Geometry and Probablity: Statistical Geometrodynamics with Holography

Transcription

1 EJTP 12, No. IYL15-34 (2015) Electronic Journal of Theoretical Phyic Geometry and Probablity: Statitical Geometrodynamic with Holography Koutubh Kabe Department of Mathematic, Gogte Intitute of Technology, Udyambag, Belgaum(Belgavi) Karnataka, India Received 25 October 2015, Accepted 29 November 2015, Publihed 15 December 2015 Abtract: The following paper make an endeavor to derive from cratch, the Eintein gravity from purely novel and ubtle et of tatitical anatz. The theory developed i that of tatitical geometrodynamic in cloe conjunction with the tandard tatitical thermodynamic, thereby extending the Eintein theory to three law involving the Holographic Principle in a totally different way. A Boltzmann-like formula i derived right after the econd law of geometrodynamic. The author hope that the following work will hed ome light on the fundamental undertanding of the underpinning of gravity and information in a nonconventional way. The ret of the latter part of the paper conit of dicuion and poible concluion that could be drawn by the author at the time of writing the paper. c Electronic Journal of Theoretical Phyic. All right reerved. Keyword: Statitical Thermodynamic, Statitical Geometrodynamic, Holographic Principle, Eintein Gravity, Bit Saturation PACS (2010): Dy, D, Mj, Lf The Holographic Principle worked out by thooft and independently by Sukind [1] ha played out a vital role in the undertanding of gravity. The work of Bouo (ee for example for a good urvey [2] and alo reference therein for an exhautive tudy of the Holographic Principle) ha given a definitive direction to the tudy of the entropy bound. Starting from the Bekentein bound ariing from the Geroch proce to the work of Sukind through hi own proce of tranforming any thermodynamic ytem into a blackhole and the application of thi tranformation to the Bekentein Generalized Second Law (GSL) yielding the more rigorou and general pherical entropy bound, one find a rather ubtle and elegant formulation through the light-heet formalim to the Bouo covariant entropy conjecture (alo look up in [2] in the reference ection under B for Bouo), now a theorem due to the proof provided by Flanagan et. al. [3]. Recently, kkabe8@gmail.com

2 84 Electronic Journal of Theoretical Phyic 12, No. IYL15-34 (2015) Verlinde ha made a revelation in gravitational phyic [4] by demontrating the emergence of Newtonian and Einteinean gravity in tep of approximation from the law of thermodynamic by the application of the Holographic Principle and bit dynamic. The work of Caticha [5](check alo reference in [5]) ha taken definitive tep in the direction of contruction of a tatitical theory of geometrodynamic. The preent work take a completely different route to derive the Eintein gravity and further extend the theory. The geometrodynamic theory i conidered from tatitical potulate applied to hypothetical fundamental contituent of curved pacetime. The theory of geometrodynamic thu derived i conidered independent of thermodynamic and yet a imilar theory in it own right with a et of quantitie analogou to entropy, thermodynamic probability, temperature, etc. There i thu no uch entropy bound here. Everything i in term of curvature and tatitical geometrodynamic. The paper i a bold attempt to pave way in the poitive direction of a fundamental undertanding of the origin of pace and time [6]. One alway arrive at the Eintein field equation and hi law of gravitation through the Principle of equivalence which aert the equivalence of acceleration and gravitation. Then, there i the Eintein-Hilbert variational principle from which alo one get the Eintein field equation of gravitation which read R μν 1 2 g μν R = T μν, (1) where, R μν i the Ricci curvature tenor, R the correponding calar, g μν i the fundamental or metric tenor and T μν i the energy momentum tenor. The above equation embodie the fact that pacetime i curved by the preence of matter and ma i made to move by the curved pacetime according to the warp of the pacetime. Thi involve tenor analytic manipulation of the Chritoffel ymbol of the econd kind thereby defining the Ricci tenor. Yet another approach i that of the Bianchi identity of the econd type which exhibit the principle of geometrodynamic that the boundary of a boundary of a boundary i zero. All thee approache involve determinitic and generically predictive propoition and interpretation. On the other hand, energymomentum tenor repreent matter and energy and thee obey quantum tatitical law. There are quantum tranition involved in the matter repreented by T μν. Thi quantum behavior hould correpondingly be accounted for by the pacetime geometry a well. If matter fluctuate tatitically then o hould geometry. If matter obey tatitical law then o hould geometry at the quantum cale. The law of phyical tatitic obeyed by T μν lead to tatitical thermodynamic. The law of phyical tatitic obeyed by the geometric part of the Eintein law viz., the l.h.. of eq (1) hould lead to tatitical geometrodynamic. Still, it i the aim of thi paper to not argue thi way by tarting with the Eintein law and the eq (1). The aim i to rather tart with a ubtle and imple et of tatitical potulate and anatz and arrive at variou reult. We begin by propoing the exitence of geomet hypothetical quantum or fundamental object of pacetime geometry which occupy different available geometrodynamic tate in the tatitical manner of peaking. The variou geometrodynamic ditribution are then

3 Electronic Journal of Theoretical Phyic 12, No. IYL15-34 (2015) arrived at by the ga of geomet. The reulting pacetime geometry i a direct conequence of the enemble of the ga of geomet occupying the geometrodynamic tate of different curvature probabilitie. Thi i our firt anatz. So, now there are two hypothetical object (i) geomet the fundamental contituent of curved pacetime and (ii) the geometrodynamic tate. Since entropy for gravity a a thermodynamic ytem i nonconcave, there are many poible geometrodynamic tate and end-tate. A uch, the geometrodynamic probability and the geometrodynamic ditribution function for a given enemble of a ga of geomet determining the curvature in the bulk of pacetime become N 2 =4 2 = 16 function and are tenor of rank two. We denote them repectively by Γ μν and f μν. Even though the number, n, of the bit available for the decription of the individual pecie of geomet i the ame (i equivalent to the number of matter or radiation particle) the non-concave nature of entropy for gravity make thee tenor. Now from our imple hypothei, we derive: (i) the Eintein law of geometrodynamic and follow it up with two additional law baed on the holographic principle in analogy with the law of thermodynamic and (ii) we derive a formula connecting the Gau-Bonnet mean curvature of the holographic urface bounding the bulk of the pacetime and the geometrodynamic probability Γ μν in the bulk. The quantity β appearing in tatitic i inverely proportional to the abolute temperature T in tatitical thermodynamic. In tatitical geometrodynamic, we define the geometrodynamic probability Γ μν by lnγ μν = f μν (n ) (2) Now, we fix the following anatz, n = N and n γ μν() = T μν (3) Here, γ μν() i the kinetic geometry of the pecie of the geomet. The kinetic geometry i defined a the geometry poeed by the geomet on account of it motion. Thi i a imple definition: moving bodie poe kinetic energy and moving element of curved pacetime geometry the geomet poe kinetic geometry. Thi fixe up an exact yet abtrue analogy between thermodynamic and geometrodynamic and further trengthen anatz (3) above. Pure tatitical geometrodynamic hould be geometrical in character. Any energy hould be tranlated into geometry and vice-vera. In fact, the firt condition of the anatz eq (3) i imply the number conervation but the econd condition of the anatz (3) i the Principle of Equivalence of Gravitation and Inertia in diguie if one think carefully. So, we fix up tenor multiplier a, α μν + βγ μν() = f μν n. (4)

4 86 Electronic Journal of Theoretical Phyic 12, No. IYL15-34 (2015) Then, δlnγ μν = f μν δn = n = α μν δn + β ( αμν + βγ μν() ) δn γ μν() δn (5) Since n i fixed by anatz (3), And from the econd part of (3), n δγ μν() + δn =0. (6) γ μν() δn = δt μν. (7) The firt term in (7) repreent the tretch in the pacetime; that i, the geometric work or in other word, the treing curvature mathematically equal to 1 2 Rg μν ;thati n δγ μν() = n γ μν() R δr = 1 2 g μνδr, (8) o that g μν = 2 γ μν() n.( (9) R Alo taking local derivative w.r.t. time we have by Hamilton Ricci flow g μν t = 2 n 2 γ μν() = 2R μν (g) ( (10) o n 2 γ μν() = R μν (g). (11) Now, the econd term γ μν() δn i the Ricci curvature which i the geometrodynamic warp accrued by the bulk of the pacetime, i.e., the curvature inide the holographic creen. Thu, δr μν = γ μν() δn. (12) From (7), (8) and (12), we have for the firt law of geometrodynamic, the Eintein field equation (1) which we rewrite here a (1) The firt law of geometrodynamic: T μν = R μν 1g 2 μνr. Alo, or n 2 γ μν() n 2 γ μν() = γ μν() δn = γ μν() δn. (13)

5 Electronic Journal of Theoretical Phyic 12, No. IYL15-34 (2015) On the other hand, δlnγ μν = β γ μν() δn = βδr μν. (14) Therefore, βδr μν i a total differential and β, the intergrating factor of the Ricci curvature. The tatitical theory lead naturally to the econd law of geometrodynamic which we enunciate a follow: (2) δr μν ha an integrating factor namely, δk μν =(1/N at )δr μν, (15) where, K μν i the Gau-Bonnet mean curvature of the horizon or the holographic creen/urface. N at i the bit aturation. Bit aturation i the ratio of the number of bit required to decribe the ytem, in the bulk of the pacetime including the bulk pacetime itelf, to the number of bit available on the holographic creen. Hence, for quantum gravitational nature of the theory and for dimenional conitency, we inert the Planck length, λ Pl,a 1 β =. (16) λ Pl N at So, or δlnγ μν = 1 λ Pl N at δr μν, (17) K μν = λ Pl lnγ μν. (18) According to Verlinde [4], N at = φ where φ repreent the Newtonian gravitational 2c 2 potential. Thi N at i a poitive number that vanihe at large ditance. So, thi coincide with the fact that the temperature alo decreae with the expanding univere. Therefore, we extend the analogy with the third law of thermodynamic and propoe that (3) The Gau-Bonnet mean curvature K μν of the evolving holographic urface tend to zero for pure gravity a the bit aturation N at tend to zero and become zero at zero bit aturation for the pure gravity ituation. Thu, we now have three law of tatitical geometrodynamic. The firt law i the Eintein law of gravitation a given by the Eintein field equation. The econd law connect the Ricci curvature (tenor) in the Eintein theory to the holographic creen/urface encompaing the Eintein bulk pacetime. And the third law i (more or le) purely about the holographic urface encloing the Eintein bulk. Another tatement for the third law of thermodynamic i that, in every irreverible thermodynamic proce, the total entropy of the univere alway increae. Similarly, (3 ) In every irreverible geometrodynamic proce, the Gau-Bonnet mean curvature of the holographic creen bounding the bulk pacetime alway increae. Now, for empty pacetime, the number of bit available in the bulk will be contant and that on the holographic creen will alo tay the ame. So, we have the zeroeth law of geometrodynamic, viz., (0) The bit aturation for a ytem of empty pacetime bulk bounded by a holographic creen, i a contant.

6 88 Electronic Journal of Theoretical Phyic 12, No. IYL15-34 (2015) Jut a temperature i a concept that hold macrocopically and break down at the individual molecular/atomic level, the bit aturation i a concept that play out a imilar role. Now, temperature i defined a the average kinetic energy of all the particle in a thermodynamic ytem. At the particle level, there i the individual kinetic energy of the moving particle. For the whole bulk of pacetime and it encloing Holographic creen, the bit aturation reduce to individual bit and thereby play out an equivalent role. The definite phyical boundarie of the object (matter) are blurred in the fine graining limit and vanih completely in the completely fine grained tructure. Similarly, the pacetime manifold defined by the l.h.. of (1) will alo diappear at the quantum cale where the tructure i ufficiently finely grained. Actually the anatz (3) omehow eem to how a fundamental equivalence between quantum geometry and quantum matter and radiation by etablihing an exact relationhip between quanta of geometry and quanta of matter. Reference [1] t Hooft, G. Dimenional Reduction in Quantum Gravity, Utrecht Preprint THU- 93/26,gr-qc/ ; Sukind, L. The World a a Hologram, hep-th/ ; t Hooft, G. The Holographic Principle, hep-th/ [2] Bouo, R. The Holographic Principle, hep-th/ [3] Flanagan, E. E., D. Marolf and R. M. Wald, Proof of Claical Verion of the Bouo Entropy Bound and of the Generalized Second Law, Phy.Rev.D62, hepth/ [4] Verlinde, E. On the Origin of Gravity and the Law of Newton, hep-th/ [5] Caticha, A. Toward a Statitical Geometrodynamic, in Decoherence and Entropy in Complex Sytem ed. By H.-T. Elze (Springer Verlag, 2004). gr-qc/ ibid., The Information Geometry of Space and Time, gr-qc/ [6] Merali, Z. The Origin of Space and Time, Nature 500, 516, 29 Aug 2013.