# Null boundary terms for Lanczos–Lovelock gravity

## Abstract

We derive boundary terms appropriate for the general Lanczos–Lovelock action on a null boundary, when Dirichlet boundary conditions are imposed. We believe that these boundary terms have been derived for the *first* time in the literature. In this derivation, we rely *only* on the structure of the boundary variation of the action for Lanczos–Lovelock gravity. We also provide the null boundary term for Gauss–Bonnet gravity separately.

## Keywords

Null surface Boundary terms Lanczos–Lovelock gravity## Notes

### Acknowledgements

Research of SC is funded by the INSPIRE Faculty Fellowship (Reg. No. DST/INSPIRE/04/2018/000893) from Department of Science and Technology, Government of India. Research of KP has been supported by the SERB-NPDF grant (No. PDF/2017/002782) from DST, Government of India; the DGAPA postdoctoral fellowship from UNAM, Mexico and the Fondecyt Postdoctoral fellowship (No. 3180421) from the Government of Chile. KP would like to thank Perimeter Institute for kind hospitality during a stay, discussions in which duration led to this project. SC thanks IIT Gandhinagar and Albert Einstein Institute, Golm, where parts of this work were done, for warm hospitality. SC and KP would like to thank T. Padmanabhan, K. Lochan, Dean Carmi and Pratik Rath for useful discussions. We would also like to thank the referee for his/her comments which have helped to improve the manuscript.

## References

- 1.Einstein, A.: Hamilton’s principle and the general theory of relativity. Sitzungsber. Preuss. Akad. Wiss. Berlin (Math.Phys.)
**1916**, 1111–1116 (1916)Google Scholar - 2.Will, C.M.: The confrontation between general relativity and experiment. Living Rev. Rel.
**17**, 4 (2014). arXiv:1403.7377 [gr-qc]CrossRefGoogle Scholar - 3.Hawking, S.W.: The path integral approach to quantum gravity. In: Hawking, S.W., Israel, W. (eds.) General Relativity: An Einstein Centenary Survey, pp. 746–789. University Press, Cambridge (1979)Google Scholar
- 4.Dyer, E., Hinterbichler, K.: Boundary terms, variational principles and higher derivative modified gravity. Phys. Rev. D
**79**, 024028 (2009). arXiv:0809.4033 [gr-qc]ADSMathSciNetCrossRefGoogle Scholar - 5.Padmanabhan, T.: Gravitation: Foundations and Frontiers. Cambridge University Press, Cambridge (2010)CrossRefGoogle Scholar
- 6.Gibbons, G., Hawking, S.: Action integrals and partition functions in quantum gravity. Phys. Rev. D
**15**, 2752–2756 (1977)ADSCrossRefGoogle Scholar - 7.York, J., James, W.: Role of conformal three geometry in the dynamics of gravitation. Phys. Rev. Lett.
**28**, 1082–1085 (1972)ADSCrossRefGoogle Scholar - 8.Katz, J.: A note on Komar’s anomalous factor. Class. Quantum Gravity
**2**(3), 423 (1985)ADSMathSciNetCrossRefGoogle Scholar - 9.Katz, J., Lerer, D.: On global conservation laws at null infinity. Class. Quantum Gravity
**14**, 2249–2266 (1997). arXiv:gr-qc/9612025 [gr-qc]ADSMathSciNetCrossRefGoogle Scholar - 10.Barth, N.H.: Second and Fourth Order Gravitational Actions on Manifolds with Boundaries. Ph.D. thesis, The University of North Carolina at Chapel Hill (1983)Google Scholar
- 11.Neiman, Y.: On-shell actions with lightlike boundary data. arXiv:1212.2922 [hep-th]
- 12.Parattu, K., Chakraborty, S., Majhi, B.R., Padmanabhan, T.: A boundary term for the gravitational action with null boundaries. Gen. Relativ. Gravit.
**48**(7), 94 (2016). arXiv:1501.01053 [gr-qc]ADSMathSciNetCrossRefGoogle Scholar - 13.Parattu, K., Chakraborty, S., Padmanabhan, T.: Variational principle for gravity with null and non-null boundaries: a unified boundary counter-term. Eur. Phys. J. C
**76**(3), 129 (2016). arXiv:1602.07546 [gr-qc]ADSCrossRefGoogle Scholar - 14.Chakraborty, S.: Boundary terms of the Einstein–Hilbert action. Fundam. Theor. Phys.
**187**, 43–59 (2017). arXiv:1607.05986 [gr-qc]MathSciNetCrossRefGoogle Scholar - 15.Lehner, L., Myers, R.C., Poisson, E., Sorkin, R.D.: Gravitational action with null boundaries. Phys. Rev. D
**94**(8), 084046 (2016). arXiv:1609.00207 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 16.Hopfmller, F., Freidel, L.: Gravity degrees of freedom on a null surface. Phys. Rev. D
**95**(10), 104006 (2017). arXiv:1611.03096 [gr-qc]ADSMathSciNetCrossRefGoogle Scholar - 17.Jubb, I., Samuel, J., Sorkin, R., Surya, S.: Boundary and corner terms in the action for general relativity. Class. Quantum Gravity
**34**(6), 065006 (2017). arXiv:1612.00149 [gr-qc]ADSMathSciNetCrossRefGoogle Scholar - 18.Aghapour, S., Jafari, G., Golshani, M.: On variational principle and canonical structure of gravitational theory in double-foliation formalism. arXiv:1808.07352 [gr-qc]
- 19.Hartle, J.B., Sorkin, R.: Boundary terms in the action for the regge calculus. Gen. Relativ. Gravit.
**13**, 541–549 (1981)ADSMathSciNetCrossRefGoogle Scholar - 20.Farhi, E., Guth, A.H., Guven, J.: Is it possible to create a universe in the laboratory by quantum tunneling? Nucl. Phys. B
**339**, 417–490 (1990)ADSMathSciNetCrossRefGoogle Scholar - 21.Brill, D.: Splitting of an extremal Reissner–Nordstrom throat via quantum tunneling. Phys. Rev. D
**46**, 1560–1565 (1992). arXiv:hep-th/9202037 [hep-th]ADSCrossRefGoogle Scholar - 22.Hayward, G.: Gravitational action for space–times with nonsmooth boundaries. Phys. Rev. D
**47**, 3275–3280 (1993)ADSMathSciNetCrossRefGoogle Scholar - 23.Brill, D., Hayward, G.: Is the gravitational action additive? Phys. Rev. D
**50**, 4914–4919 (1994). arXiv:gr-qc/9403018 [gr-qc]ADSMathSciNetCrossRefGoogle Scholar - 24.Reynolds, A., Ross, S.F.: Complexity in de sitter space. Class. Quantum Gravity
**34**(17), 175013 (2017). arXiv:1706.03788 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 25.Yang, R.-Q., Niu, C., Kim, K.-Y.: Surface counterterms and regularized holographic complexity. JHEP
**09**, 042 (2017). arXiv:1701.03706 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 26.Reynolds, A., Ross, S.F.: Divergences in holographic complexity. Class. Quantum Gravity
**34**(10), 105004 (2017). arXiv:1612.05439 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 27.Carmi, D., Myers, R.C., Rath, P.: Comments on holographic complexity. JHEP
**03**, 118 (2017). arXiv:1612.00433 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 28.Chapman, S., Marrochio, H., Myers, R.C.: Complexity of formation in holography. JHEP
**01**, 062 (2017). arXiv:1610.08063 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 29.Yang, R.-Q.: Strong energy condition and complexity growth bound in holography. Phys. Rev. D
**95**(8), 086017 (2017). arXiv:1610.05090 [gr-qc]ADSMathSciNetCrossRefGoogle Scholar - 30.Ben-Ami, O., Carmi, D.: On volumes of subregions in holography and complexity. JHEP
**11**, 129 (2016). arXiv:1609.02514 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 31.Fareghbal, R., Karimi, P.: Complexity growth in flat spacetimes. Phys. Rev. D
**98**(4), 046003 (2018). arXiv:1806.07273 [hep-th]ADSCrossRefGoogle Scholar - 32.Auzzi, R., Baiguera, S., Grassi, M., Nardelli, G., Zenoni, N.: Complexity and action for warped AdS black holes. JHEP
**09**, 013 (2018). arXiv:1806.06216 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 33.Alishahiha, M., Faraji Astaneh, A., Mohammadi Mozaffar, M.R., Mollabashi, A.: Complexity growth with Lifshitz scaling and hyperscaling violation. JHEP
**07**, 042 (2018). arXiv:1802.06740 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 34.Bolognesi, S., Rabinovici, E., Roy, S.R.: On some universal features of the holographic quantum complexity of bulk singularities. JHEP
**06**, 016 (2018). arXiv:1802.02045 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 35.Reynolds, A.P., Ross, S.F.: Complexity of the AdS soliton. Class. Quantum Gravity
**35**(9), 095006 (2018). arXiv:1712.03732 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 36.Yang, R.-Q., Niu, C., Zhang, C.-Y., Kim, K.-Y.: Comparison of holographic and field theoretic complexities for time dependent thermofield double states. JHEP
**02**, 082 (2018). arXiv:1710.00600 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 37.Carmi, D., Chapman, S., Marrochio, H., Myers, R.C., Sugishita, S.: On the time dependence of holographic complexity. JHEP
**11**, 188 (2017). arXiv:1709.10184 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 38.Yang, R.-Q.: Complexity for quantum field theory states and applications to thermofield double states. Phys. Rev. D
**97**(6), 066004 (2018). arXiv:1709.00921 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 39.Brown, A.R., Roberts, D.A., Susskind, L., Swingle, B., Zhao, Y.: Holographic complexity equals bulk action? Phys. Rev. Lett.
**116**(19), 191301 (2016). arXiv:1509.07876 [hep-th]ADSCrossRefGoogle Scholar - 40.Brown, A.R., Roberts, D.A., Susskind, L., Swingle, B., Zhao, Y.: Complexity, action, and black holes. Phys. Rev. D
**93**(8), 086006 (2016). arXiv:1512.04993 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 41.Eling, C.: Spontaneously broken asymptotic symmetries and an effective action for horizon dynamics. JHEP
**02**, 052 (2017). arXiv:1611.10214 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 42.Maltz, J.: de Sitter harmonies: cosmological spacetimes as resonances. Phys. Rev. D
**95**(6), 066006 (2017). arXiv:1611.03491 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 43.Maltz, J., Susskind, L.: de Sitter space as a resonance. Phys. Rev. Lett.
**118**(10), 101602 (2017). arXiv:1611.00360 [hep-th]ADSCrossRefGoogle Scholar - 44.Buck, M., Dowker, F., Jubb, I., Surya, S.: Boundary terms for causal sets. Class. Quantum Gravity
**32**(20), 205004 (2015). arXiv:1502.05388 [gr-qc]ADSMathSciNetCrossRefGoogle Scholar - 45.Lanczos, C.: A remarkable property of the Riemann–Christoffel tensor in four dimensions. Ann. Math.
**39**, 842–850 (1938)MathSciNetCrossRefGoogle Scholar - 46.Lovelock, D.: The Einstein tensor and its generalizations. J. Math. Phys.
**12**, 498–501 (1971)ADSMathSciNetCrossRefGoogle Scholar - 47.Dadhich, N.: Characterization of the Lovelock gravity by Bianchi derivative. Pramana
**74**, 875–882 (2010). arXiv:0802.3034 [gr-qc]ADSCrossRefGoogle Scholar - 48.Padmanabhan, T., Kothawala, D.: Lanczos–Lovelock models of gravity. Phys. Rep.
**531**, 115–171 (2013). arXiv:1302.2151 [gr-qc]ADSMathSciNetCrossRefGoogle Scholar - 49.Deruelle, N., Madore, J.: On the quasilinearity of the Einstein–’Gauss–Bonnet’ gravity field equations. arXiv:gr-qc/0305004 [gr-qc]
- 50.Chakraborty, S.: Field equations for Lovelock gravity: an alternative route. Adv. High Energy Phys.
**2018**, 6509045 (2018). arXiv:1704.07366 [gr-qc]MathSciNetCrossRefGoogle Scholar - 51.Bunch, T.: Surface terms in higher derivative gravity. J. Phys. A Math. Gen.
**14**(5), L139 (1981)ADSMathSciNetCrossRefGoogle Scholar - 52.Myers, R.C.: Higher derivative gravity, surface terms and string theory. Phys. Rev. D
**36**, 392 (1987)ADSMathSciNetCrossRefGoogle Scholar - 53.Davis, S.C.: Generalized Israel junction conditions for a Gauss–Bonnet brane world. Phys. Rev. D
**67**, 024030 (2003). arXiv:hep-th/0208205 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 54.Yale, A.: Simple counterterms for asymptotically AdS spacetimes in Lovelock gravity. Phys. Rev. D
**84**, 104036 (2011). arXiv:1107.1250 [gr-qc]ADSCrossRefGoogle Scholar - 55.Miskovic, O., Olea, R.: Counterterms in dimensionally continued AdS gravity. JHEP
**10**, 028 (2007). arXiv:0706.4460 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 56.Deruelle, N., Merino, N., Olea, R.: Einstein–Gauss–Bonnet theory of gravity: the Gauss–Bonnet–Katz boundary term. Phys. Rev. D
**97**(10), 104009 (2018). arXiv:1709.06478 [gr-qc]ADSCrossRefGoogle Scholar - 57.Deruelle, N., Merino, N., Olea, R.: Chern–Weil theorem, Lovelock Lagrangians in critical dimensions and boundary terms in gravity actions. arXiv:1803.04741 [gr-qc]
- 58.Chakraborty, S., Parattu, K., Padmanabhan, T.: A novel derivation of the boundary term for the action in Lanczos–Lovelock gravity. Gen. Relativ. Gravit.
**49**(9), 121 (2017). arXiv:1703.00624 [gr-qc]ADSMathSciNetCrossRefGoogle Scholar - 59.Cano, P.A.: Lovelock action with nonsmooth boundaries. Phys. Rev. D
**97**(10), 104048 (2018). arXiv:1803.00172 [gr-qc]ADSCrossRefGoogle Scholar - 60.Padmanabhan, T.: A short note on the boundary term for the Hilbert action. Mod. Phys. Lett. A
**29**, 1450037 (2014)ADSMathSciNetCrossRefGoogle Scholar - 61.Lovelock, D., Rund, H.: Tensors, Differential Forms, and Variational Principles. Dover Publications, Mineola (1989)zbMATHGoogle Scholar
- 62.Frankel, T.: The Geometry of Physics: An Introduction, 2nd edn. Cambridge University Press, Cambridge (2006)zbMATHGoogle Scholar
- 63.Carter, B.: ch. 6: The general theory of the mechanical, electromagnetic and thermodynamic properties of black holes. In: Hawking, S.W., Israel, W. (eds.) General Relativity: An Einstein Centenary Survey, pp. 294–369. Cambridge University Press, Cambridge (1979)Google Scholar
- 64.Poisson, E.: A Relativist’s Toolkit: The Mathematics of Black–Hole Mechanics, 1st edn. Cambridge University Press, Cambridge (2007)zbMATHGoogle Scholar
- 65.Chakraborty, S., Padmanabhan, T.: Thermodynamical interpretation of the geometrical variables associated with null surfaces. Phys. Rev. D
**92**(10), 104011 (2015). arXiv:1508.04060 [gr-qc]ADSMathSciNetCrossRefGoogle Scholar - 66.Hájíček, P.: Exact models of charged black holes. Commun. Math. Phys.
**34**(1), 53–76 (1973)ADSCrossRefGoogle Scholar - 67.Gourgoulhon, E., Jaramillo, J.L.: A 3+1 perspective on null hypersurfaces and isolated horizons. Phys. Rep.
**423**, 159–294 (2006). arXiv:gr-qc/0503113 [gr-qc]ADSMathSciNetCrossRefGoogle Scholar - 68.Corichi, A., Reyes, J.D., Vukainac, T.: Weakly isolated horizons: first order actions and gauge symmetries. Class. Quantum Gravity
**34**(8), 085005 (2017). arXiv:1612.01462 [gr-qc]ADSMathSciNetCrossRefGoogle Scholar - 69.Cano, P.A., Hennigar, R.A., Marrochio, H.: Complexity Growth Rate in Lovelock Gravity. arXiv:1803.02795 [hep-th]
- 70.Vega, I., Poisson, E., Massey, R.: Intrinsic and extrinsic geometries of a tidally deformed black hole. Class. Quantum Gravity
**28**, 175006 (2011). arXiv:1106.0510 [gr-qc]ADSMathSciNetCrossRefGoogle Scholar - 71.Yang, R.-Q., Ruan, S.-M.: Comments on joint terms in gravitational action. Class. Quantum Gravity
**34**(17), 175017 (2017). arXiv:1704.03232 [gr-qc]ADSMathSciNetCrossRefGoogle Scholar - 72.Gourgoulhon, E.: 3+1 Formalism in General Relativity: Bases of Numerical Relativity, vol. 846. Springer, Berlin (2012)zbMATHGoogle Scholar
- 73.Wald, R.M.: General Relativity, 1st edn. The University of Chicago Press, Chicago (1984)CrossRefGoogle Scholar
- 74.Moncrief, V., Isenberg, J.: Symmetries of cosmological cauchy horizons. Commun. Math. Phys.
**89**(3), 387–413 (1983). https://doi.org/10.1007/BF01214662 ADSMathSciNetCrossRefzbMATHGoogle Scholar - 75.Friedrich, H., Racz, I., Wald, R.M.: On the rigidity theorem for space–times with a stationary event horizon or a compact Cauchy horizon. Commun. Math. Phys.
**204**, 691–707 (1999). arXiv:gr-qc/9811021 [gr-qc]ADSCrossRefGoogle Scholar - 76.Racz, I.: Stationary black holes as holographs. Class. Quantum Gravity
**24**, 5541–5572 (2007). arXiv:gr-qc/0701104 [gr-qc]ADSMathSciNetCrossRefGoogle Scholar - 77.Morales, E.M.: On a Second Law of Black Hole Mechanics in a Higher Derivative Theory of Gravity, Master’s thesis, Institut für Theoretische Physik der Georg-August-Universität zu Göttingen (2008). http://www.theorie.physik.uni-goettingen.de/forschung/qft/theses/dipl/Morfa-Morales.pdf
- 78.Baumgarte, T.W., Shapiro, S.L.: Numerical Relativity: Solving Einstein’s Equations on the Computer. Cambridge University Press, Cambridge (2010)CrossRefGoogle Scholar