Introduction Quadratic-Curvature Gravity in 4D ADT energy Iyer-Wald Charges and Topological Regularization Conclusions and future directions Energy in Higher-Derivative Gravity via Topological Regularization D. Rivera-Betancour Universidad Andrés Bello based on Anastasiou, Olea and D.R.B (UNAB), PLB 788, 021 (2018), Giribet (NYU), Miskovic (PUCV), Olea and D.R.B (UNAB), PRD 98,044046 (2018) and arXiv:1904.XXXX. April 4, 2019 D. Rivera-Betancour Introduction Quadratic-Curvature Gravity in 4D ADT energy Iyer-Wald Charges and Topological Regularization Conclusions and future directions Overview 1 Introduction 2 Quadratic-Curvature Gravity in 4D 3 ADT energy 4 Iyer-Wald Charges and Topological Regularization 5 Conclusions and future directions D. Rivera-Betancour Introduction Quadratic-Curvature Gravity in 4D ADT energy Iyer-Wald Charges and Topological Regularization Conclusions and future directions Topology and its applications Topology plays a fundamental role in Mathematics and Physics. Some examples are Four color theorem Topological Networks Quantum Hall effect Topological Insulators D. Rivera-Betancour Introduction Quadratic-Curvature Gravity in 4D ADT energy Iyer-Wald Charges and Topological Regularization Conclusions and future directions Topology and gravity How to use top. terms in gravity? D. Rivera-Betancour Introduction Quadratic-Curvature Gravity in 4D ADT energy Iyer-Wald Charges and Topological Regularization Conclusions and future directions Topology and gravity Problems with the application of Noether theorem to gravity. 2M G ds = − 1 − r 2 2M G −1 2 dt + 1 − dr + rdΩ2 r 2 The Komar charge gives Q[∂t ] = Incorrect overall factor. D. Rivera-Betancour M . 2 Introduction Quadratic-Curvature Gravity in 4D ADT energy Iyer-Wald Charges and Topological Regularization Conclusions and future directions Topology and gravity In the presence of a negative cosmological constant −1 2M G r2 2M G r2 2 ds = − 1 − + 2 dt + 1 − + 2 dr2 +rdΩ2 . r ` r ` 2 We have Q[∂t ] = M + ∞. 2 The energy is divergent. Background subtraction Holographic renormalization [S. de Haro, K. Skenderis and S. Solodukhin, CMP 788, 217 (2000)] D. Rivera-Betancour Introduction Quadratic-Curvature Gravity in 4D ADT energy Iyer-Wald Charges and Topological Regularization Conclusions and future directions Topology and gravity The inclusion of topological terms to the Einstein-Hilbert action with a negative cosmological constant render the variational problem well-posed and regularizes the Noether charges. Z Z 1 4 √ d x −g (R − 2Λ) + γ E4 , (1) I= 16πG M where γ = `2 /(64πG). [ R. Aros et al., PRL 84, 1647 (2000)] D. Rivera-Betancour M Introduction Quadratic-Curvature Gravity in 4D ADT energy Iyer-Wald Charges and Topological Regularization Conclusions and future directions Quadratic-Curvature Gravity QCG action is defined by a combination of quadratic couplings in the curvature added on top to the Einstein-Hilbert action. Z √ 1 IQCG = d4 x −g R − 2Λ + αRµν Rµν + βR2 16πG where α y β are coupling constants. The inclusion of the curvature squared terms improve the UV behavior. [ K.S. Stelle, PRD 16, 953 (1977)] This theory has a massive scalar field and massive and massless spin-2 fields. Higher-Derivative gravity introduces new couplings and sources at the boundary. D. Rivera-Betancour Introduction Quadratic-Curvature Gravity in 4D ADT energy Iyer-Wald Charges and Topological Regularization Conclusions and future directions Field equations and surface terms Taking arbitrary variations of the action one gets Z √ 1 δI = d4 x −g (E µν δgµν + ∇α Θα ) , 16πG where E µν = Gµν + α (g µν − ∇µ ∇ν ) R + αGµν + 1 µν σρ 1 µν µσνρ µν +2α R − g R Rσρ + 2βR R − g R + 4 4 +2β (g µν − ∇µ ∇ν ) R. D. Rivera-Betancour Introduction Quadratic-Curvature Gravity in 4D ADT energy Iyer-Wald Charges and Topological Regularization Conclusions and future directions Field equations For the vacuum state of the theory that satisfies µν Rαβ =− 1 [µν] δ , 2 `ef f [αβ] the relation between the cosmological constant and the AdS radius is not affected by the quadratic curvature contribution. D. Rivera-Betancour Introduction Quadratic-Curvature Gravity in 4D ADT energy Iyer-Wald Charges and Topological Regularization Conclusions and future directions Abbott-Deser-Tekin mass Consider the linearization the EOMs through gµν = ḡµν + hµν δ(Gµν + E µν ) = T µν J µ = T µν ξ¯ν ¯ ν q µν Jµ = ∇ ADT The charge is written as a surface integral of the 2-form as Z 1 α αβ dSβ qADT QADT [ξ] = 8πG Σ [ S. Deser and B. Tekin, PRD 67, 084009 (2003) ] D. Rivera-Betancour Introduction Quadratic-Curvature Gravity in 4D ADT energy Iyer-Wald Charges and Topological Regularization Conclusions and future directions Abbott-Deser-Tekin mass Expressing the full prepotential in terms of α y β for QCG action we obtain αβ ¯ [α hβ]λ + ξ¯[α ∇ ¯ β] h + hλ[α ∇ ¯ β] ξ¯λ − qADT = (1 + 2Λ (4β + α)) ξ¯λ ∇ ! 1 ¯ α ¯β [α ¯ β]λ ¯ ¯ β] RL + RL ∇ ¯ α ξ¯β − + (α + 2β) 2ξ¯[α ∇ − ξ ∇λ h + h∇ ξ 2 ¯ [α Gβ]λ + 2Gλ[α ∇ ¯ β] ξ¯λ . −α 2ξ¯λ ∇ L L For the Schwarzschild-AdS black hole the mass is M = m [1 + 2Λ(α + 4β)] . D. Rivera-Betancour Introduction Quadratic-Curvature Gravity in 4D ADT energy Iyer-Wald Charges and Topological Regularization Conclusions and future directions Critical Gravity QCG suffers from having ghosts (Massive spin-2 field). This can be eliminate at the points α = −3β and β = −1/2Λ (Critical Gravity). The massive spin-2 field becomes massless and the scalar mode is decoupled. The mass of the black hole and the energy of the propagating modes vanish. The linearized EOMs degenerates and, Additional propagating modes appear (Logarithmic modes). D. Rivera-Betancour Introduction Quadratic-Curvature Gravity in 4D ADT energy Iyer-Wald Charges and Topological Regularization Conclusions and future directions Critical Gravity Can we trust linearized charges? D. Rivera-Betancour Introduction Quadratic-Curvature Gravity in 4D ADT energy Iyer-Wald Charges and Topological Regularization Conclusions and future directions Iyer-Wald Charges For a Lagrangian of the type, L (gµν , Rµναβ ), the conserved current reads αβ µ ν αβ ν J α = 2∇β Eµν ∇ ξ + 2∇µ Eµν ξ + (EOM )αβ ξβ , αβ where Eµν correspond to derivatives of the Lagrangian with respect to the Riemann tensor αβ Eµν = D. Rivera-Betancour ∂L µν . ∂Rαβ Introduction Quadratic-Curvature Gravity in 4D ADT energy Iyer-Wald Charges and Topological Regularization Conclusions and future directions Iyer-Wald Charges For QCG the Iyer charge reads 1 QIW [ξ] = 8πG α Z αβ dSβ qIW , Σ for the Iyer-Wald prepotential αβ αβ µ ν αβ ν = Eµν ∇ ξ + 2∇µ Eµν ξ . qIW The mass of Schwarzchild-AdS black hole is divergent and has the incorrect overall factor D. Rivera-Betancour Introduction Quadratic-Curvature Gravity in 4D ADT energy Iyer-Wald Charges and Topological Regularization Conclusions and future directions Topological invariants and conserved quantities In Gauss-Bonnet we trust! Consider now the addition of the Euler density in four dimensions by an arbitrary coupling. √ µν αβ E4 = −g Rαβ Rµν − 4Rµν Rµν + R2 . This does not affect the EOMs, but It does modify the surface terms. Z √ [αβγδ] σλ δItot = d4 x −g · · · + γ g νλ δ[µνσλ] Rγδ δΓµβλ , M D. Rivera-Betancour Introduction Quadratic-Curvature Gravity in 4D ADT energy Iyer-Wald Charges and Topological Regularization Conclusions and future directions Topological invariants and conserved quantities The minimum requirement for finite conserved charges is a finite energy for the vacumm. Demanding that δItot = 0 for the vacuum solution (well-posed variational principle) one can fix the coupling γ γ= `2 [1 + 2Λ (4β + α)] . 4 D. Rivera-Betancour Introduction Quadratic-Curvature Gravity in 4D ADT energy Iyer-Wald Charges and Topological Regularization Conclusions and future directions Topological invariants and conserved quantities The prepotential now is αβ qtop = i 1 µ ν h [αβ] [α β] [αβ] [αβγδ] σλ ∇ ξ δ[µν] + αR[µ δν] + 2βRδ[µν] + γδ[µνσλ] Rγδ + 2 [α β] [αβ] + ∇µ αR[µ δν] + 2βRδ[µν] ξ ν , and the conserved charges read Qα top [ξ] 1 = 8πG Z Σ D. Rivera-Betancour αβ dSβ qtop . Introduction Quadratic-Curvature Gravity in 4D ADT energy Iyer-Wald Charges and Topological Regularization Conclusions and future directions Topological invariants and conserved quantities For the Schwarzchild-AdS black hole the mass reads Qt [∂t ] = m [1 + 2Λ (4β + α)] . Coming back to Critical Gravity, Noether charges are written as Z `2 1 [γδ] [µνσλ] α β γδ γδ µ Q [ξ] = dSν δ[αβγδ] ∇ ξ Rσλ + 2 δ[σλ] − Wσλ . 64πG ` Σ All Einstein spaces give a vanishing Noether charge. [G. Anastasiou, R. Olea and D.R.B, PLB 788 (2018)]. D. Rivera-Betancour Introduction Quadratic-Curvature Gravity in 4D ADT energy Iyer-Wald Charges and Topological Regularization Conclusions and future directions Non-Einstein solutions Considering the non-Einstein space that solves the EOM and represent the propagation of gravitational waves ds2 = `2 −(1 + F (z))dt2 + 2dtdu + dz 2 + dx2 , 2 z where F is a function that does not depend on the lightlike coordinate u. The only non-vanishing part of the EOM is ( − m̃2 )F = 0 , with the effective mass parameter m̃2 = 6(α + 4β) − `2 . α D. Rivera-Betancour Introduction Quadratic-Curvature Gravity in 4D ADT energy Iyer-Wald Charges and Topological Regularization Conclusions and future directions Non-Einstein solutions Two sectors appear from the above equation : F (z) = cte and F (z) = z 3 (Einstein). F (z) = z k (non-Einstein). q 2 Here k± = 32 ± 94 + 6(α+4β)−` . [E. A. Beato, G. Giribet and M. α Hassaine, arXiv:1207.0475 (2012)] D. Rivera-Betancour Introduction Quadratic-Curvature Gravity in 4D ADT energy Iyer-Wald Charges and Topological Regularization Conclusions and future directions Non-Einstein solutions For the non-Einstein sector of the solution one finds the prepotential [αβ] αβ = z k+1 `−4 k(αk 2 − 3αk + `2 − 6α − 24β) δ[uz] . qtop It vanishes for k = k± . This is in total agreement with the ADT method. D. Rivera-Betancour Introduction Quadratic-Curvature Gravity in 4D ADT energy Iyer-Wald Charges and Topological Regularization Conclusions and future directions Conclusions It was provided a new energy definition for Higher-Derivative Gravity that does not come from linearization. Topological regularization contributes for gravity theories with non-linear contribution on the curvature. Our method works for Einstein and non-Einstein spaces. D. Rivera-Betancour Introduction Quadratic-Curvature Gravity in 4D ADT energy Iyer-Wald Charges and Topological Regularization Conclusions and future directions Future directions Check this energy definition for other massive non-Einstein solutions in QCG, for example non-Einstein black holes in Critical Gravity [R. Svarc, J. Podolsky, V. Pravda and A. Pravdova, PRL 121, 231104 (2018)] . Extend this definition to higher dimensions. [arXiv:1904.XXXX]. Prove that for certain cases that the charge is proportional to the Weyl tensor (Ashtekar-Magnon-Das mass) in QCG [ Y. Pang, PRD 81, 087501 (2011)]. D. Rivera-Betancour
