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higher order field theories; boundary terms
It is shown that when in a higher order variational principle one fixes fields at the boundary leaving the field derivatives unconstrained, then the variational principle (in particular the solution space) is not invariant with respect to the addition of boundary terms to the action, as it happens instead when the correct procedure is applied. Examples are considered to show how leaving derivatives of fields unconstrained affects the physical interpretation of the model. This is justified in particular by the need of clarifying the issue for the purpose of applications to relativistic gravitational theories, where a bit of confusion still exists. On the contrary, as it is well known for variational principles of order $k$, if one fixes variables together with their derivatives (up to order $k-1$) on the boundary then boundary terms leave solution space invariant.
[1] Capozziello, S., Laurentis, M. De, Faraoni, V.: A bird’s eye view of $f(R)$-gravity. (2009), arXiv:0909.4672
[2] Carrol, S. M.: Lecture Notes on General Relativity. arXiv:gr-qc/9712019v1
[3] Fatibene, L., Ferraris, M., Francaviglia, M.: Augmented Variational Principles and Relative Conservation Laws in Classical Field Theory. Int. J. Geom. Methods Mod. Phys. 2 (3) 2005 373-392 DOI 10.1142/S0219887805000557 | MR 2152166 | Zbl 1133.70335
[4] Fatibene, L., Ferraris, M., Francaviglia, M., Raiteri, M.: Noether charges, Brown-York quasilocal energy, and related topics. J. Math. Phys. 42 (3) 2001 1173-1195 DOI 10.1063/1.1336514 | MR 1814443 | Zbl 1053.83009
[5] Ferraris, M., Francaviglia, M.: Covariant first-order Lagrangians, energy-density and superpotentials in general relativity. GRG 22 (9) 1990 965-985 DOI 10.1007/BF00757808 | MR 1067612 | Zbl 0713.53055
[6] Hawking, S.W., Ellis, G.F.R., Landshoff, P.V., Nelson, D.R., Sciama, D.W., Weinberg, S.: The Large Scale Structure of Space-Time. Cambridge University Press, New York 1973 MR 0424186
[7] Hawking, S.W., Hunter, C.J.: Gravitational Entropy and Global Structure. Phys. Rev. D59 1999 044025 arXiv:hep-th/9808085v2 MR 1683086
[8] Hinterbichler, K.: Boundary Terms, Variational Principles and Higher Derivative Modified Gravity. Phys. Rev. D79 2009 024028 arXiv:0809.4033 MR 2491179 | Zbl 1222.83142
[9] Nojiri, S., Odintsov, S.D.: Unified cosmic history in modified gravity: from $F(R)$ theory to Lorentz non-invariant models. arXiv:1011.0544v4 [gr-qc]
[10] Sotiriou, T.P., Faraoni, V.: $f(R)$ theories of gravity. arXiv:0805.1726 MR 2629610 | Zbl 1205.83006
[11] Hooft, G.’t: Introduction to General Relativity. Rinton Press 2001 MR 1921289 | Zbl 1008.83001
[12] Wald, R.M.: General Relativity. University of Chicago Press 1984 MR 0757180 | Zbl 0549.53001
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