# Variational principle (in gravity) ## Related - [[0102 Hayward term]] - [[0006 Higher-derivative gravity]] - [[0047 Renyi at finite n for higher derivative gravity]] - [[0127 Black hole thermodynamics]] ## Refs - [[2008#Dyer, Hinterbichler]] - [[SmolicTaylor2013]] - [[MadsonBarrow1989]] - [[GuarnizoCastanedaTejeiro2010]] f(R) - [[BuenoCanoLassoRamirez2016]] f(Lovelock) - [[TeimouriTalaganisEdholmMazumdar2016]] - [[BuenoCanoRuiperez2018]] Einstein cubic gravity - [[2022#Erdmenger, Hess, Matthaiakakis, Meyer]]: general prescription - in AdS - [[2005#Hollands, Ishibashi, Marolf]] - [[HohmTonni2010]]: arXiv:1001.3598 - [[2023#Cassani, Ruiperez, Turetta]] gauged [[0332 Supergravity|SUGRA]] ## Why boundary term - in Hamiltonian method, it is needed to give the right ADM energy [[1995#Hawking, Horowitz]] - ensure correct composition properties of PI - see [[GibbonsHawking1977]] - semiclassical BH entropy in Euclidean calculation comes solely from this term - see [[1992#Brown, York (a)]] ## Einstein - $S=S_{\mathrm{EH}}+S_{\mathrm{GHY}} =\frac{1}{16\pi G} \int d^{4} x \sqrt{-g} R+\frac{1} {8\pi G}\oint d^{3} x \sqrt{|h|} K$ - Neumman BC: [[KrishnanRaju2016]][](https://arxiv.org/pdf/1605.01603.pdf) - Robin BC: [[KrishnanMaheshwariSubramanian2017]][](https://arxiv.org/pdf/1702.01429.pdf) - separation of bulk and boundary action: [[Bhattacharya2022]][](https://arxiv.org/pdf/2207.08199.pdf) ## $f(R)$ - [[2008#Dyer, Hinterbichler]] - $S\sim\int d^{4} x \sqrt{|g|} F(R)+2 \oint d^{3} x \sqrt{|h|} F^{\prime}(R) K$ - straightforward generalisation of Einstein - boundary term needed to make the correspondence to scalar-tensor theory hold even at the boundary - see [[0006 Higher-derivative gravity]] - [[2019#Harlow, Wu]] ## $f(Ricci)$ - [[WangZhao2018]]@[](https://arxiv.org/pdf/1812.01854.pdf) - $\mathcal{L}_{B}=f^{\prime \mu \nu}\left(K_{\mu \nu}+n_{\mu} n_{\nu} K\right)$ - using $I_{b u l k}=\int_{\mathcal{M}} \mathrm{d}^{n} x \sqrt{g}\left[f\left(\phi_{\mu \nu \rho \sigma}, g_{\mu \nu}\right)-\psi^{\mu \nu \rho \sigma}\left(\phi_{\mu \nu \rho \sigma}-R_{\mu \nu \rho \sigma}\right)\right]$ - or $\mathcal{L}_{B}=n_{\mu}\left(f^{\prime \rho \mu} \Gamma_{\nu \rho}^{\nu}-f^{\prime \rho \nu} \Gamma_{\rho \nu}^{\mu}\right)$ - using $I_{b u l k}=\int_{\mathcal{M}} \mathrm{d}^{n} x \sqrt{g}\left[f\left(\phi_{\mu \nu}, g_{\mu \nu}\right)-\psi^{\mu \nu}\left(\phi_{\mu \nu}-R_{\mu \nu}\right)\right]$ ## $f(Riem)$ - boundary action $\bar{S}=-\oint_{\partial \mathcal{M}} \mathrm{d} \Sigma_{\mu} n^{\mu} \Psi \cdot K$ - n.b. [[2009#Deruelle, Sasaki, Sendouda, Yamauchi]] defines $d\Sigma_\mu$ w.r.t. inward normal for timelike and outward for spacelike, whereas [[2018#Jiang, Zhang]] uses outward normal for all cases ## GB - for 4-deriv theories, the only combination that allows good variational principle with one BC is GB - shown in [[Bunch1981]] - see also [[0341 Lovelock gravity]] and [[0425 Gauss-Bonnet gravity]] ## Theorems and general understanding - [[Gieres2022]][](https://arxiv.org/abs/2205.01459): Improvement of a conserved current density v.s. adding a total derivative to a Lagrangian density