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Journal of the Korean Physical Society, Vol. 57, No. 3, September 2010, pp. 578∼581

Emergent Universe from Noncommutative Spacetime

Jungjai Lee∗

Department of Physics, Daejin University, Pocheon 487-711

Hyun Seok Yang†

Institute for the Early Universe, Ewha Womans University, Seoul 120-750

(Received 4 February 2010)

The Big Bang, which was the birth of our Universe, happened at the Planck epoch. It was

not an event that developed in a pre-existing space-time. Rather, it was a cosmological event

simultaneously generating space-time as well as all other matter fields.

describe the origin of our Universe, it is necessary to have a background-independent theory for

quantum gravity in which no space-time structure is a priori assumed, but is defined from the

theory. The emergent gravity based on noncommutative gauge theory provides such a background-

independent formulation of quantum gravity, and the emergent space-time leads to a novel picture

of the dynamical origin of space-time. We address some issues about the origin of our Universe and

discuss the implications to cosmology of the emergent gravity.

Therefore, in order to

PACS numbers: 02.40.Gh, 11.10.Nx, 98.80.-k

Keywords: Emergent gravity, Noncommutative space-time, Cosmology

DOI: 10.3938/jkps.57.578

I. INTRODUCTION

Gravity is a mysterious force. In Newtonian gravity,

it is sourced by a mass mG. The fact that gravity is

generated by a mass is a mundane feature also shared

by other forces: an electric charge for the electromag-

netic force, isospins for the weak force and color charges

for the strong force. A mysterious and clandestine fea-

ture of gravity now arises from the fact that the mass

mG as a charge generating gravitational force, the so-

called gravitational mass, is equal to the inertial mass

mIappearing in the Newton’s law of motion, F = mIa.

The equivalence principle stating that mG= mIimplies

that gravity can be interpreted as an inertial force and is

clearly a universal force because every (massive) objects

must satisfy the Newton’s law, F = mIa, so they must

be subject to the gravitational force. Gravity influences

and is influenced by everything that carries a mass.

The importance of the equivalence principle was beau-

tifully perceived by Einstein. He realized that it is always

possible to locally eliminate gravitational force by a co-

ordinate transformation, i.e., by a local inertial frame.

That immediately leads to the remarkable picture that

gravity has to describe a space-time geometry rather

than a force immanent in space-time. Furthermore, it

turns out that any object carrying an energy should feel

∗E-mail: jjlee@daejin.ac.kr

†E-mail: hsyang@ewha.ac.kr

the gravitational force; thus, even a massless particle,

e.g., a photon, cannot be exempt from gravity because a

massless particle cannot be at rest, but carries a definite

momentum p or an energy E = |p|c.

Before Einstein, space-time only served as a stage

where physical events occurred. It never appeared as

an actor. However, the equivalence principle again guar-

antees the universality of gravity and, as coined by John

A. Wheeler, the conspiracy between matters and gravity

continues such that matter tells space-time how to curve,

and space-time tells matter how to move. The spacetime,

therefore, has to serve as a stage for the electromagnetic,

the weak and the strong forces as well as an actor for the

dynamical evolution of the stage (spacetime) itself. If

gravity is a fundamental force as many still think, why

is it so different from the other fundamental forces, and

what is meant by the “fundamental”?

We usually refer to a physical entity (force or field) as

being “fundamental” when it does not have any super-

ordinate substructure, but gravity reveals in many ways

that it may not be a fundamental phenomenon in the

above sense. It is quite amazing to notice that this pic-

ture was already inherent in the Cartan formulation of

gravity.

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Emergent Universe from Noncommutative Spacetime – Jungjai Lee and Hyun Seok Yang -579-

II. EMERGENT GRAVITY

In general relativity, the gravitational force is repre-

sented by a Riemannian metric of the curved spacetime

manifold M:

?∂

Cartan showed that the metric in Eq. (1) could be de-

fined by the tensor product of two vector fields Ea =

Eµ

∂s

?2

= gµν(x)

∂

∂xµ⊗

∂

∂xν.

(1)

a(x)

∂

∂xµ ∈ χ(TM) as follows:

?∂

Here, the vector fields Ea ∈ χ(TM) are smooth sec-

tions of the tangent bundle TM → M which are dual

to the vector space Ea= Ea

?Eb,Ea? = δb

avatar of gravity that a spin-two graviton might arise

as a composite of two spin-one vector fields. In other

words, Eq. (2) can be abstracted by using the relation

(1⊗1)S= 2⊕0. Incidentally, both mathematicians and

physicists are using the same word, vector field, in spite

of the slightly different meanings. (Mathematically, a

vector field X on a smooth manifold M is a derivation

of the algebra C∞(M).)

Equation (2) suggests that we need gauge fields taking

values in the Lie algebra of diffeomorphisms in order to

realize a composite graviton from spin-one vector fields.

To be precise, the vector fields Ea= Eµ

will be identified with “0-dimensional” gauge fields sat-

isfying the Lie algebra [1]

∂s

?2

= ηabEa⊗ Eb.

(2)

µ(x)dxµ∈ χ(T∗M); i.e.,

a. The expression in Eq. (2) glimpses the

a(x)

∂

∂xµ ∈ χ(TM)

[Ea,Eb] = −fabcEc.

Of course, the Standard Model does not have such kinds

of gauge fields, but it turns out [1–5] that the desired vec-

tor fields arise from electromagnetic fields living in non-

commutative (NC) space-time. Thus, a NC space-time

will allow a novel unification between electromagnetism

and Einstein gravity in a completely different context

from the Kaluza-Klein unification.

If gravity can arise from electromagnetism in NC

space-time, what does that mean physically?

means that gravity is not a fundamental force; instead,

it is an emergent phenomenon from gauge fields in NC

space-time. Emergence usually means arising of novel

and coherent structures, patterns and properties through

collective interactions of more fundamental entities: for

example, the superconductivity in a condensed matter

system or the organization of life in biology. The emer-

gence of gravity, if any, should lead to a novel and radical

picture about the dynamical origin of space-time [1].

According to Einstein, gravity is the dynamics of

space-time geometry. Therefore the emergence of grav-

ity necessarily requires the emergence of space-time it-

self. That is, space-time is not given a priori as usual,

(3)

That

but should be defined by the fundamental ingredients in

quantum gravity theory, say, “space-time atoms.” How-

ever, for consistency, the entire space-time, including a

flat space-time, must be emergent. In other words, the

emergent gravity should necessarily be background in-

dependent, where no space-time structure is a priori as-

sumed, but is defined from the theory. Furthermore, if

space-time is emergent, then all fields supported in this

space-time must also be emergent.

The question is how everything, including space-time,

gauge fields and matter fields, could be emergent col-

lectively. We know emergent phenomena in condensed

matter arise due to a very coherent condensation in vac-

uum, so in order to realize all these emergent phenomena,

the emergent space-time should originate from an ex-

tremely coherent vacuum, which is the lesson we learned

from condensed matter. This turns out [1–3] to be the

case if a flat space-time is emergent from a NC algebra,

such as quantum harmonic oscillators. As illustrated by

quantum mechanics, NC algebras, compared to commu-

tative ones, admit a much greater variety of algebraic

and topological structures. Likewise, when space-time

at a fundamental level is replaced by a NC algebra, al-

gebraic and topological structures in the NC space-time

become extremely rich and coherent, which will be re-

sponsible for emergent properties, e.g., diffeomorphisms,

gauge symmetries and matter fields [1,2].

Now, we will briefly sketch how emergent gravity can

achieve the background-independent formulation and re-

veal a radically different picture of the dynamical origin

of space-time [1]. The emergent space-time picture will

suggest very interesting implications to cosmology.

Consider the 0-dimensional IKKT matrix model whose

action is given by

SIKKT= −1

4Tr?[Xa,Xb][Xa,Xb]?.

(4)

Since the action in Eq. (4) is 0-dimensional, it does not

assume the prior existence of any space-time structure.

There are only a bunch of N × N Hermitian matrices

Xa(a = 1,··· ,2n), which are subject to a couple of

algebraic relations given by

[Xa,[Xa,Xb]] = 0,

[Xa,[Xb,Xc]] + [Xb,[Xc,Xa]] + [Xc,[Xa,Xb]] = 0. (6)

(5)

In order to consider fluctuations around a vacuum of

the matrix theory in Eq. (4), first one has to specify

a vacuum of the theory where all fluctuations are sup-

ported. Of course, the vacuum solution itself should also

satisfy Eqs. (5) and (6). Suppose that the vacuum so-

lution is given by Xa

Moyal NC space defined by

vac= ya. In the limit N → ∞, the

[ya,yb]?= iθab,

(7)

where θabis a constant matrix of rank 2n, definitely sat-

isfies the equations of motion, Eq. (5), as well as the

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Jacobi identity, Eq. (6). Furthermore, in this case, the

matrix algebra (MN,[·,·]) defining the action in Eq. (4)

can be mapped to the NC ?-algebra (Aθ,[·,·]?) defined

by the NC space in Eq. (7). To be explicit, we expand the

large N matrices Xa≡ θab?Dbaround the Moyal vacuum

?Da(y) = Babyb+? Aa(y),

the originally background-independent matrix model in

Eq. (4) reduces to the NC electromagnetism in 2n-

dimensions for the fluctuating fields in Eq. (8) around

the vacuum (7):

?

One can show that the NC ?-algebra (Aθ,[·,·]?) can

be mapped to the space of generalized vector fields by

considering an adjoint action of NC gauge fields?Da(y) ∈

ad? Da[?f](y) ≡ −i[?Da(y),?f(y)]?

∂yν

≡ Va[f](y) + O(θ3).

The leading term in Eq. (10) recovers the usual vector

fields, and the vector field Va(y) = Vµ

takes values in the Lie algebra of volume-preserving dif-

feomorphisms because ∂µVµ

mentioned before, the vector fields Va∈ χ(TM) are ex-

actly related to the orthonormal frames (vielbeins) Ea

in Eq. (2) by Va= λEa, where λ2= detVµ

we see that the NC space-time in Eq. (7) implements a

deep principle to realize a Riemannian manifold as an

emergent geometry from NC gauge fields through the

correspondence in Eq. (10) whose metric is given by

in Eq. (7) as follows:

(8)

where Bab = (θ−1)ab. Then, it is easy to see [3] that

?SIKKT=

1

4g2

Y M

d2ny??F −B?ab???F −B?

ab. (9)

Aθas follows [1–4]:

= −θµν∂Da(y)

∂f(y)

∂yµ

+ ··· ,

(10)

a(y)

∂

∂yµ ∈ χ(TM)

a = 0 by definition. As we

a. Therefore,

ds2= ηabEa⊗ Eb= λ2ηabVa

where Ea= λVa∈ χ(T∗M) are dual one-forms.

Using the correspondence in Eq. (10), if one confines

oneself to the leading order in Eq. (10) where one re-

covers the usual vector fields, one can show [1] that the

Jacobi identity in Eq. (6) is equivalent to the first Bianchi

identity for Riemann tensors, i.e., R[abc]d= 0, and that

the equations of motion in Eq. (5) for NC gauge fields

are mapped to the Einstein equations, Rab−1

8πGTab, for the emergent metric in Eq. (11).

One can trace the emergent metric in Eq. (11) back

to see where the flat space-time comes from. As we re-

marked before, the flat space-time is emergent from the

uniform condensation of gauge fields, giving rise to the

NC spacetime in Eq. (7). This is a tangible difference

from Einstein gravity, for which the flat space-time is a

completely empty space. Furthermore, because gravity

µVb

νdyµ⊗ dyν,

(11)

2gabR =

emerges from NC gauge fields, the parameters, g2

|θ| defining a NC gauge theory should be related to the

Newton constant G in emergent gravity. A simple di-

mensional analysis shows that

dimensions, this relation immediately leads to the fact

that the energy density of the vacuum in Eq. (7) is ρvac∼

|Bab|2∼ M4

Planck mass. Therefore, the emergent gravity reveals

a remarkable picture in which the huge Planck energy

MP is actually the origin of a flat space-time. This is

very surprising, but it should be expected from the back-

ground independence of the emergent gravity that a flat

space-time is not free gratis but is a result of Planck en-

ergy condensation in a vacuum. Hence, a vacuum energy

does not gravitate unlike Einstein gravity. It was argued

in Ref. 6 that this emergent space-time picture would be

essential to resolve the cosmological constant problem,

to understand the nature of dark energy and to explain

why gravity is so weak compared to other forces.

Y Mand

G?2

c2

∼ g2

Y M|θ|. In four

P, where MP= (8πG)−1/2∼ 1018GeV is the

III. UNIVERSE IN EMERGENT

SPACE-TIME

We have started with a background-independent ma-

trix theory in which no space-time structure was intro-

duced. A specific space-time background, viz., a flat

space-time, was defined by specifying the vacuum in Eq.

(7) of the theory. In this sense, the flat space-time is

emergent from the vacuum algebra in Eq. (7) induced by

a uniform condensation of gauge fields in vacuum. We

observed that the dynamical scale of the vacuum con-

densate is of the Planck scale. This result will lead to a

noble picture of the dynamical origin of space-time and

will have very interesting implications to cosmology.

In addition to the remarkable solution of the notorious

problems in theoretical physics already mentioned above,

our picture for the emergent space-time implies that the

global Lorentz symmetry should be a perfect symmetry

up to the Planck energy because the flat space-time was

emergent from the Planck energy condensation in vac-

uum - the maximum energy in our Universe. The huge

vacuum energy ρvac ∼ |Bab|2∼ M4

to make a flat space-time and so, surprisingly, does not

gravitate [1,6]! Actually, the vacuum algebra in Eq. (7)

describes an extremely coherent condensation because it

is equal to the Heisenberg algebra of an n-dimensional

quantum harmonic oscillator. As a consequence, the NC

algebra in Eq. (7) should describe a zero-entropy state in

spite of the involvement of Planck energy. It is very mys-

terious, but it should be the case, because a flat space-

time emergent from the algebra in Eq. (7) is a completely

empty space from the viewpoint of Einstein gravity and

so has no entropy.

If the vacuum algebra in Eq. (7) describes a zero-

entropy state, it can have a very important implication to

cosmology. According to our picture for emergent space-

Pwas simply used

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Emergent Universe from Noncommutative Spacetime – Jungjai Lee and Hyun Seok Yang -581-

time, a flat space-time is emergent from the Planck en-

ergy condensation in vacuum; thus, the time scale for the

condensate will be roughly of the Planck time. There-

fore, it is natural to expect that the explosive inflation

era that lasted roughly 10−33seconds at the beginning of

our Universe corresponds to a dynamical process for the

instantaneous condensation of vacuum energy ρvac∼ M4

to enormously spread out a flat space-time. Of course,

it is not clear how to microscopically describe this dy-

namical process by using the matrix action in Eq. (4).

Nevertheless, it is quite obvious that cosmological in-

flation should be a dynamical condensation in vacuum

for the emergence of (flat) space-time according to our

emergent gravity picture.

The emergence of space-time was caused by accumu-

lating Planck energy in vacuum, but Planck energy con-

densation caused the underlying space-time to be NC,

which introduced an uncertainty relation between space-

time. Therefore, a further accumulation of energy over

the NC space-time will be subject to the UV/IR mixing

[1]. This reasoning implies that the condensation of vac-

uum energy ρvac∼ M4

the same reason, cosmic inflation should take place only

once. Thereby, eternal inflation and cyclic universe seem

to be inconsistent with our picture [7].

Interestingly, the emergent gravity based on NC geom-

etry provides a natural concept of “emergent time” [1]

because a symplectic manifold (M,B) always admits a

Hamiltonian dynamical system on M defined by a Hamil-

tonian vector field XH, i.e., ιXHB = dH, described by

df

dt= XH(f) = {f,H}θfor any f ∈ C∞(M). The emer-

gent time can be generalized to the NC space in Eq. (7)

by considering the adjoint derivation in Eq. (10) instead

P

Phappens at most only once. By

of the Poisson bracket {f,H}θ. Time being emergent

in this way also implies a very interesting consequence.

Note that Bnis a volume form of symplectic manifold

(M,B). Therefore, the symplectic structure B also spec-

ifies an orientation of the spacetime manifold. Because

the time evolution of the manifold M is defined by the

Poisson structure θ = B−1, the overall time evolution

of spacetime will have a direction depending on the ori-

entation dt ∧ Bneven though a local time evolution has

time reversal symmetry. If gravity is emergent from elec-

tromagnetism supported on a symplectic manifold as we

have devised so far, it may be possible to explain the “ar-

row of time” in the cosmic evolution of our Universe - the

most notoriously difficult problem in quantum gravity.

ACKNOWLEDGMENTS

This work was supported by the Daejin University

Special Research Grants in 2010.

REFERENCES

[1] H. S. Yang, J. High Energy Phys. 05, 012 (2009).

[2] H. S. Yang, Int. J. Mod. Phys. A 23, 2181 (2009).

[3] H. S. Yang, Eur. Phys. J. C 64, 445 (2009).

[4] H. S. Yang, Europhys. Lett. 88, 31002 (2009).

[5] H. S. Yang and M. Sivakumar, arXiv:0908.2809.

[6] H. S. Yang, arXiv:0711.2797; arXiv:0902.0035.

[7] J. Lee and H. S. Yang, Universe or Multiverse?, in prepa-

ration.