is the (3 + 1) − d nature of the universe a thermodynamic

7/25/2019 Is the (3 + 1) − d nature of the universe a thermodynamic

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Is the (3 + 1) − d nature of the universe a thermodynamic

necessity?

Julian Gonzalez-Ayala and F Angulo-Brown.

Departamento de Física, Escuela Superior de Física y Matemáticas, Instituto Politécnico Nacional,Edificio 9, CP 07738, México D. F.

E-mail: [email protected], [email protected]

Abstract

It is well established that at early times, long before the time of radiation-matter equality, the universe could havebeen well described by a spatially flat, radiation only model. In this letter we consider the whole primeval universe asa black body radiation (BBR) system in an n−dimensional Euclidean space. We propose that the (3 + 1) − d nature

of the universe could be the result of a kind of thermodynamic selection principle stemming from the second law of thermodynamics. In regard the three spatial dimensions we suggest that they were chosen by means of the minimizationof the Helmholtz free energy per hypervolume unit, while the time dimension, as it is well known was the result of theprinciple of increment of entropy for closed systems; that is, the so-called arrow of time.

In 1989, Landsberg and de Vos [1] proposed a spatialn−dimensional generalization of the Planck distribution,the Wien displacement, and the Stefan-Boltzmann laws forblack-body radiation (BBR) for a zero curvature space.Later, Menon and Agrawal [2] modified the n dimensionalStefan-Boltzmann constant found by Landsberg and de Vosby using the appropriate spin-degeneracy factor of the pho-ton without affecting normalized Planck spectrum given by

Landsberg and de Vos. Shortly thereafter, Barrow andHawthorne investigated the behavior of matter and radi-ation in thermal equilibrium in an n−dimensional space inthe early universe, in particular they calculated the numberof particles N , the pressure p and the energy density u [3].More recently, Gonzalez-Ayala et al. [4] calculated severalthermodynamic potentials for BBR such as the Helmholtzpotential F , the enthalpy H , the Gibbs potential G and theentropy S by means of the generalized Planck distributionfor an n−dimensional Euclidean space. Moreover, they cal-culated the corresponding densities per hypervolume unitfor these potentials; that is f , h, g and s respectively. As itis expected, g and G resulted equal to zero for any spatialdimensionality n, such as it happens for n = 3. The restof the potential densities s, f , and also the internal energydensity u are functions depending on the absolute temper-ature T and the dimensionality n. When all these func-tions were plotted against T and n they seemingly did notshow any interesting feature [4]. However, when a zoom wasmade in the region of very high temperatures (of the orderof Planck’s temperature T P ) and low dimensionality (in theinterval nǫ(1, 10)), a very surprising behavior was observed[4]. On the other hand, it is well established that at early

times, long before the time of radiation-matter equality, theuniverse could have been well described by a spatially flat,radiation-only model [5]. However, in times very close tothe Big Bang within the Planck scale, it is believed that therealm of string theory [6] or loop quantum gravity (LQG) [7]is found, where relativistic quantum gravity effects can bevery important. In the present letter we analyze some ther-modynamic properties of a universe dominated only by radi-

ation within a flat spatial geometry, beginning at the Planckscale and until times around t = 1011s, the time separatingthe radiation-dominated epoch from the matter-dominatedepoch [8]. Our analysis is based on an n-dimensional Eu-clidean space filled with BBR. As it was shown in [4], thiskind of system exhibits convexities for certain thermody-namic potential densities only in the region of very hightemperatures and low dimensionality. In particular, the in-ternal energy density u, and the entropy density s show inthe first place (as n increases) a local maximum and thena local minimum while in the Helmholtz free energy den-sity first a local minimum and then a local maximum. It isvery interesting that the first function that presents a criti-cal point is the Helmholtz potential density. These extremepoints are found within the context of isothermal processesin a plane of the corresponding potential density against thedimensionality n. Remarkably, the form of the isotherms isreminiscent of the isotherms found in some first order phasetransitions in gas state equations of the Van der Waals type,for example (see Fig. 2). If we start at Planck temperatureT P , the density f finds its minimum value at T ∗ = 0.93T P ,and the other two densities (u and s) present their extremepoints after this value. If n is considered as an integer, the

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minimum of f is located at n = 3. For any later moment,after the time corresponding to T ∗ = 0.93T P , the spatial di-mension n = 3 remains “frozen”. This extreme point occursin the unique isotherm corresponding to a saddle point inthe plane f vs n. Below this point a continuous transitionfrom n = 3 to any n = 3 is forbidden by the second law of thermodynamics and for all T > T ∗ = 0.93T P , any value of n is permitted (allowing for proposals of the type of stringtheories [6] or even of the type of the so-called vanishing di-mensions models [9, 10, 11]). This kind of thermodynamicbehavior offers a possible mechanism for determining the3 + 1 dimensional nature of the space-time.

The question of why is space 3−dimensional goes back toancient Greece [12]. In modern times this question was firstraised by Kant in 1746 [13]. Later, Ehrenfest in 1917 bymeans of the stable orbits argument showed that n = 3[14]. In 1983, Barrow brought forward a very interestingapproach to the dimensionality problem [12]. Since then,many authors have worked on this problem extended to thecase (3 + 1); that is, including time. Such is the case of

Brandenberger and Vafa [15], that in 1989 proposed a nat-ural mechanism for explaining why there are 3 large spacedimensions in the context of string gas cosmology [15, 16].Within the context of the (3 + 1) problem Tegmark pub-lished an enlightening article summarized through his Fig.1 [17]. For a deeper discussion on this issue one can also seethe following works [18, 17, 19, 20].

For a black body in an n-dimensional space it is known[1, 2, 3, 4, 21, 22] that the number of modes per unit fre-quency interval is,

2 (n − 1) πn

2 V ν n−1

Γn2

cn

, (1)

where V is the hypervolume and n the dimensionality of the space. The multiplication of the above equation by theBose-Einstein factor, which is the probability of finding aphoton with a frequency ν gives the number dN of photonsin the frequency interval ν to ν + dν ,

dN = 2 (n − 1) π

n

2 V ν n−1

Γn2

cn

ehν/kT − 1dν.

The integration of hνdN/V for all the frequencies gives theenergy density (per hypervolume unit) u (T, n), which is

[3, 4],

u (T, n) = 2 (n− 1) π

n

2 (kT )n+1 ζ (n + 1) Γ (n + 1)

cnhnΓn2

. (2)

The Helmholtz function F , the entropy S , the pressure pand the Gibbs free energy G are given by the following ex-pressions [4]

F = −V

nu, S =

n + 1

n

V u

T , p =

1

n

u, G = 0, (3)

respectively. Since the Helmholtz potential density will be-come very important in our analysis, the complete form willbe given [4],

f (T, n) = −2 (n− 1) π

n

2 (kT )n+1

ζ (n + 1)Γ (n + 1)

n cnhnΓn2

< 0.

The corresponding thermodynamic potential densities areimmediately obtained by dividing each quantity of Eq. (3)by the hypervolume V . All these results agree with the well-known 3-dimensional cases [23].

In a previous work [4] we showed that the thermodynamicpotential densities have regions of minima and maxima whenwe consider isothermal processes. In Figure 1 the criticalpoints of u (n, T ), f (n, T ) and s (n, T ) with respect to thedimension n (∂ nu , f , s = 0) are shown. These critical pointscan be minima, maxima or saddle points. Notice that foreach function there is a certain critical temperature T ∗ atwhich for any T > T ∗ there are no more extreme pointsof the corresponding thermodynamic potential density. The

region of transition between such convexities (from maximato minima or viceversa) occurs at dimensionalities in theinterval from 3 to 5 and at very high temperatures (nearlythe Planck temperature). In each case (see Fig. 1) theboundary between the maxima and the minima is the sad-dle point corresponding to the unique temperature T ∗. InFig. 1 it can be seen that by decreasing the temperaturefrom T = T P , the first function that reaches a critical pointis f (n, T ). The isotherm that passes by this point is thatwith T ∗ ≈ 0.93T P . For temperatures below this value thereare maxima and minima of the Helmholtz density functionand it is possible to talk about thermodynamic optimizationcriteria. That being said, the first function to optimize was

the Helmholtz free energy density and it happens in a regionnear the value n = 3. This might not be a mere coincidenceas will be pointed out later.

su

f

2 3 4 5 6 7

0.4

0.5

0.6

0.7

0.8

0.9

1.T

n

T

Figure 1:

∂

∂n f,s ,u = 0. The regions of maxima and minimaare well defined. The red continuous lines are the minimaand the dotted lines are the maxima of f , s and u. The bluevertical dashed lines are located at the unique saddle pointof each potential.

The separations between the regions of maxima and minimafor u (n, T ), f (n, T ) and s (n, T ) are well defined (see Fig. 1and it is also summarized in Table 1 of Ref. [ 4]).

Since the works by Kaluza and Klein [24, 25], many pro-posals about universe models with dimensionality different

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from three have been published [1, 2, 3, 9, 10, 11, 15, 16,24, 25, 26, 27, 28, 29, 30, 31]. Remarkably this has beenthe case of results stemming from the string, D-branes andgauge theories [15, 16, 26, 27, 28, 29, 30]. However, nowa-days we only have evidence for a universe with three spaceand one time dimensions.

Temperatures as high as 1032K are only possible in veryearly times in the evolution of the universe. It is knownthat this period was dominated by energy in the form of ra-diation, which means, that in this period the universe couldhave been a flat space [32]. Thus, considering the wholeprimeval universe as a black body radiation system in anEuclidean space is in principle a reasonable approach. Itis considered [33, 34] that the critical energy density at thePlanck epoch (t ≈ 1tP ≈ 10−44s, a Planck time) was aroundu ≈ 1E P /V P ≈ 109J/V P (an energy of Planck in a Planckvolume) and the temperature around T ≈ 1T P ≈ 1032K

. The temperature and energy density obtained when themaximum of the Helmholtz density is located at n = 3 (infact n ≈ 3.3, see Fig 2) are surprisingly close to these val-ues. For example, according to string theories at some point(at the end of the Planck epoch) the rest of the dimensionscollapsed (or they simply stayed at the same size) and onlythe 3−dimensional space grew bigger [6, 32]. The remainingquestion is: why did only 3 dimensions expand? If thermo-dynamic laws were born with the universe, then, thermo-dynamic analysis could give us a clue that a maximum ora minimum criteria would have been fulfilled at the earliestperiod of our universe. The conditions of energy and tem-perature in that epoch were probably suitable to maximize

some kind of thermodynamic function.

In Figure 2 we depict the isotherm curves corresponding tothe negative of the Helmholtz potential density. In fact, fromFigure 1, as the temperature decreases from T = 1T P tozero, the first potential that reaches a critical point (a saddlepoint) is the Helmholtz energy density. The first isothermthat reaches a critical point is the one with T ∗ = 0.93T P .Between T = 1T P and T ∗ there are no other critical points.After this moment, that is, for temperatures in the intervalfrom T < T ∗ to T = 0, there is always first a maximum

and then a minimum of this function. This kind of behav-ior resembles the form of the isotherms of a gas-liquid phasetransition. As is well known the critical isotherm (CI) corre-sponds to a saddle point which divides the pressure-volumeplane in two regions: above CI, the transitions do not occurand below CI, the transitions are permitted. However, as weshall see in the present case, in the region below the criti-cal isotherm T ∗ restrictions imposed by 2nd law appear andtransitions between dimensions are forbidden. With this inmind, in Figure 2 we can see the isotherms as possible tran-sitions in the space dimensionality.

timeT 1 T P

T 0.93 T P

T 0.854 T P

T 0.695 T P

T 1.03 T P

1 2 3 4 5 7 9 110.00

0.05

0.10

0.15

0.20

0.25

0.30

n

f T , n

Figura 2: The continuous line are the isothermal processes,and in dotted line the critical points. The highest extremepoint corresponds to the saddle point located at the isothermT = T ∗, above it there are no more extreme points.

It is very interesting to observe that the critical isotherm T ∗

in Fig. 2 divides the plane f − n in two regions; above T ∗

there are no restrictions over any particular value of n. How-ever, as mentioned before, below T ∗ there will be restrictionson transitions from n = 3 to any other dimensionality.

A well known theorem derived from 2nd law linked with theHelmholtz potential F is the Helmholtz potential minimumprinciple [35]. This theorem implies that for isothermal pro-cesses

∂F

∂V

T

≤ 0. (4)

where the equality is fulfilled by reversible processes. Let’s

consider a hypercube with volume V = Rn as the blackbody system, where R is the length of the edge. The aboveequation can be expressed as,

∂F

∂V =

∂f V

∂V = f + V

∂ f

∂n

∂n

∂V ≤ 0,

where it has been used the consideration that ∂f ∂R = 0. Then,

∂ (−f )

∂n ≥ f ln R. (5)

In an adiabatic expansion (no heat exchange between the

universe and the “exterior”, a reasonable assumption alreadyconsidered in standard cosmology), the condition showed in

[4] is pV n+1

n = constant. By using Eq. (2) and the ex-pression for p in Eq. (3), the adiabatic condition for twodifferent moments is

R = T 0R0

T =

T P lP T

, (6)

where R0 and T 0 are some initial conditions that, in ourcase, are very near the Planck epoch and lP ≈ 1.6×10−35mis the Planck length. This is a result that agrees with typical

3



treatments in cosmology (see for example [33]). Then, Eq.(5) is now (in Planck units)

∂ (−f )

∂n ≥ −f ln T. (7)

The difference between the dot-dashed line in Fig. 2 andthe restriction given by Eq. (7) (red dotted line in Fig. 2) is

very small. According to Eq. (7) for an isothermal processthe region below the dotted line is prohibited. That is, oncethe integer value n = 3 is reached at T ∗, for any T < T ∗

the space dimension of the BBR system remains frozen atthis dimension because, as it can be seen in Fig. (2), anytransition between n = 3 and n = 3 is not permitted by 2ndlaw.

As mentioned before, the first objective function reaching anextreme point was f (n, T ), but is it an adequate quantity tooptimize? Notice that when the internal energy is minimizedit is possible to have an infinite number of configurations,some with more order than others for the same value of the

energy. On the other hand, by maximizing the entropy itis possible to obtain an endless number of energetic config-urations, thus, by themselves neither the maxima nor theminima of s and u, respectively, determine an advantageousand unambiguous ob jective function. This is not the casefor f = u − T s, which gives a kind of trade off between en-tropy (organization) and energetic content, and this couldbe indeed a more meaningful optimization criterion. Whenmaximized, it offers the better configuration that gives theless amount of disorder for a given energy. Lets recall thatin stability analysis for closed systems, the stability pointsare found in the maximum value of entropy or in minimumenergy configurations. Each one on their own do not es-

tablishes an optimum criterion, because a restriction over u(or s) allows an infinite number of configurations of s (oru, respectively). Then, the minimum of the Helmholtz freeenergy gives a good commitment between the organizationof a system and its energetic content. In the scenario wherethe universe cools down through an adiabatic free expan-sion, the first potential whose optimization is reached is the−f > 0 function. The first maximum appears at n = 3.We emphasize the fact that the only isotherm with a saddlepoint (∂ nf = 0 and ∂ nnf = 0) is that with T = .93T P ,which could have fixed globally or locally the dimensional-ity of the black body system to n = 3. After this point, theadiabatic cooling process continues due to the expansion.

Nevertheless, any isothermal dimensionality “phase” tran-sition is forbidden as soon as the temperature diminishes,because for small times δt after, the change from n = 3 toany other value of n is inside the forbidden zone. It is possi-ble to see in Fig. 1.b that the saddle point of f (n, T ) is notlocated exactly in n = 3. This could be a consequence of having a simplified model. A possible correction using theresults of quantum gravity can be incorporated by meansof the modified dispersion relation [7] stemming from LQGtheories. In ref. [4] we found the modified Helmholtz freeenergy density f by taking into account the modified dis-

persion relation given in [7], obtaining the following result[4],

f (n, T ) = −2π

n

2 (n− 1) T n+1

Γn2

Γ (n + 1) ζ (n + 1)

n +

3

2 + n−1

2

αT 2Γ (n + 3) ζ (n + 3)

n + 2 +

α2

−

5

8 +

3 (n − 1)

4

+

α′

5

2 +

n − 1

2

T 4Γ (n + 5) ζ (n + 5)

n + 4

, (8)

where α and α′ take different values depending on the de-tails of the quantum gravity candidates. By manipulatingthe value of the parameters α and α′ in Eq. (8), which de-pend on the LQG model, it is possible to modify the value atwhich the saddle point occurs. In Fig. 3, it is depicted theanalysis of maxima and minima for the Helmholtz potentialdensity incorporating the LQG corrections. A similar anal-ysis of critical points reveal that the corresponding saddlepoint T ∗ decreases in one order of magnitude.

2.0 2.5 3.0 3.5 4.0

0.04

0.08

0.12

0.16

0.2

n

T

n

T

Figure 3: Maxima (dotted lines) and minima (continuouslines) of f incorporating LQG corrections. In the caseα = 0.1 and α′ = 2.28 (these values reproduce the curves of the spectral energy density showed in [7]) the saddle point

is located at n ≈ 2.44. In the case α = 0.15 and α′ = .001the saddle point is at n ≈ 3 (see vertical shaded lines).

In this way it has been proposed a model based on simplesuppositions to study scenarios that put the space dimen-sionality n ≈ 3 as a convenient candidate to optimize athermodynamic quantity: the Helmholtz potential density.After picking this value, later transitions were prohibitedby second law restrictions and the dimensionality remained

“frozen” at n = 3. In regard the 1 in (3 + 1)−d, is generallyaccepted that the arrow of time was imposed by the princi-ple of entropy increment for closed systems. Thus, possibly

the laws of thermodynamics selected the dimensions of theuniverse.

Acknowledgement. We want to thank partial support fromCOFAA-SIP-EDI-IPN and SNI-CONACYT, MÉ XICO.

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