# Higher-derivative gravity It is sometimes interesting to study higher-derivative gravity. One big motivation (for me) comes from [[0001 AdS-CFT|holography]]. In holography, the bulk theory is dual to the boundary theory, and in particular, one should be able to decode the bulk dynamics solely from boundary data. Adding higher-derivative terms adds another (actually infinitely many) dimension(s) to the game, and the game is to find out how the boundary theory knows about them. As an example, higher-derivative corrections change the [[0004 Black hole entropy|black hole entropy]] of a two-sided black hole, a quantity that should be computable on the boundary (as [[0145 Generalised area|an entanglement entropy]] between two copies of CFT). Besides, higher-derivative gravity can also be used to test things, such as methods of computing the entropy formula. Most methods of deriving the [[0004 Black hole entropy|entropy formula]] for general higher-derivative theories cannot fix the [[0018 JKM ambiguity|JKM ambiguity]], except the [[0005 Black hole second law|second law]] method (see [[2015#Wall (Essay)]]) and the [[0145 Generalised area|LM method]] (see [[2013#Lewkowycz, Maldacena]] and [[2013#Dong]]). Phenomenologically, higher derivatives generically appear when one integrates out the UV degrees of freedom in any UV-complete theories. Alternatively, even without knowing a UV-complete model, higher derivatives appear in any effective field theory just from dimensional analysis. In String Theory, at energy scales much smaller than the Planck mass, the effective theory takes the form of Einstein gravity plus infinitely many higher-derivative corrections organised in powers of $\alpha^\prime$. At energy scales comparable to $1/\sqrt{\alpha^\prime}$, there are also infinitely many higher spin fields, but they decouple at lower energy scales. ## Facts - cannot have consistent holographic theory of quantum gravity in which Gauss-Bonnet term is finite as $\hbar \rightarrow 0$, unless there is an infinite tower of spin fields as in string theory. - [[2014#Camanho, Edelstein, Maldacena, Zhiboedov]] ## Related topics - [[0118 Causality constraints for gravity]] - [[0005 Black hole second law]] - [[0004 Black hole entropy]] - [[0138 Variational principle]] - [[0102 Hayward term]] - [[0047 Renyi at finite n for higher derivative gravity]] - [[0145 Generalised area]] ## Properties ### Ghosts - a brief explanation around eq.3.7 of [[2010#Myers, Sinha (Nov)]] ### Solutions - order $\alpha^\prime$: [[ChernicoffGiribetOlivaOyarzo2022]][](https://arxiv.org/pdf/2207.13214.pdf) (both $R^2$ and dilaton) ## Examples ### 4-derivative - $f= \alpha\, R^2 + \beta\, R_{\mu\nu}R^{\mu\nu}$ - equivalent to Einstein gravity via a [[0355 Field redefinitions|field redefinition]] - identification of d.o.f. in [[Stelle1978]] - generically 6 (massive spin-0 and massive spin-2 (a ghost of negative energy)) - when $\beta=-4\alpha$ (square of Weyl tensor): reduces to 5 - recast in first-order form and constraints analysed in [[Boulware in Christensen1984]] and [[BuchbinderLyakhovich1987]] - $f= \alpha\, R^2 + \beta\, R_{\mu\nu}R^{\mu\nu}+\gamma\,R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$ - [[MagnanoFerrarisFrancaviglia1990]] ### Lovelock gravity - see [[0341 Lovelock gravity]] - EOM is second-order ### Einstein cubic gravity - original: [[2016#Bueno, Cano]] ### f(R) - see [[0561 f(R) gravity]] ### f(Riemann) - [[2009#Deruelle, Sasaki, Sendouda, Yamauchi]] - summarised in [[2018#Jiang, Zhang]] ### With covariant derivatives - [[EdelsteinSanchezRodriguez2022]][](https://arxiv.org/pdf/2204.13567.pdf) ## Special ones - general refs - [[2013#Grumiller, Riedler, Rosseel, Zojer]] - New Massive Gravity [[BergshoeffHohmTownsend2021]][](https://arxiv.org/abs/0901.1766) - 3d -> kinetic term okay because no propagation possible - string effective action - expansion in $\alpha^\prime$ for a restricted class of metrics to all orders - [[2020#Wang, Wu, Yang, Ying]] - [[HohmZwiebach2019]]