Based on constructive conversations with
Prof. Jungjai Lee (Daejin University).

Ether and Special Relativity
Remember that the 19 th century physicists introduced the Ether due to a deep trust of Newtonian mechanics when Maxwell theory was completed.
Maxwell’s electromagnetism predicts the existence of a universal velocity 𝑐 which contradicts the Galilean relativity 𝑣 ′ = 𝑣 + 𝑢
. Newtonian space and time needs a single universal frame of reference , Ether, to account the universal velocity.
For Einstein, the Maxwell theory implies a conceptual change about space and time instead of Ether: The concept of space or time is not absolute, but can differ depending on the observer's velocity and only the single entity, spacetime , has an invariant meaning.

Multiverse and Emergent Spacetime
I want to remark that the 20 th century physicists introduced the Multiverse picture due to a deep trust of general relativity when string theory was formulated.
There are basically two approaches to formulate a quantum theory of gravity. The first treats gravity as a fundamental interaction which it attempts to quantise. In the second approach gravity is not fundamental but an emergent phenomenon. String theory falls into the second category.
It has the gratifying feature that not only gravity but also the gauge interactions which are mediated by a spin one gauge boson are emergent. String theory thus provides a unifying framework of all elementary particles and their interactions:
………
M. Blau and S. Theisen, GRG 41 (2009) 743

Emergent Gravity
Basic idea
Gravity is not a fundamental force but a collective or emergent phenomenon from NC or large N matrices:
1 ⊗ 1
𝑆
⇄ 2 ⊕ 0 or ⊂ ⊗ ⊃ ⇄
○
In a market
(→): AdS/CFT correspondence, Matrix models, open-closed string duality, KLT relation, etc.
(←): Kaluza-Klein theory, String compactifications
I will be more radical: Generate spacetime as well as matter fields from NC or large N matrices.

Everything Is Emergent !
Gravity is emergent ⇒ Spacetime should be emergent too.
If spacetime is emergent, every structure supported on spacetime should be emergent too !
Matter fields and non-Abelian gauge fields (weak and strong forces) are emergent together with spacetime.
We need extra dimensions where a matter field is a stable localized geometry (i.e., a topological object in
K-theory) in extra dimensions.

1: Cosmological constant problem with the anthropic principle
2: Chaotic and eternal inflation
3: String landscape
I want to emphasize that the multiverse picture has been invited to cure some problems arising from the conventional (old) spacetime picture or derived from an effective theory based on the general relativity and QFT defined in a preexisting spacetime.
But I will assume that the birth of our Universe was a cosmological event happened at the Planck epoch simultaneously generating spacetime as well as all other matter fields.

Big-Bang: Arena of Quantum Gravity
Now I will borrow Hawking’s words (The Nature of Space and Time,
Princeton Univ. Press, 1996):
Unlike the black hole pair creation, one couldn’t say that the de
Sitter universe was created out of field energy in a preexisting space. Instead, it would quite literally be created out of nothing: not just out of the vacuum, but out of absolutely nothing at all, because there is nothing outside the universe.
If so, in order to pose the origin of our Universe, it is necessary to have a background-independent theory for quantum gravity in which any spacetime structure is not a priori assumed but defined by the theory itself.
Therefore, I want to test the multiverse idea with the assumption that spacetime as well as matter fields are emergent .

Basic Slogan
(
H. Elvang and J. Polchinski: hep-th/0209104
)
The emergence of gravity requires the emergence of spacetime itself.
Spacetime is not given a priori but defined by more fundamental ingredients of the underlying theory.
Emergent gravity should be BACKGROUND INDEPENDENT.

Background Independent Quantum Gravity
0-dimensional IKKT Matrix model: No a priori spacetime structure
𝑆 = −
1
8𝜋𝑔 𝑠
𝑇𝑟 𝑋 𝑎
, 𝑋 𝑏
2
Algebraic relations:
𝑋 𝑎
, 𝑋 𝑏
, 𝑋 𝑐
+ cyclic = 0,
𝑋 𝑎
, 𝑋 𝑎
, 𝑋 𝑏
= 0.
Specify a vacuum:
𝑋
⇔
𝐵 𝑎𝑏 𝑣𝑎𝑐
= 𝜃 −1 𝑎𝑏 𝑣𝑎𝑐 𝑎
≡ 𝐵 𝑎𝑏 𝑦 𝑏 satisfying 𝑦 𝑎 , 𝑦 𝑏 = 𝑖 𝜃 𝑎𝑏 where 𝑎, 𝑏 = 1, ⋯ , 𝑑.
Consider fluctuations around the vacuum
𝑋 𝑎
= 𝐵 𝑎𝑏 𝑦 𝑏 𝐴 𝑎
(𝑦)

Noncommutative Spacetime as Quantum Geometry
IKKT Matrix model = NC U(1) gauge theory on
ℝ 𝑑
𝑁𝐶
𝑆
𝑁𝐶
=
4𝑔
1
2
𝑌𝑀
∫ 𝑑 𝑑
𝐹 𝑎𝑏
− 𝐵 𝑎𝑏
2 where
𝐹 𝑎𝑏
= 𝜕 𝑎
𝐴 𝑏
− 𝜕 𝑏
𝐴 𝑎
𝐴 𝑎
𝐴 𝑏
.
NC spacetime: Quantize a symplectic (or Poisson) manifold
(𝑀, 𝐵)
à la Dirac 𝑓 𝑦 , 𝑔 𝑦 𝜃 𝜃
+ 𝑂 𝜃 3
∃ isomorphism between noncommutative ∗-algebra
𝐴 𝜃 generalized vector fields
Γ(𝑇𝑀)
: and
For
𝑋 𝑎 𝑦 = 𝐵 𝑎𝑏 𝑦 𝑏 𝐴 𝑎 𝑦 ∈ 𝐴 𝜃
,
𝑉 𝑎
𝑋 𝑎 𝑓 = 𝑉 𝑎 𝑓 + 𝑂 𝜃 3 .

“Lie algebra” homomorphism:
−𝑖 𝑋 𝑎 𝑦 , 𝑋 𝑏 𝑦 = −𝐵 𝑎𝑏
𝐹 𝑎𝑏 𝑦 ∈ 𝐴 𝜃
,
⇔ 𝑉
𝐹 𝑎𝑏
𝑉 𝑎
𝑉 𝑏 where
𝑉 𝑎 𝑋 𝑎
,
𝑉 𝑏
𝑋 𝑎 𝑦 , [𝑋 𝑏 𝑦 , 𝑋 𝑐
𝐷 𝑎
𝐹 𝑏𝑐 𝑦 ∈ 𝐴 𝜃
,
𝑋 𝑏
.
⇔ 𝑉
𝐷 𝑎
𝐹 𝑏𝑐
𝑉 𝑎
𝑉 𝑏
𝑉 𝑐
]
.
Einstein gravity arises from the leading order of NC gauge fields: 𝑑𝑠
2
= 𝐸 𝑎
⊗ 𝐸 𝑎
= 𝜆
2 𝛿 𝑎𝑏
𝑉 𝜇 𝑎
𝑉 𝜈 𝑏 𝑑𝑥 𝜇 𝑑𝑥 𝜈 where 𝑑𝑒𝑡
−1
𝑉 𝜇
𝑉 𝑎 𝑎
.
= 𝜆 𝐸 𝑎
∈ Γ 𝑇𝑀 , 𝐸 𝑎 = 𝜆𝑉 𝑎 ∈ Γ 𝑇 ∗ 𝑀 , 𝑉 𝜇 𝑎 𝑉 𝑏 𝜇
= 𝛿 𝑎 𝑏 and 𝜆 2 =
𝑋 𝑎
, 𝑋 𝑏
, 𝑋 𝑐
+ cyclic = 0 ⇔
𝑅 𝑎𝑏𝑐 𝑑
+ 𝑂 𝜃
3
= 0,
𝑋 𝑎 , 𝑋 𝑎
, 𝑋 𝑏
= 0 ⇔ 𝑅 𝑎𝑏
−
1
2 𝛿 𝑎𝑏
𝑅 + 𝑂 𝜃 3 = 8𝜋𝐺 𝑇 𝑎𝑏
+ 𝑂 𝜃 3 where 𝐺ℏ
2 𝑐 2
~ 𝑔
2
𝑌𝑀 𝜃
.



if space emerges, so should time.
Expect:
◦ Time is not fundamental.
◦ Approximate
(classical) notion of macroscopic time
◦ Time is fuzzy
(ill defined) near singularities.






We have
of derived time.
in time is more puzzling because of the relation to
Physics is about predicting the outcome of an experiment
it is performed
(causality). What do we do without time?
How can things
without time?
How is a
theory formulated?

Emergent Time from Noncommutative Space
Time in quantum mechanics: 𝑥 𝑖 , 𝑝 𝑗
= 𝑖ℏ 𝛿 𝑗 𝑖 𝑑𝑡
↔ 𝑈 𝑡 + 𝑓 𝑡 where
𝑈 𝑡 = exp ℏ
.
(Particle) Time is an inner automorphism of noncommutative algebra
𝐴 ℏ
.
Note that we have introduced a noncommutative space 𝑦 𝑎 , 𝑦 𝑏 = 𝑖 𝜃 𝑎𝑏 → 𝑌 𝑎 𝑦 , 𝑌 𝑏 𝑦 = 𝑖 Θ 𝑎𝑏
↔ 𝑈 𝑦, 𝑡 + 𝑓 𝑦, 𝑡 where
𝑈 𝑦, 𝑡 = exp ℏ 𝑑𝑡
.
(Space) Time is an inner automorphism of noncommutative algebra
𝐴 𝜃
. 𝑎𝑏
.

Anatomy of Spacetime
If gravity and spacetime are emergent, it says that
𝑋 𝑣𝑎𝑐 𝑎
= 𝐵 𝑎𝑏 𝑦 𝑏 →
𝑉 𝑎 𝑣𝑎𝑐
= 𝜕 𝑎
→ 𝑑𝑠
2
= 𝜂 𝜇𝜈 𝑑𝑥 𝜇
⊗ 𝑑𝑥 𝜈
.
Flat spacetime is coming from the vacuum
𝐵 𝑎𝑏 the vacuum does not gravitate.
Gauge/gravity correspondence: 𝐺ℏ 𝑐 2
2
~ 𝑔
2
𝑌𝑀 𝜃
.
= 𝜃
−1 𝑎𝑏 and
So 𝜌 𝑣𝑎𝑐
~ 𝐵
2 𝑎𝑏
~ 𝑀 4
𝑃 where
𝑀 2
𝑃
=
1
8𝜋𝐺 in 4 dimensions.
Flat spacetime is NOT free gratis BUT a result of Planck energy condensation in vacuum.
The flat spacetime and Lorentz symmetry must be very robust against any perturbations and gravitational fields should be very weak:
𝑋 𝑎
(𝑦) = 𝐵 𝑎𝑏 𝑦 𝑏 𝐴 𝑎
(𝑦)
→
𝑉 𝑎
= 𝑉 𝑎 𝜇
𝜕 𝜇
→ 𝑔 𝜇𝜈
= 𝜂 𝜇𝜈
+ ℎ 𝜇𝜈 𝑑𝑥 𝜇 ⊗ 𝑑𝑥 𝜈 .

No Cosmological Constant Problem !
Shift symmetry
:
= 𝑣𝑎𝑐 𝑎
1
4𝑔
2
𝑌𝑀
𝐵 𝑎𝑏 𝑦 𝑏
𝐵
2 𝑎𝑏
→
→ ds 2
′
= 𝜂
1
= + 2 Λ = 𝜈
4𝑔
2
𝑌𝑀 𝜇𝜈 𝑑𝑥 𝜇 ⊗ 𝑑𝑥
𝐵
2 𝑎𝑏 𝑎 𝜇
= Λ 𝑎 𝑏 𝑉 𝑏 𝜇
⇔
Global automorphism of 𝐴 𝜃
→ 𝐴 = Global Lorentz transformation .
Planck energy consensation in vacuum
⇒
NC space 𝑦 𝑎 , 𝑦 𝑏 = 𝑖𝜃 𝑎𝑏
→ Spacetime uncertainty principle
→ ∃
UV/IR mixing
⇒ Exclusion principle of spacetime
We cannot further accumulate Planck energy over NC space and so a vacuum condensation for spacetime creation can happen only once !

Dark Energy
Recall that a vacuum energy itself does not couple to gravity since it is used to create a flat spacetime in our picture. So let us calculate “vacuum fluctuation” energy with the largest possible wavelength 𝐿
𝐻 coming from
UV/IR mixing. 𝜌 = 𝜌 𝑣𝑎𝑐
+ 𝛿𝜌 =
1
4𝑔 2
𝑌𝑀
𝐵 𝑎𝑏
𝐹 𝑎𝑏 𝑦
2
=
1
4𝑔 2
𝑌𝑀
𝐵 2 𝑎𝑏
~ 𝑀 4
𝑃
𝐿
𝐿
2
𝑃
2
𝐻
2
~ 𝑀 4
𝑃
1
1 + + where we assumed that 𝐹(𝑦)~
𝐿
2
𝑃
𝐿
2
𝐻
1
𝐿
2
𝐻 due to a dimensional analysis.
Note that the vacuum fluctuation energy 𝛿𝜌 ~
1
2𝑔
2
𝑌𝑀
𝐵 𝑎𝑏
𝐹 𝑎𝑏 𝑦 ~
1
𝐿
2
𝑃
𝐿
2
𝐻 is a total derivative term and so a boundary term on a hypersurface of radius 𝐿
𝐻
. If 𝐿
𝐻
= the size of cosmic horizon, 𝛿𝜌 = 10 −3 𝑒𝑉 4 = dark energy ?
2

Inflation ?
I don’t know how to describe the dynamical process of the vacuum energy condensate to generate an extremely large spacetime.
But we know that the dynamical scale of the vacuum condensate is around the Planck energy
~ 𝑀
𝑃 and the time scale for the condensate is roughly
10 −44 ≤ 𝛿𝑡
𝐼
≤ 10 −33 .
A simple energy consideration for the condensate leads to the expansion rate
1
2 𝑣
𝐼
2 −
4𝜋𝐺 𝜌 𝑣𝑎𝑐
3
𝑅 2 = 0
⇔ 𝑣
𝐼
= 𝐻
𝐼
𝑅 where
𝐻
𝐼
=
8𝜋𝐺𝜌 𝑣𝑎𝑐
3 and 𝜌 𝑣𝑎𝑐
= 𝑀 4
𝑃
.
The dynamical process of the vacuum condensate may correspond to a cosmic inflation to generate an extremely large spacetime
~ 10 30 𝐿
𝐻
~ 𝑒 60 𝐿
𝐻
.

Personal Conclusion ?
The emergent spacetime picture seems to show a completely new perspective about the cosmological constant problem.
A background independent quantum gravity, if any, may cripple all the rationale to introduce the multiverse and there seems to be no reason to rely on the picture to explain physical laws.
Nevertheless a background independent theory for emergent gravity seems to imply an extremely large homogeneous and isotropic space but the observable (our Universe) is a tiny part
(
10 −30 ~ 10 −26 ) of entire spacetime.
(Maybe Level I multiverse according to M. Tegmark)