100 Years after Einstein A New Approach to Quantum Gravity

Transcription

1 100 Years after Einstein A New Approach to Quantum Gravity Y. M. Cho Administration Building 310-4, Konkuk University and School of Physics and Astronomy College of Natural Science, Seoul National University Korea December 21, 2015 Y. M. Cho (Seoul National University) Restricted Gravity December 21, / 58

2 Introduction A. What have we learned during the last 100 years? From the observational and phenomenological point of view there has been a huge progress, in particular in cosmology. From the pure theoretical point of view we have had the following reformalisms of Einstein s theory 1. Twistor formalism: Penrose (1960) provides a deeper understanding, but so far has added not much. Y. M. Cho (Seoul National University) Restricted Gravity December 21, / 58

3 2. Gauge formalism: Utiyama (1957), Kibble (1961), Cho (1976), Heyl (1976),... (a) Gauge theory of Lorentz invariant translation: PRD 14, 2521 (1976) shows that Einstein s theory can be viewed as a gauge theory of translation which is Lorentz invariant, which justifies the teleparallelism. (b) Gauge theory of Lorentz group: PRD 14, 3335 (1976) provides a new (and the best) way to re-discover the Einstein s theory. tells that the fermions creates torsion, and generalizes Einstein s theory to Einstein-Cartan theory. Y. M. Cho (Seoul National University) Restricted Gravity December 21, / 58

4 And the following generalizations 1. Higher-dimensional unification: Kaluza (1921), Jordan (1959), Cho (1975) 2. Supergravity: Zumino (1974), Superstring: Schwartz (1982), Witten (1983),... teaches that Einstein s theory should be generalized to include the scalar graviton, the Jordan-Brans-Dicke dilaton, which generates the fifth force. suggests that the dilaton could play the role of quintessence or inflaton, and explain the dark matter (even the dark energy). But the dilaton physics and the supersymmetric Einstein s theory are not complete yet! Y. M. Cho (Seoul National University) Restricted Gravity December 21, / 58

5 But during the last 10 years we had two remarkable developments which revealed the hidden structure of Einstein s theory fiber bundle formalism: Yoon (2004) treats Einstein s theory as an infinite-dimensional gauge theory of Diff S 2 defined on a 2-dimensional space-time manifold. Figure : The 2-2 fiber bundle. Y. M. Cho (Seoul National University) Restricted Gravity December 21, / 58

6 2. Abelian decomposition: Cho (2009) provides the anatomy of Einstein s theory. reveals that the core dynasmics of Einstein s theory is described by the restricted gravity (RG) obtained by the Abelian projection of Einstein s theory, which establishes the Abelian dominance in Einstein s theory. describes the gravity by an Abelian gauge potential, which suggests that the graviton could be a massless spin-one particle. Y. M. Cho (Seoul National University) Restricted Gravity December 21, / 58

7 On the quantum side Feynman (1967) started the quantum gravity identifying the metric as the quantum field of gravity. In this scheme the graviton is described by the massless spin-two field. But this approach has a critical defect because it is the tetrad (4 spin-one fields), not the metric, which couples to the fermionic matter field L int ( ψγ a µ ψ) e µ a. This strongly implies that the metric may not be the quantum field of gravity. Y. M. Cho (Seoul National University) Restricted Gravity December 21, / 58

8 B. What is the quantum field of gravity? Classically the tetrad and the metric describe the same physics. But in quantum theory they are totally different: the tetrad describes a spin-one graviton but the metric represents the spin-two graviton. On the other hand the tetrad may not be the ideal quantum field of gravity. Moreover, technically the tetrad is not easy to quantize. So the problem is not how to quantize the metric, but what is the correct quantum field of gravity. To answer this we need to understand the skeleton structure of Einstein s theory first. Y. M. Cho (Seoul National University) Restricted Gravity December 21, / 58

9 Questions and Plan 1. Is Einstein s theory the simplest possible generally invariant theory? No! 2. What is the simpler theory? Restricted gravity. 3. How can we obtain such gravity? Making the Abelian projection. 4. How can we describe the graviton in this theory? By a spin-one Abelian gauge field. Y. M. Cho (Seoul National University) Restricted Gravity December 21, / 58

10 Strategy 1 Treat Einstein s theory as a gauge theory of Lorentz group, and make the Abelian decomposition of the connection to the restricted and valence parts. 2 Make the Abelian projection to obtain the restricted gravity. 3 Express the restricted gravity by an Abelian gauge theory, and show that the graviton can be described by a massless spin-one gauge field. 4 Recover Einstein s theory adding the valence part, and establish the Abelian dominance in Einstein s theory. Example: QCD Y. M. Cho (Seoul National University) Restricted Gravity December 21, / 58

11 Einstein s Theory: Gauge Theory of Lorentz Group In the presence of spinor field physics must have the local Lorentz invariance. This necessitates a gauge theory of Lorentz group, where the tetrad plays the fundamental role. Einstein s theory can be viewed as a gauge theory of Lorentz group. Constructing a gauge theory of Lorentz group is a natural way to re-discover Einstein s theory. This demonstrates that the tetrad is more fundamental than the metric. Y. M. Cho (Seoul National University) Restricted Gravity December 21, / 58

12 Introduce a coordinate basis µ and an orthonormal basis e a [ µ, ν ] = 0, [e a, e b ] = fab c e c, e a = e a µ µ, µ = eµ a e a. (µ, ν; a, b, c = 0, 1, 2, 3) Let J ab = J ba be the generators of Lorentz group, [J ab, J cd ] = η ac J bd η bc J ad + η bd J ac η ad J bc = f mn ab,cd J mn, where η ab = diag ( 1, 1, 1, 1) is the Minkowski metric. Y. M. Cho (Seoul National University) Restricted Gravity December 21, / 58

13 With the 3-dimensional rotation and boost generators L i and K i we have [L i, L j ] = ɛ ijk L k, [L i, K j ] = ɛ ijk K k, [K i, K j ] = ɛ ijk L k. 1. The Lorentz group is non-compact, so that the invariant metric is indefinite. 2. The Lorentz group allows the dual transformation, because it has the totally antisymmetric invariant tensor ɛ abcd. 3. The Lorentz group has two commuting generators, because it has rank two. 4. The Lorentz group has two (two-dimensional) maximally Abelian subgroups. Y. M. Cho (Seoul National University) Restricted Gravity December 21, / 58

14 Let p (p ab ) be an adjoint representation of Lorentz group. Express p by two triplets, the magnetic m and the electric e which represents rotation and boost, p = 1 2 pab I ab = I ( m e ab cd ), p ab = p I ab = 1 2 pcd I ab = ( δc a δd b δ b c δd a Introduce the Lorentz covariant 4-index metric g ab µν g µν = gµν ab I ab = eµ a eν b I ab, gµν ab = (eµ a eν b eν a eµ b ) = eµ c eν d Icd ab, ). cd, Y. M. Cho (Seoul National University) Restricted Gravity December 21, / 58

15 Express the connection (potential) Γµ ab and the curvature (field strength) Rµν ab by Lorentz sextets Γ µ and R µν. Identify the Einstein-Hilbert action by the simplest Lorentz invariant action S[eµ a 1, Γ µ ] = e (g µν R µν ) d 4 x (e = Det eµ a ). 16πG N R µν = µ Γ ν ν Γ µ + Γ µ Γ ν. It is g µν which makes the Einstein-Hilbert action different from the Yang-Mills action. In this first order formalism we have two equations δe a µ : g µν R νa = 0, δγ µ : D µ g µν = 0 (D µ = µ + Γ µ ). Y. M. Cho (Seoul National University) Restricted Gravity December 21, / 58

16 The second equation is nothing but the metric compatibility of the connection, which follows from D µ g µν = 0 α g µν = 0, D µ e a ν = µ e a ν Γ α µν e a α + Γ a µb e b ν = 0. With this the first equation describes the Einstein s equation. It is the Lorentz covariant 4-index metric g µν which allows the action to be linear in R µν. Y. M. Cho (Seoul National University) Restricted Gravity December 21, / 58

17 Remember that 1. In the gauge formalism of Einstein s theory the connection Γ µ corresponds to the potential Aµ, and the curvature tensor R µν corresponds to the field strength F µν in gauge theory. 2. In Einstein s theory the metric g µν (not Γ µ ) propagates, but in gauge theory the potential Aµ propagates. 3. The Einstein-Hilbert action is linear in R µν, but in gauge theory the Yang-Mills action is quadratic in F µν. So the dynamics is different. Y. M. Cho (Seoul National University) Restricted Gravity December 21, / 58

18 Abelian Projection: SU(2) QCD A. Abelian projection What is the Abelian subdynamics of QCD, and how can we project out the Abelian part from the non-abelian part gauge independently? This is a very important question to understand the dynamics of QCD, in particular the confinement mechanism in QCD. Consider SU(2) QCD for simplicity. To make the Abelian decomposition choose ˆn to be the Abelian direction. Y. M. Cho (Seoul National University) Restricted Gravity December 21, / 58

19 Impose the isometry D µˆn = µˆn + g A µ ˆn = 0, A µ µ = A µˆn 1 g ˆn µˆn = A µ + C µ, A µ = A µˆn, C µ = 1 g ˆn µˆn. 1.  µ is made of two parts, the non-topological Maxwell part A µ and the topological Dirac part C µ. 2.  µ projects out the monopole part gauge independently, since ˆn represents the monopole topology Π 2 (S 2 ). Abelian (Cho) projection Y. M. Cho (Seoul National University) Restricted Gravity December 21, / 58

20  µ is essentially Abelian, but has a dual structure ˆF µν = µ  ν ν  µ + gâµ Âν = (F µν + H µν )ˆn, F µν = µ A ν ν A µ, H µν = 1 g ˆn ( µˆn ν ˆn) = µ C ν ν C µ, C µ = 1 g ˆn 1 µˆn 2. So ˆF µν is described by two Abelian potentials, the Maxwellian electric A µ and the Diracian magnetic C µ. Y. M. Cho (Seoul National University) Restricted Gravity December 21, / 58

21 With this we have the Cho-Duan-Ge (CDG) decomposition, also known as the Cho-Faddeev-Niemi (CFN) decomposition, A µ = A µˆn 1 g ˆn µˆn + X µ, (ˆn X µ = 0). Under the infinitesimal gauge transformation we have δâµ = 1 g ˆD µ α, δ X µ = α X µ. 1.  µ has the full SU(2) gauge degrees of freedom, and represents the neutral binding gluons (the neurons ). 2. Xµ transforms covariantly, and describes the colored valence gluons (the chromons ). Two Types of Gluons!!! Y. M. Cho (Seoul National University) Restricted Gravity December 21, / 58

22 B. Restricted QCD (RCD) and Extended QCD (ECD) Define RCD which describes the Abelian sub-dynamics of QCD by L RCD = 1 4 ˆF 2 µν = (F µν + H µν ) 2 = 1 4 F 2 µν + 1 2g F µν ˆn ( µˆn ν ˆn) 1 4g 2 ( µˆn ν ˆn) 2. It has the full non-abelian gauge invariance and thus inherits all topological properties of QCD. In particular, it has the monopole degrees explicitly. Abelian dominance: RCD is responsible for the confinement because the valence gluons are the colored source which have to be confined. Y. M. Cho (Seoul National University) Restricted Gravity December 21, / 58

23 From F µν = ˆF µν + ( ˆD µ Xν ˆD ν Xµ ) + g X µ X ν we have ECD L QCD = 1 4 F 2 µν = 1 4 ˆF 2 µν g 2 ˆF µν ( X µ X ν ) 1 4 ( ˆD µ Xν ˆD ν Xµ ) 2 g2 4 ( X µ X ν ) 2. So QCD becomes RCD which has the valence gluons as colored source. ECD has two gauge symmetries, the classical (slow) and quantum (fast) gauge symmetries. To fix the quantum gauge we can impose ˆD µ Xµ = 0. Y. M. Cho (Seoul National University) Restricted Gravity December 21, / 58

24 C. Monopole condensation and color confinement The recent KEK-Chiba and SNU-Konkuk lattice calculations based on the CDG decomposition proves that it is the monopole which confines the color in QCD. More importantly, it has been shown that the one-loop effective action of QCD generates a stable monopole condensation which establishes the color confinement. The Abelian dominance can not tell what is the confinement mechanism, because RCD is made of two potentials. Y. M. Cho (Seoul National University) Restricted Gravity December 21, / 58

25 = 8.52 log(< W[R,T]>)/T U eld T= Û eld T= M 1000 eld T= R Figure : The monopole dominance: SNU-Konkuk lattice calculation of confining potential based on the gauge independent Abelian projection. Y. M. Cho (Seoul National University) Restricted Gravity December 21, / 58

26 Vacuum Decomposition A. Vacuum potential How can we find the most general vacuum potential which satisfies F µν = 0? Impose the vacuum isometry i D µˆn i = ( µ + g A µ ) ˆn i = 0, i [D µ, D ν ] ˆn i = g F µν ˆn i = 0 F µν = 0. Construct the most general vacuum potential A µ ˆΩ µ = C k µ ˆn k = 1 2g ɛ k ij (ˆn i µˆn j ) ˆn k. Y. M. Cho (Seoul National University) Restricted Gravity December 21, / 58

27 With S 3 compactification of R 3, we have the multiple vacua n classified by the Hopf invariant Π 3 (S 3 ) Π 3 (S 2 ) which represents the knot topology of ˆn = ˆn 3, n = g3 96π 2 ɛ αβγ ɛ ijk CαC i j β Ck γ d 3 x. (α, β, γ = 1, 2, 3) With ˆΩ µ, the restricted potential  µ admits further decomposition  µ = ˆΩ µ + B µ, Bµ = (A µ + C µ ) ˆn, δ ˆΩ µ = 1 g D (0) µ α, δ B µ = α B µ, (D (0) µ = µ + g ˆΩ µ ). So ˆΩ µ (just like µ) forms its own SU(2) connection space. Y. M. Cho (Seoul National University) Restricted Gravity December 21, / 58

28 Figure : The structure of non-abelian connection space: It has two proper subspaces made of the restricted potentials µ and the vacuum potentials ˆΩ µ which form their own non-abelian connection spaces. Y. M. Cho (Seoul National University) Restricted Gravity December 21, / 58

29 B. Vacuum tunneling The multiple vacua n are physically (as well as topologically) inequivalent, but are unstable under the quantum fluctuation. They are connected by thevacuum tunneling through the instantons. The vacuum tunneling assures the existence of the θ-vacuum in QCD Ω = n e inθ n. The SU(2) results directly applies to Einstein s theory because SU(2) is the rotation subgroup of Lorentz group. Y. M. Cho (Seoul National University) Restricted Gravity December 21, / 58

30 Abelian Decomposition of Einstein s Theory Let A 2 be the maximal Abelian subgroup made of L 3 and K 3, and B 2 be the one made of (L 1 + K 2 )/ 2 and (L 2 K 1 )/ 2. Let one of the Abelian direction be p, and impose the isometry D µ p = ( µ + Γ µ ) p = 0. This automatically assures D µ p = ( µ + Γ µ ) p = 0. So we need to select only one isometry. Y. M. Cho (Seoul National University) Restricted Gravity December 21, / 58

31 The isometry is described by two Casimir invariants α and β, α = p p = m 2 e 2, β = p p = 2 m e, and we can always choose (α, β) to be either (±1, 0) or (0, 0). The A 2 isometry has (±1, 0), so that it can be called the rotation-boost (or non-lightlike) isometry. But the B 2 isometry has (0, 0), so that it can be called the null (or lightlike) isometry. Y. M. Cho (Seoul National University) Restricted Gravity December 21, / 58

32 A. A 2 isometry Express the A 2 isometry by ( ˆn l = l 3 = 0 ) ( 0, l = k3 = ˆn D µ l = 0, D µ l = 0, ), and find (α, β) = (1, 0). Find the restricted connection ˆΓ µ ˆΓ µ = Γ µ l Γ µ l l µ l = Γ µ l Γ µ l 1 2 (l µl l µ l), Γ µ = l Γ µ, Γµ = l Γ µ. Y. M. Cho (Seoul National University) Restricted Gravity December 21, / 58

33 The restricted curvature ˆR µν is given by ˆR µν = µ ˆΓν ν ˆΓµ + ˆΓ µ ˆΓ ν = (Γ µν + H µν ) l ( Γ µν + H µν ) l, Γ µν = µ Γ ν ν Γ µ, H µν = l ( µ l ν l) = µ Cν ν Cµ, Γ µν = µ Γν ν Γµ, Hµν = l ( µ l ν l) = 0, so that we have ˆR µν ab = (Γ µν + H µν ) l ab Γ lab µν. Y. M. Cho (Seoul National University) Restricted Gravity December 21, / 58

34 A 2 decomposition: Express the full connection of Lorentz group by Γ µ = ˆΓ µ + Z µ, l Z µ = l Z µ = 0, where Z µ is the valence connection. With this the full curvature tensor R µν is given by R µν = µ Γ ν ν Γ µ + Γ µ Γ ν = ˆR µν + Z µν, Z µν = ˆD µ Z ν ˆD ν Z µ + Z µ Z ν, ˆD µ = µ + ˆΓ µ. Y. M. Cho (Seoul National University) Restricted Gravity December 21, / 58

35 B. B 2 isometry Express the B 2 isometry by j = eλ 2 (l 1 + k 2 ) = eλ 2 ( ˆn1 ˆn 2 e j λ ( = (l 2 k 1 ) = eλ 2 2 D µ j = 0, D µ j = 0, ˆn2 ˆn 1 ), ), where λ is an arbitrary function. Find (α, β) = (0, 0). Y. M. Cho (Seoul National University) Restricted Gravity December 21, / 58

36 Let k = e λ 2 (l 1 k 2 ), k = e λ 2 (l 2 + k 1 ), l = j k, l = j k. With this find the restricted connection ˆΓ ˆΓ µ = Γ µ j Γ µ j k µ j = Γ µ j Γ µ j 1 2 (k µj k µ j) Γ µ = k Γ µ, Γµ = k Γ µ. Y. M. Cho (Seoul National University) Restricted Gravity December 21, / 58

37 The restricted curvature tensor ˆR µν is given by ˆR µν = µ ˆΓν ν ˆΓµ + ˆΓ µ ˆΓ ν = (Γ µν + H µν )j ( Γ µν + H µν ) j, Γ µν = µ Γ ν ν Γ µ, Γµν = µ Γν ν Γµ, H µν = k ( µ j ν k ν j µ k) = µ H ν ν H µ, H µν = k ( µ j ν k ν j µ k) = µ Hν ν Hµ. B 2 decomposition: Obtain the full connection and curvature tensor adding the valence part Z µ to ˆΓ µ, Γ µ = ˆΓ µ + Z µ, k Z µ = k Z µ = 0. R µν = ˆR µν + Z µν, Z µν = ˆD µ Z ν ˆD ν Z µ + Z µ Z ν. Y. M. Cho (Seoul National University) Restricted Gravity December 21, / 58

38 Restricted Gravity A. Restricted metric Decompose the Lorentz covariant metric g µν to the restricted and valence parts ĝ µν and h µν g µν = ĝ µν + h µν. Require h µν to be the part which is orthogonal to ˆR αβ h µν ˆR αβ = 0. This assures that only ĝ µν interact with the restricted gravity. Y. M. Cho (Seoul National University) Restricted Gravity December 21, / 58

39 B. A 2 gravity Impose the A 2 isometry and put Z µ = 0. Find ĝ µν = G µν l + G µν l, G µν = e a µ e b ν l ab, Gµν = e a µ e b ν l ab. Let = e Ŝ = e { ĝ µν ˆR µν} d 4 x { G µν (Γ µν + H µν ) G µν Γµν } d 4 x, Γ µν + H µν = µ A ν ν A µ, A µ = Γ µ + C µ. Γ µν = µ B ν ν B µ, (B µ = Γ µ ). Y. M. Cho (Seoul National University) Restricted Gravity December 21, / 58

40 Find the Maxwell-type equation of motion of A 2 gravity G µν ( ν A ρ ρ A ν ) G µν ( ν B ρ ρ B ν ) = 0, µ G µν = 0, µ Gµν = 0, ˆD µ ĝ µν = 0. Remarkably the restricted metric G µν admit gravitational potential G µ G µν = µ G ν ν G µ = µ G ν ν G µ. Compare this with Einstein s equation R µν 1 2 R g µν = 0. Y. M. Cho (Seoul National University) Restricted Gravity December 21, / 58

41 C. B 2 gravity Impose the B 2 isometry and put Z µ = 0. Find ĝ µν = J µν k J µν k, J µν = e a µ e b ν j ab, Jµν = e a µ e b ν j ab. Let = e Ŝ = e { ĝ µν ˆR µν} d 4 x { J µν (Γ µν + H µν ) J µν ( Γ µν + H } µν ) d 4 x, Γ µν + H µν = µ K ν ν K µ, K µ = Γ µ + H µ, Γ µν + H µν = µ Kν ν Kµ, Kµ = Γ µ + H µ. Y. M. Cho (Seoul National University) Restricted Gravity December 21, / 58

42 Find the Maxwell-type equation of motion of B 2 gravity J µν ( ν K ρ ρ K ν ) J µν ( ν Kρ ρ Kν ) = 0, µ J µν = 0, µ J µν = 0, ˆD µ ĝ µν = 0. Again the restricted metric J µν admits the gravitational potential J µ J µν = µ J ν ν J µ = µ J ν ν J µ. Y. M. Cho (Seoul National University) Restricted Gravity December 21, / 58

43 So the restricted metric becomes a composite field of the massless spin one potential, just as the Einstein s metric is a composite field of the tetrad g µν = e a µ e b ν η ab, g µν = e a µ e b ν I ab, ĝ µν = G µν l + G µν l, ĝµν = J µν k J µν k, G µν = µ G ν ν G µ, J µν = µ J ν ν J µ, µ G µν = 0, µ J µν = 0. The B 2 gravity describes the Einstein-Rosen-Bondi gravitational wave, which implies that the gravitational potential of the restricted metric can be interpreted as the quantum field of gravity. Massless spin-one graviton?! Y. M. Cho (Seoul National University) Restricted Gravity December 21, / 58

44 Notice that 1. Restricted gravity is generally invariant, but simpler than Einstein s gravity. 2. It describes the Abelian (dual) core dynamics of Einstein s gravity, with massless spin-one graviton. 3. It inherits all topological properties of Einstein s gravity. 4. In particular the restricted gravity and Einstein s gravity have identical vacuum. Abelian Dominance Y. M. Cho (Seoul National University) Restricted Gravity December 21, / 58

45 Question: It has been asserted that the spin one field can not describe the gravity, because it must generate both attractive and repulsive force. Is this true? 1. The gluons in QCD are spin one fields, but they generate the confining (attractive) force. 2. Einstein s theory is described by tetrad, the 4 spin one fields, but they generate only the attractive force. 3. So it is the dynamics, NOT the spin, which determines what force the field generates. Y. M. Cho (Seoul National University) Restricted Gravity December 21, / 58

46 Topology of Vacuum Space-time How can one obtain the most general vacuum space-time? Solving the vacuum Einstein s equation R µν 1 2 R g µν = 0 will not help, because we need the vacuum of quantum gravity (the flat space-time) R µν = 0. Impose the vacuum isometry and construct the most general vacuum connection. Classify the classical vacua using the isometry. Y. M. Cho (Seoul National University) Restricted Gravity December 21, / 58

47 Let ( ) ( ) ˆni 0 l i =, k 0 i = = l ˆn i, i ˆn 1 ˆn 2 = ˆn 3, (i = 1, 2, 3) and impose the vacuum isometry (the maximal isometry) i D µ l i = 0, i D µ k i = 0. Notice that D µ l i = 0, D µ k i = 0. Y. M. Cho (Seoul National University) Restricted Gravity December 21, / 58

48 Let p = ( m e ) ( Aµ, Γ µ = B µ ), and find in 3-d notation D µ p = 0 is written as D µ m = B µ e, D µ e = B µ m. So the vacuum isometry i D µ l i = 0 (and i D µ k i = 0) is written as i D µˆn i = B µ ˆn i, D µˆn i = B µ ˆn i, or equivalently i D µˆn i = 0, Bµ = 0! Y. M. Cho (Seoul National University) Restricted Gravity December 21, / 58

49 Obtain the most general vacuum connection ( ) ˆΩµ Γ µ Ω µ = 0 ˆΩ µ = 1 2 ɛ k ij (ˆn i µˆn j )ˆn k. This tells that the flat space-time has Π 3 (S 2 ) topology of the SU(2) QCD vacuum. This is nothing but the topology of Π 3 (SO(3, 1)) Π 3 (SO(3)). Knot Topology of Vacuum Space-time Y. M. Cho (Seoul National University) Restricted Gravity December 21, / 58

50 Physical Interpretation Consider a flat R 4 with a global coordinate basis µ. Choose the Minkowski metric g µν = η µν and require µ to be parallel to each other (i.e., Γ α µν = 0), µ ν = Γ α µν α = 0. Find the trivial connection Γ α µν = 0 is metric compatible and torsionless, α η µν = 0, C α µν = Γ α µν Γ (0)α µν = 0, where C α µν and Γ (0)α µν are the contortion and the Levi-Civita connection. Y. M. Cho (Seoul National University) Restricted Gravity December 21, / 58

51 Introduce a local orthonormal frame (i.e., tetrad) e a e i = e α i e 0 = e0 α α = 0 (e0 α = δ 0 α), α = ˆn i α α (ˆn i 0 = 0), (i = 1, 2, 3). Express the trivial connection Γµν α = 0 in the orthonormal basis. Find the corresponding Γµ ab becomes the vacuum connection, Γ ab µ = η αβ 2 ( e aα µ e bβ e bα µ e aβ) = Ω ab µ, Γ ij µ = 1 2 ˆni µˆn j, Γ 0i µ = 0, Γ µ = Ω µ Y. M. Cho (Seoul National University) Restricted Gravity December 21, / 58

52 So the flat connection Γ α µν = 0, in the orthonormal basis, becomes identical to the SU(2) vaccum potential. This confirms that the torsionless Minkowski space-time with flat connection has a non-trivial Π 3 (S 2 ) topology. It is the tetrad (i.e., the spin structure), not the metric, which describes the knot topology of the vacuum space-time. Y. M. Cho (Seoul National University) Restricted Gravity December 21, / 58

53 Knot is everywhere! 1 Non-linear sigma model (Faddeev and Niemi, Nature 1998) 2 Plasma (Faddeev and Niemi, PRL 1999) 3 Skyrme theory (Cho, PRL 2002) 4 Condensed matter Two-component BEC (Cho, PRA 2003) Two-gap SC (Babaev, PRL 2003; Cho, PRB 2004) 5 QCD Knot glueball (Cho, PLB 2005) QCD vacuum (Cho, PLB 2006) 6 Einstein s theory Vacuum space-time Knot in gravity? Y. M. Cho (Seoul National University) Restricted Gravity December 21, / 58

54 Space-time tunneling: Gravito-instantons are proposed, but never confirmed. With the tunneling, we can define the θ-vacuum in Einstein s theory. The restricted gravity could be very useful in describing the space-time of gravito-magnetic monopole. 1. Π 2 (S 2 ) topology 2. Energy quantization (cf. charge quantization) 3. Gravitational Bohm-Ahronov effect Y. M. Cho (Seoul National University) Restricted Gravity December 21, / 58

55 Einstein s Gravity Reactivate the valence connection Z µ in the restricted gravity to recover the full Einstein s theory. Find that Einstein s gravity is nothing but the restricted gravity which has the valence connection as a gauge covariant gravitational source. Conclude that the restricted gravity describes the skeleton structure and the core dynamics of Einstein s theory. Establish the Abelian dominance in Einstein s theory. Y. M. Cho (Seoul National University) Restricted Gravity December 21, / 58

56 Discussion Anatomy of Einstein s theory: Dissect and decompose it to the skeleton and the flesh. Find that the flesh (the valence connection) can not move (has no dynamical role). The skeleton can fly, and describes a restricted gravity which is much simpler than Einstein s gravity but has the full general invariance. Moreover it becomes Abelian, g µν J µν J µ (J µν = µ J ν ν J µ ), R µν 1 2 R g µν = 0 J µν ( ν K ρ ρ K ν ) J µν ( ν Kρ ρ Kν ) = 0. Y. M. Cho (Seoul National University) Restricted Gravity December 21, / 58

57 The restricted metric, the Abelian part of the metric, describes the restricted gravity. This establishes the Abelian dominance (of a different type) in Einstein s theory. Π 3 (S 2 ) topology of the tetrad (the spin structure) describes the topology of the vacuum space-time. Challenge: Identify the potential of the restricted metric as the quantum field of gravity, and quantize it to have the massless spin-one graviton. Y. M. Cho (Seoul National University) Restricted Gravity December 21, / 58

58 References 1. Y.M. Cho, PRD 14, 2521 (1976); 3335 (1976). 2. J. H. Yoon, PRD 70, (2004); CQG 31, (2014). 3. Y.M. Cho, PRD 21, 1080 (1980); PRL 44, 1115 (1980). 4. Y.M. Cho, PLB 644, 208 (2006). 5. Y.M. Cho, IJMPA 24, 3327 (2009); Y.M. Cho, S.H. Oh, and S.W. Kim, CQG 29, (2012); Y.M. Cho, Franklin H. Cho, and J.H. Yoon, CQG 30, (2013). Y. M. Cho (Seoul National University) Restricted Gravity December 21, / 58