Effective Dynamics of Loop Quantum Gravity: Bouncing Black Holes and - - PowerPoint PPT Presentation

Effective Dynamics of Loop Quantum Gravity: Bouncing Black Holes and Gravitational Phonons

Andrea Dapor*

Louisiana State University Jurekfest, Warsaw, September 20th 2019

• Phys. Lett. B 785, 506-510 (2018)
• Phys. Rev. Lett. 121, 081303 (2018)

arXiv:1906.05315 arXiv:1908.05756

*with Mehdi Assanioussi, Wojciech Kami´ nski, Klaus Liegener and Tomasz Paw lowski

This work is supported by the National Science Foundation and the Hearne Institute for Theoretical Physics at LSU

introduction QG on a graph cosmology BH’s GW’s conclusion

• utline

1

introduction: what is effective dynamics?

2

semiclassical states in quantum gravity on a graph

3

example: homogeneous cosmology

4

example: spherical black hole interior

5

example: linearized Einstein equations

6

conclusion

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introduction QG on a graph cosmology BH’s GW’s conclusion

• utline

1

introduction: what is effective dynamics?

2

semiclassical states in quantum gravity on a graph

3

example: homogeneous cosmology

4

example: spherical black hole interior

5

example: linearized Einstein equations

6

conclusion

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introduction QG on a graph cosmology BH’s GW’s conclusion

A “semiclassical state” in QM: ψ(xo,po)(x) = 1

• ǫ√π

e− (x−xo )2

2ǫ2

+ipo(x−xo)

Peakedness: ψ(xo,po)| ˆ X|ψ(xo,po) = xo, ψ(xo,po)|ˆ P|ψ(xo,po) = po and δX :=

• ∆X 2

ˆ X2 = ǫ √ 2xo ≪ 1, δP :=

• ∆P2

ˆ P2 = 1 √ 2ǫpo ≪ 1 which is achieved if po ≫ ǫ−1 ≫ 1. For any polynomial operator ˆ A = A( ˆ X, ˆ P, [·, ·]), we have ψ(xo,po)|ˆ A|ψ(xo,po) = A(xo, po, i{·, ·}) [1 + O(δX + δP)]

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introduction QG on a graph cosmology BH’s GW’s conclusion

What do we meen by “effective dynamics”? Quantum dynamics: ψt

(xo,po)|ˆ

A|ψt

(xo,po)

where ψt

(xo,po) := e−i ˆ Htψ(xo,po)

“Classical” dynamics on phase space: ˙ a(t) = {Heff, a(t)}, a(0) = A(xo, po) where Heff(x, p) := ψ(x,p)|ˆ H|ψ(x,p) Effective Dynamics ψt

(xo,po)|ˆ

A|ψt

(xo,po) = a(t)[1 + O(δX + δP)]

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introduction QG on a graph cosmology BH’s GW’s conclusion

a good example (free particle) and a bad one (just everything else)

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introduction QG on a graph cosmology BH’s GW’s conclusion

• utline

1

introduction: what is effective dynamics?

2

semiclassical states in quantum gravity on a graph

3

example: homogeneous cosmology

4

example: spherical black hole interior

5

example: linearized Einstein equations

6

conclusion

7 / 29

introduction QG on a graph cosmology BH’s GW’s conclusion

Effective dynamics in LQG so far: Successfull in LQC [Ashtekar, Pawlowski and Singh, 2006; Taveras, 2008] Conjectured in QRLG [Alesci, Bahrami, Botta, Cianfrani, Luzi, Pranzetti, Stagno] What about the full theory (at least on a fixed graph)? Fix Γ (e.g., cubic lattice) with N edges: kinematical Hilbert space H = L2(SU(2)N, dg1..dgN) with operators, ˆ he (multiplication by ge) and ˆ E I

e (right-invariant v.f. RI e)

discrete “geometry”: Collection (ue, ξe) ∈ SU(2) × su2 for every edge e. Inspired on complexifier coherent states [Sahlmann, Thiemann and Winkler, 2002; Bahr

and Thiemann, 2007], we construct a class of “generalized coherent states”.

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introduction QG on a graph cosmology BH’s GW’s conclusion

Generalized coherent state: Ψ(u,ξ) ∈ H given by Ψ(u,ξ)(g1, .., gN) =

N

• e=1

ψ(ue,ξe)(ge), ψ(ue,ξe)(g) = 1 Ne fe(g)e−Se(g)/ǫ where Se satisfies: Re(Se) has single minimum at ge,o Hessian RI

eRJ e Se is non-degenerate at ge,o

⇒ “approximate peakedness” wrt ˆ he and ˆ E I

e:

Ψ(u,ξ)|ˆ he|Ψ(u,ξ) = ue

• 1 + O(ǫ)
• ,

Ψ(u,ξ)|ˆ E I

e|Ψ(u,ξ) = ξI e

• 1 + O
• 1

ǫ|ξe|2

• δhe ∝ √ǫ[1 + O(ǫ)],

δE I

e ∝

1 √ǫ|ξe|

• 1 + O
• 1

√ǫ|ξe|

• where ue = ge,o and ξI

e = 1

ǫ Im(RI

eSe)(ge,o).

So, if we take |ξe|2 ≫ ǫ−1 ≫ 1, we find that δhe, δE I

e ≪ 1, and hence Ψ(u,ξ) is

peaked on the discrete “geometry” (ue, ξe).

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introduction QG on a graph cosmology BH’s GW’s conclusion

Effective dynamics conjecture [AD, Kami´

nski and Liegener]

Two observations: If ˆ A is a pdo1 with principal symbol a, then Ψ(u,ξ)|ˆ A|Ψ(u,ξ) = a(u, ξ)[1 + O(δh + δE)] Egorov’s Theorem: Let ˆ B be a positive, self-adjoint, elliptic pdo. Then ˆ A is a pdo = ⇒ ˆ At := eit ˆ

B ˆ

Ae−it ˆ

B is a pdo

If true, then Ψt

(u,ξ)|ˆ

A|Ψt

(u,ξ) = Ψ(u,ξ)|ˆ

At|Ψ(u,ξ) = at(u, ξ)

• 1 + O(δh + δE)
• with at principal symbol of ˆ
• At. It follows

d dt at(u, ξ) ≈ d dt Ψt

(u,ξ)|ˆ

A|Ψt

(u,ξ) = iΨ(u,ξ)|[ˆ

B, ˆ At]|Ψ(u,ξ) ≈ −{b(u, ξ), at(u, ξ)} with b ≈ Ψ(u,ξ)|ˆ B|Ψ(u,ξ) principal symbol of ˆ B.

1See e.g. L. H¨

• rmander, The Analysis of Linear Partial Differential Operators, (1987).

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introduction QG on a graph cosmology BH’s GW’s conclusion

Summary of the conjecture at given by Ψt

(u,ξ)|ˆ

A|Ψt

(u,ξ) = at(u, ξ)

• 1 + O(δh + δE)
• satisfies

˙ at = {at, b}, a0 = Ψ(u,ξ)|ˆ A|Ψ(u,ξ)

• 1 + O(δh + δE)
• with

b = Ψ(u,ξ)|ˆ B|Ψ(u,ξ)

• 1 + O(δh + δE)
• This is exactly effective dynamics with effective Hamiltonian b!

Expectation value of ˆ HLQG [Giesel, Thiemann, 2006] on generalized coherent state: Heff(u, ξ) := Ψ(u,ξ)|ˆ HLQG|Ψ(u,ξ) = Ψ(u,ξ)|HLQG(ˆ h, ˆ E, [·, ·])|Ψ(u,ξ) = = Hµ

GR(u, ξ, i{·, ·})[1 + O(δh + δE)]

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introduction QG on a graph cosmology BH’s GW’s conclusion

Examples: 1 homogeneous spacetimes (cosmology) 2 spherical black hole interior (BH’s) 3 linearized Einstein equations (GW’s)

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introduction QG on a graph cosmology BH’s GW’s conclusion

• utline

1

introduction: what is effective dynamics?

2

semiclassical states in quantum gravity on a graph

3

example: homogeneous cosmology

4

example: spherical black hole interior

5

example: linearized Einstein equations

6

conclusion

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introduction QG on a graph cosmology BH’s GW’s conclusion

ds2 = −dt2 + p(t)[dx2 + dy 2 + dz2] Ashtekar-Barbero variables: AI

a = cδI a and E a I = pδa I

fix the graph: cubic lattice embedded in space along the coordinate axes read off the classical holonomy and flux on each edge: ue = e−cµτe, ξI

e = δI eαµ2p

with µ the coordinate length of each edge By construction, Ψ(u,ξ) is peaked on this “geometry”: Ψ(u,ξ)|ˆ he|Ψ(u,ξ) ≈ e−cµτe, Ψ(u,ξ)|ˆ E I

e|Ψ(u,ξ) ≈ δI eµ2p

Note – Scale µ is independent of (c, p): we are in µo-scheme.

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introduction QG on a graph cosmology BH’s GW’s conclusion

Effective Hamiltonian [AD and Liegener, 2017]: Heff = − 3 8πGγ2µ2 √p

• sin2(µc) − (1 + γ2) sin4(µc) + O(δh + δE)
• ≈ Heff

LQC

• 1 − (1 + γ2) sin2(µc)
• If it were ¯

µ-scheme, then: example of general LQC effective Hamiltonian [Engle and Vilensky, 2018] Equations of motion analytically solvable [Assanioussi, AD, Liegener and

Paw lowski, 2018 and 2019]

Volume: p

3 2 (φ) ∝ 1 + γ2 cosh2(

√ 12πGφ) | sinh( √ 12πGφ)|

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introduction QG on a graph cosmology BH’s GW’s conclusion

• 1.0
• 0.5

0.5 1.0 ϕ 2 4 6 8 10 V

Pre-bounce branch: contracting de Sitter with effective cosmological constant Λeff = 3 ∆(1 + γ2) Notes: quantum LQC-like model first proposed in [Yang, Ding and Ma, 2009] effective dynamics consistent with quantum dynamics [Assanioussi, AD,

Liegener and Paw lowski, 2018]

inclusion of inflaton [Agullo, 2018; Li, Singh and Wang, 2018]

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introduction QG on a graph cosmology BH’s GW’s conclusion

Generalization to Bianchi I [Garc´

ıa-Quismondo and Mena Marug´ an, 2019]

Heff = 1 8πG p2p3 p1 sin(c2µ2) µ2 sin(c3µ3) µ3

• 1 − 1 + γ2

4γ2 A(c3, c1)A(c1, c2)

• + cyclic

where A(ci, cj) := cos(ciµi) + cos(cjµj).

• 200
• 100

100 200 t

• 2
• 1

1 2 k1

• 20
• 10

10 20 t 0.1 0.2 0.3 0.4 0.5 0.6 0.7 k1

Notes: Vacuum case: only small differences between LQC and the new model. Matter case: no Kasner transition, pre-bounce dS phase (with same Λeff)

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introduction QG on a graph cosmology BH’s GW’s conclusion

• utline

1

introduction: what is effective dynamics?

2

semiclassical states in quantum gravity on a graph

3

example: homogeneous cosmology

4

example: spherical black hole interior

5

example: linearized Einstein equations

6

conclusion

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introduction QG on a graph cosmology BH’s GW’s conclusion

Studied within loop context from various perspecives [Alesci, Ashtekar, Bodendorfer,

Boehmer, Bojowald, Campiglia, Corichi, Gambini, Mele, Modesto, M¨ unch, Olmedo, Pranzetti, Pullin, Saini, Singh, Vandersloot, ...]

Schwartzschild interior = Kantowski-Sachs ds2 = −dT 2 + pb(T)2 4|pa(T)|dR2 + |pa(T)|dΩ2 Ashterkar-Barbero variables written in terms of canonical variables a, b, pa, pb. After an appropriate choice of graph, we construct the discrete geometry (ue, ξe). For example, holonomies are uR = exp[γaτ1µ1], uθ = exp[γbτ2µ2], uφ = exp[(γbτ3 sin θ − τ1 cos θ)µ3] The state Ψ(u,ξ) is peaked on this geometry, and we can compute Heff.

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introduction QG on a graph cosmology BH’s GW’s conclusion

After solving the equations of motion [Assanioussi, AD and Liegener, 2019] ...

5 5 10 Logpa 5 10 15 20 pb

horizons: pb = 0 and pa = 4M2 singularity: pa = 0 In all cases, singularity replaced by BH → WH transition.

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introduction QG on a graph cosmology BH’s GW’s conclusion

MWH as a function of M:

5 10 15 20 LogMbh 5 10 15 20 LogMwh

For M large enough, MWH ∝ ∼ M in all cases.

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introduction QG on a graph cosmology BH’s GW’s conclusion

• utline

1

introduction: what is effective dynamics?

2

semiclassical states in quantum gravity on a graph

3

example: homogeneous cosmology

4

example: spherical black hole interior

5

example: linearized Einstein equations

6

conclusion

22 / 29

introduction QG on a graph cosmology BH’s GW’s conclusion

ds2 = −dt2 + [1 + h+(t, z)]dx2 + [1 − h+(t, z)]dy 2 + 2h×(t, z) dxdy + dz2 Lattice with N nodes along z direction. Repeat construction... arrive at Heff. From ˙ hA = {hA, Heff}, we find hA(t, z) = 1 N

• k

eikz˜ hA,k(t), where ¨ ˜ hA,k + ω2

k˜

hA,k = 0 with ωk = | sin(kµ)| µ

• (1 + γ2) cos(kµ) − γ2

Modified dispersion relation due to lattice. Reminiscent of [Sahlmann, Thiemann, 2002].

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introduction QG on a graph cosmology BH’s GW’s conclusion

ωk = | sin(kµ)| µ

• (1 + γ2) cos(kµ) − γ2

0.2 0.4 0.6 0.8 1.0 1.2 1.4 k μ 0.1 0.2 0.3 0.4 0.5 0.6 ωkμ

Notes: µ → 0 recovers the classical dispersion relation ωk → |k| not all modes propagate: for k > ko, ωk is imaginary restricing to k ∈ [0, ko], the high-k behavior is different from phonons

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introduction QG on a graph cosmology BH’s GW’s conclusion

Toy model: tensor modes in cosmology Classical cosmology: ˆ hA(η, z) ∝

• k
• ˆ

bkξ(k, η)eikz + ˆ b†

kξ∗(k, η)e−ikz

For background ds2 = a2(η)[−dη2 + dx2 + ..], mode functions ξ satisfy ξ′′ +

• k2 − a′′

a

• ξ = 0

Generalization to our case: ξ′′ +

• ω2

k − a′′

a

• ξ = 0

⇒ amplification for low-k and high-k modes!

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introduction QG on a graph cosmology BH’s GW’s conclusion

Number nk of k-particles in the “far” future (t ≈ tP × 104) starting with vacuum at bounce (t = 0):

0.2 0.4 0.6 0.8 1.0 1.2 1.4 k μ

• 2
• 1

1 2 log10(nk) 26 / 29

introduction QG on a graph cosmology BH’s GW’s conclusion

• utline

1

introduction: what is effective dynamics?

2

semiclassical states in quantum gravity on a graph

3

example: homogeneous cosmology

4

example: spherical black hole interior

5

example: linearized Einstein equations

6

conclusion

27 / 29

introduction QG on a graph cosmology BH’s GW’s conclusion

Done Generalized coherent state Ψ(u,ξ) peaked on discrete “geometry”. Dynamical conjecture for Heff. Applications:

∗ cosmology: non-symmetric bounce, pre-bounce dS (with Λeff); Bianchi I deviates from LQC only if matter is present. ∗ BH’s: singularity replaced by BH → WH transition with MWH ∝ MBH ∗ GW’s: modified ωk; particle creation at both ends of k-spectrum

Next different regularizations ⇒ different physics. Role of µ? role of matter: why Bianchi I different from LQC only if φ present? if toy model on tensor modes is serious: observable effects in CMB?

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Thank you! Happy birthday prof. Lewandowski!