Fractal universe and the speed of light: Revision of the universal constants. Antonio Alfonso-Faus

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1 Fratal universe and the speed of light: Revision of the universal onstants Antonio Alfonso-Faus E.U.I.T. AeronÄutia Plaza Cardenal Cisneros 40, 8040 Madrid, Spain E-ail: Abstrat. We apply the property of selfsiilarity that orresponds to the onept of a fratal universe, to the diension of tie. It follows that any interval of tie, given by any tik of any lok, is proportional to the age of the universe. The fratality of tie gives the fratality of spae and ass. First onsequene is that the speed of light dereases inversely proportional to tie, sae as the Hubble paraeter. We then revise the universal onstants and, at the osologial sale, they are all of order one, as Dira proposed. We find three different sales, eah one separated by a fator of about 5x10^60: the universe, the Plank sale and what we all the sub Plank sale. Integration of the Einstein osologial equations, for this fratal universe, gives the solution of a non-expanding universe with the present value of the observed nuerial paraeters. The red shift easured fro the distant galaxies is interpreted here as due to the dereasing speed of light in a fratal universe. Key words: fratality, self-siilarity, speed of light, osology, red shift. 1.-Introdution Muh work has been done on the fratality of spae-tie. See for exaple Mandelbrot (198), Ord (198), Nottale (199), El Nashie (004) et. Here we start with the assuption of the fratality of tie and apply its property of self-siilarity. This eans that any tie interval of any lok is proportional to the age of the universe. In that ase the ratio of these two ties is a onstant, a universal onstant that defines the partiular lok with its partiular tiking. We analyse a few loks and find out that spae is fratal too. Adding the usual assuption that Plank s onstant is a true universal onstant (as a onsequene of the onstany of angular oentu), we arrive at the iportant onlusion that all lengths in the universe are onstants, inluding the size of the universe. Then there is no expansion. Obviously the speed of light, as any other speed, dereases with tie due to the tiks of the loks being proportional to the age of the universe. We have presented elsewhere the proportionality between the speed of light and the Hubble paraeter, Alfonso-Faus (008). As one would expet fro relativity it turns out that spae is also fratal. The onstany of Plank s onstant is also a plausible assuption as a result of the 1

2 lifetie found for beta deay: this interval of tie is proportional to the 7 th power of so that a sall variation in tie for would ertainly have to be notied. Then we get the validity of the onstany of two well known physial properties (if no interation is present): the angular oentu and the linear oentu. At a osologial sale this orresponds to the osologial priniple of hoogeneity and isotropy. Zel dovih (1967) forulation for the osologial onstant and Weinberg (197) relation between quantu and osologial paraeters have been shown to be one and the sae thing, Alfonso-Faus (008). They have also been shown to be a result of Newton s laws, Alfonso-Faus (008). We note that these relations are valid for the osologial sale, the quantu sale, as well as for the Plank and sub Plank sales. Finally we integrate the Einstein osologial equations for the ase we are dealing with. We use the ost urrent values for the paraeters, = 1/, = / and w = -1 for the equation of state..-fratal tie Fratality iplies the property of self-siilarity. Applied to the diension of tie this eans that any tie interval t is proportional to the age of the universe t, t/t = diensionless onstant. This defines a sale and t defines the tiking of a lok, any type of lok. We establish the proportionality with t hoosing the sales of Plank, atoi, inverse photon frequeny and gravitational period (Kepler): G 5 1/ a G 1/ t (1) We now take the usual assuption that is onstant. We also take the assuption that the linear oentu is onstant. Then fro (1) we obtain t onst G onst a onst ()

3 We know that in order to derive the Einstein s field equations fro the ation priniple, the fators in fro of the integrals ust be onstant. These are G/ and. The onstany of, the linear oentu, is a very well known law. The onstany of the ass rate /G is not so well known and its physial ipliations should not be overlooked, Alfonso-Faus (008)..-Fratal spae Having analysed the onsequenes of the assuption of a fratal tie, we now see that the fratality of spae is inevitable too. All we have to do is to ultiply the diension tie in (1) by the speed of light, and of ourse note that fro () the produt t is a onstant. Then Plank s length is onstant, so is the size of the atoi sizes, as well as the wavelengths of photons. Fratality of spae iplies a onstant size for everything in the universe. And the iediate onlusion is that all speeds, inluding, derease inversely proportional to the osologial tie. And that all asses inrease proportionally to t, the Mass-Boo effet, Alfonso-Faus (008). In addition there is no expansion of the universe. The observed Hubble red shift is seen here as a result of the tie variation of the speed of light. 4.-Fratal ass The onstany of the linear oentu, together with the linear derease of the speeds, iplies the linear inrease of all asses. This eans that the ratio of two different asses is a universal onstant. This is a self-siilarity property of fratality. If M is the ass of the universe and t its age then the ratio t/m is a universal onstant. Clearly it is the universal onstant G/ in (), an expression of Mah s priniple. We then have a fratal universe: fratal spae-tie and fratal ass. 5.-Universal onstants One expets, as Dira notied in 197, that all universal onstants at the osologial level ust be of order one. We an define the size of the universe, its total oentu ontent averaged over t, its total angular oentu and the ass rate as universal onstants of order one: t 1 M 1 M. t 1 H G 1 M t u ()

4 All these properties refer to the sale of the universe. We reover the Plank s sale by applying to the ass, length and tie the dividing onstant fator 5x By dividing one ore using the sae fator we get the sub Plank sale with ass gr., length and tik se. Zel dovih (1967)and Weinberg (197) relations hold in these sales and the quantu one. 6.-Cosologial equations The Einstein osologial equations for a onstant size universe and a total pressure p = w plus p Q are given by Gp k 8 R 8 k G R (4) With the usual definitions these equations beoe w k Q k (5) whih have the solution w = -1, = 1/, k = 1, = Q = /. The paraeter Q ay be identified for exaple to an eletrial outward pressure due to the fat that the Debye length for the universe is a bit larger than its size t. An overall net harge Q, of the order of 5x10 60 e, ay be present in the universe, e the harge of the eletron. 7.-Conlusions Muh work has been done on the fratality of spae-tie. Here we add the fratality of ass. The universal onstants at the osologial level are of order one and the physial properties of ass and tie ay be onsidered as idential. Then the universe is fratal. All speeds, inluding the speed of light, derease with the osologial tie, and the length travelled in eah tik of the lok is onstant. Then the size of the universe as given by t is onstant. The horizon proble is not present in this theory, neither the entropy nor the osologial onstant proble, et. The so alled oinidenes are just an effet of the saling fators. 4

5 8.-Referenes Alfonso-Faus, A., (008), Astrophysis and Spae Siene, 15, 5 and next issue. Also in the arxiv.org arhives. El Nashie, M.S., (004), in Chaos, Solitons and Fratals Vol 19, is.1, January. Mandelbrot, Benoit B, (198), The fratal geoetry of nature, W.H. Freean and Co. Nottale L., (199), Fratal Spae-Tie and Mirophysis, World Sientifi Pub. Co. In., May. Ord, G.N., (198), Phys. A : Math. Gen16, Weinberg, S., (197), Gravitation and Cosology, John Wiley and Sons, New York. Zel dovih, Ya.B., (1967), Pis a Zh. Eksp. Teor. Fiz., vol 6, p