# On the classification of consistent boundary conditions for \( f ( R )\)-gravity

## Abstract

Using a completely covariant approach, we discuss the role of boundary conditions (BCs) and the corresponding Gibbons–Hawking–York (GHY) terms in \( f ( R ) \)-gravity in arbitrary dimensions. Following the Ostrogradsky approach, we can introduce a scalar field in the framework of Brans–Dicke formalism to the system to have consistent BCs by considering appropriate GHY terms. In addition to the Dirichlet BC, the GHY terms for both Neumann and two types of mixed BCs are derived. We show the remarkable result that the \(f( R )\)-gravity is itself compatible with one type of mixed BCs, in *D* dimension, i.e. it doesn’t require any GHY term. For each BC, we rewrite the GHY term in terms of Arnowit–Deser–Misner (ADM) variables.

## 1 Introduction

Since the theory of general relativity (GR) is a classical field theory of gravitation, the choice of BCs is of great importance. The role of surface integrals in GR has been investigated first in Dewitt and Dirac’s papers [1, 2] and then was covered deeply in the works of York [3] and Regge and Teitelboim [4]. Trying to quantize GR in path integral formalism, Gibbons and Hawking [5] showed that, a boundary term should be added to the Einstein–Hilbert (EH) action, in order to have a well-defined variational principle with Dirichlet BC, i.e. \( \delta g_{ab}|_{\textit{ Boundary}}=0 \). Such terms, added to the EH action, or the action of generalized theories of gravity [6, 7, 8, 9, 10, 11, 12], are called GHY terms which are used for for finite volume spaces. For non-compact spaces like asymptotically Minkowski or Ads, the variational problem needs extra boundary terms in addition to the GHY term since it depends on the specific asymptotic form of the space-time [13].

The theory of GR is described by a degenerate Lagrangian, i.e. it can be written as the sum of a quadratic part in the first derivatives of metric and a total derivative term. There are two approaches to deal with GR. The first one is the well-known ADM formalism which uses the Gauss–Codazzi equation to get rid of the second derivative terms of the Lagrangian [5, 7, 8, 14]. The second one, which is more covariant, manifests the quadratic Lagrangian by subtracting a suitable boundary term which can be removed by adding a GHY term [15, 16].

For modified gravity models such as \(f( R )\)-gravity, which is non-degenerate, one needs to use the so-called Ostrogradsky approach [17], by introducing enough number of fields to the theory such that the whole Lagrangian includes at most the first derivatives of the fields. In this way one is able to go through a canonical approach and at the same time introduce consistent BCs. For \(f( R )\)-gravity without considering additional fields, one needs to consider the extrinsic curvature variation \(\delta K_{ij}\), as well as \(\delta g_{ij}\) to vanish on the boundary, which is inconsistent since extrinsic curvature \( K_{ij}\), includes derivatives of the metric. However, by adding the famous GHY term \( - 2\int _{\partial {\mathcal {M}}}d^{D-1}y\epsilon \sqrt{h} f'(R)\delta K \) [*h* and *K* are the trace of induced metric and the extrinsic curvature respectively, \( \epsilon =\pm 1 \) depending on the timelike or spacelike nature of the boundary \( \partial {\mathcal {M}} \) and \( f'(R)=\frac{\partial f}{\partial R} \)] to the action, the BCs reduce to vanish \(\delta R\) on the boundary simultaneously with the Dirichlet BC. But calculating \(\delta R\) (see Appendix C) shows that the main problem is not resolved since *R* is not an independent field and its variation includes again variations of the derivatives of the metric.

In this paper we try in Sect. 2 to change *f*(*R*) Lagrangian into a degenerate one by Ostrogradsky approach. To do so, we write the \( f( R ) \)-gravity in the Jordan frame of the Brans–Dicke action [18, 19, 20]. We find that the action of the theory is degenerate. Hence, by adding appropriate GHY terms, Dirichlet or other BCs can be achieved. Writing the boundary terms of the action in terms of fields and momentum fields, in a foliation independent approach, enables us to introduce the consistent GHY term for Dirichlet, Neumann and two types of mixed BCs in arbitrary dimensions. For one type of mixed BC, the GHY term vanishes. This may be interpreted that the \(f( R )\)-gravity is more consistent with this mixed type of BC in *D* dimension.

In this paper the Latin indices are used to show the space-time coordinates and the Greek ones are used to denote the space coordinates. The calculations are done in arbitrary dimensions of space-time and the signature of metric is (\(-,+,+,+\)).

## 2 \(f( R )\)-gravity

^{1}Varying the above action and integrating by part, without implying any BC, we have

^{2}

^{3}we can get rid of the first surface integral in the above equation. To remove the second surface integral, the usual GHY boundary term can be added to the action (3) as follows

*N*and \( N^{\alpha }.\)

^{4}In the last equality we have used the following two identities

### 2.1 Dirichlet BC

### 2.2 Neumann BC

### 2.3 Mixed BC

*f*(

*R*)-gravity: \( \delta {\bar{P}}_{ab}|_{\textit{ Boundary}}=\delta \phi |_{\textit{ Boundary}}=0\) or \( \delta {\bar{P}}_\phi |_{\textit{ Boundary}}=\delta g_{ab}|_{\textit{ Boundary}}=0\). We begin with the first one. Using the variation of

*f*(

*R*)-gravity action, (24) or (32), the first type mixed BC would be consistent if we have added the following GHY term to the action

*f*(

*R*) -gravity. Comparing the above result with that obtained in Eq. (27), we see that

*f*(

*R*)-gravity with the above type of mixed BC is self-consistent with no need to any GHY term in

*D*dimension. To clarify this point better, let us return to the relation (4) in which the boundary terms are written in the ADM formalism. One can write these terms in term of the momenta conjugate to \( \phi \), \( h_{ij} \),

*N*and \( N^\alpha \). These are derived in details in appendix D and are as follows

*D*dimension without any GHY term.

## 3 Conclusion

In this paper it is shown that unlike GR, the action of \( f( R ) \)-gravity is not holographic such as the action of Lanczos–Lovelock theory. Moreover, the Lagrangian of \( f( R ) \)-gravity can not be expressed as the sum of quadratic and total derivative terms. So \( f( R ) \) Lagrangian is not degenerate. Following the Ostrogradsky approach, since \( f( R ) \)-gravity is a theory with higher order derivatives of metric, it carries a single additional degree of freedom, which is the scalar field of equivalent Brans–Dicke action. Introducing this field, leads to a degenerate Lagrangian which is used to develop the problem of BC and the corresponding GHY terms in \( f( R ) \)-gravity [7, 8].

Here we have followed a foliation independent approach to find the GHY boundary terms in \( f( R ) \)-gravity, required to make the BC variation problem well-defined. We have shown that in addition to the Dirichlet BC, the Neumann BC and two types of the mixed BCs can be introduced for the \(f( R )\)-gravity. The remarkable point which is one of the main results of this paper is about the mixed BCs. We have shown that one of the mixed BC: \( \delta {\bar{P}}_{ab}|_{\textit{ Boundary}}=\delta \phi |_{\textit{ Boundary}}=0\) is reduced to the Neumann BC in the case of GR. This BC together the other mixed BC: \(\delta {\bar{P}}_\phi |_{\textit{ Boundary}}=\delta g^{ab}|_{\textit{ Boundary}}=0\) are self-consistent BCs, i.e. these do not need to any GHY term to be consistent with the theory, the first one for GR and the second one for \(f( R )\)-gravity, both in *D* dimension.

## Footnotes

- 1.
This point is explicitly shown in Appendix A.

- 2.
For a detailed calculations see Appendix B.

- 3.
In GR, Dirichlet BC means, \( \delta g_{ab}|_{\textit{ Boundary}}=0 \). Since \(g_{ab}= h_{ab}+\epsilon n_a n_b\), it follows that \( \delta h_{ab}|_{\textit{ Boundary}}= \delta n^i|_{\textit{ Boundary}}=0 \).

- 4.
Thus in the ADM formulation of \(f( R )\) gravity, the Dirichlet BC means \( \delta h^{ab}|_{\textit{ Boundary}}= \delta N^\mu |_{\textit{ Boundary}}=\delta N|_{\textit{ Boundary}}=\delta \phi |_{\textit{ Boundary}}=0 \).

## Notes

### Acknowledgements

The authors would like to thank the Iran National Science Foundation (INSF) for supporting this research under grant number 97015575. F. Shojai is grateful to the University of Tehran for supporting this work under a grant provided by the university research council.

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