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Nonlinear dynamics in the Einstein–Friedmann equation
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Chaos, Solitons and Fractals 41 (2009) 533–549 www.elsevier.com/locate/chaos
Nonlinear dynamics in the Einstein–Friedmann equation Yosuke Tanaka a,*, Yuji Mizuno b, Shigetoshi Ohta a, Keisuke Mori c, Tanji Horiuchi d b
a Department of Physics, Faculty of Engineering, Kyushu Kyoritsu University, Kita-Kyushu 807, Japan Department of Information Systems, Graduate School of Information Technology, Kobe Institute of Computing, Kobe 650, Japan c Department of Information Systems, Faculty of Engineering, Kyushu Kyoritsu University, Kita-Kyushu 807, Japan d Department of Sports Science, Faculty of Sports Science, Kyushu Kyoritsu University, Kita-Kyushu 807, Japan
Accepted 18 February 2008
Abstract We have studied the gravitational field equations on the basis of general relativity and nonlinear dynamics. The space component of the Einstein–Friedmann equation shows the chaotic behaviours in case the following conditions are satisfied: (i) the expanding ratio: h ¼ x_ =x < 0, (ii) the curvature: f = 1, and (iii) the cosmological constant: k < jp. In this paper, we have studied the k-dependence of solutions in the Einstein–Friedmann equation and found the upper limit of k (kmax = +0.14) for the occurrence of the chaotic behaviours in the Einstein–Friedmann equation (0 5 k 5 +0.14). The numerical calculations are performed with the use of the Microsoft EXCEL(2003), and the results are shown in the following cases; k = 2b = +0.06 and +0.14. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction Recently, the space–time properties have been discussed on the basis of general relativity and nonlinear dynamics by several authors [1–6]. Especially, El Naschie has studied the solitonic solution in the gravitational field equation with the use of the Eguchi–Hanson metric [1]. Rugh has analyzed the extended Robertson–Walker metric and discussed the chaos in the Einstein equation [3]. By using the compact Robertson–Walker metric, we have studied the chaotic behaviours in the Friedmann equation [4–6]. According to the general theory of relativity, the gravitational field equation is given as follows [7–11]:
*
Corresponding author. Tel.: +81 93 693 3050; fax: +81 93 603 8186. E-mail address: [email protected] (Y. Tanaka).
0960-0779/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2008.02.027
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Gik þ jT ik þ kgik ¼ 0; 8pK j ¼ 2 ¼ 1:86 1027 ðcm=gÞ; c where symbols Gik and Tik denote the Einstein tensor and the energy-momentum tensor, respectively. The third term is the cosmological term, and symbols k and gik show the cosmological constant and the metric tensor, respectively [9,11]. We call the above equation as ‘‘the Einstein equation” in this paper. In the previous papers [5,6], with the use of the fixed value of the cosmological constant (k = 0), we have studied the initial value dependence of solutions in the Friedmann equation (k = 0). In this paper, we study the k-dependence of solutions in the Einstein–Friedmann equation (k – 0), and show that the cosmological constant (k) and the curvature (f) are important to discuss the chaotic behaviours in the Einstein–Friedmann equation (k – 0). In Section 2, we make a brief review of ‘‘the Einstein–Friedmann equation”, following the textbook of Einstein [11]. In Section 3, we give numerical results and discuss the chaotic behaviours in the Einstein–Friedmann equation (k – 0). In Section 4, we give some discussions on the related problems. Summary is given in Section 5.
2. Einstein–Friedmann equation and chaotic behaviours In this section, we review briefly the Einstein–Friedmann equation, following the text book of Einstein [11]. In 1922, Friedmann proposed the model of universe, where the three dimensional space is isotropic and homogeneous. The metric is given as follows: ds2 ¼ ðdx4 Þ2 G2 A2 fðdx1 Þ2 þ ðdx2 Þ2 þ ðdx3 Þ2 g; 1 ; ðf ¼ 1; 0; þ1Þ: A¼ 1 þ 4f r2 The function G = G(t) satisfies the following equations: 1 _ 2 þ 2GGg € þ jp k ¼ 0; ff þ ðGÞ G2 3 _ 2 g þ jq þ k ¼ 0; 2 ff þ ðGÞ G where symbol p and q show the pressure and the density, respectively. These equations are the space component and the time component of the Einstein equation, respectively. By eliminating the constant k from these equations, we obtain € G_ 2 f j G 2 þ ðp þ qÞ ¼ 0: 2 G G G Also, by eliminating f, we have € j k G þ ð3p þ qÞ ¼ 0: 3 G 6 Among these four equations, two of them are independent, and we discuss the properties of these equations except for the second equation (the time-component of the Einstein equation). Here, we use the variable x instead of G; x¼
1 : G
Then, the above three equations are transformed into the following form [4–6]: €x þ a
ð_xÞ2 þ bx þ cx3 ¼ d F ðp; q; xÞ; x
where the function F(p, q, x) is defined as follows: 9 8 ½Case 1 > > = < px; F ðp; q; xÞ ¼ ðp þ qÞx; ½Case 2 : > > ; : ð3p þ qÞx; ½Case 3
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Table 1 The values of coefficients a, b, c and d in the Einstein–Friedmann equationa
Case 1 Case 2 Case 3
a
b
c
d
Comment
52
k 2
2f
1 2
0
f 0
k 2 k 2 k 6
F(p, q, x) = px F(p, q, x) = (p + q)x F(p, q, x) = (3p + q)x
k 3
See the text for details. a The Einstein–Friedmann equation is given as follows:
€x þ að_xÞ2 =x þ bx þ cx3 ¼ d F ðp; q; xÞ: The coefficients a, b, c and d are listed in Table 1. We call above differential equation as ‘‘the Einstein–Friedmann equation” in this paper. At this stage, we have two independent field equations (the space- and time-component of the Einstein equation) and three unknowns (p, q and x = 1/G). To obtain the solutions of these equations, we must have one more equation. Here, we use the conservation law q_ ¼ 3
G_ ðp þ qÞ; G
which is obtained by the covariant derivative T ik;i ¼ 0; ði; k ¼ 1; 2; 3; 4Þ: Then, we have the standard form of dynamical equations [4–6]. 8 > < x_ ¼ y; 2 y_ ¼ a yx bx cx3 þ d F ðp; q; xÞ; > : q_ ¼ 3 yx ðp þ qÞ: We call these equations as the standard form of ‘‘the Einstein–Friedmann equation” in this paper. The expansion rate of phase space volume is given as follows: x_ V_ ¼ ð3 2aÞ ; x V where the symbol V denotes the phase space volume. Thus, there may occur chaos in case h ¼ x_ =x < 0, and there may occur no chaos in case h ¼ x_ =x=0, as far as we use the value a < 0 in Table 1.
3. Numerical results We have analyzed the Einstein–Friedmann equation with the use of the Microsoft EXCEL (2003) and obtained the chaotic behaviours for the open universe (f = 1) in Case 1. Following the name of ‘‘the Lorenz chaos” [12] and that of ‘‘the Ro¨ssler chaos” [13], we may call these chaotic behaviours as ‘‘the Einstein–Friedmann chaos”. In this section, we show the numerical results of the following values; k = +0.06 and k = +0.14. 3.1. Results in case k = 2b = +0.06 The numerical results in case k = +0.06 (b = +0.03) are given in Table 2, where only the first 30 lines are shown for convenience. The units of time (t), space (x) and density (q) are arbitrary. We show the graphical solutions in the Einstein–Friedmann equation [Case 1] in Fig. 1, where the figures are demonstrated in the t–x plane, t–_x plane and t–q plane, respectively, for f = 1 (c = +0.5). The other parameters are fixed and given in the figure caption. These figures show the typical nonlinear vibration mode. Only in case c = +0.5(f = 1), there occurs chaos in the Einstein– Friedmann equation [Case 1], as far as we use parameters in Table 1. We show the parameter c dependence of solutions in the Einstein–Friedmann equation [Case 1] in Figs. 2 and 3, where the figures are shown in the x–_x plane and the x–q plane for c = 0.5, 0.0 and +0.5, respectively. The other parameters are fixed and given in each figure captions. As noted before, only in case c = +0.5 (f = 1), there occurs chaos in the Einstein–Freidmann equation [Case 1].
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Table 2 The numerical solutions in the Einstein–Friedmann equation [Case 1: the damping type] t 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29
x
y
q
0.608
0.3
0.3
0.62 0.632456077 0.645391751 0.658832012 0.672803414 0.687334179 0.702454306 0.718195689 0.734592228 0.751679958 0.769497171 0.788084544 0.807485265 0.827745155 0.848912785 0.871039582 0.894179914 0.918391152 0.943733699 0.97027097 0.998069307 1.027197814 1.057728085 1.089733792 1.123290101 1.158472866 1.195357545 1.234017775 1.274523525
0.311401917 0.323391866 0.336006529 0.349285038 0.363269113 0.378003192 0.393534564 0.40991348 0.427193252 0.445430329 0.464684327 0.485018018 0.506497246 0.52919076 0.553169925 0.578508287 0.605280946 0.633563683 0.663431775 0.694958421 0.728212676 0.763256771 0.800142676 0.83890773 0.879569121 0.922116973 0.966505764 1.012643743 1.060379997
0.335526316 0.373830338 0.415176055 0.459856605 0.508197889 0.560562664 0.617355202 0.679026569 0.746080637 0.819080907 0.89865827 0.985519831 1.080458927 1.184366504 1.298244023 1.423218066 1.560556844 1.711688793 1.878223449 2.061974775 2.264987044 2.489563361 2.738296732 3.014103444 3.320258252 3.660430469 4.038719536 4.459687885 4.928387924
b = 0.03. For convenience, only the first 30 lines are shown.
Also, we show the chaotic behaviours in the Einstein–Friedmann equation [Case 1] in Figs. 4 and 5, where the figures are demonstrated in the x–_x plane and x–q plane for p = 0.28, 0.30, 0.32 and 0.34, respectively. The other parameters are fixed and given in each figure captions. These figures show the self-similarity, which is one of the typical characters of the chaotic behaviours [12,13]. 3.2. Results in case k = 2b = +0.14 The numerical results in case k = +0.14(b = +0.07) are given in Table 3, where only the first 30 lines are shown for convenience. The units of time (t), space (x) and density (q) are arbitrary. We show the graphical solutions in the Einstein–Friedmann equation [Case 1] in Fig. 6, where the figures are demonstrated in the t–x plane, t–_x plane and t–q plane, respectively, for f = 1 (c = +0.5). The other parameters are fixed and given in the figure caption. These figures show the typical nonlinear vibration mode. Only in case c = +0.5 (f = 1), there occurs chaos in the Einstein–Friedmann equation [Case 1], as far as we use parameters in Table 1. We show the parameter c dependence of solutions in the Einstein–Friedmann equation [Case 1] in Figs. 7 and 8, where the figures are shown in the x–_x plane and x q plane for c = 0.5, 0.0 and +0.5, respectively. The other parameters are fixed and given in each figure captions. As stated before, only in case c = +0.5 (f = 1), there occurs chaos in the Einstein–Friedmann equation [Case 1]. Also, we show the chaotic behaviours in the Einstein–Friedmann equation [Case 1] in Figs. 9 and 10, where the figures are demonstrated in the x–_x plane and x q plane for p = 0.28, 0.30, 0.32 and 0.34, respectively. The other parameters are fixed and given in each figure captions. These figures show the self-similarity, which is one of the typical properties of chaotic behaviours [12,13].
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Fig. 1. The numerical solutions in the Einstein–Friedmann equation [Case 1: the damping type]. b = 0.03. Figures are shown in the t–x plane, the t–_x plane and the t–q plane, from the upper to the lower, respectively. The parameters are given as follows: a = 2.5, b = 0.03, c = 0.5, d = 0.25, p = 0.3, x0 = 0.608, x_ 0 ¼ 0:3, and q0 = 0.3.
4. Discussions In this section, we give some discussions on the related problems. 4.1. The Higgs-type W-shape potential and the classification of solutions In case p = const., by integrating the Einstein–Friedmann equation [Case 1], we have obtained the potential equation ( 1 ð_xÞ2 þ U ðxÞ ¼ const:0 ; 2 U ðxÞ ¼ 14 ðk jpÞx2 8f x4 ;
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Fig. 2. The parameter c dependence of solutions in the Einstein–Friedmann equation [Case 1: the damping type]. b = 0.03. Figures are shown in the x–_x plane for c = 0.5, 0 and +0.5, respectively. The other parameters are fixed and given as follows: a = 2.5, b = 0.03, d = 0.25, p = 0.3, x0 = 0.608, x_ 0 ¼ 0:3, and q0 = 0.3.
where const.0 is defined as follows: Z 5 ð_xÞ3 const:0 ¼ const: þ dt: x 2 The W-shape function U = U(x) may be considered as the effective potential in the Einstein–Friedmann field [14]. With the use of this W-shape function, we have classified the solutions in the Einstein–Friedmann equation. Here, we show the non-trivial case [Case 1: f = 1]. (i) k < jp (a) const.0 > 0; vibrational but unphysical, (b) const.0 = 0; critical, (c) 1/8(k jp)2 < const.0 < 0; vibrational and chaotic,
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Fig. 3. The parameter c dependence of solutions in the Einstein–Friedmann equation [Case 1: the damping type]. b = 0.03. Figures are shown in the x–q plane for c = 0.5, 0 and +0.5, respectively. The other parameters are fixed and given as follows: a = 2.5, b = 0.03, d = 0.25, p = 0.3, x0 = 0.608, x_ 0 ¼ 0:3, and q0 = 0.3.
(d) (e)
const.0 = 1/8(k jp)2; critical, const.0 < 1/8(k jp)2; no solutions.
(ii) k = jp (a) const.0 > 0; vibrational but unphysical, (b) const.0 = 0; critical, (c) const.0 < 0; no solutions. In this paper, we have checked these classifications of solutions by the numerical calculations. Especially, we have used the values of parameter k = 0.06 and 0.14, which satisfy the condition k < j p = 0.5 0.3 = 0.15. More details on the theoretical classification of solutions will be discussed elsewhere.
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Fig. 4. Chaotic behaviors in the Einstein–Friedmann equation [Case 1: the damping type]. b = 0.03. Figures are shown in the x–_x plane. The parameter p is changed from p = 0.28 to 0.34. The other parameters are fixed and given as follows: a = 2.5, b = 0.03, c = +0.5, d = 0.25, x0 = 0.608, x_ 0 ¼ 0:3, and q0 = 0.3.
4.2. The role of the cosmological constant (k) In 1917, Einstein introduced the cosmological constant (k) and discussed the role of its constant [9]. Since then, many authors have discussed this problem [15,16]. In this subsection, we comment on the cosmological constant (k). (1) Experimentally, the cosmological constant (k) is observed to be very small [17]; kðexpÞ 3:0 1057 cm2 Þ: With the use of the values given in the works by Einstein [9,11], the cosmological constant is calculated to be as follows: jq 3:3 1055 ðcm2 Þ: kðtheorÞ ¼ 2 pffiffiffiffiffiffiffiffi Following the Einstein’s expressions [9,11], k = 0 means q = 2k/j = 0 and R ¼ 1=k ¼ 1, where the symbol R denotes the averaged radius of the universe. Inversely, if R = 0 then k = 1 and q = 1. It is important to note that the observed value of the cosmological constant (k) is very small but not zero. (2) From theoretical point of view, as shown in the previous subsection, the cosmological constant (k) plays an important role in introducing the Higgs-type W-shape potential [14] and in discussing the chaotic behaviours in the gravitational field. As discussed in the previous subsection, the effective potential curve (U) in the Einstein–Friedmann field is classified in the three types; (i) k < jp, (ii) k = jp, and (iii) k > j p. This classification may correspond to that of the Higgs-type W-shape potential curves given by Perkins [14]; (i) l2 < 0, (ii) l2 = 0, and (iii) l2 > 0, where the symbol l denotes the particle mass, and the Higgs Lagrangian is given as follows:
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Fig. 5. Chaotic behaviors in the Einstein–Friedmann equation [Case 1: the damping type]. b = 0.03. Figures are shown in the x–q plane. The parameter p is changed from p = 0.28 to 0.34. The other parameters are fixed and given as follows: a = 2.5, b = 0.03, c = +0.5, d = 0.25, x0 = 0.608, x_ 0 ¼ 0:3, and q0 = 0.3.
(
L ¼ 12 ðom /Þðom /Þ Uð/Þ; U ð/Þ ¼ 12 l2 /2 þ 14 K/4 ;
Here, the symbol / and K show the scalar field and the constant. With the use of this W-shape potential, Perkins has discussed the Higgs mechanism [18] and the mass of elementary particles [14]. (3) Einstein pointed out the gravitational field plays an essential role in the structure of elementary particles [10] but the complete theory of the gravitational field is not yet determined [11]. In the previous subsection (Section 4.1), we have shown that the Einstein–Friedmann equation leads to the Higgstype W-shape potential [14], which plays an essential part in the Higgs mechanism [18]. 4.3. Comments on the model and dimension of the universe Recently, El Naschie has studied the hierarchy structure of the string model and obtained the hierarchy dimension formula of the Heterotic string [19]; 4m 1 Gm ¼ N SM ð/Þm ¼ 10 ; ðm ¼ 0; 1; 2 Þ; / where the symbols N SM ¼ a0 =2 and / denote the elementary particle number in the standard model and the golden ratio, respectively. In case m = 0, 1 and 2, this formula gives the coupling constant; G0 ¼ 10ð1=/Þ4 ¼ a0 =2, G1 ¼ 10ð1=/Þ3 ¼ ag and G2 ¼ 10ð1=/Þ2 ¼ ags . Moreover, in case m = 6 above formula gives the classical dimension of the universe G6 ¼ 10ð/Þ2 ¼ 4 k ¼ Dð4Þ ;
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Table 3 The numerical solutions in the Einstein–Friedmann equation [Case 1: the damping type] t 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29
x
y
q
0.5
0.2
0.2
0.508 0.516224 0.524680032 0.533376427 0.542321817 0.551525145 0.560995666 0.570742946 0.580776868 0.591107623 0.60174571 0.612701928 0.623987356 0.635613347 0.647591497 0.659933621 0.672651715 0.685757912 0.699264426 0.713183489 0.727527267 0.742307771 0.757536737 0.773225503 0.789384847 0.806024811 0.823154492 0.8407818 0.85891318
0.2056 0.211400804 0.217409859 0.223634758 0.230083211 0.236763013 0.243682007 0.250848035 0.258268879 0.265952194 0.273905427 0.282135715 0.290649771 0.299453753 0.308553099 0.317952347 0.327654915 0.337662858 0.347976571 0.358594465 0.369512583 0.380724164 0.392219145 0.403983603 0.415999109 0.428242023 0.440682683 0.453284517 0.466003059
0.224 0.249449071 0.276449902 0.305113289 0.335558811 0.367915558 0.402322917 0.438931426 0.477903681 0.519415316 0.563656052 0.610830812 0.661160911 0.714885307 0.772261927 0.833569046 0.899106717 0.969198243 1.044191665 1.12446125 1.210408936 1.302465709 1.401092846 1.506782957 1.620060755 1.74148344 1.871640575 2.011153301 2.160672714
b = 0.07. For convenience, only the first 30 lines are shown.
where symbols D(N) and k = /3(1 /3) show the dimension and the constant, respectively. If neglecting the fractal part (k/4 = 0.045 1), we have D(4) 4. As is well known, the Kalza–Klein theory is a model which attempts to unify the two fundamental forces of the gravitation and the electromagnetism. This theory is extended by replacing the electromagnetic field by the more general gauge field, which is specified by the Lie group [17]. One of the extended models, which unify the strong interaction and weak-electro interaction is obtained by using the unitary groups; SU(3) SU(2) U(1). In this case, the dimension of the universe is 10; Dim = 4 + 6 = 10. On the other hand, with the use of the hierarchy dimension formula, El Naschie has obtained the relation Dð4Þ þ Dð6Þ ¼ ð4 kÞ þ ð6 þ kÞ ¼ 10 ¼ Dð10Þ : The special role of D(10) = 10 has been discussed by El Naschie. We summarize these models and dimensions of the universe as follows: (1) (2) (3) (4)
Dim = 4 1 = 3; Newton model. Dim = 4; Einstein model. Dim = 4 + 1 = 5; Kalza–Klein model. Dim = 4 + 6 = 10; Super string model.
Thus, by using the E-infinity theory of El Naschie [19], we can systematically understand the model and dimension of the universe, on the basis of Einstein model (Dim = 4).
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Fig. 6. The numerical solutions in the Einstein–Friedmann equation [Case 1: the damping type]. b = 0.07. Figures are shown in the t–x plane, the t–_x plane and the t–q plane, from the upper to the lower, respectively. The parameters are given as follows: a = 2.5, b = 0.07, c = 0.5, d = 0.25, p = 0.3, x0 = 0.5, x_ 0 ¼ 0:2, and q0 = 0.2.
4.4. Comments on the Hausdorff dimension of E-infinity space In this subsection, we comment on the Hausdorff dimension (DH) of E-infinity space. (1) The dimension of the universe As stated in the previous subsection (Section 4.3), El Naschie has obtained the Hausdorff dimension (DH) of E-infinity space [19]; DH ¼
1 X n¼0
nð/Þn ¼
1 ð/Þ3
¼ 4 þ ð/Þ3 ¼ 5 2ð/Þ2 :
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Fig. 7. The parameter c dependence of solutions in the Einstein–Friedmann equation [Case 1: the damping type]. b = 0.07. Figures are shown in the x–_x plane for c = 0.5, 0 and +0.5, respectively. The other parameters are fixed and given as follows: a = 2.5, b = 0.07, d = 0.25, p = 0.3, x0 = 0.5, x_ 0 ¼ 0:2, and q0 = 0.2.
This leads to the dimension of the Einstein space (Dim = 4) and also to that of the Kalza–Klein space (Dim = 5), by neglecting the fractal part in the above equation. (2) The Cabibbo angle (hC) Recently, El Naschie has studied the electro-weak theory and obtained the Weinberg angle expression sin2 hW ðtheorÞ ¼ cos
3p ¼ 0:2225; 7
which is in nice agreement with the experiment [20]. sin2 hW ðexpÞ ¼ 0:2229 0:0004: Also, these values are almost equal to the Cabibbo angle [17] sin hC ðexpÞ ¼ 0:224951:
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Fig. 8. The parameter c dependence of solutions in the Einstein–Friedmann equation [Case 1: the damping type]. b = 0.07. Figures are shown in the x–q plane for c = 0.5, 0 and +0.5, respectively. The other parameters are fixed and given as follows: a = 2.5, b = 0.07, d = 0.25, p = 0.3, x0 = 0.5, x_ 0 ¼ 0:2, and q0 = 0.2.
From E-infinity theoretical point of view [19], these values are given approximately by the inverse of the Hausdorff dimension of E-infinity space (DH) 1 ¼ ð/Þ3 ¼ 0:236067: DH Here, with the use of E-infinity theory [19], we discuss the Cabibbo angle. We start from the identity of E-infinity theory with the Fibonacci sequence {an}; an1 ð/Þnþ1 þ an ð/Þn ¼ 1; and 1 an ¼ pffiffiffi 5
ðn ¼ 1; 2; 3 Þ;
( ) nþ1 1 n nþ1 ; þ ð1Þ ð/Þ /
ðn ¼ 0; 1; 2 Þ:
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Fig. 9. Chaotic behaviors in the Einstein–Friedmann equation [Case 1: the damping type]. b = 0.07. Figures are shown in the x–_x plane. The parameter p is changed from p = 0.28 to 0.34. The other parameters are fixed and given as follows: a = 2.5, b = 0.07, c = +0.5, d = 0.25, x0 = 0.5, x_ 0 ¼ 0:2, and q0 = 0.2.
These expressions are proved by the mathematical induction method. By using above relations, we have the angle formula sin2 hn an1 ð/Þnþ1 ¼ sin2 h1 f1 þ ð1Þn1 ð/Þ2n g; where the value sin2h1 is given as follows: 1 sin2 h1 ¼ lim sin2 hn ¼ pffiffiffi ð/Þ ¼ 0:276393: n!1 5 The Cabibbo angle (hC) is found to be given approximately by the angle h2 sin hC ðtheorÞ sin2 h2 ¼ ð/Þ3 ¼ 0:236067: More precisely, with the use of the correction formula D ¼ 47 sin hC ðtheorÞ ¼ ð/Þ3 1 ð/Þ6 ; ¼ 0:224825; 55
P
bn ð/Þn , we obtain the relation
which is in nice agreement with the experiment [17] cited before. The theoretical calculation of the Cabibbo angle (hC) is an interesting and an open problem at this stage [17]. Here, we have discussed the Cabibbo angle (hC) with the use of the identity of E-infinity theory [an1(/)n+1 + an (/)n = 1]. In this sense, we may say that E-infinity theory gives the theoretical foundation of the Cabibbo angle. More details on this problem will be discussed elsewhere. (3) The two-slit problem in quantum mechanics Recently, El Naschie has studied the two-slit experiment and discussed the foundation of E-infinity theory [21–23]. In the simple case, he has derived the equations
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Fig. 10. Chaotic behaviors in the Einstein–Friedmann equation [Case 1: the damping type]. b = 0.07. Figures are shown in the x–q plane. The parameter p is changed from p = 0.28 to 0.34. The other parameters are fixed and given as follows: a = 2.5, b = 0.07, c = +0.5, d = 0.25, x0 = 0.5, x_ 0 ¼ 0:2, and q0 = 0.2.
p1 þ p2 ¼ 1; p1 p2 ¼ p1 p2 ;
where symbols p1 and p2 denote the probability of passing through the slit 1 and slit 2, respectively. From these equations, he has obtained the solutions p1 / ¼ ; ð/Þ2 p2 and
p1 p2
2
3 /1 4 ¼ 2 5 : 1 /
The corresponding total probabilities are given as follows: P ¼ jp1 p2 j ¼ ð/Þ3 ¼
1 ; DH
and P ¼ jp1 p2 j ¼
3 1 ¼ DH : /
Moreover, the solutions (p1, p2) and ðp1 ; p2 Þ are related in the following way;
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p1 p2
"
¼
0
1 DH
D1H
0
#
p1 : p2
These are the results of the gedanken experiment of El Naschie [21–23]. It is important to note that the two-slit experiment leads to the golden ratio (/) and the Hausdorff dimension (DH) of E-infinity space. Thus, the Hausdorff dimension (DH) of E-infinity space plays important roles not only in the space–time physics but also in the elementary particle physics and quantum mechanics.
5. Summary Finally, we summarize our work as follows: With the use of the Higgs-type W-shape potential [U = (1/ 4)(k jp)x2 (1/8) fx4] [14], which has been derived by integrating the Einstein–Friedmann equation, we have analyzed the chaotic behaviours in the Einstein–Friedmann equation. In this paper, we have studied the k- dependence of solutions in the Einstein–Friedmann equation and found the upper limit of k(kmax = +0.14) for the occurrence of the chaotic behaviours in the Einstein–Friedmann equation. This upper limit satisfies the condition k < j p = 0.5 0.3 = 0.15, where the values j and p are given arbitrarily for convenience. We have shown the numerical and graphical solutions in the following cases; k = 2b = +0.06 and +0.14. We have discussed the space–time properties with the use of E-infinity theory [19]. Especially, by using the dimension formula of El Naschie Gm = 10(/)m4, we have discussed the model and dimension of the universe. We have also discussed the Cabibbo angle (hC) by using the identity of E-infinity theory.
Acknowledgements We would like to thank the Information Processing Center of Kyushu Kyoritsu University for using the Microsoft EXCEL(2003) which is very powerful for analyzing nonlinear equations. Thanks are also to Ms. H. Shiroshita for typing the manuscript of this paper. We would also like to thank Prof. Y. Takesue for reading the manuscript of this paper.
References [1] El Naschie MS. Gravitational instantons in Hilbert space and the mass of high energy elementary particles. Chaos, Solitons & Fractals 2004;20:917–23. [2] ‘t Hooft G. A physical interpretation of gravitational instantons. Nucl Phys B 1989;315:512–27. [3] Rugh SE. Chaos in the Einstein equations. In: Quantum mechanics diffusion and chaotic fractals. Oxford: Pergamon Press; 1995. p. 161–82. [4] Tanaka Y. Relativistic field theory and chaotic dynamics. Chaos, Solitons & Fractals 2005;23:33–41. [5] Tanaka Y, Mizuno Y, Kado T. Chaotic dynamics in the Friedmann equation. Chaos, Solitons & Fractals 2005;24:407–22. [6] Tanaka Y, Mizuno Y, Kado T, Zhao HA. Nonlinear dynamics in the relativistic field equation. Chaos, Solitons & Fractals 2007;31:1054–75. [7] Zwiebach B. A first course in string theory. Cambridge: Cambridge University Press; 2005. [8] Nottale L. Fractal space–time and microphysics. Singapore: World Scientific Publishing Co. Pte. Ltd; 1993. [9] Einstein A. Cosmological considerations on the general theory of relativity. In: Perrett W, Jeffery GB, translators. The principle of relativity. New York: Dover Publications; 1952. p. 175–88. [10] Einstein A. Do gravitational fields play an essential part in the structure of the elementary particles of matter? In: Perrett W, Jeffery GB, translators. The principle of relativity. New York: Dover Publications; 1952. p. 189–98. [11] Einstein A. The meaning of relativity. 5th ed. New Jersey: Princeton University Press; 1995. Appendix. [12] Lorenz EN. Deterministic nonperiodic flow. J Atmos Sci 1963;20:130–41. [13] Ro¨ssler OE. An equation for continuous chaos. Phys Lett 1976;57A(5):397–8. [14] Perkins D. Introduction to high energy physics. 4th ed. Cambridge: Cambridge University Press; 2005. p. 267. [15] Dolgov AD, Zeldvich YB. Cosmology and elementary particles. Rev Mod Phys 1981;53:1–41. [16] Guth AH. Inflation universe: a possible solution to the horizon and flatness problems. Phys Rev D 1981;23:347–56. [17] Parker SP. McGraw-Hill encyclopedia of physics. 2nded. New York: McGraw-Hill; 1993. [18] Higgs PW. Spontaneous symmetry breakdown without massless bosons. Phys Rev 1966;145:1156–63. [19] El Naschie MS. A review of E-infinity theory and the mass of high energy particle physics. Chaos, Solitons & Fractals 2004;19:209–36.
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[20] Eidelman S et al. The review of particle physics. Phys Lett B 2004;592:1.. [21] El Naschie MS. From Hilbert space to the number of Higgs particles via the quantum two-slit experiment. In: Weibel P, Ord G, Ro¨ssler OE, editors. Space time physics and fractality. Wien and New York: Springer; 2005. [22] El Naschie MS. The two-slit experiment as the foundation of E-infinity high energy physics. Chaos, Solitons & Fractals 2005;25:509–14. [23] El Naschie MS. Hilbert, Fock and Cantorian spaces in the quantum two-slit gedanken experiment. Chaos, Solitons & Fractals 2006;27:39–42.
Chaos, Solitons and Fractals 41 (2009) 533–549 www.elsevier.com/locate/chaos
Nonlinear dynamics in the Einstein–Friedmann equation Yosuke Tanaka a,*, Yuji Mizuno b, Shigetoshi Ohta a, Keisuke Mori c, Tanji Horiuchi d b
a Department of Physics, Faculty of Engineering, Kyushu Kyoritsu University, Kita-Kyushu 807, Japan Department of Information Systems, Graduate School of Information Technology, Kobe Institute of Computing, Kobe 650, Japan c Department of Information Systems, Faculty of Engineering, Kyushu Kyoritsu University, Kita-Kyushu 807, Japan d Department of Sports Science, Faculty of Sports Science, Kyushu Kyoritsu University, Kita-Kyushu 807, Japan
Accepted 18 February 2008
Abstract We have studied the gravitational field equations on the basis of general relativity and nonlinear dynamics. The space component of the Einstein–Friedmann equation shows the chaotic behaviours in case the following conditions are satisfied: (i) the expanding ratio: h ¼ x_ =x < 0, (ii) the curvature: f = 1, and (iii) the cosmological constant: k < jp. In this paper, we have studied the k-dependence of solutions in the Einstein–Friedmann equation and found the upper limit of k (kmax = +0.14) for the occurrence of the chaotic behaviours in the Einstein–Friedmann equation (0 5 k 5 +0.14). The numerical calculations are performed with the use of the Microsoft EXCEL(2003), and the results are shown in the following cases; k = 2b = +0.06 and +0.14. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction Recently, the space–time properties have been discussed on the basis of general relativity and nonlinear dynamics by several authors [1–6]. Especially, El Naschie has studied the solitonic solution in the gravitational field equation with the use of the Eguchi–Hanson metric [1]. Rugh has analyzed the extended Robertson–Walker metric and discussed the chaos in the Einstein equation [3]. By using the compact Robertson–Walker metric, we have studied the chaotic behaviours in the Friedmann equation [4–6]. According to the general theory of relativity, the gravitational field equation is given as follows [7–11]:
*
Corresponding author. Tel.: +81 93 693 3050; fax: +81 93 603 8186. E-mail address: [email protected] (Y. Tanaka).
0960-0779/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2008.02.027
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Gik þ jT ik þ kgik ¼ 0; 8pK j ¼ 2 ¼ 1:86 1027 ðcm=gÞ; c where symbols Gik and Tik denote the Einstein tensor and the energy-momentum tensor, respectively. The third term is the cosmological term, and symbols k and gik show the cosmological constant and the metric tensor, respectively [9,11]. We call the above equation as ‘‘the Einstein equation” in this paper. In the previous papers [5,6], with the use of the fixed value of the cosmological constant (k = 0), we have studied the initial value dependence of solutions in the Friedmann equation (k = 0). In this paper, we study the k-dependence of solutions in the Einstein–Friedmann equation (k – 0), and show that the cosmological constant (k) and the curvature (f) are important to discuss the chaotic behaviours in the Einstein–Friedmann equation (k – 0). In Section 2, we make a brief review of ‘‘the Einstein–Friedmann equation”, following the textbook of Einstein [11]. In Section 3, we give numerical results and discuss the chaotic behaviours in the Einstein–Friedmann equation (k – 0). In Section 4, we give some discussions on the related problems. Summary is given in Section 5.
2. Einstein–Friedmann equation and chaotic behaviours In this section, we review briefly the Einstein–Friedmann equation, following the text book of Einstein [11]. In 1922, Friedmann proposed the model of universe, where the three dimensional space is isotropic and homogeneous. The metric is given as follows: ds2 ¼ ðdx4 Þ2 G2 A2 fðdx1 Þ2 þ ðdx2 Þ2 þ ðdx3 Þ2 g; 1 ; ðf ¼ 1; 0; þ1Þ: A¼ 1 þ 4f r2 The function G = G(t) satisfies the following equations: 1 _ 2 þ 2GGg € þ jp k ¼ 0; ff þ ðGÞ G2 3 _ 2 g þ jq þ k ¼ 0; 2 ff þ ðGÞ G where symbol p and q show the pressure and the density, respectively. These equations are the space component and the time component of the Einstein equation, respectively. By eliminating the constant k from these equations, we obtain € G_ 2 f j G 2 þ ðp þ qÞ ¼ 0: 2 G G G Also, by eliminating f, we have € j k G þ ð3p þ qÞ ¼ 0: 3 G 6 Among these four equations, two of them are independent, and we discuss the properties of these equations except for the second equation (the time-component of the Einstein equation). Here, we use the variable x instead of G; x¼
1 : G
Then, the above three equations are transformed into the following form [4–6]: €x þ a
ð_xÞ2 þ bx þ cx3 ¼ d F ðp; q; xÞ; x
where the function F(p, q, x) is defined as follows: 9 8 ½Case 1 > > = < px; F ðp; q; xÞ ¼ ðp þ qÞx; ½Case 2 : > > ; : ð3p þ qÞx; ½Case 3
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Table 1 The values of coefficients a, b, c and d in the Einstein–Friedmann equationa
Case 1 Case 2 Case 3
a
b
c
d
Comment
52
k 2
2f
1 2
0
f 0
k 2 k 2 k 6
F(p, q, x) = px F(p, q, x) = (p + q)x F(p, q, x) = (3p + q)x
k 3
See the text for details. a The Einstein–Friedmann equation is given as follows:
€x þ að_xÞ2 =x þ bx þ cx3 ¼ d F ðp; q; xÞ: The coefficients a, b, c and d are listed in Table 1. We call above differential equation as ‘‘the Einstein–Friedmann equation” in this paper. At this stage, we have two independent field equations (the space- and time-component of the Einstein equation) and three unknowns (p, q and x = 1/G). To obtain the solutions of these equations, we must have one more equation. Here, we use the conservation law q_ ¼ 3
G_ ðp þ qÞ; G
which is obtained by the covariant derivative T ik;i ¼ 0; ði; k ¼ 1; 2; 3; 4Þ: Then, we have the standard form of dynamical equations [4–6]. 8 > < x_ ¼ y; 2 y_ ¼ a yx bx cx3 þ d F ðp; q; xÞ; > : q_ ¼ 3 yx ðp þ qÞ: We call these equations as the standard form of ‘‘the Einstein–Friedmann equation” in this paper. The expansion rate of phase space volume is given as follows: x_ V_ ¼ ð3 2aÞ ; x V where the symbol V denotes the phase space volume. Thus, there may occur chaos in case h ¼ x_ =x < 0, and there may occur no chaos in case h ¼ x_ =x=0, as far as we use the value a < 0 in Table 1.
3. Numerical results We have analyzed the Einstein–Friedmann equation with the use of the Microsoft EXCEL (2003) and obtained the chaotic behaviours for the open universe (f = 1) in Case 1. Following the name of ‘‘the Lorenz chaos” [12] and that of ‘‘the Ro¨ssler chaos” [13], we may call these chaotic behaviours as ‘‘the Einstein–Friedmann chaos”. In this section, we show the numerical results of the following values; k = +0.06 and k = +0.14. 3.1. Results in case k = 2b = +0.06 The numerical results in case k = +0.06 (b = +0.03) are given in Table 2, where only the first 30 lines are shown for convenience. The units of time (t), space (x) and density (q) are arbitrary. We show the graphical solutions in the Einstein–Friedmann equation [Case 1] in Fig. 1, where the figures are demonstrated in the t–x plane, t–_x plane and t–q plane, respectively, for f = 1 (c = +0.5). The other parameters are fixed and given in the figure caption. These figures show the typical nonlinear vibration mode. Only in case c = +0.5(f = 1), there occurs chaos in the Einstein– Friedmann equation [Case 1], as far as we use parameters in Table 1. We show the parameter c dependence of solutions in the Einstein–Friedmann equation [Case 1] in Figs. 2 and 3, where the figures are shown in the x–_x plane and the x–q plane for c = 0.5, 0.0 and +0.5, respectively. The other parameters are fixed and given in each figure captions. As noted before, only in case c = +0.5 (f = 1), there occurs chaos in the Einstein–Freidmann equation [Case 1].
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Table 2 The numerical solutions in the Einstein–Friedmann equation [Case 1: the damping type] t 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29
x
y
q
0.608
0.3
0.3
0.62 0.632456077 0.645391751 0.658832012 0.672803414 0.687334179 0.702454306 0.718195689 0.734592228 0.751679958 0.769497171 0.788084544 0.807485265 0.827745155 0.848912785 0.871039582 0.894179914 0.918391152 0.943733699 0.97027097 0.998069307 1.027197814 1.057728085 1.089733792 1.123290101 1.158472866 1.195357545 1.234017775 1.274523525
0.311401917 0.323391866 0.336006529 0.349285038 0.363269113 0.378003192 0.393534564 0.40991348 0.427193252 0.445430329 0.464684327 0.485018018 0.506497246 0.52919076 0.553169925 0.578508287 0.605280946 0.633563683 0.663431775 0.694958421 0.728212676 0.763256771 0.800142676 0.83890773 0.879569121 0.922116973 0.966505764 1.012643743 1.060379997
0.335526316 0.373830338 0.415176055 0.459856605 0.508197889 0.560562664 0.617355202 0.679026569 0.746080637 0.819080907 0.89865827 0.985519831 1.080458927 1.184366504 1.298244023 1.423218066 1.560556844 1.711688793 1.878223449 2.061974775 2.264987044 2.489563361 2.738296732 3.014103444 3.320258252 3.660430469 4.038719536 4.459687885 4.928387924
b = 0.03. For convenience, only the first 30 lines are shown.
Also, we show the chaotic behaviours in the Einstein–Friedmann equation [Case 1] in Figs. 4 and 5, where the figures are demonstrated in the x–_x plane and x–q plane for p = 0.28, 0.30, 0.32 and 0.34, respectively. The other parameters are fixed and given in each figure captions. These figures show the self-similarity, which is one of the typical characters of the chaotic behaviours [12,13]. 3.2. Results in case k = 2b = +0.14 The numerical results in case k = +0.14(b = +0.07) are given in Table 3, where only the first 30 lines are shown for convenience. The units of time (t), space (x) and density (q) are arbitrary. We show the graphical solutions in the Einstein–Friedmann equation [Case 1] in Fig. 6, where the figures are demonstrated in the t–x plane, t–_x plane and t–q plane, respectively, for f = 1 (c = +0.5). The other parameters are fixed and given in the figure caption. These figures show the typical nonlinear vibration mode. Only in case c = +0.5 (f = 1), there occurs chaos in the Einstein–Friedmann equation [Case 1], as far as we use parameters in Table 1. We show the parameter c dependence of solutions in the Einstein–Friedmann equation [Case 1] in Figs. 7 and 8, where the figures are shown in the x–_x plane and x q plane for c = 0.5, 0.0 and +0.5, respectively. The other parameters are fixed and given in each figure captions. As stated before, only in case c = +0.5 (f = 1), there occurs chaos in the Einstein–Friedmann equation [Case 1]. Also, we show the chaotic behaviours in the Einstein–Friedmann equation [Case 1] in Figs. 9 and 10, where the figures are demonstrated in the x–_x plane and x q plane for p = 0.28, 0.30, 0.32 and 0.34, respectively. The other parameters are fixed and given in each figure captions. These figures show the self-similarity, which is one of the typical properties of chaotic behaviours [12,13].
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Fig. 1. The numerical solutions in the Einstein–Friedmann equation [Case 1: the damping type]. b = 0.03. Figures are shown in the t–x plane, the t–_x plane and the t–q plane, from the upper to the lower, respectively. The parameters are given as follows: a = 2.5, b = 0.03, c = 0.5, d = 0.25, p = 0.3, x0 = 0.608, x_ 0 ¼ 0:3, and q0 = 0.3.
4. Discussions In this section, we give some discussions on the related problems. 4.1. The Higgs-type W-shape potential and the classification of solutions In case p = const., by integrating the Einstein–Friedmann equation [Case 1], we have obtained the potential equation ( 1 ð_xÞ2 þ U ðxÞ ¼ const:0 ; 2 U ðxÞ ¼ 14 ðk jpÞx2 8f x4 ;
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Fig. 2. The parameter c dependence of solutions in the Einstein–Friedmann equation [Case 1: the damping type]. b = 0.03. Figures are shown in the x–_x plane for c = 0.5, 0 and +0.5, respectively. The other parameters are fixed and given as follows: a = 2.5, b = 0.03, d = 0.25, p = 0.3, x0 = 0.608, x_ 0 ¼ 0:3, and q0 = 0.3.
where const.0 is defined as follows: Z 5 ð_xÞ3 const:0 ¼ const: þ dt: x 2 The W-shape function U = U(x) may be considered as the effective potential in the Einstein–Friedmann field [14]. With the use of this W-shape function, we have classified the solutions in the Einstein–Friedmann equation. Here, we show the non-trivial case [Case 1: f = 1]. (i) k < jp (a) const.0 > 0; vibrational but unphysical, (b) const.0 = 0; critical, (c) 1/8(k jp)2 < const.0 < 0; vibrational and chaotic,
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Fig. 3. The parameter c dependence of solutions in the Einstein–Friedmann equation [Case 1: the damping type]. b = 0.03. Figures are shown in the x–q plane for c = 0.5, 0 and +0.5, respectively. The other parameters are fixed and given as follows: a = 2.5, b = 0.03, d = 0.25, p = 0.3, x0 = 0.608, x_ 0 ¼ 0:3, and q0 = 0.3.
(d) (e)
const.0 = 1/8(k jp)2; critical, const.0 < 1/8(k jp)2; no solutions.
(ii) k = jp (a) const.0 > 0; vibrational but unphysical, (b) const.0 = 0; critical, (c) const.0 < 0; no solutions. In this paper, we have checked these classifications of solutions by the numerical calculations. Especially, we have used the values of parameter k = 0.06 and 0.14, which satisfy the condition k < j p = 0.5 0.3 = 0.15. More details on the theoretical classification of solutions will be discussed elsewhere.
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Fig. 4. Chaotic behaviors in the Einstein–Friedmann equation [Case 1: the damping type]. b = 0.03. Figures are shown in the x–_x plane. The parameter p is changed from p = 0.28 to 0.34. The other parameters are fixed and given as follows: a = 2.5, b = 0.03, c = +0.5, d = 0.25, x0 = 0.608, x_ 0 ¼ 0:3, and q0 = 0.3.
4.2. The role of the cosmological constant (k) In 1917, Einstein introduced the cosmological constant (k) and discussed the role of its constant [9]. Since then, many authors have discussed this problem [15,16]. In this subsection, we comment on the cosmological constant (k). (1) Experimentally, the cosmological constant (k) is observed to be very small [17]; kðexpÞ 3:0 1057 cm2 Þ: With the use of the values given in the works by Einstein [9,11], the cosmological constant is calculated to be as follows: jq 3:3 1055 ðcm2 Þ: kðtheorÞ ¼ 2 pffiffiffiffiffiffiffiffi Following the Einstein’s expressions [9,11], k = 0 means q = 2k/j = 0 and R ¼ 1=k ¼ 1, where the symbol R denotes the averaged radius of the universe. Inversely, if R = 0 then k = 1 and q = 1. It is important to note that the observed value of the cosmological constant (k) is very small but not zero. (2) From theoretical point of view, as shown in the previous subsection, the cosmological constant (k) plays an important role in introducing the Higgs-type W-shape potential [14] and in discussing the chaotic behaviours in the gravitational field. As discussed in the previous subsection, the effective potential curve (U) in the Einstein–Friedmann field is classified in the three types; (i) k < jp, (ii) k = jp, and (iii) k > j p. This classification may correspond to that of the Higgs-type W-shape potential curves given by Perkins [14]; (i) l2 < 0, (ii) l2 = 0, and (iii) l2 > 0, where the symbol l denotes the particle mass, and the Higgs Lagrangian is given as follows:
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Fig. 5. Chaotic behaviors in the Einstein–Friedmann equation [Case 1: the damping type]. b = 0.03. Figures are shown in the x–q plane. The parameter p is changed from p = 0.28 to 0.34. The other parameters are fixed and given as follows: a = 2.5, b = 0.03, c = +0.5, d = 0.25, x0 = 0.608, x_ 0 ¼ 0:3, and q0 = 0.3.
(
L ¼ 12 ðom /Þðom /Þ Uð/Þ; U ð/Þ ¼ 12 l2 /2 þ 14 K/4 ;
Here, the symbol / and K show the scalar field and the constant. With the use of this W-shape potential, Perkins has discussed the Higgs mechanism [18] and the mass of elementary particles [14]. (3) Einstein pointed out the gravitational field plays an essential role in the structure of elementary particles [10] but the complete theory of the gravitational field is not yet determined [11]. In the previous subsection (Section 4.1), we have shown that the Einstein–Friedmann equation leads to the Higgstype W-shape potential [14], which plays an essential part in the Higgs mechanism [18]. 4.3. Comments on the model and dimension of the universe Recently, El Naschie has studied the hierarchy structure of the string model and obtained the hierarchy dimension formula of the Heterotic string [19]; 4m 1 Gm ¼ N SM ð/Þm ¼ 10 ; ðm ¼ 0; 1; 2 Þ; / where the symbols N SM ¼ a0 =2 and / denote the elementary particle number in the standard model and the golden ratio, respectively. In case m = 0, 1 and 2, this formula gives the coupling constant; G0 ¼ 10ð1=/Þ4 ¼ a0 =2, G1 ¼ 10ð1=/Þ3 ¼ ag and G2 ¼ 10ð1=/Þ2 ¼ ags . Moreover, in case m = 6 above formula gives the classical dimension of the universe G6 ¼ 10ð/Þ2 ¼ 4 k ¼ Dð4Þ ;
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Table 3 The numerical solutions in the Einstein–Friedmann equation [Case 1: the damping type] t 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29
x
y
q
0.5
0.2
0.2
0.508 0.516224 0.524680032 0.533376427 0.542321817 0.551525145 0.560995666 0.570742946 0.580776868 0.591107623 0.60174571 0.612701928 0.623987356 0.635613347 0.647591497 0.659933621 0.672651715 0.685757912 0.699264426 0.713183489 0.727527267 0.742307771 0.757536737 0.773225503 0.789384847 0.806024811 0.823154492 0.8407818 0.85891318
0.2056 0.211400804 0.217409859 0.223634758 0.230083211 0.236763013 0.243682007 0.250848035 0.258268879 0.265952194 0.273905427 0.282135715 0.290649771 0.299453753 0.308553099 0.317952347 0.327654915 0.337662858 0.347976571 0.358594465 0.369512583 0.380724164 0.392219145 0.403983603 0.415999109 0.428242023 0.440682683 0.453284517 0.466003059
0.224 0.249449071 0.276449902 0.305113289 0.335558811 0.367915558 0.402322917 0.438931426 0.477903681 0.519415316 0.563656052 0.610830812 0.661160911 0.714885307 0.772261927 0.833569046 0.899106717 0.969198243 1.044191665 1.12446125 1.210408936 1.302465709 1.401092846 1.506782957 1.620060755 1.74148344 1.871640575 2.011153301 2.160672714
b = 0.07. For convenience, only the first 30 lines are shown.
where symbols D(N) and k = /3(1 /3) show the dimension and the constant, respectively. If neglecting the fractal part (k/4 = 0.045 1), we have D(4) 4. As is well known, the Kalza–Klein theory is a model which attempts to unify the two fundamental forces of the gravitation and the electromagnetism. This theory is extended by replacing the electromagnetic field by the more general gauge field, which is specified by the Lie group [17]. One of the extended models, which unify the strong interaction and weak-electro interaction is obtained by using the unitary groups; SU(3) SU(2) U(1). In this case, the dimension of the universe is 10; Dim = 4 + 6 = 10. On the other hand, with the use of the hierarchy dimension formula, El Naschie has obtained the relation Dð4Þ þ Dð6Þ ¼ ð4 kÞ þ ð6 þ kÞ ¼ 10 ¼ Dð10Þ : The special role of D(10) = 10 has been discussed by El Naschie. We summarize these models and dimensions of the universe as follows: (1) (2) (3) (4)
Dim = 4 1 = 3; Newton model. Dim = 4; Einstein model. Dim = 4 + 1 = 5; Kalza–Klein model. Dim = 4 + 6 = 10; Super string model.
Thus, by using the E-infinity theory of El Naschie [19], we can systematically understand the model and dimension of the universe, on the basis of Einstein model (Dim = 4).
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Fig. 6. The numerical solutions in the Einstein–Friedmann equation [Case 1: the damping type]. b = 0.07. Figures are shown in the t–x plane, the t–_x plane and the t–q plane, from the upper to the lower, respectively. The parameters are given as follows: a = 2.5, b = 0.07, c = 0.5, d = 0.25, p = 0.3, x0 = 0.5, x_ 0 ¼ 0:2, and q0 = 0.2.
4.4. Comments on the Hausdorff dimension of E-infinity space In this subsection, we comment on the Hausdorff dimension (DH) of E-infinity space. (1) The dimension of the universe As stated in the previous subsection (Section 4.3), El Naschie has obtained the Hausdorff dimension (DH) of E-infinity space [19]; DH ¼
1 X n¼0
nð/Þn ¼
1 ð/Þ3
¼ 4 þ ð/Þ3 ¼ 5 2ð/Þ2 :
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Fig. 7. The parameter c dependence of solutions in the Einstein–Friedmann equation [Case 1: the damping type]. b = 0.07. Figures are shown in the x–_x plane for c = 0.5, 0 and +0.5, respectively. The other parameters are fixed and given as follows: a = 2.5, b = 0.07, d = 0.25, p = 0.3, x0 = 0.5, x_ 0 ¼ 0:2, and q0 = 0.2.
This leads to the dimension of the Einstein space (Dim = 4) and also to that of the Kalza–Klein space (Dim = 5), by neglecting the fractal part in the above equation. (2) The Cabibbo angle (hC) Recently, El Naschie has studied the electro-weak theory and obtained the Weinberg angle expression sin2 hW ðtheorÞ ¼ cos
3p ¼ 0:2225; 7
which is in nice agreement with the experiment [20]. sin2 hW ðexpÞ ¼ 0:2229 0:0004: Also, these values are almost equal to the Cabibbo angle [17] sin hC ðexpÞ ¼ 0:224951:
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Fig. 8. The parameter c dependence of solutions in the Einstein–Friedmann equation [Case 1: the damping type]. b = 0.07. Figures are shown in the x–q plane for c = 0.5, 0 and +0.5, respectively. The other parameters are fixed and given as follows: a = 2.5, b = 0.07, d = 0.25, p = 0.3, x0 = 0.5, x_ 0 ¼ 0:2, and q0 = 0.2.
From E-infinity theoretical point of view [19], these values are given approximately by the inverse of the Hausdorff dimension of E-infinity space (DH) 1 ¼ ð/Þ3 ¼ 0:236067: DH Here, with the use of E-infinity theory [19], we discuss the Cabibbo angle. We start from the identity of E-infinity theory with the Fibonacci sequence {an}; an1 ð/Þnþ1 þ an ð/Þn ¼ 1; and 1 an ¼ pffiffiffi 5
ðn ¼ 1; 2; 3 Þ;
( ) nþ1 1 n nþ1 ; þ ð1Þ ð/Þ /
ðn ¼ 0; 1; 2 Þ:
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Fig. 9. Chaotic behaviors in the Einstein–Friedmann equation [Case 1: the damping type]. b = 0.07. Figures are shown in the x–_x plane. The parameter p is changed from p = 0.28 to 0.34. The other parameters are fixed and given as follows: a = 2.5, b = 0.07, c = +0.5, d = 0.25, x0 = 0.5, x_ 0 ¼ 0:2, and q0 = 0.2.
These expressions are proved by the mathematical induction method. By using above relations, we have the angle formula sin2 hn an1 ð/Þnþ1 ¼ sin2 h1 f1 þ ð1Þn1 ð/Þ2n g; where the value sin2h1 is given as follows: 1 sin2 h1 ¼ lim sin2 hn ¼ pffiffiffi ð/Þ ¼ 0:276393: n!1 5 The Cabibbo angle (hC) is found to be given approximately by the angle h2 sin hC ðtheorÞ sin2 h2 ¼ ð/Þ3 ¼ 0:236067: More precisely, with the use of the correction formula D ¼ 47 sin hC ðtheorÞ ¼ ð/Þ3 1 ð/Þ6 ; ¼ 0:224825; 55
P
bn ð/Þn , we obtain the relation
which is in nice agreement with the experiment [17] cited before. The theoretical calculation of the Cabibbo angle (hC) is an interesting and an open problem at this stage [17]. Here, we have discussed the Cabibbo angle (hC) with the use of the identity of E-infinity theory [an1(/)n+1 + an (/)n = 1]. In this sense, we may say that E-infinity theory gives the theoretical foundation of the Cabibbo angle. More details on this problem will be discussed elsewhere. (3) The two-slit problem in quantum mechanics Recently, El Naschie has studied the two-slit experiment and discussed the foundation of E-infinity theory [21–23]. In the simple case, he has derived the equations
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Fig. 10. Chaotic behaviors in the Einstein–Friedmann equation [Case 1: the damping type]. b = 0.07. Figures are shown in the x–q plane. The parameter p is changed from p = 0.28 to 0.34. The other parameters are fixed and given as follows: a = 2.5, b = 0.07, c = +0.5, d = 0.25, x0 = 0.5, x_ 0 ¼ 0:2, and q0 = 0.2.
p1 þ p2 ¼ 1; p1 p2 ¼ p1 p2 ;
where symbols p1 and p2 denote the probability of passing through the slit 1 and slit 2, respectively. From these equations, he has obtained the solutions p1 / ¼ ; ð/Þ2 p2 and
p1 p2
2
3 /1 4 ¼ 2 5 : 1 /
The corresponding total probabilities are given as follows: P ¼ jp1 p2 j ¼ ð/Þ3 ¼
1 ; DH
and P ¼ jp1 p2 j ¼
3 1 ¼ DH : /
Moreover, the solutions (p1, p2) and ðp1 ; p2 Þ are related in the following way;
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p1 p2
"
¼
0
1 DH
D1H
0
#
p1 : p2
These are the results of the gedanken experiment of El Naschie [21–23]. It is important to note that the two-slit experiment leads to the golden ratio (/) and the Hausdorff dimension (DH) of E-infinity space. Thus, the Hausdorff dimension (DH) of E-infinity space plays important roles not only in the space–time physics but also in the elementary particle physics and quantum mechanics.
5. Summary Finally, we summarize our work as follows: With the use of the Higgs-type W-shape potential [U = (1/ 4)(k jp)x2 (1/8) fx4] [14], which has been derived by integrating the Einstein–Friedmann equation, we have analyzed the chaotic behaviours in the Einstein–Friedmann equation. In this paper, we have studied the k- dependence of solutions in the Einstein–Friedmann equation and found the upper limit of k(kmax = +0.14) for the occurrence of the chaotic behaviours in the Einstein–Friedmann equation. This upper limit satisfies the condition k < j p = 0.5 0.3 = 0.15, where the values j and p are given arbitrarily for convenience. We have shown the numerical and graphical solutions in the following cases; k = 2b = +0.06 and +0.14. We have discussed the space–time properties with the use of E-infinity theory [19]. Especially, by using the dimension formula of El Naschie Gm = 10(/)m4, we have discussed the model and dimension of the universe. We have also discussed the Cabibbo angle (hC) by using the identity of E-infinity theory.
Acknowledgements We would like to thank the Information Processing Center of Kyushu Kyoritsu University for using the Microsoft EXCEL(2003) which is very powerful for analyzing nonlinear equations. Thanks are also to Ms. H. Shiroshita for typing the manuscript of this paper. We would also like to thank Prof. Y. Takesue for reading the manuscript of this paper.
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