d 4 x g [ RF 1 ( ) R + RF Towards Ghost free and singularity - - PowerPoint PPT Presentation

Anupam Mazumdar

Lancaster University

Towards Ghost free and singularity free construction of gravity

Warren Siegel, Tirthabir Biswas

Alex Koshelev, Sergei Vernov, Erik Gerwick, Tomi Koivisto, Aindriu Conroy, Spyridon Talaganis, Ali Teimouri

⇤ d4x⇥g[RF1(⇤)R + RF RλσF5(⇤)⇤µ⇤σ⇤ν⇤λRµν + Rρ

λF8(⇤)⇤µ⇤σ⇤ν⇤ρRµνλσ

RµνλσF10(⇤)Rµνλσ + Rρ

µνλF

Rν1ρ1σ1

µ

F13(⇤)⇤ρ1⇤σ1⇤ν1⇤

• Phys. Rev. Lett. (2012), JCAP (2012, 2011), JCAP (2006)

Class.Quant. Grav. (2013), Phys. Rev. D (2014), 1412.3467, 1503.05568 (Phys. Rev. Lett. 2015)

Einstein’s GR is well behaved in IR, but UV is Pathetic; Aim is to address the UV aspects of Gravity

Born, Enfeld, Utiyama, Eifimov, Tseytlin, Siegel, Grisaru, Biswas, Krasnov, Antoniadis, Anselmi, DeWitt, Desser, Stelle, Witten, Sen, Zwiebach, Kostelecky, Motola, Samuel, Frampton, Okada, Olson, Freund, Tomboulis, Talaganis, Khoury, Modesto, Page, Bravinsky, Koivisto, Mazur, Frolov, Cline, Chiba, Barnaby, Kamran, Woodard, Vernov, Kapusta, Daffayet, Arefeva, Dvali, Arkani-Hamed, Koshelev, Mukhoyama, Conroy, Craps, Sagnotti, Rubakov, Yukawa, …

Many Contributors

Many are present in this room …

see also: Valarie Frolov and Ilia Shapiro’s talks in this workshop

UV is Pathological, IR Part is Safe

Classical Singularities

⇤ d4x⇥g[RF1(⇤)R + RF2 RλσF5(⇤)⇤µ⇤σ⇤ν⇤λRµν + Rρ

λF8(⇤)⇤µ⇤σ⇤ν⇤ρRµνλσ

RµνλσF10(⇤)Rµνλσ + Rρ

µνλF

Rν1ρ1σ1

µ

F13(⇤)⇤ρ1⇤σ1⇤ν1⇤

S = Z √−gd4x ✓ R 16πG + · · · ◆

S = Z √−gd4x ✓ R 16πG ◆

What terms shall we add such that gravity behaves better at small distances and at early times ?

While keeping the General Covariance

Different approach from string theory, but there could be some connection with closed string field theory

Motivations

(1) Resolution of Blackhole Singularity

Information loss paradox : is it really a fundamental problem of nature?

(2) Resolution of Cosmological Big Bang Singularity

Classical and Quantum initial conditions for Inflationary cosmology

(3) Understanding UV aspects of gravity and see how its connected to other approaches of Quantum Gravity

F5(⇤)⇤µ⇤σ⇤ν⇤λR +

ρ λF8(⇤)⇤µ⇤σ⇤ν⇤ρRµνλσ + µνλσF10(⇤)Rµνλσ + Rρ µνλF ν1ρ1σ1 µ

F13(⇤)⇤ρ1⇤σ1⇤ν1⇤ν

Bottom-up approach

Higher derivative gravity & ghosts Covariant extension of higher derivative ghost-free gravity Singularity free theory of gravity - “Classical Sense” Divergence structures in 1 and 2-loops in a scalar Toy model

⇤ d4x⇥g[RF1(⇤)R + RF RλσF5(⇤)⇤µ⇤σ⇤ν⇤λRµν Rρ

λF8(⇤)⇤µ⇤σ⇤ν⇤ρRµνλ

RµνλσF10(⇤)Rµνλσ + Rρ

µνλ

Rν1ρ1σ1

µ

F13(⇤)⇤ρ1⇤σ1⇤ν1

Mp M

GR is a good approximation in IR Corrections in UV becomes important

4th Derivative Gravity & Power Counting renormalizability

I =

• d4x √g
• λ0 + k R + a RµνRµν − 1

3 (b + a)R2

Utiyama, De Witt (1961), Stelle (1977)

Massive Spin-0 & Massive Spin-2 ( Ghost ) Stelle (1977)

D ∝ 1 k4 + Ak2 = 1 A ✓ 1 k2 − 1 k2 + A ◆

Modification of Einstein’s GR

Modification

• f Graviton

Propagator

Extra propagating degree of freedom

Challenge: to get rid of the extra dof

Higher Order Derivative Theory Generically Carry Ghosts ( -ve Risidue ) with real “m”( No- Tachyon)

Propagator with first

• rder poles

Ghosts

Ghosts cannot be cured order by order, finite terms in perturbative expansion will always lead to Ghosts !!

No extra states other than the

• riginal dof.

Tomboulis (1997), Siegel (2003), Biswas, Grisaru, Siegel (2004), Biswas, Mazumdar, Siegel (2006)

Higher order action of Gravity

S = SE + Sq

Sq =

• d4x√−gRµ1ν1λ1σ1Oµ1ν1λ1σ1

µ2ν2λ2σ2Rµ2ν2λ2σ2

Covariant derivatives

Unknown Infinite Functions of Derivatives

gµν = ηµν + hµν

R ∼ O(h)

Sq = Z d4x√−g [R....O....

....R.... + R....O.... ....R....O.... ....R.... + R....O.... ....R....O.... ....R....O.... ....R.... + · · · ]

Redundancies & Form Factors

= Z d4x√−g ⇥ R + RF1(⇤)R + RµνF2(⇤)Rµν + RµναβF3(⇤)Rµναβ⇤

Sq =

• d4x√−g[RF1()R + RF2()∇µ∇νRµν + RµνF3()Rµν + Rν

µF4()∇ν∇λRµλ

+ RλσF5()∇µ∇σ∇ν∇λRµν + RF6()∇µ∇ν∇λ∇σRµνλσ + RµλF7()∇ν∇σRµνλσ + Rρ

λF8()∇µ∇σ∇ν∇ρRµνλσ + Rµ1ν1F9()∇µ1∇ν1∇µ∇ν∇λ∇σRµνλσ

+ RµνλσF10()Rµνλσ + Rρ

µνλF11()∇ρ∇σRµνλσ + Rµρ1νσ1F12()∇ρ1∇σ1∇ρ∇σRµρνσ

+ Rν1ρ1σ1

µ

F13()∇ρ1∇σ1∇ν1∇ν∇ρ∇σRµνλσ + Rµ1ν1ρ1σ1F14()∇ρ1∇σ1∇ν1∇µ1∇µ∇ν∇ρ∇σRµνλσ

(1) GR (2) Weyl Gravity (3) F(R) Gravity (4) Gauss-Bonnet Gravity (5) Ghost free Gravity

Complete Field Equations

Biswas, Conroy, Koshelev, Mazumdar 1308.2319 Class.Quant. Grav. (2014)

Rµνλσ = 1 2(∂[λ∂νhµσ] − ∂[λ∂µhνσ]) Rµν = 1 2(∂σ∂(νhσ

µ) − ∂ν∂µh − ⇤hµν)

R = ∂ν∂µhµν − ⇤h

Sq = − ⇤ d4x ⌅1 2hµνa(⇤)⇤hµν + hσ

µb(⇤)∂σ∂νhµν

(3) + hc(⇤)∂µ∂νhµν + 1 2hd(⇤)⇤h + hλσ f(⇤) ⇤ ∂σ∂λ∂µ∂νhµν⇧ .

a + b = 0 c + d = 0 b + c + f = 0

a() = 1 − 1 2F2() − 2F3() b() = −1 + 1 2F2() + 2F3() c() = 1 + 2F1() + 1 2F2() d() = −1 − 2F1() − 1 2F2() f() = −2F1() − F2() − 2F3().

F3(⇤) is redundant

= Z d4x√−g ⇥ R + RF1(⇤)R + RµνF2(⇤)Rµν + RµναβF3(⇤)Rµναβ⇤

gµν = ηµν + hµν

Linearised Equations of Motion

Graviton Propagator

Π−1

µν λσhλσ = ⇤⇧µν

a(⇤)⇤hµν + b(⇤)∂σ∂(νhσ

µ) + c(⇤)(ηµν∂ρ∂σhρσ + ∂µ∂νh)

+ηµνd(⇤)⇤h + 1 4f(⇤)⇤−1∂σ∂λ∂µ∂νhλσ = κτµν

κτ⇧µτ µ

ν = 0 = (c + d)⇤∂νh + (a + b)⇤hµ ν,µ + (b + c + f)hαβ ,αβν

= 0 = 0 = 0

a + b = 0 c + d = 0 b + c + f = 0

Bianchi Identity

h = hT T + hL + hT

Π = P 2 ak2 + P 0

s

(a − 3c)k2

Spin projection operators

Biswas, Koivisto, AM 1302.0532

For this action, see:

.

• P. Van Nieuwenhuizen,

Nucl.Phys. B60 (1973), 478. 


Tree level Graviton Propagator Π = P 2 ak2 + P 0

s

(a − 3c)k2

No new propagating degree of freedom

• ther than the massless Graviton

a() = c() ⇒ 2F1() + F2() + 2F3() = 0

S = Z d4x√−g R 2 + RF1(⇤)R − 1 2RµνF2(⇤)Rµν

Well known actions of Gravity

a() = 1 − 1 2F2() − 2F3() b() = −1 + 1 2F2() + 2F3() c() = 1 + 2F1() + 1 2F2() d() = −1 − 2F1() − 1 2F2() f() = −2F1() − F2() − 2F3().

(1) GR: (2) F(R) Gravity: (3) GB Gravity: (4) Weyl Gravity:

Π = ΠGR L = R − 1 m2 C2

C2 = RµνρσRµνρσ − 2RµνRµν + 1 3R2 Biswas, Koivisto, AM 1302.0532

(1) Gravitational Entropy

SW = −8π I

r=rH, t=const

✓ ∂L ∂Rrtrt ◆ q(r)dΩ2

ds2 = −f(r)dt2 + f(r)−1dr2 + r2dΩ2

SW = Area 4G [1 + α (2F1 + F2 + 2F3) R]

= 0

Holography is an IR effect Higher order corrections yield zero entropy “Ground State of Gravity”

Conroy, Mazumdar, Teimouri, 1503.05568, hep-th ( Phys. Rev. Lett. 2015)

Wald (1990, 1993), Iyer, Wald (1993)

(2) Newtonian Limit

Π = P 2 ak2 + P 0

s

(a 3c)k2

a(⇤) = c(⇤) = e−⇤/M2

ds2 = −(1 − 2Φ)dt2 + (1 + 2Ψ)dr2

Φ = Ψ = Gm r erf ✓rM 2 ◆

Biswas, Gerwick, Koivisto, Mazumdar, Phys. Rev. Lett. (2012) (gr-qc/1110.5249)

S =

• d4x √−g
• R

2 + R

• e

− M2 − 1

• R − 2Rµν
• e−

M2 − 1

• Rµν
• above action contains only the graviton as physical degrees of freedom as

Non-singular static solution

Out[37]=

2 4 6 8 10 0.2 0.4 0.6 0.8 1.0 1.2

r → 0, erf(r) → r

ds2 = ✓ 1 − 2Gm r erf(rM/2) ◆ dt2 − dr2

• 1 − 2Gm

r

erf(rM/2)

• Φ(r) → const.

r → ∞, erf(r) → 1

Φ(r) → 1 r

mM ⌧ M 2

p =

) m ⌧ Mp

Biswas, Gerwick, Koivisto, AM, PRL (2012) (gr-qc/1110.5249)

V(r)

r

Frolov (2015), Frolov, et.al., (2015), Modesto, et.al., (2014), Shapiro, et.al. (2014)

Rotational Motion of

Where would you expect the modifications?

Singularity is capped at the scale of non-locality M ≤ Mp

(3) Gravitational Waves

r = ⇒ 0, No Singularity ¯ hjk ≈ Gω2(ML2) r ¯ hjk ≈ Gω2(ML2) r erf ✓rMP 2 ◆

Large r limit

Biswas, Gerwick, Koivisto, AM,

• Phys. Rev. Lett. (gr-qc/1110.5249)

(4) Non-singular cosmological solutions

h ⇤ diag(0, A sin ⇧t, A sin ⇧t, A sin ⇧t) with A ⌅ 1

Non- Singular Bouncing, Homogeneous & Isotropic Universe

Such a solution is not possible in GR

Biswas, Gerwick, Koivisto, Mazumdar,

• Phys. Rev. Lett. (gr-qc/1110.5249)

S =

• d4x √−g
• R

2 + R

• e

− M2 − 1

• R − 2Rµν
• e−

M2 − 1

• Rµν
• above action contains only the graviton as physical degrees of freedom as

(4) Cosmological non-singular solution

S = Z d4x√−g " R 2 + R " e

−⇤ M2 −1

⇤ # R + Λ #

a(t) = cosh r1 2 t

• Biswas, Mazumdar, Siegel, JCAP (2006)

a(t)

t

Stay tuned: details of the Singularity theorem due to “Hawking-Penrose” in this context will arrive sometime this summer … ( a very nasty computation! )

RabN aN b = 8πTabN aN b ≥ 0

General Relativity Non-local extension of GR

ρ + p ≥ 0

RabN aN b 6= 8πTabN aN b

RabN aN b ≤ 0, dθ dτ + 1 2θ2 ≥ 0

Conroy, Koshlev, Mazumdar, PRD (2014)(gr-qc/1408.6205)

Revisiting Hawking-Penrose Singularity Theorems

Revisiting Singularity Theorems

Rµνkµkν ≤ 0, Tµνkµkν ≥ 0 → (ρ + p ≥ 0)

dθ dτ + 1 2θ2 ≥ 0

(5) Quantum aspects

• Superficial degree of divergence goes as

E = V − I. Use Topological relation : L = 1 + I − V

E = 1 − L

E < 0, for L > 1

• At 1-loop, the theory requires counter term, the 1-

loop, 2 point function yields divergence

• At 2-loops, the theory does not give rise to

additional divergences, the UV behaviour becomes finite, at large external momentum, where dressed propagators gives rise to more suppression than the vertex factors Λ4

Talaganis, Biswas, Mazumdar, (2014)

gµν → Ω gµν

Toy model based on Symmetries

GR e.o.m :

Construct a scalar field theory with infinite derivatives whose e.o.m are invariant under

Sfree = 1 2 Z d4x(φ⇤a(⇤)φ)

→ (1 + ✏) + ✏

hµν → (1 + ✏)hµν + ✏⌘µν

Sint = 1 Mp Z d4x ✓1 4φ∂µφ∂µφ + 1 4φ⇤φa(⇤)φ − 1 4φ∂µφa(⇤)∂µφ ◆

Around Minkowski space the e.o.m are invariant under:

a(⇤) = e−⇤/M2

Π(k2) = − i k2e¯

k2

Conclusions

• We have constructed a Ghost Free & Singularity Free

Theory of Gravity

• If we can show all order loops are finite then it is a

great news -- this is what we have shown up to 2 loops

• Studying singularity theorems, positive energy

theorems, Hawking radiation, Non-Singular Bouncing Cosmology , ....., many interesting problems can be studied in this framework

• Holography is not a property of UV, becomes part of

an IR effect.

Remnants of stringy Gravity

L10d ∼ R + R4 + · · ·

Perturbative string theory has α0 & gs corrections

κ2 = g2

s(α0)4

For all orders : String field theory 1 − loop in gs all orders in α0

Mp ms mW mKK

L4d ∼ R + X

i

ciR ✓ ⇤ mkk ◆i R + · · ·

Witten (1998) , Tseytlin (1995), Zwiebach (2000), Sigel (1998, 2003), …

Extra Slides

Loop quantum gravity

• r

CDT approach Wilson loops Non-local objects

It would be interesting to establish the connection

Conjecture

S =

• d4x √−g
• R

2 + R

• e

− M2 − 1

• R − 2Rµν
• e−

M2 − 1

• Rµν
• above action contains only the graviton as physical degrees of freedom as

Absence of Cosmological and Blackhole Singularities

S = Z d4x√−g R 2 + α0(R, Rµν) + α1(R, Rµν)RF1(⇤)R +α2(R, Rµν)RµνF2(⇤)Rµν + α3(R, Rµν)CµνλσF3Cµνλσ⇤

Conjecture : The Form of Most General Action