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

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d 4 x g [ RF 1 ( ) R + RF Towards Ghost free and singularity free construction of gravity Anupam Mazumdar R F 5 ( ) R + Lancaster University R Warren Siegel, Tirthabir Biswas

  1. d 4 x ⇥� g [ RF 1 ( ⇤ ) R + RF ⇤ Towards Ghost free and singularity free construction of gravity Anupam Mazumdar R λσ F 5 ( ⇤ ) ⇤ µ ⇤ σ ⇤ ν ⇤ λ R µ ν + Lancaster University R ρ Warren Siegel, Tirthabir Biswas λ F 8 ( ⇤ ) ⇤ µ ⇤ σ ⇤ ν ⇤ ρ R µ νλσ Alex Koshelev, Sergei Vernov, Erik Gerwick, Tomi Koivisto, Aindriu Conroy, Spyridon Talaganis, Ali Teimouri R µ νλσ F 10 ( ⇤ ) R µ νλσ + R ρ Phys. Rev. Lett. (2012), JCAP (2012, 2011), JCAP (2006) µ νλ F Class.Quant. Grav. (2013), Phys. Rev. D (2014), 1412.3467, 1503.05568 (Phys. Rev. Lett. 2015) R ν 1 ρ 1 σ 1 F 13 ( ⇤ ) ⇤ ρ 1 ⇤ σ 1 ⇤ ν 1 ⇤ Einstein’s GR is well behaved in IR, but UV is Pathetic; Aim is to address the UV aspects of Gravity µ

  2. Many Contributors 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 are present in this room … see also: Valarie Frolov and Ilia Shapiro’s talks in this workshop

  3. d 4 x ⇥� g [ RF 1 ( ⇤ ) R + RF 2 ⇤ Classical Singularities UV is Pathological, IR Part is Safe R λσ F 5 ( ⇤ ) ⇤ µ ⇤ σ ⇤ ν ⇤ λ R µ ν + Z √− gd 4 x ✓ ◆ R S = 16 π G + · · · R ρ λ F 8 ( ⇤ ) ⇤ µ ⇤ σ ⇤ ν ⇤ ρ R µ νλσ What terms shall we add such that gravity behaves better at small distances and at early times ? R µ νλσ F 10 ( ⇤ ) R µ νλσ + R ρ µ νλ F While keeping the General Covariance Different approach from string Z √− gd 4 x ✓ ◆ R ν 1 ρ 1 σ 1 R theory, but there could be F 13 ( ⇤ ) ⇤ ρ 1 ⇤ σ 1 ⇤ ν 1 ⇤ S = some connection with closed 16 π G string field theory µ

  4. F 5 ( ⇤ ) ⇤ µ ⇤ σ ⇤ ν ⇤ λ R + Motivations λ F 8 ( ⇤ ) ⇤ µ ⇤ σ ⇤ ν ⇤ ρ R µ νλσ + ρ (1) Resolution of Blackhole Singularity Information loss paradox : is it really a fundamental µ νλσ F 10 ( ⇤ ) R µ νλσ + R ρ problem of nature? µ νλ F (2) Resolution of Cosmological Big Bang Singularity ν 1 ρ 1 σ 1 F 13 ( ⇤ ) ⇤ ρ 1 ⇤ σ 1 ⇤ ν 1 ⇤ ν Classical and Quantum initial conditions for Inflationary µ cosmology (3) Understanding UV aspects of gravity and see how its connected to other approaches of Quantum Gravity

  5. d 4 x ⇥� g [ RF 1 ( ⇤ ) R + RF Bottom-up approach ⇤ Higher derivative gravity & ghosts Covariant extension of higher derivative ghost-free gravity R λσ F 5 ( ⇤ ) ⇤ µ ⇤ σ ⇤ ν ⇤ λ R µ ν Singularity free theory of gravity - “Classical Sense” R ρ λ F 8 ( ⇤ ) ⇤ µ ⇤ σ ⇤ ν ⇤ ρ R µ νλ Divergence structures in 1 and 2-loops in a scalar Toy model R µ νλσ F 10 ( ⇤ ) R µ νλσ + R ρ µ νλ Corrections in GR is a good M M p UV becomes R ν 1 ρ 1 σ 1 F 13 ( ⇤ ) ⇤ ρ 1 ⇤ σ 1 ⇤ ν 1 approximation in IR important µ

  6. 4th Derivative Gravity & Power Counting renormalizability � � 3 ( b + a ) R 2 � d 4 x √ g λ 0 + k R + a R µ ν R µ ν − 1 I = ✓ 1 ◆ k 4 + Ak 2 = 1 1 1 D ∝ k 2 − k 2 + A A Massive Spin-0 & Massive Spin-2 ( Ghost ) Stelle (1977) Utiyama, De Witt (1961), Stelle (1977) Modification of Einstein’s GR Extra propagating Modification degree of freedom of Graviton Propagator Challenge: to get rid of the extra dof

  7. Ghosts Higher Order Derivative Theory Generically Carry Ghosts ( -ve Risidue ) with real “m”( No- Tachyon) Propagator with first order poles Ghosts cannot be cured order by order, finite terms in perturbative expansion will always lead to Ghosts !! No extra states other than the original dof. Tomboulis (1997), Siegel (2003), Biswas, Grisaru, Siegel (2004), Biswas, Mazumdar, Siegel (2006)

  8. Higher order action of Gravity S = S E + S q Z .... R .... + R .... O .... .... R .... + R .... O .... .... R .... + · · · ] d 4 x √− g [ R .... O .... S q = .... R .... O .... .... R .... O .... .... R .... O .... R ∼ O ( h ) g µ ν = η µ ν + h µ ν � d 4 x √− gR µ 1 ν 1 λ 1 σ 1 O µ 1 ν 1 λ 1 σ 1 µ 2 ν 2 λ 2 σ 2 R µ 2 ν 2 λ 2 σ 2 S q = Unknown Infinite Covariant derivatives Functions of Derivatives

  9. Redundancies & Form Factors d 4 x √− g [ RF 1 ( � ) R + RF 2 ( � ) ∇ µ ∇ ν R µ ν + R µ ν F 3 ( � ) R µ ν + R ν � µ F 4 ( � ) ∇ ν ∇ λ R µ λ S q = + R λσ F 5 ( � ) ∇ µ ∇ σ ∇ ν ∇ λ R µ ν + RF 6 ( � ) ∇ µ ∇ ν ∇ λ ∇ σ R µ νλσ + R µ λ F 7 ( � ) ∇ ν ∇ σ R µ νλσ λ F 8 ( � ) ∇ µ ∇ σ ∇ ν ∇ ρ R µ νλσ + R µ 1 ν 1 F 9 ( � ) ∇ µ 1 ∇ ν 1 ∇ µ ∇ ν ∇ λ ∇ σ R µ νλσ + R ρ + R µ νλσ F 10 ( � ) R µ νλσ + R ρ µ νλ F 11 ( � ) ∇ ρ ∇ σ R µ νλσ + R µ ρ 1 νσ 1 F 12 ( � ) ∇ ρ 1 ∇ σ 1 ∇ ρ ∇ σ R µ ρνσ F 13 ( � ) ∇ ρ 1 ∇ σ 1 ∇ ν 1 ∇ ν ∇ ρ ∇ σ R µ νλσ + R µ 1 ν 1 ρ 1 σ 1 F 14 ( � ) ∇ ρ 1 ∇ σ 1 ∇ ν 1 ∇ µ 1 ∇ µ ∇ ν ∇ ρ ∇ σ R µ νλσ + R ν 1 ρ 1 σ 1 µ Z R + R F 1 ( ⇤ ) R + R µ ν F 2 ( ⇤ ) R µ ν + R µ ναβ F 3 ( ⇤ ) R µ ναβ ⇤ d 4 x √− g ⇥ = (1) GR (2) Weyl Gravity (3) F(R) Gravity (4) Gauss-Bonnet Gravity (5) Ghost free Gravity

  10. Complete Field Equations Biswas, Conroy, Koshelev, Mazumdar 1308.2319 Class.Quant. Grav. (2014)

  11. Linearised Equations of Motion Z R + R F 1 ( ⇤ ) R + R µ ν F 2 ( ⇤ ) R µ ν + R µ ναβ F 3 ( ⇤ ) R µ ναβ ⇤ d 4 x √− g ⇥ = g µ ν = η µ ν + h µ ν ⇤ ⌅ 1 2 h µ ν a ( ⇤ ) ⇤ h µ ν + h σ µ b ( ⇤ ) ∂ σ ∂ ν h µ ν d 4 x S q = − (3) + hc ( ⇤ ) ∂ µ ∂ ν h µ ν + 1 2 hd ( ⇤ ) ⇤ h + h λσ f ( ⇤ ) ∂ σ ∂ λ ∂ µ ∂ ν h µ ν ⇧ . ⇤ a ( � ) = 1 − 1 1 2 F 2 ( � ) � − 2 F 3 ( � ) � = 2( ∂ [ λ ∂ ν h µ σ ] − ∂ [ λ ∂ µ h νσ ] ) R µ νλσ b ( � ) = − 1 + 1 1 2 F 2 ( � ) � + 2 F 3 ( � ) � 2( ∂ σ ∂ ( ν h σ = µ ) − ∂ ν ∂ µ h − ⇤ h µ ν ) R µ ν c ( � ) = 1 + 2 F 1 ( � ) � + 1 ∂ ν ∂ µ h µ ν − ⇤ h 2 F 2 ( � ) � = R d ( � ) = − 1 − 2 F 1 ( � ) � − 1 a + b = 0 2 F 2 ( � ) � f ( � ) = − 2 F 1 ( � ) � − F 2 ( � ) � − 2 F 3 ( � ) � . c + d = 0 b + c + f = 0 F 3 ( ⇤ ) is redundant

  12. Graviton Propagator µ ) + c ( ⇤ )( η µ ν ∂ ρ ∂ σ h ρσ + ∂ µ ∂ ν h ) b ( ⇤ ) ∂ σ ∂ ( ν h σ a ( ⇤ ) ⇤ h µ ν + 1 4 f ( ⇤ ) ⇤ − 1 ∂ σ ∂ λ ∂ µ ∂ ν h λσ = � κτ µ ν + η µ ν d ( ⇤ ) ⇤ h + = 0 = 0 = 0 ν ,µ + ( b + c + f ) h αβ ν = 0 = ( c + d ) ⇤ ∂ ν h + ( a + b ) ⇤ h µ � κτ ⇧ µ τ µ , αβν a + b = 0 Bianchi Identity c + d = 0 b + c + f = 0 λσ h λσ = ⇤⇧ µ ν Π − 1 h = h T T + h L + h T µ ν Π = P 2 P 0 ak 2 + s ( a − 3 c ) k 2

  13. Spin projection operators P. Van Nieuwenhuizen, . Nucl.Phys. B60 (1973), 478. 
 For this action, see: Biswas, Koivisto, AM 1302.0532

  14. Tree level Graviton Propagator Π = P 2 P 0 ak 2 + s ( a − 3 c ) k 2 No new propagating degree of freedom other than the massless Graviton a ( � ) = c ( � ) ⇒ 2 F 1 ( � ) + F 2 ( � ) + 2 F 3 ( � ) = 0  R 2 + R F 1 ( ⇤ ) R − 1 � Z d 4 x √− g S = 2 R µ ν F 2 ( ⇤ ) R µ ν

  15. Well known actions of Gravity (1) GR: a ( � ) = 1 − 1 2 F 2 ( � ) � − 2 F 3 ( � ) � b ( � ) = − 1 + 1 2 F 2 ( � ) � + 2 F 3 ( � ) � c ( � ) = 1 + 2 F 1 ( � ) � + 1 (2) F(R) Gravity: 2 F 2 ( � ) � d ( � ) = − 1 − 2 F 1 ( � ) � − 1 2 F 2 ( � ) � f ( � ) = − 2 F 1 ( � ) � − F 2 ( � ) � − 2 F 3 ( � ) � . (3) GB Gravity: (4) Weyl Gravity: L = R − 1 C 2 = R µ νρσ R µ νρσ − 2 R µ ν R µ ν + 1 m 2 C 2 3 R 2 Π = Π GR Biswas, Koivisto, AM 1302.0532

  16. (1) Gravitational Entropy ds 2 = − f ( r ) dt 2 + f ( r ) − 1 dr 2 + r 2 d Ω 2 ✓ ◆ I ∂ L q ( r ) d Ω 2 S W = − 8 π ∂ R rtrt r = r H , t =const Wald (1990, 1993), Iyer, Wald (1993) S W = Area 4 G [1 + α (2 F 1 + F 2 + 2 F 3 ) 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)

  17. (2) Newtonian Limit Π = P 2 P 0 a ( ⇤ ) = c ( ⇤ ) = e − ⇤ /M 2 s ak 2 + ( a � 3 c ) k 2 − � e − � � � � � � � M 2 − 1 M 2 − 1 d 4 x √− g R e � R µ ν S = 2 + R R − 2 R µ ν � � above action contains only the graviton as physical degrees of freedom as ds 2 = − (1 − 2 Φ ) dt 2 + (1 + 2 Ψ ) dr 2 ✓ rM ◆ Φ = Ψ = Gm r erf 2 Biswas, Gerwick, Koivisto, Mazumdar, Phys. Rev. Lett. (2012) (gr-qc/1110.5249)

  18. Non-singular static solution V(r) dr 2 ✓ ◆ 1 − 2 Gm ds 2 = dt 2 − erf ( rM/ 2) 1.2 1 − 2 Gm � � r erf ( rM/ 2) r 1.0 mM ⌧ M 2 ) m ⌧ M p p = 0.8 Out[37]= 0.6 0.4 0.2 r 2 4 6 8 10 Φ ( r ) → const . r → 0 , erf( r ) → r Biswas, Gerwick, Koivisto, AM, Φ ( r ) → 1 PRL (2012) erf( r ) → 1 (gr-qc/1110.5249) r → ∞ , r Frolov (2015), Frolov, et.al., (2015), Modesto, et.al., (2014), Shapiro, et.al. (2014)

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