# Algorithmic Complexity in Cosmology and Quantum Gravity

^{1}

^{2}

^{*}

## Abstract

**:**

## 1 Introduction

- The cosmological appearance of a Universe with certain physical laws.
- The quantum fluctuations of physical laws at the level of the spacetime foam (e.g. at the Planck scale).

## 2 Kolmogorov’s algorithmic complexity

## 3 A toy model for the birth of Minkowski space

^{3}; $d{l}_{2}^{2}=d{u}^{2}+d{v}^{2}+d{w}^{2}$ is the metric of the extra dimensions (ED) which are also a flat E

^{3}space. The 7D Lagrangian is [4]

_{ED}= 0; G is the determinant of the 7D metric; R

_{7D}and R

_{4D}are the scalar curvature of the 7D and 4D spaces respectively. The Einstein vacuum field equations have the following form

_{1}= b

_{1}= l

_{Pl}(l

_{Pl}is the Planck length). This represents a collapsing 7D spacetime. The scalar curvature is

_{0,1}and b

_{0,1}the Ricci scalar is R ≈ 1/${l}_{Pl}^{2}$ when |t|≈ t

_{Pl}.

_{Pl}) we will assume that quantum fluctuations between spacetimes of different dimensions is more likely. Thus there should be some likelihood of a spontaneous transition from a 7D to a 4D spacetime, so that three of the extra spatial dimensions of the 7D spacetime become non-dynamical. Mathematically this is written as

^{2}= dt

^{2}− a

^{2}(t)dl

^{2}

- First, for t < 0 we have an empty 7D Kazner Universe (−∞ < t < 0) evolving according to (5). This solution is collapsing toward a singularity at t = 0.
- Second, at time |t| ≈ t
_{Pl}a quantum fluctuation of the dimensionality of spacetime takes place. This results in a quantum splitting off of the ED, i.e. three of the six spatial dimensions from the 7D Universe become non-dynamical resulting in an effective 4D Universe. - Third, the linear 4D scales (a
_{0}for 3D space and b_{0}for the other three EDs) become fixed, classical variables whose values are determined by the values they took just before the splitting off of the EDs. Thus we have a static, 4D, Minkowski Universe with three nondynamical EDs.

_{1,2}are respectively the AC of the multidimensional and the 4D Universes described by the algorithms (system of equations) (5) and (10). Since the system (5) is larger (i.e. more complex) than the system (10) we will assume ${K}_{1}\gg {K}_{2}$ (even in simple cases the detailed calculation of AC is a very complicated problem). Thus Eq.(12) can be approximate as follows:

## 4 Fluctuation of the metric signature

^{A}are the coordinates on the total space of the principal bundle with a structural group $\mathcal{G}$, and C, D are the multidimensional (MD) coordinate indices. The metric on the total space of the principal bundle (we will consider gravity on the principal bundle) can be rewritten

_{1,2}= 0.

#### 4.1 The 5D Fluctuating Universe

^{1}fibre over an S

^{2}base. In the 5-bein formalism we have

_{µν}is the 4D metric on the base; ${A}_{\mu}^{a}$ are the gauge fields of the group G (the off-diagonal components of the multidimensional metric); dl

^{2}= σ

^{a}σ

_{a}is the symmetric metric on the fibre a = 5, ··· , dimG is the index on the fibre and µ = 0, 1, 2, 3 is the index on the base.

^{−S}). Eq. (28) is similar, but with the AC replacing the action. The denominator normalizes the probability (it is a sum rather than integral since we are dealing with a discrete variable).

**Mixed system of the equations**Under the approximation where the probability associated with each of the equations in (24) is p ≈ 0 or 1 the mixed system of equations which describe a Universe fluctuating between Euclidean and Lorentzian modes

**The mixed origin of the Universe**The following model for the quantum birth of Universe has been advanced by Hawking : one begins with an Euclidean space of the Planck size (R

^{4}, S

^{4}or some other smooth non-singular Euclidean space); then a Lorentzian Universe emerges from a boundary of this initial Euclidean piece. In this scenario the Euclidean and Lorentzian spaces are connected by a hypersurface with a mixed signature.

^{+}and p

^{−}are the probabilities for the scalar curvature with σ = +1 and σ = −1 respectively. Using

_{Pl}). In our model the spacetime is M

^{4}× S

^{1}, with M

^{4}being a space with fluctuating metric signature: Euclidean ↔ Lorentzian. At some point a quantum transition to the Lorentzian mode occurs, and at the same or later time the 55 component of the metric becomes a non-dynamical quantity. Thus the fluctuation of the metric signature of the original Planck scaled, 5D Universe leads to a 4D Lorentzian Universe and a “frozen” or non-dynamical 5

^{th}dimension.

#### 4.2 The 7D Fluctuating Universe

_{1,2}are the MD and 4D Λ-constants; $N=dim\left(\mathcal{G}\right)$ . The MD action of Eq. (45) has several points in common with the 4D EYM action considered in Ref. [17] (non-zero cosmological constants and effective SU(2) “Yang-Mills” gauge fields). Eq. (45) also has a connection to the action for the Non-gravitating Vacuum Energy Theory [18]. In Ref. [18] Guendelman considers an action which has degrees of freedom which are independent of the metric, with the resulting action having two measures of integration (involving metric and nonmetric degrees of freedom). Eq. 45 incorporates two distinct degrees of freedom : the continuous variables, ${h}_{B}^{\overline{A}}$, and the discrete variables, ${\eta}_{\overline{A}\overline{B}}$. In Ref. [18] both the metric and non-metric degrees of freedom were continuous.

^{α}). ${e}_{b}^{\overline{a}}$ is defined as

^{b}are the coordinates on the group $\mathcal{G}$; ${\omega}^{\overline{a}}$ are the 1-forms satisfying

^{µ}).

^{3}sphere and x

^{0}= t, x

^{1}= χ, x

^{2}= θ, x

^{3}= φ, x

^{5}= α, x

^{6}= β, x

^{7}= γ. (α, β, γ are the Euler angles for the SU(2) group)

_{0}= const

**Mixed system of equations**The mixed system of equations for the 7D spacetime with fluc-tuating metric signature is

- Eq. (64a) implies a flat 4D Einstein spacetime that is not effected by matter.
- Eq. (64b) implies a Polyakov - ’t Hooft instanton gauge field configuration which is not effected by gravity.
- Eq. (64c) implies a frozen ED.
- Eqs. (64d)-(64e) imply that the dynamical equations uniquely determine the Λ
_{1,2}-constants.

_{1}and Λ

_{2}/b

^{3/2}) are inversely proportional to the size of the ED, b

_{0}. Thus in order to have a small cosmological constant term one needs to have a large ED. This could be seen as supporting the large extra dimensions scenarios [14].

#### 4.3 Physical applications of the solutions

#### 4.3.1 Regular Universe

#### 4.3.2 Non-singular birth of the Universe

_{Pl}) of our model 5D/7D Universe as a quantum birth of the regular 4D Universe. In contrast to other scenarios this origin has a metric signature trembling between Euclidean and Lorentzian modes. Further we postulate that on a boundary of this spacetime there occurs

- a quantum transition to only one Lorentzian mode with a fixed metric signature.
- a splitting off the ED so that the metric on the fibres (${h}_{b}^{\overline{a}}$) becomes a non-dynamical variable. After this splitting off the linear size of the gauge group remains constant yielding ordinary 4D Einstein-Yang-Mills gravity.

_{00}and the splitting off of the ED).

^{−S }

## 5 Algorithmic complexity applied to non-cosmological systems

#### 5.1 A composite 5D wormhole as the sum of Holographic principle and the AC idea

^{th}coordinate. The metric is symmetric around r = 0. The 5D vacuum Einstein equations are

_{0}> 0 and q are constants.

_{+}is the event horizon for the Reissner - Nordström solution. The quantities with the (0) subscript are evaluated at r = ±r

_{0}. Note that G

_{tt}(±r

_{0}) = 0 and ds

^{2}= 0 on the surfaces r = ±r

_{0}. Hypersurface such as r = ±r

_{0}have been called T−horizons by Bronnikov [11].

_{χt}can be connected to the 4D electric field by examining the 5D (R

_{χt}= 0) and 4D Maxwell equations

_{r}is the 4D electric field. These two equations are essentially Gauss’s law; they indicate that some quantity multiplied by an area is conserved. In 4D this quantity is the 4D Maxwell electric field. We can naturally join this 4D, Reissner - Nordström electric field, ${E}_{RN}=e/{r}_{+}^{2}$, with the Kaluza - Klein, “electrical” field, ${E}_{KK}={\omega}^{\prime}{\Delta}^{2}$, on the event and T−horizons

_{g}) = 0, where r

_{g}is the radius of the event horizon. Hence in this case we see that on the event horizon

_{g}). Condition (77) tells us that the derivative of the metric on the event horizon is expressed through the value of the metric on the event horizon. This shows that the Holographic principle applies in this case since the spacetime can be determined from information on some surface (the event horizon).

_{0}) = 0, and therefore from Eq. (74) we have

_{0}) where the T−horizons are located. This also indicates that the Holographic principle applies to the T−horizons.

#### 5.2 The AC of the Schwarzschild black hole.

_{g}is radius at the event horizon. Thus the metric on the whole Schwarzschild spacetime is defined by the value of the G

_{θθ}component of metric at the origin. The AC for the Schwarzschild metrics can be written as the sum of two quantities. The first quantity is connected with some Lorentz-invariant number which is related to the event horizon (the surface t = 0, R = 0). The second quantity is connected with the Einstein equations. We take the first quantity to be related to the area of the event horizon (4$\pi {r}_{g}^{2}$). We will divide this by 4$\pi {l}_{Pl}^{2}$ in order to obtain a dimensionless number. The second quantity is taken as the length of the program for calculating the metric. Thus the AC of the Schwarzschild black hole is given by the following expression :

_{g}/l

_{Pl})

^{2}] is the program length for the definition of the dimensionless number ${r}_{g}^{2}/{l}_{Pl}^{2}$ which is determined from some universal machine. L

_{Einstein}is the program length of the solution of Einstein’s differential equations using some universal machine, for example, the Turing machine. Finding an exact expression for the length, L, for determining the number (r

_{g}/l

_{Pl})

^{2}is a difficult problem. As a rough approximation we assume that each Planck sized cell, ${l}_{Pl}^{3}$ can contain one bit so that L[(r

_{g}/l

_{pl})

^{2}] ≈ (r

_{g}/l

_{pl})

^{2}. With this approximation we can compare the first term of Eq. (83) with the Bekenstein-Hawking equation

#### 5.3 Algorithmic complexity and the path integral

_{µν}is some arbitrary metric; K[g] is the AC for the metric g; Z[J] is a generating functional for quantum gravity.

^{2}- theory, Euclidean theory, etc.) have a smaller AC in comparison with random metrics. In this sense one can take gravitational instantons as the simplest gravitational objects: they are symmetrical spaces, with the corresponding metrics possessing the same symmetry group. One way of understanding why instantons have a small AC is that they can be determined via their topological charges rather than by the field equations. This greatly reduces the AC of such configurations.

^{2}- theories, multidimensional theories etc. The larger the AC of a given configuration the larger the order of approximation at which it contributes to the path integral. An interesting point is that for quantum gravity based on the integral (85) the Universe can contain different regions where different gravitational equations hold. An example of this is the composite wormhole discussed above.

## 6 Conclusions

## 7 Acknowledgment

## A Gravitational equations

_{1}, ${\Lambda}_{2}^{\prime}$ are the MD and 4D λ-constants. This Lagrangian is correct if the coordinate transformations conserve the topological structure of the total space (i.e. does not mix the fibres)

^{µ}and y

^{b}are the coordinates along the base and fibres respectively; (Greek indices)= 0, 1, 2, 3 and (Latin indices)= 5, 6, ··· , N; $\overline{A}=\overline{a}$, $\overline{\mu}$ is the viel-bein index; ${\eta}_{\overline{A}\overline{B}}$ = {±1, ±1, ··· , ±1} is the signature of the MD metric; ${\omega}^{\overline{a}}$ are the 1-forms satisfying to the structural equations

^{α}) [4]. All functions depend only on the point x

^{µ}on the base of the principal bundle as a consequence of the symmetry of the fibres.

^{N}det ${e}_{b}^{\overline{a}}$ is the volume element on the fibre and $\sqrt{G}=\sqrt{\left|g\right|}\sqrt{\left|\gamma \right|}$ is a consequence of the following structure of the MD metric

^{a}

^{α}) leads to

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Dzhunushaliev, V.; Singleton, D.
Algorithmic Complexity in Cosmology and Quantum Gravity. *Entropy* **2002**, *4*, 3-31.
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Dzhunushaliev V, Singleton D.
Algorithmic Complexity in Cosmology and Quantum Gravity. *Entropy*. 2002; 4(1):3-31.
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Dzhunushaliev, V., and D. Singleton.
2002. "Algorithmic Complexity in Cosmology and Quantum Gravity" *Entropy* 4, no. 1: 3-31.
https://doi.org/10.3390/e4010003