Abstract
The main goal of this paper is to get in a straightforward form the field equations in metric f (R) gravity, using elementary variational principles and adding a boundary term in the action, instead of the usual treatment in an equivalent scalar–tensor approach. We start with a brief review of the Einstein–Hilbert action, together with the Gibbons–York–Hawking boundary term, which is mentioned in some literature, but is generally missing. Next we present in detail the field equations in metric f (R) gravity, including the discussion about boundaries, and we compare with the Gibbons–York–Hawking term in General Relativity. We notice that this boundary term is necessary in order to have a well defined extremal action principle under metric variation.
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References
Will C.M.: Theory and Experiment in Gravitational Physics. Cambridge University Press, Cambridge (1993)
Misner C.W., Thorne K.S., Wheeler J.H.: Gravitation. W.H. Freeman and Company, Reading (1973)
Wald R.M.: General Relativity. The University of Chicago Press, Chicago (1984)
Poisson E.: A Relativist’s Toolkit—The Mathematics of Black-Hole Mechanics. Cambridge University Press, Cambridge (2004)
Padmanabhan T.: Gravitation: Foundations and Frontiers. Cambridge University Press, Cambridge (2010)
Weinberg S.: Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. Wiley, New York (1972)
Carroll S.M.: Spacetime and Geometry: An Introduction to General Relativity. Addison Wesley, San Francisco (2004)
Hawking S.W., Ellis J.F.R.: The Large Scale Structure of Space–Time. Cambridge University Press, Cambridge (1973)
Gibbons G.W., Hawking S.W.: Action integrals and partition functions in quantum gravity. Phys. Rev. D 15, 27–52 (1977)
Hawking S.W., Horowitz G.T.: The gravitational Hamiltonian, action, entropy and surface terms. Class. Quantum. Grav. 13, 1487–1498 (1996) arXiv:gr-qc/9501014
Schmidt H.-J.: Variational derivatives of arbitrarily high order and multi-inflation cosmological models. Class. Quantum Grav. 7, 1023–1031 (1990)
Wands D.: Extended gravity theories and the Einstein–Hilbert action Class. Quantum Grav. 11, 269–280 (1994) arXiv:gr-qc/9307034
Farhoudi M.: On higher order gravities, their analogy to GR, and dimensional dependent version of Duff’s trace anomaly relation. Gen. Relativ. Gravit. 38, 261–1284 (2006) arXiv:physics/0509210v2
Querella, L.: Variational principles and cosmological models in higher-order gravity. PhD thesis (1998). arXiv:gr-qc/9902044v1
Nojiri S., Odintsov S.D.: Modified gravity as an alternative for Lambda-CDM cosmology. J. Phys. A 40, 6725–6732 (2007) arXiv:hep-th/0610164
Sami, M.: Dark energy and possible alternatives (2009). arXiv:0901.0756
Borowiec, A., Godlowski, W., Szydlowski, M.: Dark matter and dark energy as a effects of modified gravity. ECONF C0602061, 09 (2006). arXiv:astro-ph/0607639v2
Durrer R., Maartens R.: Dark energy and dark gravity: theory overview. Gen. Relativ. Gravit. 40, 301–328 (2008)
Carroll S.M., Duvvuri V., Trodden M., Turner M.S.: Is cosmic speed-up due to new gravitational physics?. Phys. Rev. D 70, 043528 (2004) arXiv:astro-ph/0306438
Nojiri, S., Odintsov, S.D.: Introduction to modified gravity and gravitational alternative for dark energy. ECONF C0602061, 06 (2006). arXiv:hep-th/0601213
Capozziello S., Francaviglia M.: Extended theories of gravity and their cosmological and astrophysical applications. Gen. Relativ. Gravit. 40, 357–420 (2008) arXiv:0706.1146v2
Faraoni, V.: f (R) gravity: successes and challenges (2008). arXiv:0810.2602
Sotiriou, T.P.: 6+1 lessons from f (R) gravity. J. Phys. Conf. Ser. 189, 012039 (2009). arXiv:0810.5594
Capozziello, S., De Laurentis, M., Faraoni, V.: A bird’s eye view of f (R)-gravity (2009). arXiv:0909.4672
Sotiriou, T.P., Faraoni, V.: f (R) Theories of gravity (2008). arXiv:0810.2602
Buchdahl H.A.: Non-linear Lagrangians and cosmological theory. Mon. Notices R. Astron. Soc. 150, 1–8 (1970)
Barth N.H.: The fourth-order gravitational action for manifolds with boundaries. Class. Quantum Grav. 2, 497–513 (1985)
Nojiri S., Odintsov S.D.: Brane-world cosmology in higher derivative gravity or warped compactification in the next-to-leading order of AdS/CFT correspondence. JHEP 0007, 049 (2000) arXiv:hep-th/0006232
Nojiri S., Odintsov S.D.: Is brane cosmology predictable? Gen. Relativ. Gravit. 37, 1419–1425 (2005) arXiv:hep-th/0409244
Madsen M.S., Barrow J.D.: De Sitter ground states and boundary terms in generalized gravity. Nucl. Phys. B 323, 242–252 (1989)
Fatibene L., Ferraris M., Francaviglia M.: Augmented variational principles and relative conservation laws in classical field theory. Int. J. Geom. Methods Mod. Phys. 2, 373–392 (2005) arXiv:math-ph/0411029v1
Francaviglia M., Raiteri M.: Hamiltonian, energy and entropy in general relativity with non-orthogonal boundaries. Class. Quant. Grav. 19, 237–258 (2002) arXiv:grqc/ 0107074v1
Casadio R., Gruppuso A.: On boundary terms and conformal transformations in curved spacetimes. Int. J. Mod. Phys. D 11, 703–714 (2002) arXiv:gr-qc/0107077
Balcerzak, A., Dabrowski, M.P.: Gibbons-Hawking boundary terms and junction conditions for higher-order brane gravity models (2008). arXiv:0804.0855
Nojiri S., Odintsov S.D.: Finite gravitational action for higher derivative and stringy gravities. Phys. Rev. D 62, 064018 (2000) arXiv:hep-th/9911152
Dabrowski, M.P., Balcerzak, A.: Higher-order brane gravity models (2009). arXiv:0909.1079
Nojiri S., Odintsov S.D., Ogushi S.: Finite action in d5 gauged supergravity and dilatonic conformal anomaly for dual quantum field theory. Phys. Rev. D 62, 124002 (2000) arXiv:hep-th/0001122v4
Dyer E., Hinterbichler K.: Boundary terms, variational principles and higher derivative modified gravity. Phys. Rev. D. 79, 024028 (2009) arXiv:0809.4033
Sotiriou, T.P.: Modified actions for gravity: theory and phenomenology. PhD thesis, International School for Advanced Studies (2007). arXiv:0710.4438v1
Sotiriou T.P., Liberati S.: Metric-affine f(R) theories of gravity. Ann. Phys. 322, 935–966 (2007) arXiv:gr-qc/0604006v2
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Guarnizo, A., Castañeda, L. & Tejeiro, J.M. Boundary term in metric f (R) gravity: field equations in the metric formalism. Gen Relativ Gravit 42, 2713–2728 (2010). https://doi.org/10.1007/s10714-010-1012-6
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DOI: https://doi.org/10.1007/s10714-010-1012-6