Towards a ghost- and singularity-free theory of gravity · curvature is power-counting renormalizable: = 1 16𝜋 න 4 T− ℛ+ Ùℛ2+ Úℛ ℛ • Unitarity is violated at the

Towards a ghost- and singularity-free

theory of gravity

Luca Buoninfante

International School of Subnuclear physics57° Course – Erice – Sicily

21-30 June 2019

Introduction

• Einstein’s general relativity (GR) has been tested to veryhigh precision in the IR regime;

• Despite the great succes of GR, there are still openproblems that make the theory incomplete in the UVregime, e.g. Blackhole and Cosmological singularities,non-renormalizability [‘t hooft, Veltmann (1974); Goroff, Sagnotti (1985)]

• To what extent is GR valid in the UV?

• The inverse-square law of Newton’s potential has beentested only up to 5.6 × 10−5meters with torsion balanceexperiments.

𝑆 =1

16𝜋𝐺න𝑑4𝑥 −𝑔ℛ

Introduction

Our knowledge of the gravitational interaction

at short distances is really limited!

4th order gravity

• The 4th order gravitational action quadratic in the curvature is power-counting renormalizable:

𝑆 =1

16𝜋𝐺න𝑑4𝑥 −𝑔 ℛ + 𝛼ℛ2 + 𝛽ℛ𝜇𝜈ℛ

𝜇𝜈

• Unitarity is violated at the tree-level:

• Conflict: Unitarity VS Renormalizability!

Spin-2 ghost degree of freedom

[Stelle, 1977, PRD]

Π 𝑘 = Π𝐺𝑅 +1

2

𝒫0

𝑘2+𝑚02 −

𝒫2

𝑘2+𝑚22 ,

𝑚0 ≔ 3𝛼 +𝛽 −1/2

𝑚2 ≔ −1

2𝛽

−1/2

Unitarity VS Renormalizability

• Einstein’s GR is unitary but non-renormalizable, while4th order quadratic gravity is power-countingrenormalizable but non-unitary!

Several (recent) attempts:

• Asymptotically safe gravity [Eichhorn’s and Schiffer’s talks…]

• 4th order gravity with Fakeons [Anselmi & Piva 2017+]

• Lee-Wick gravity theories [Modesto & Shapiro 2016+; Anselmi & Piva 2017+]

• Nonlocal gravity theories [Efimov, Krasnikov, Kuz’min, Tomboulis, Biswas, Mazumdar, Koshelev, Modesto, Rachwal….]

• Local (polynomial) Lagrangians:

ℒ𝐿 ≡ ℒ𝐿 𝜙, 𝜕𝜙, 𝜕2𝜙,… , 𝜕𝑛𝜙

• Nonlocal (non-polynomial) Lagrangians:

ℒ𝑁𝐿 ≡ ℒ𝑁𝐿 𝜙, 𝜕𝜙, 𝜕2𝜙,… , 𝜕𝑛𝜙,… , 𝑒⧠𝜙, ln ⧠ 𝜙,1

⧠𝜙,…

Local VS Nonlocal

• Finite order derivative theory:

ℒ𝐿 =1

2𝜙⧠ ⧠+𝑚2 𝜙 ⟹ Π𝐿(𝑝) =

1

𝑚2

1

𝑝2−

1

𝑝2 +𝑚2

• Infinite order derivative theory:

Π𝑁𝐿 𝑝 =𝐹 𝑝2

𝑝2 +𝑚2=

𝑛=0

∞

𝐹𝑛𝑝2𝑛

𝑝2 +𝑚2

• Is there any higher derivative operator 𝐹 𝑝2 such that the propagator is ghost-free? YES!

Local VS Nonlocal

GHOST!

• Nonlocal scalar field:

ℒ =1

2𝜙𝑒−𝛾 ⧠/𝑀𝑠

2⧠+𝑚2 𝜙 + 𝑉 𝜙 ,

𝛾 ⧠/𝑀𝑠2 =

𝒏=𝟎

∞

𝛾𝒏⧠

𝑀𝑠2

𝒏

• Ghost-free propagator:

Π 𝑝 =𝑒−𝛾 −𝑝2/𝑀𝑠

2

𝑝2 +𝑚2

Renormalizability for some specific entire functions

[Modesto 2011; Modesto & Rachwal 2014]

Perturbative unitarity (optical theorem and Cutkosky rules)[Pius & Sen 2015; Briscese & Modesto 2018; Chin & Tomboulis 2018]

Causality violation at microscopic scales (acausal Green functions and local commutativity violation)[Tomboulis 2015; LB, Lambiase, Mazumdar 2018]

Exponential of entire functions

Entire function: No extra poles!

Infinite Derivative Gravity (IDG)

• Non-local higher derivative theories can be ghost-free:

• Ghost-free propagator:

• Entire function, e.g.

[Kuz’min, 1989; Tomboulis 1997; Biswas et al., 2006,2011; Modesto et al. 2011+]

𝑒−𝛾 ⧠/𝑀𝑠2= 𝑒−⧠/𝑀𝑠

2

Π𝜇𝜈𝜌𝜎 𝑘 = 𝑒𝛾(−𝑘2/𝑀𝑠

2)Π𝜇𝜈𝜌𝜎𝐺𝑅 𝑘 = 𝑒𝛾(−𝑘

2/𝑀𝑠2)

𝒫𝜇𝜈𝜌𝜎2

𝑘2−𝒫𝜇𝜈𝜌𝜎0

2𝑘2

𝑆 =1

16𝜋𝐺න𝑑4𝑥 −𝑔 ℛ + 𝐺𝜇𝜈𝐹(⧠)ℛ

𝜇𝜈 , 𝐹 ⧠ =𝑒−𝛾 ⧠/𝑀𝑠

2− 1

⧠

• Linearized metric for a static point-like source:

𝑑𝑠2 = − 1 + 2𝜙 𝑟 𝑑𝑡2 + 1 − 2𝜙 𝑟 (𝑑𝑟2 + 𝑟2𝑑Ω2)

[Tseytlin, 1995 PLB; Biswas et al., 2006 JCAP; Modesto, 2012 PRD; Biswas et al., 2012, PRL]

Singularity-free!

IR UV

𝑒−∇2

𝑀𝑠2∇2𝜙 Ԧ𝑟 = 4π𝐺𝑚𝛿(3) Ԧ𝑟 ⟹ 𝜙 𝑟 = −

𝐺𝑚

𝑟𝐸𝑟𝑓

𝑀𝑠𝑟

2

𝜙 𝑟 ~ −𝐺𝑚

𝑟𝜙 𝑟 ~−

𝐺𝑚𝑀𝑠

𝜋< ∞

𝐸𝑟𝑓 𝑥 ≔2

𝜋න0

𝑥

𝑒−𝑡2𝑑𝑡


• Schwarzschild solution in Einstein’s GR:

• IDG case:

ℛ =𝐺𝑚𝑀𝑠

3𝑒−𝑀𝑠2𝑟2/4

𝜋

• Non-singular Kretschmann invariant!

• Smearing of the (delta-) source due to non-locality!

• The UV/short distances behavior is ameliorated!

“NON-VACUUM”SOLUTION!

[LB, et al. JCAP]


𝜙 𝑟 = −𝐺𝑚

𝑟⟹ ℛ𝐺𝑅 = 8𝜋𝐺𝑚𝛿(3) Ԧ𝑟

• Linearized metric for a rotating ring source in IDG:

• Stress-energy tensor:

• Differential equations:

[LB, et al. PRD]


𝑑𝑠2 = − 1 + 2𝜙 𝑟 𝑑𝑡2 + 2ℎ ∙ 𝑑 Ԧ𝑟𝑑𝑡 + 1 − 2𝜙 𝑟 (𝑑𝑟2 + 𝑟2𝑑Ω2)


• Non-singular solutions:

[LB, et al. PRD]


Summary and Outlook

• Non-local higher derivative Lagrangians can be ghost-free.

• Nonlocality can improve the high-energy/short-distance behavior.

• In the linear regime the spacetime metrics can be made non-singularin IDG: non-local gravitational interaction smears out the delta-source.

Open Problems

• Quantification of the causality violation?

• UV behavior of the tree-level amplitudes in nonlocal gravity?

• Huge arbitrarity on the choice of the entire function?

• Non-linear (non-singular) spacetime metric solutions?

Thank you

for

your attention!

• Nonlocal action:

𝑆 = න𝑑4𝑥𝑑4𝑦𝜙 𝑥 𝐾 𝑥 − 𝑦 𝜙 𝑦

= න𝑑4𝑥𝑑4𝑦𝜙 𝑥 න𝑑4𝑘

2𝜋 4𝐹 −𝑘2 𝑒𝑖𝑘∙ 𝑥−𝑦 𝜙 𝑦

= න𝑑4𝑥𝑑4𝑦 𝜙 𝑥 𝐹 ⧠ න𝑑4𝑘

2𝜋 4𝑒𝑖𝑘∙ 𝑥−𝑦 𝜙 𝑦

= න𝑑4𝑥𝑑4𝑦 𝜙 𝑥 𝐹 ⧠ 𝜙 𝑥

Extra Slides: nonlocality

• NO time-ordered propagator:

𝑒−𝛾 ⧠ ⧠− 𝑚2 Π 𝑥 = 𝑖𝛿 4 𝑥 ,

Π 𝑥 = Π𝑐 𝑥 + Π𝑛𝑐 𝑥 , Π𝑛𝑐 𝑥 = 𝑖

𝑞=1

∞𝑖𝑞−1

𝑞!𝜕𝑥0(𝑞−1)

[𝑊 𝑞 𝑥 −𝑊 𝑞 −𝑥 ]

𝑊 𝑞 𝑥 = න𝑑4𝑘

2𝜋 4 𝑒𝑖𝑘∙𝑥𝜃(𝑘0)𝛿(𝑘2 +𝑚2)

𝜕(𝑞)𝑒−𝛾 −𝑘2

𝜕𝑘0(𝑞)

Extra slides: causality violation

[LB, Lambiase, Mazumdar, NPB]

Extra slides: causality violation

• Acausal Green function:

𝑒−⧠𝑀𝑠2

2𝑛

⧠− 𝑚2 𝐺𝑅(𝑥 − 𝑦) = 𝑖𝛿 4 𝑥 − 𝑦 ,

[LB, Lambiase, Mazumdar, NPB]

𝐺𝑅 𝑥 − 𝑦 ≠ 0, 𝑓𝑜𝑟 (𝑥 − 𝑦)2 > 0

Extra slides: unitarity

• Optical theorem:

• Ghost and unitarity violation:

• Nonlocal propagator:

𝑆+𝑆 = 1, 𝑆 = 1 + 𝑖𝑇 ⟹ 2𝐼𝑚 𝑇 = 𝑇+𝑇 > 0

𝑆+𝑆 = 1, 𝐼𝑚−1

𝑝2 − 𝑖𝜖= −

𝜖

𝑝2 + 𝜖2= −π𝛿 4 𝑝2 < 0

𝐼𝑚𝑒−𝑝

2/𝑀𝑠2

𝑝2 − 𝑖𝜖=𝑒−𝑝

2/𝑀𝑠2𝜖

𝑝2 + 𝜖2= π𝛿 4 𝑝2 > 0

• Locality, causality, unitarity , renormalizability… too manyrequirements?

• Beyond higher order derivatives? Diffeomorphism invarianceallows more…

• General quadratic gravitational action, parity-invariant and torsion-free:

Extra slides: most general quadratic action

• Analytic form-factors:

• Propagator:

• Ghost-free condition:

• Exponential of an entire function: NO extra poles!

[Tomboulis 1997; Modesto, 2012 PRD; Biswas et al., 2012, PRL]

Extra slides: ghost-free propagator

Extra slides: spin projection operators

• Tomboulis’ entire function:

𝛾 𝑧 = Γ 0, 𝑃 𝑧 + 𝛾𝐸 + 𝑙𝑜𝑔 𝑃 𝑧

• UV behavior:

z → ∞ ⟹ 𝑒𝛾 𝑧 ⟶ 𝑒𝛾𝐸𝑃 𝑧

Extra Slides: Renormalizable nonlocal gravity

[Tomboulis 1977]

• All curvature invariants are singularity-free at r=0!

• Kretschmann invariant:

Extra slides: Non-singular curvature invariants

[LB, Koshelev, Lambiase, Mazumdar, JCAP. , LB, Koshelev, Lambiase, Marto, Mazumdar, JCAP]


• Solutions:

[LB, et al. PRD]

Extra slides: metric for rotating ring


• Non-singular solutions:

[LB, et al. PRD]

Extra slides: metric for rotating ring

Multipole expansion in IDG

[L.B., et al., PRD]

Introduction

Our knowledge of the gravitational interaction

at short distances is really limited!

• Gravitational force/acceleration:

Extra slides: gravitational force

• Linearized metric for a static charged point-like source in IDG:

[LB, Harmsen, Maheswari, Mazumdar, 1804.09624]

Singularity-free!

IR UV

Extra slides: metric for an electric charge

• Trace of the field equations:

[Biswas, Conroy, Koshelev, Mazumdar, 2013, CQG]

Extra slides: trace eqution of IDG

Extra slides: most general ghost-free choice for a(k) and c(k)

Exponentials of entire functions!

Extra slides: non-local scale

• Fourth order order gravity has curvature singularity:

• Theories with derivatives of order higher than 4 are singularity-free and conformally-flat at the origin:

Extra slides: Local vs Non-local

[Giacchini, Paula Netto, EPJC]

• Non-local higher derivative theories can be made ghost-free at the tree-level.

• Is the Schwarzschild metric a solution of the full non-lineartheory?



• Non-local theories:

• Example of non-point support:

[L.B, Koshelev, Lambiase, Marto, Mazumdar, JCAP (2018)]



• Local theories: point-support!

• Non-local theories:

[L.B, Koshelev, Lambiase, Marto, Mazumdar, JCAP (2018)]

Non-point-support


• Schwarzschild metric as a solution coupled to a non-delta source distribution?

Hint from the linear regime

• In local theorie YES:

• In non-local theories (a-priori) NO:


• Stress-energy tensor for Schwarzschild metric in Kerr-Schild coordinates:

• Stress-energy tensor for Kerr metric in Kerr-Schild coordinates:

[H. Balasin, H. Nachbagauer, PLB 1993; CQG 1993.]

Extra slides: sources for Schwarzschild and Kerr metrics

Extra slides: different nonlocal models

• P-adic string: [Freund, Witten, Frampton…1987+]

• Infinite Derivative Theories: [Krasnikov, Biswas, Mazumdar, Modesto…]

• Nonlocal model with complex conjugate poles: [LB, Lambiase, Yamaguchi 2018]

ℒ𝑝−𝑎𝑑𝑖𝑐 = −𝑚𝑠4

2𝑔𝑠2

𝑝2

𝑝 − 1𝜙𝑝

−⧠𝑚𝑠2𝜙, Π𝑝−𝑎𝑑𝑖𝑐 𝑘 = 𝑝

−𝑘2

𝑚𝑠2= 𝑒

−ln(𝑝)𝑘2

𝑚𝑠2

ℒ𝐼𝐷𝑇 =1

2𝜙𝑒−⧠/𝑀𝑠

2⧠𝜙, Π𝐼𝐷𝑇 𝑘 =

𝑒−𝑘2/𝑀𝑠

2

𝑘2

ℒ = −𝑀𝑠2

2𝜙(𝑒−⧠/𝑀𝑠

2− 1)𝜙, Π 𝑘 =

1

𝑀𝑠2(𝑒𝑘

2/𝑀𝑠2− 1)

Documents

Towards a ghost- and singularity-free theory of gravity · curvature is power-counting renormalizable: = 1 16𝜋 න 4 T− ℛ+ Ùℛ2+ Úℛ ℛ • Unitarity is violated at the