nr π T R2 DFe Now let us introduce the “big” angular momentum of The total amount of energy Etot entered inward (or reflected by) the observer during the period of time T the Hubble sphere measured along any given direction is: :
2 MRc (34) ETmctot in tot (26) Thus, multiplying both sides of the Equation (28) by where m is a total mass which our observer should tot cN , we would propose the exact Largest number law in have at present time if he absorbs (or reflects) the U the following form: incoming energy flux in completely. Thus we can write down the following equation: NU π (35) 2 2 The total sum of N fundamental quanta of the Mc mctot U π (27) angular momentum in universe equals exactly to the Rr22 0 angular momentum of the Hubble sphere multiplied by Now, making a comparison of the Equations (27) and π . This is a direct consequence of geometrical Lambert’s (19) one would ultimately conclude that if the coefficient cosine law and rotational symmetry of space (conser- 2 nx in the (19) equals exactly to π then mrtot 0 must vation of angular momentum). 2 The following expression, as well as expressions (29) be equal to mree . Thus: and (35), represent just another form of this law: M 1 m e (28) 22 NNN π (36) Rr π UMDR e universe electron With the help of large numbers ratios and Largest Using (4), (11) and (28), we immediately obtain a number law, one can get various representations of the noteworthy ratio between Weyl-Eddington-Dirac large Newton constant of gravitation G using initial expres- numbers : sion: 2 2 11NDR qe NNDRM π (29) G (37) 24N π m2 Hence, we can obtain the formula for the universe M 0 e mass via the Hubble time: The most elegant cases, in our opinion, are: 2 mR22 mc q M eeT 2 (30) G e (38) 22 m2 8 N ππrree e 0 DR The last one represents the proportional relation and: 2 between M and T which was hypothesized by P. 22 23 2 ππrceeh rc Dirac almost 80 years ago. G (39) 22mR 8πmR3 mT As we see from (29), both fundamental constants eee and π are deeply involved in large number relations The last one represents the reverse proportional re- and thus we can assume that it is an evidence that lation between G and R , which was hypothesized by cosmological and quantum parameters of our model of H.Weyl and between G and T which was hypo- the universe are closely connected through geometry. thesized by P. Dirac.
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By means of (39), one can get representation of the The ratios of large numbers described in the previous cosmological constant : sections enable to link the Hubble volume radius and 8πG 63GM 6 πN other constants with the length dimensions, including U M (40) such parameters as the electron gravitational radius and ccRNRR4232222 DR the Planck’s length. All such constants can be calculated one from the other, via the fine structure constant , EU where U —energy density of the universe. Weyl-Eddington-Dirac large numbers N or N V DF DR U and π . The Tables 4 and 5 show the mutual conversion The last one represents the de Sitter space—a va- ratios of the constants—from the Compton wave length cuum solution of Einstein’s equation with cosmological of the universe to the Hubble sphere radius. To obtain the constant-for the 4-dimensional case. Hence, we can ob- value of the top line parameter, one should multiply the tain the formula for the cosmological constant via initial parameter in the respective column by the formula the fine structure constant , Weyl-Eddington-Dirac in the cell at their crossing. large number N , classical radius of electron r and DF e In Table 5, one can find one more elegant expre- π : ssion-the ratio between largest and smallest distances in 12 universe-radius of the Hubble sphere and the Compton 2 (41) NrDFπ e wave length of the entire universe. It is proportional to the Largest number: 4. Calculations R 1 2 NU (42) The ratios between large numbers as described in Sec- U 2π tions 2 to 3 enable us to calculate many cosmological parameters in the proposed model. The calculations has 5. Examples of Applying the Large Number been carried out (by using the constants ,,rmcGee ,,,π ) Ratios as follows: hN,, E, then N , then DFe DR The previous sections contain a rather simple derivation NN,,,, RHT, then M ,,NE and then E . UM WW U of the inter-dependence of all main large numbers. The The results of calculations of the constants based on Largest number law that links the Weyl-Eddington-Dirac referential data and information from the most recent numbers enable validation of whether the known hypo- measurements are shown in Table 3. thetic equations and inequations conform to the large
Table 3. Proposed values based on the the Largest number numbers combination we proposed or not. Below are law. several examples. Example 1. J. Teller proposed [16] an interesting ratio Parameter Known value Proposed value between Planck’s values, the fine structure constant and km km H 67.80 77 68.7457 82 Hubble’s cosmological parameter: sMpc sMpc
18 1 18 1 H 1 H 2.197 25 10 s 2.22789 27 10 s mHcpp 8π tH exp (43) lp T 4.551 52 1017 s 4.48853 54 1017 s T 14.43 16 Gyr 14.2330 17 Gyr 8πG where 4 —Einstein constant, mltp ,,pp—Planck 52 52 c M 9.188 10 10 kg 9.0606 22 10 kg units. 26 26 R 1.364 15 10 m 1.345629 16 10 m Having very large magnitude 1060 , the left and 42 N DF 4.16589 50 10 - right parts of the formula give us values which differ
40 40 from each other by only 1.5%. It is really remarkable but N DR 4.842 55 10 4.77522 57 10 it is about 70 times more uncertain than other values N 1.135 39 10122 1.08888 39 10122 U calculated by us earlier: 83 82 NM 1.009 11 10 9.9465 24 10 1132 π3 N 3.564 81 10121 3.4660 12 10121 W 8πtHp e e 0.98588 12 (44) NU 10 J 10 J UUU EV 7.65 26 10 7.97879 95 10 m3 m3 Teller’s formula cannot be recognized exact and EMc 2 8.257 94 1069 J 8.1433 19 1069 J U expressing a fundamental physical law as it does not 52 52 EW 2.317 26 10 J 2.34947 28 10 J provide any strict derivation yet. A. Eddington’s contem-
52 1 52 1 plative assumptions on the quantity of particles in the 1.612 35 10 1.656801 39 10 256 m2 m2 universe, which is supposedly equal to precisely 2 ,
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Table 4. Calculating parameters with a length dimension via N DF .
U lp rg re rb e R
33 N 2 22N N 2 N 2 π 33πN NDF DF DF DF DF DF U 1 64 8 8 8 4 16
2 64 N 4π N π 33 l N DF DF N p 33 1 DF 3 DF NDF N DF 2
8 N N 2πN π N 2 r DF N DF DF DF g 2 1 DF 2 NDF 2
8 1 1 1 2π π N r DF e 22 1 2 NDF N DF N DF 2
8 3 2 3 2 π N DF rb 2 2π N N 1 DF N DF DF 2
4 1 2 N DF e 2 2 1 N DFπ 4π NDF 2πN DF 2π 2π 4
16 4 2 2 2 4
R 33 32 3 2 3 2 1 πN DF π N DF π N DF π N DF π N DF NDF where lp—Planck length, rb—Bohr radius.
Table 5. Calculating parameters with a length dimension via N DR .
U lp rg re rb e R
3 N N 2 N 2 N 2 N 3 N DR DR DR DR DR DR U 1 8π3 4π 2π2 2 22π π 2π2
8π3 2 π 2 2 8πN 2 l 1 DR 3 p 3 N DR 4 N DR 2 N DR N DR 2N DR π π π
4π 2N 2N 2N 4N 2N 2 r DR 1 DR DR DR DR g 2 3 2 N DR π π π π
2π2 π π 1 2π r N e 2 1 2 DR N DR 2N DR 2N DR
2 22π π 4 3π r 2 1 2π 2 N b N 2 2N DR DR 2N DR DR
π 2 2 1 N 1 DR e N 2 4N DR 8πN DR DR 2π 2π 2π
2π2 π π 1 1 2π
R 3 3 2 2 1 N DR 2N DR 2N DR N DR N DR N DR are even further from the reality. expected 1.0 : Example 2. There is known Eddington-Weinberg 2 H 0.000264629 45 1 (46) approximate relation [17]: 3 Gcmp 23H Gcm (45) p Thus we can conclude that Eddington-Weinberg 27 where mp 1.672621777 17 10 kg —proton mass approximate identity is not confirmed in our model. But, (CODATA 2010). we should say that Eddington-Weinberg formula does Using the approximation of H calculated above have physical meaning in a nutshell. One can get the
(Table 3) one can get the ratio, which is quite far from exact identity by replacing mp by me and applying
Open Access JMP Y. RYAZANTSEV 1653 the large number ratios to this hypothetical formula: article provide the powerful means for finding relations 2 among various information and physical parameters of H 2 33 (47) our model universe. However, it does not help answer the Gcme π main question: where do these enormous numbers come Example 3. from in physics? Hopefully, the law and hypotheses J. Bekenstein proposed [18] universal entropy bound proposed in this article will let find the correct answer in for a complete physical system whose total mass-energy the foreseeable future. The ratios we suggested impose rather many stringent (in our case) is EU , and which fits inside a sphere of radius R . Applying the large numbers ratios, we can get limitations on the way physical constants may change the universal entropy bound value: over time. We must note that the sharply tuned combi- nation of large numbers, including the mentioned appro- 2πkEUU R E SB 2π k2π kNWU2 kN (48) ximations for Hubble time T , mass of the universe M cEW and Hubble limit R correspond to the values of the Where k is Boltzmann constant. very precisely measured physical constants of quantum On the other hand, the Bekenstein-Hawking entropy scale. Looking at (39) and (40), it is obvious that the bound for the Black Hole with radius R is: gravity is closely connected with the Hubble radius, Hubble time and the properties of electron. However, 32 πkc R there are rather reliable measurements that establish a SBH (49) G very low limit for the Newton constant change rate in the By dividing (48) by (49) and using (8) one can get: long-term <1% . Thus, if Hubble parameters TR, varies with time then the corresponding variation of SB 2EGU 2MG 2 or/and electron energy mce should preserve the cons- 421 (50) SBH cR cR tancy of G . If the entire universe energy E comprises (or once The last one says that both Bekenstein universal U comprised) quantums of the minimum energy E , we entropy bound and Bekenstein-Hawking Black Hole W entropy bound have the same value in our model and should make the conclusion that number (14) is a natural equal exactly to 2kN : number. The same is true about (15). Hence, we can U establish the hypothesis that all large numbers in the real J Sk2.17776 78 10122 3.0067 11 1099 (51) (not infinite) universe should be naturals or rationals. B K However, the denominator and the numerator in these Using the large number ratios it is also easy to express fractional numbers are so great that one can confidently presume the real large numbers approximate some per- the Hawking radiation EH of our Black Hole with fect limit of fundamental importance infinitely closely. energy EU via Largest number NU or Hubble parameter: Taking into account the transcendentalism of (29), (31)
3 and (35), we dare to express one more hypothesis: all cHEEUW Large numbers are infinitely approaching their limits ETkHH (52) 8πGM44 NU π 4π and these very limits are transcendent mathematical where T is Hawking radiation temperature. constants. H The Schwarzschild solution is one of the well known 6. Discussion and Conclusions models of our universe, representing a Black Hole. Avoiding creation of new essences, we just consider the The two currently prevailing physical theories, i.e. quan- interaction between internal radiation and a small sphe- tum mechanics and the general relativity, describe the rical observer. It appears that such a simple model re- reality very precisely, each in its range of energy and veals a lot of interesting identities and conservation laws. 2 spatial scale. It is presumed that sometimes in the future, For example, the value of mrtot 0 should be conserved the value of NDF will be obtained in theory directly as for a sphere with a radius r0 . Furthermore we found that a direct result of consolidation of gravitation with other identity mmtot e is valid for a sphere with classical known interactions, strong and electroweak. electron radius re . The fruitless attempts at explaining the proximity of The last derivation allowed us to propose a list of
NDF and NDR resulted in a broad application of the exact ratios between well known Eddington-Weyl-Dirac term of “coincidence” that somehow highlights the large numbers which are listed in Table 2. The ratios (30) randomness of the event. As shown in previous sections and (39) reveals the exact formulae supporting P. Dirac’s it is not random. hypothesis and propositions which were announced many As we see, the large number ratios proposed in the years ago and never were written in a clear mathematical
Open Access JMP 1654 Y. RYAZANTSEV form. versity Press, Cambridge, 1946. Using these large numbers ratios we have claimed the [6] P. C. W. Davies, “The Accidential Universe,” Cambridge new Largest number law (35) which is based on Lam- University Press, Cambridge, 1982. bert’s cosine law and the rotational symmetry of space. It [7] S. Ray, U. Mukhopadhyay and P. P. Ghosh, Large Num- is quite important that this law precisely unites the major ber Hypothesis: A Review, 2007. cosmological and quantum parameters of our universe. It [8] K. A. Tomilin, Issledovaniya po Istorii Fiziki i Mekhaniki, could be interpreted as follows: our universe comprises Vol. 141, 1999. of the mathematically determined number of elementary [9] P. A. M. Dirac, Nature, Vol. 139, 1937, p. 323. spatial clusters which can be matched up to energy quan- http://dx.doi.org/10.1038/139323a0 ta and angular momentum quanta. We should note that [10] P. A. M. Dirac, Proceedings of the Royal Society A, Vol. basic parameters of our model universe have been ob- 165, 1938, p. 199. tained form the properties of electron, speed of light and [11] P. A. M. Dirac, Nature, Vol. 1001, 1937. fine structure constant. [12] E. A. Milne, Proceedings of the Royal Society A, Vol. 158, If Equations (13), (28) and (31) are valid inside the 1937, p. 324. Hubble sphere, which is very probable, we should make [13] P. J. Mohr, B. N. Taylor and D. B. Newell, The 2010 a conclusion that all cosmological parameters are fully CODATA Recommended Values of the Fundamental Phy- and unambiguously determined by quantum and mathe- sical Constants, National Institute of Standards and Tech- matical constants. Simply put, all of us are very likely to nology, Gaithersburg, 2011. live in an extremely precisely self-tuned quantized uni- http://physics.nist.gov/constants verse. [14] C. Planck, P. A. R. Ade, N. Aghanim, C. Armitage-Ca- plan, M. Arnaud, et al., Planck 2013 Results, Cosmologi- cal parameters, 2013. REFERENCES [15] J. Casado, Connecting Quantum and Cosmic Scales by a [1] H. Weyl, Annalen der Physik, Vol. 359, 1917, pp. 117- Decreasing-Light-Speed Model, 2004. 145. http://dx.doi.org/10.1002/andp.19173591804 [16] E. Teller, Physical Review, Vol. 73, 1948, pp. 801-802. [2] H. Weyl, “The Open World Yale,” Oxbow Press, Oxford, http://dx.doi.org/10.1103/PhysRev.73.801 1989. [17] S. Weinberg, “Gravitation and Cosmology,” Wiley, New [3] H. Weyl, “Space Time Matter,” Methuen, Dover, 1952. York, 1972. [4] A. S. Eddington, “The Math. Theory of Relativity,” Cam- [18] J. D. Bekenstein, Physical Review D, Vol. 23, 1981, pp. bridge University Press, Cambridge, 1924. 287-298. http://dx.doi.org/10.1103/PhysRevD.23.287 [5] A. S. Eddington, “Fundamental Theory,” Cambridge Uni-
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