Eur. Phys. J. C (2016) 76:552DOI 10.1140/epjc/s10052-016-4394-0

Regular Article - Theoretical Physics

Solving higher curvature gravity theories

Sumanta Chakraborty 1,a, Soumitra SenGupta2,b

1 IUCAA, Post Bag 4, Ganeshkhind, Pune University Campus, Pune 411 007, India2 Theoretical Physics Department, Indian Association for the Cultivation of Science, Kolkata 700032, India

Received: 31 August 2016 / Accepted: 20 September 2016 / Published online: 8 October 2016© The Author(s) 2016. This article is published with open access at Springerlink.com

Abstract Solving field equations in the context of highercurvature gravity theories is a formidable task. However, inmany situations, e.g., in the context of f (R) theories, thehigher curvature gravity action can be written as an Einstein–Hilbert action plus a scalar field action. We show that not onlythe action but the field equations derived from the actionare also equivalent, provided the spacetime is regular. Wealso demonstrate that such an equivalence continues to holdeven when the gravitational field equations are projected on alower-dimensional hypersurface. We have further addressedexplicit examples in which the solutions for Einstein–Hilbertand a scalar field system lead to solutions of the equiva-lent higher curvature theory. The same, but on the lower-dimensional hypersurface, has been illustrated in the reverseorder as well. We conclude with a brief discussion on thistechnique of solving higher curvature field equations.

1 Introduction

The energy scales in particle physics are arranged in a hierar-chical manner. While the scale of the weak interaction corre-sponds to E ∼ 103 GeV, the strong interaction at a scale ofE ∼ 1016 GeV exceeds the weak scale by a factor of 1013.This large difference leads to a fine tuning problem in thescheme of renormalization – known as the gauge hierarchyproblem. This fine tuning is absolutely necessary to renor-malize the mass of the Higgs boson, which recently has beendetected with a mass of 127 GeV. At face value, this fine tun-ing is what nature prefers and the question is why? Hence itis natural to ask, is there a more fundamental principle fromwhich this fine tuning would appear naturally?

There has been a large amount of work to address thehierarchy problem, and a few candidates have emerged out ofit – supersymmetry, technicolor, and extra dimensions. In this

a e-mail: sumantac.physics@gmail.comb e-mail: tpssg@iacs.res.in

work we will be concerned solely with the third alternative,i.e., we will assume the actual spacetime has more than threespatial dimensions (commonly referred to as the bulk), whilethe spacetime we live in is a four-dimensional hypersurfacein the bulk (commonly referred to as the brane). The twoimmediate observable consequences are a change of 1/r2

gravitational force law at the length scale of extra dimensions,and the existence of a massive graviton through the Kaluza–Klein tower [1–9].

To probe these extra dimensions one needs to have a highenough energy or a high enough curvature, such that the rel-evant energy scale of the problem comes close to the Planckscale. General Relativity, described by the Einstein–Hilbertaction is considered to be an effective theory of gravity,valid far below the fundamental Planck scale [10]. Onceenergies approach the Planck scale, one not only expectsto observe deviations from the Einstein–Hilbert action butalso signatures of the extra dimensions. This is particu-larly relevant, since future colliders will probe higher andhigher energies such that aspects beyond general relativityshould become apparent. Since the ultraviolet behavior of thetrue gravity theory is yet unknown one hopes that in thesehigh energy/high curvature regimes deviations from stan-dard model or deviations from Einstein gravity may appearthrough the existence of extra dimensions. To capture someof the aspects of “quantum gravity” one is tempted to con-sider how the presence of higher curvature (and higher deriva-tive) invariants in the higher-dimensional gravitational actionmodifies the well-known results [11–15].

The higher derivative terms that one can add to theEinstein–Hilbert action are not unique. However, many ofthese terms can lead to a linear instability, called the Ostro-gradski instability, leading to the appearance of ghost fieldsand hence will not be considered in this work. Among thehigher curvature theories, the Lanczos–Lovelock gravity andthe f (R) gravity are of much importance. The Lanczos–Lovelock gravity is special in the sense that the field equa-tions derived from the Lanczos–Lovelock action contain only

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second order derivatives of the metric and have a natural ther-modynamic interpretation [16–20]. On the other hand, f (R)

gravity was first introduced to explain both early and latetime exponential expansion of the universe without invokingadditional matter components, e.g., dark energy [21–27]. Butonly addressing cosmological observations does not lead toa viable model, for that the f (R) theories should pass thelocal gravity tests – perihelion precession of Mercury andbending angle of light – as well. It turns out that solar systemexperiments do not exclude the viability of f (R) theories atscales shorter than the cosmological ones, but they provideconstraints on f ′′(R) and hence constraints on the parame-ters of the model. Thus it is can be affirmed that extendedgravity theories cannot be ruled out, definitively, using SolarSystem experiments [28–31].

It is also well known that f (R) gravity theories can berelated to scalar–tensor theories by a conformal transforma-tion at the action level [22–24,32–38]. Thus it is important toconsider the following situation – obtaining field equationsfrom the scalar–tensor representation and from the f (R)

gravity representation. Since the two actions are related bya conformal transformation, the field equations should alsobe equivalent. However, the situation is not trivial, since themetric in scalar–tensor representation depends on the con-formal factor, its variation can potentially lead to variousadditional terms, which must cancel other terms exactly inorder to arrive at the equivalence. If the equivalence exists,we can use it to solve field equations for scalar–tensor the-ory and obtain the solution corresponding to f (R) actionand vice versa. This would be advantageous, since in gen-eral solving the field equations for f (R) gravity, where R isnot a constant, is difficult1 [22,41–47]. The correspondingscalar–tensor solution could in principle be much simpler.The same should work on the brane hypersurface as well.The effective field equations on the brane derived through theGauss–Codazzi formalism in the f (R) representation [48–53] should be equivalent to the same but derived from thescalar–tensor representation. The non-triviality of this resultoriginates from the quadratic combination of energy momen-tum tensor and extrinsic curvature appearing in the effectivefield equations.

As an aside, we should mention that the conformal trans-formation is well motivated only when the spacetime doesnot have singularities. Such singularity free spacetimes havebeen obtained earlier in the context of cosmology with mod-uli dependent loop corrections of the gravitational part ofsuperstring effective action with orbifold compactifications[54]. However, to obtain a singularity free description it was

1 Note that in the cosmological context one can solve the field equationsfor f (R) gravity by a trick, known as the reconstruction method [39,40].We will have an occasion to comment on this method when we compareour technique introduced in this work with the reconstruction scheme.

necessary that the stress energy tensor associated with themodulus should violate the strong energy condition. In cir-cumstances where the energy conditions are obeyed, oneobtains singular solutions in general. For singular space-times, viz., cosmological spacetimes near the Big Rip or BigCrunch the transformation can break down. In those contextsit can exhibit peculiar behavior, e.g., the Big Rip singular-ity, which may appear in some versions of f (R) gravity caneither map itself to the infinite past or future, or it can bereplaced by a Big Crunch singularity [35,55]. Another pointrequires clarification at this stage; this has to do with thephysical non-equivalence of the two frames. All the com-ments phrased above have to do with mathematical equiva-lence, but the physical solutions can be very different [55].This is evident, since the conformal factor can change thecomplete structure of the spacetime. This fact was pointedout earlier in [56] by showing that through a conformal trans-formation one can create matter and as a result, one frame isempty while another has matter, and clearly they are phys-ically non-equivalent. This should not come as a surprise,since the Schwarzschild metric under conformal transforma-tion no longer satisfies Einstein’s equations. Further in thecosmological context for f (R) gravity model it was explic-itly demonstrated [55,57] that neither the Hubble parameternor the deceleration parameter matches in Jordan and Ein-stein frame, showing the physical non-equivalence. In viewof the above, the phrase “equivalence” in the following sec-tions should be understood in a mathematical sense, not ina physical sense. Further, we will content ourselves, withonly those spacetimes (or regions of spacetimes) which areregular, such that the conformal transformation between thetwo frames is well defined throughout the region of inter-est.

The paper is organized as follows: In Sect. 2 we presenta brief review of the equivalence between the f (R) grav-ity and scalar–tensor theory in five dimensions and hencethe equivalence between the bulk field equations as well.Section 3 is devoted to show the equivalence between theeffective field equations on the brane. The application ofthe bulk equivalence is presented in Sect. 5. There we havestarted from scalar–tensor theory and have solved the bulkequations, from which the solution in f (R) representa-tion is obtained. In Sect. 6 we consider brane spacetimewhere the solution in scalar–tensor representation startingfrom the f (R) representation is derived as another explicitexample. We conclude with a brief discussion of this tech-nique.

We have set the fundamental constants c and h to unityand shall work with mostly the positive signature of themetric. The Latin indices, a, b, . . . runs over the full space-time indices, while Greek indices, μ, ν, . . . stand for four-dimensional spacetime.

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2 Equivalence of gravitational field equations in thebulk

The starting point for any fundamental theory corresponds tocorrect identification of the dynamical variable and the asso-ciated field equations. A useful trick to obtain the field equa-tions is to introduce an action principle, extremizing whichone can obtain the field equations. Along similar lines, ingravitational theories as well one considers the metric gab asdynamical. Given an appropriate action, when arbitrary vari-ations of gab are considered, the gravitational action reachesan extremum value only if gab satisfies the gravitational fieldequations. For example, Einstein’s equations follow fromvariation of the Einstein–Hilbert action, which is the Ricciscalar. At high enough curvature (or energy) the Einstein–Hilbert action is most likely to be supplemented by highercurvature corrections. Among many such viable modifica-tions, f (R) theories are of particular interest. The action forthe f (R) gravity model (also known as the Jordan frame) infive spacetime dimensions reads

SJ ≡∫

d5x√−g

[f (R)

2κ25

+ Lm

]

=∫

d5x√−g

[(f ′R2κ2

5

− f ′R − f

2κ25

)+ Lm

], (1)

where κ5 is the five-dimensional gravitational constant, Lm

is the matter Lagrangian, and f ′ stands for d f/dR. A con-formal transformation of the Jordan frame metric gab resultsin gab = �2gab, where � is the conformal factor. The result-ing action, written in terms of gab, can be obtained using thetransformation properties of the curvature tensor yielding

S =∫

d5x√−g

[�−5

{f ′

2κ25

(�2R + 8�2� ln �

−12gab∇a�∇b�)

−U ( f )

}+ Lm

]. (2)

Here U ( f ) stands for ( f ′R − f )/2κ25 and Lm is the mat-

ter Lagrangian in the conformally transformed action. Notethat, with our conventions, all the metric dependent quantitiesoriginating from a conformal transformation of the Jordanframe are boldfaced. We will follow this convention through-out this work. Under the following identifications:

�3 = f ′; κ5φ = 2√3

ln f ′ = 2√

3 ln �, (3)

the action presented in Eq. (2) reduces to the following form(known as the Einstein frame action):

SE =∫

d5x√−g

(R

2κ25

− 1

2gab∇aφ∇bφ − V (φ) + Lm

)

+∫

d4x√h

4

κ25

nc∇c ln �. (4)

The Einstein frame action can be divided into a bulk termand a surface term, as evident from Eq. (4). If we are onlyinterested in a variation of the action and a derivation of thefield equations, the boundary term can be safely ignored.However, while concentrating on brane dynamics, which is aboundary effect, the surface term will be important. Further,the potential term V (φ) appearing in Eq. (4) corresponds to

V (φ) = f ′(φ)R(φ) − f (φ)

2κ25 f ′(φ)5/3

, (5)

which can be obtained by solving Eq. (3). Extremizing theEinstein frame action presented in Eq. (4) with respect toarbitrary variations of the metric gab, we readily obtain thefield equations in the Einstein frame,

Gab ≡ Rab − 1

2Rgab

= κ25

[∇aφ∇bφ − gab

(1

2gcd∇cφ∇dφ+V (φ)

)]+ κ2

5Tab.

(6)

The energy momentum tensorTab is obtained from the matteraction Lm by variation of the Einstein frame metric gab.Using the conformal transformation, the energy momentumtensor Tab in the Einstein frame can be related to the energymomentum tensor Tab in the Jordan frame by

Tab ≡ − 2√−g

δ(√−gLm

)δgab

= − 2√−g�5

δ(√−gLm

)�−2δgab

= 1

f ′ Tab . (7)

Since the Jordan frame action SJ is derivable from the Ein-stein frame action SE through a conformal transformation,the bulk equations derived from them should coincide. Theabove statement, though physically well motivated is by nomeans trivial. This has to do with the fact that while derivingthe Einstein frame equations one should vary the Einsteinframe metric gab. This in turn leads to an arbitrary variationof the Jordan frame metric gab and the conformal factor �.Since the conformal factor can be written in terms of the cur-vature tensor due to the identification in Eq. (3), it can leadto various additional correction terms. It is not clear a priorihow these terms combine and yield correct field equationsin the Jordan frame. Since there exists no explicit derivationof the same, it is worthwhile to explicitly demonstrate theequivalence.

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In order to prove the same we will start from Eq. (6) andshall try to write every curvature tensor components in termsof the Jordan frame metric gab. This can be done using thetransformation properties of curvature tensors between thetwo frames related by a conformal transformation, leadingto

Rab = Rab − ∇a∇b f ′

f ′ + 4

3

∇a f ′∇b f ′

f ′2 − gab� f ′

3 f ′ (8)

and

gabR = gabR − 8

3gab

� f ′

f ′ + 4

3gab

gcd∇c f ′∇d f ′

f ′2 . (9)

Having expressed both the Ricci tensor and the Ricci scalarin the Einstein frame in terms of the conformal factor and thecorresponding curvature components in the Jordan frame, theEinstein tensor in the Einstein frame can be expressed as

Gab = Gab − ∇a∇b f ′

f ′ + 4

3

∇a f ′∇b f ′

f ′2

− 2

3gab

gcd∇c f ′∇d f ′

f ′2 + gab� f ′

f ′ . (10)

Further the contribution from the scalar field present in theright hand side of Eq. (6) can be written in terms of f (R)

and its various derivatives as

∇aφ∇bφ − gab

(1

2gcd∇cφ∇dφ + V (φ)

)

= 1

κ25

[4

3

∇a f ′∇b f′

f ′2 − 2

3gab

gcd∇c f ′∇d f ′f ′2 −gab

f ′R − f

2 f ′

].

(11)

Using these relations between the Einstein frame and the Jor-dan frame, the field equations in the Einstein frame presentedin Eq. (6) can be written as

f ′Gab + f ′R − f

2gab − ∇a∇b f

′ + gab� f ′ = κ25Tab,

(12)

which are precisely the field equations one would haveobtained by extremizing Eq. (1) for an arbitrary variationof the Jordan frame metric gab. Hence follows the equiv-alence. As a consequence if one can solve for gab startingfrom the field equations in the Einstein frame, the solution ina Jordan frame can be obtained through a conformal transfor-mation and vice versa. We should emphasize that the abovestatement, though mathematically correct, practically mightrequire suitable approximations for inverting various func-tional relations connecting the two frames. We will provide

detailed comments on this aspect later on, while providingconcrete examples.

Even though we have used the metric formalism to arriveat the equivalence, one can also use another method knownas the Palatini method. For discussions of the same we referthe reader to [23,58–61].

3 Equivalence of effective field equations on the brane

In the previous section, we have shown the equivalencebetween the bulk field equations derived from the Einsteinand the Jordan frames. However, from the perspective ofbrane world, governed by effective field equations derivedfrom the bulk action, the equivalence of the effective fieldequations is more important. The effective field equationsinvolve various quadratic combinations of the extrinsic cur-vature and the matter energy momentum tensor. Thus all theadditional terms with their appropriate factors present in theEinstein frame must cancel each other such that effectivefield equations in the Jordan frame are obtained. In this con-nection we would like to highlight that in most of the worksrelated to f (R) gravity the surface term in the Einstein frameis ignored; however, in order to prove the equivalence on thebrane this term is absolutely necessary. Hence the equiv-alence of the effective field equations too, is a non-trivialstatement. In this section we will explicitly demonstrate this.

The bulk field equations in the Einstein frame involve theenergy momentum tensor of the scalar field along with anyother matter fields which may be present in the bulk. Thebulk energy momentum tensor T bulk

ab (the trace is denotedby T bulk) will induce an effective brane energy momentumtensor T brane

ab as

κ24T

braneμν = 2

3κ2

5

[T bulkab eaμe

bν + hμν

(T bulkab nanb − 1

4T bulk

)].

(13)

Here κ5 is the five-dimensional (i.e., bulk) gravitational con-stant while κ4 is the four-dimensional (i.e., brane) gravita-tional constant. Moreover, the object eaμ stands for ∂xa/∂yμ,where yμ corresponds to the brane coordinates and xa arethe bulk coordinates. The normal to the brane hypersurfaceis na such that the induced metric on the brane hypersurfacebecomes hμν = eaμe

bν(gab − nanb) [62].

To obtain the effective field equations in the Jordan frame,we need to express the scalar field in terms of f (R) and itsderivatives. Thus, the bulk energy momentum tensorT(φ)

ab forthe scalar field reads

κ25T

(φ)ab = 4

3∇a ln f ′∇b ln f ′ − 2

3gab

(gcd∇c ln f ′∇d ln f ′)

− gabf ′R − f

2 f ′ . (14)

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Using this the following results can be obtained:

κ25T

(φ)ab n

anb = 1

�2

[4

3

(na∇a ln f ′)2 − 2

3∇a ln f ′∇a

× ln f ′ − f ′R − f

2 f ′]

, (15)

κ25T

(φ) = 1

�2

[4

3∇a ln f ′∇a ln f ′ − 10

3∇a ln f ′∇a

× ln f ′ − 5f ′R − f

2 f ′]. (16)

From these expressions of the bulk energy momentum tensorof the scalar field, the brane energy momentum tensor, aftersome simplifications (using Eq. (13) in particular), leads to

κ24T

(φ)braneμν = 8

9∇μ ln f ′∇ν ln f ′ − 5

9hμν∇a ln f ′∇a ln f ′

− 3

4hμν

f ′R − f

3 f ′ + 8

9hμν

(na∇a ln f ′)2

.

(17)

Let us now work through the last bit of this analysis regardingthe Einstein tensor. Using the transformation properties of theRiemann tensor and the Ricci scalar we immediately obtainthe following result for the induced Einstein tensor:

Gμν = Gμν −2∇μ∇ν�

�+2hμν

hαβ∇α∇β�

�+4

∇μ�∇ν�

�2

− hμν

hαβ∇α�∇β�

�2 . (18)

In this particular case, the conformal factor � is related tod f/dR through the relation � = f ′1/3. Using this relation,after some straightforward manipulation and simplification,we arrive at

Gμν = Gμν − 2

3

∇μ∇ν f ′

f ′ + 8

9

∇μ f ′∇ν f ′

f ′2

− 5

9hμνh

αβ ∇α f ′∇β f ′

f ′2 + hμνhαβ 2∇α∇β f ′

3 f ′ . (19)

The only remaining part corresponds to the electric partEμν of the bulk Weyl tensor Cabcd . From the transforma-tion property of the Weyl tensor it immediately follows thatEμν = Eμν . Combining all these, in the Einstein frame theeffective gravitational field equations on the brane take theform

Tbraneμν = Gμν −

{KKμν − Kα

μKνα − 1

2hμν

(K2 − KμνKμν

)}

+ Eμν − κ24T

(φ)μν

= Gμν −{KKμν −K α

μKνα− 1

2hμν

(K 2−KμνK

μν)}+Eμν

− 2

3

∇μ∇ν f ′

f ′ + hμν

f ′R − f

4 f ′ + hμνhαβ 2∇α∇β f ′

3 f ′ , (20)

which are precisely the effective gravitational field equa-tions in the Jordan frame with the identification T brane

μν =(1/ f ′)Tbrane

μν . Hence the equivalence works at the level ofeffective field equations as well. However, the practicalimplementation of the above result again requires inversionof complicated functional forms and hence invites approxi-mations.

4 A comparison with reconstruction methods in f(R)gravity

In the above two sections we have shown the equivalence ofgravitational field equations both in the bulk and in the brane,respectively. In this section we will present a comparison ofour method with an existing well-known method in f (R)

gravity, the reconstruction method. As already emphasized,due to the presence of higher derivatives in the field equa-tions for f (R) gravity, obtaining a straightforward solutionin a general case is very difficult. Even for systems with alarge number of symmetries, e.g., in cosmology which hasa single unknown function a(t), solving the field equationsdirectly in the Jordan frame is very complicated. This leads tothe reconstruction method, which we will briefly summarize[63–66].

In the reconstruction method one assumes that the expan-sion history of the universe is known exactly and by invert-ing the field equations one can determine what class of f (R)

theories can give rise to the observed universe. For example,power law solutions for the scale factor singles out Rn to bethe gravitational action. Since the scale factor a(t) is known,the Ricci scalar is also known as R(t). This can be invertedto get t = g(R) and hence the Hubble parameter is known tobe a function of the Ricci scalar. This, when used in the fieldequations, leads to a differential equation for f (R), whichcan be solved to find the gravity model [63–65]. There havebeen other variants of this model, e.g., assuming every physi-cal quantity to be a function of the scale factor or a function ofthe Hubble parameter, which ultimately leads to a differentialequation for the gravity model [66]. The essential ingredientsremain the same but one particular case may be convenientin comparison to the other in a particular situation. Let usnow explicitly point out the advantages and disadvantages ofthe reconstruction scheme as well as our approach.

• An important limitation of the reconstruction method isthat only very simple cosmic histories, e.g. simple powerlaw behaviors, can be connected to f (R) theory in anexact way. Our method has similar disadvantages. Eventhough the field equations can be exactly solved in the

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Einstein frame for a few cases of interest, the inversionof the potential to f (R) theory can be performed only insimple situations.

• The reconstruction method is adapted to cosmologi-cal spacetimes only, since the cosmic history is knownthrough experiments. However, the situation we are inter-ested in corresponds to higher-dimensional physics in thepresence of higher curvature gravity, the possible behav-ior of the warp factor, and the brane separation. Sincethere is no experimental backdrop for extra dimensionsit is not possible to come up with a physical ansatz. Thusone needs to solve the field equations at face value, whichcan be efficiently done using our method as we have illus-trated in the next sections.

• The utility of the reconstruction scheme lies in its quickand straightforward analysis. Given a phenomenologicalscale factor a(t) one needs to solve a single differentialequation to get f (R), given the inversion t = t (R). Inour method one first need to solve the gravity plus scalarfield system to get the solution in Einstein frame, whichitself is a formidable task. Then one needs to invert thepotential V (φ) to get f (R) and finally the conformaltransformation will yield the solution in the Jordan frame.

Thus both methods have their own advantages and disadvan-tages. The reconstruction method is very simple, applicableeven in the presence of singularity and useful in a cosmo-logical context, while it is not so useful when applied toother scenarios, e.g. extra dimensions. On the other hand,the method introduced in this work, even though it requiresa regular spacetime region and involves more steps to arriveat the solution, is very robust. It will work for any regu-lar spacetime region, from cosmological scenarios to extradimensions as explained in later examples.

5 Einstein to Jordan frame in the bulk: explicitexamples

We will now illustrate through simple examples how onemight obtain solutions to bulk field equations in f (R) grav-ity, which involve higher derivative terms, by exploiting theequivalence with scalar tensor theory depicted in the previ-ous sections. As emphasized before, due to the occurrenceof higher derivative terms it is difficult to solve for the bulkequations of f (R) gravity. On the other hand, solving a set ofcoupled equations of gravity plus scalar field system is muchsimpler. Hence, through the equivalence shown earlier, if wecan obtain a solution for the bulk metric gab in the Einsteinframe, the corresponding solution in the Jordan frame willdiffer only by a conformal factor.

Before we jump into detailed calculations it is worthwhileto sketch the flowchart we are going to follow – (a) We will

start with the bulk action in the Einstein frame. (b) For somesuitable potential, we will find the metric describing the bulkspacetime, by solving the bulk field equations. (c) We willmatch the potential in the Einstein frame with the corre-sponding f (R) theory in the Jordan frame. (d) Finally theconformal transformation will yield the corresponding bulkmetric in the Jordan frame. In the examples to follow we willexplicitly illustrate all the four steps mentioned above.

Bulk field equations in Einstein frame We start by solving thefield equations of gravity and the scalar field in the Einsteinframe. We assume that the branes are flat, viz., ημν is thespacetime metric on the brane. Further, the two branes areassumed to be separated by the stabilized value of the radionfield rc [67,68] such that the metric ansatz turns out to be(this ansatz is useful, particularly in the context of the gaugehierarchy problem)

ds2 = e−2A(y)ημνdxμdxν + r2c dy2, (21)

where A(y) is the warp factor and is dependent on the extraspacetime coordinate alone. From the above metric ansatzthe non-zero components of the Ricci tensor are immediate,

Rμν = e−2A

r2c

(A′′ − 4A′2) ημν; Ryy = 4A′′ − 4A′2.

(22)

Note that we are following the previously mentioned con-vention: all the metric dependent quantities in the Einsteinframe are boldfaced. From the components of the Ricci ten-sor, a straightforward computation of the Ricci scalar leadsto

R = 1

r2c

(8A′′ − 20A′2) . (23)

Given the Ricci tensor and the Ricci scalar one can furthercompute the non-zero components of the Einstein tensor:

Gμν = 3e−2A

r2c

ημν

(−A′′ + 2A′2) ; Gyy = 6A′2, (24)

such that the bulk gravitational field equations (see Eq. (6))in the Einstein frame become

Gab = −�gab + κ25

[∂aφ∂bφ − gab

{1

2gcd∂cφ∂dφ + V (φ)

}],

(25)

where � is the bulk cosmological constant. From Eq. (23) itis clear that the bulk curvature depends only on the extra coor-dinate y, since A depends on y only. Thus logical consistencyof the field equations demand that φ should also depend onlyon the extra-dimensional coordinate y. In this case Eq. (25)

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reduces to the following three coupled equations for gravityand scalar field:

−3A′′ + 6A′2 = −�r2c − κ2

5

[1

2φ′2 + r2

c V (φ)

], (26)

A′2 = −�

6r2c + κ2

5

12φ′2 − r2

c κ25

6V (φ), (27)

φ′′ − 4A′φ′ = r2c∂V

∂φ, (28)

where a prime denotes the derivative with respect to y. Elim-inating A′ from the first two equations we obtain

A′′ = κ25

3φ′2. (29)

In general the solution to the above coupled equations canbe obtained by introducing a super-potential W (φ) whichsatisfies the following differential equation:

1

9W 2 = −�

6r2c + 1

12κ25

(∂W

∂φ

)2

− r2c κ2

5

6V (φ). (30)

The above differential equation for W (φ) can be inverted andthe potential V (φ) gets determined in terms of W (φ) by

V (φ) = − �

κ25

+ 1

2κ45r

2c

(∂W

∂φ

)2

− 2

3κ25r

2c

W 2. (31)

In such a scenario the coupled equations become separable.One thus obtains separate differential equations for the metricfunction A and the scalar field φ in the following form:

A′ = 1

3W ; φ′ = 1

κ25

∂W

∂φ. (32)

On the other hand, if one postulates the separability of thecoupled field equations, then also the expression of the poten-tial V (φ) in terms of the super-potential W (φ) follows. Thischoice for A and φ also satisfies the field equation for φ aswell, as one can easily check. The only remaining one cor-responds to Eq. (29), i.e., the A′′ equation.

Solutions in Einstein frame We have set the stage; it is nowtime to act. Having obtained the field equations in a sensibleform, let us now solve for the bulk metric. In order to satisfyEq. (29) one requires two possible choices for W (φ) – (i)W (φ) = cφ, (ii) W (φ) = a− bφ2 for arbitrary choices of a,b, and c. Solving the field equations in both cases separatelyleads to the following:

• The super-potential is linear in φ, i.e., W (φ) = cφ. Thenfrom Eq. (32) one obtains A′ = (b/3)φ, such that A′′ =(c/3)φ′, while φ′ = c/κ2

5 . This set identically satisfies

Eq. (29). The corresponding potential V (φ) turns out tobe

V (φ) = − �

κ25

+ c2

2κ45r

2c

− 2c2

3κ25r

2c

φ2, (33)

with the following solution for A(y) and φ(y):

φ(y) = φ0 + c

κ25

y , (34)

A(y) = A0 + cφ0

3y + c2

6κ25

y2 . (35)

• The super-potential is quadratic in φ, i.e., W (φ) = a −bφ2. From Eq. (32) we get A′ = (1/3)(a − bφ2), hencethis yields A′′ = −(2b/3)φφ′, with φ′ = −(2b/κ2

5 )φ.These expressions can easily be manipulated to show thatEq. (29) is indeed satisfied. Then the potential becomes

V (φ) =(

− �

κ25

− 2a2

3r2c κ2

5

)

+(

b2

2r2c κ4

5

+ 4ab

3r2c κ2

5

)φ2 − 2b2

3r2c κ2

5

φ4, (36)

with the following solutions for A(y) and φ(y):

φ(y) = φ0 exp

(− b

κ25

y

), (37)

A(y) = A0 +√

−�r2c

6y + κ2

5

6φ2

0 exp

(−2

b

κ25

y

). (38)

Thus we have exactly solved the bulk gravitational field equa-tions in the Einstein frame for two choices of the scalar fieldpotential. One of them is quadratic, i.e., V (φ) = a + bφ2,while the other corresponds to quartic potential, V (φ) =a+ bφ2 + cφ4. Now we need to execute the last two steps inour flowchart, namely, (a) one should identify a f (R) model,which gives rise to the potentials obtained above and (b) oneneeds to find the conformal factor relating the two framesand hence the metric can be obtained in the Jordan frame.

Connecting Jordan and Einstein frame So far we have beenworking solely in the Einstein frame. In order to obtain therespective solution in the Jordan frame, we have to connectthe scalar field to a f (R) model, which can be done throughthe relations

κ5φ = 2√3

ln f ′(R); κ25V = f ′R − f

2 f ′5/3. (39)

These relations follow from the original connection betweenEinstein and Jordan frame discussed in Eqs. (3) and (5),

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552 Page 8 of 13 Eur. Phys. J. C (2016) 76 :552

respectively. In principle one should start with the quadraticand quartic potentials obtained previously and hence obtainthe corresponding f (R) theory in the Jordan frame by usingEq. (39). However, we will take the opposite route, i.e., wewill start with some f (R) model and arrive at the respectivepotentials in the Einstein frame using Eq. (39) and map it tothose obtained earlier.

• The simplest model is always the best to start with. Forf (R) gravity this corresponds to a situation, where theEinstein–Hilbert term receives a quadratic correction,2

i.e., f (R) = R+αR2. For this particular model of f (R)

gravity, the scalar field and the potential can be writtenin terms of the Ricci scalar in the Jordan frame as (usingEq. (39))

κ5φ = 2√3

ln (1 + 2αR) ; R = 1

2α

[exp

(√3

2κ5φ

)− 1

],

(40)

V (φ) = 1

8ακ25

[exp

(κ5φ

2√

3

)− 2 exp

(−κ5φ√

3

)

+ exp

(− 5

2√

3κ5φ

)]. (41)

The minimum of the potential corresponds to ∂V/∂φ =0, with ∂2V/∂φ2 > 0. Both conditions can be satis-fied, provided eκ5φ = 1, or φ = 0. Finally, expandingaround the minimum one obtains the following form forthe potential:

V (φ) = 1

8ακ25

[(1 + κ5φ

2√

3+ 1

2

κ25 φ2

12

)− 2

(1 − κ5φ√

3+ 1

6κ2

5 φ2)

+(

1 − 5

2√

3κ5φ + 1

2

25κ25 φ2

12

)]= 3

32αφ2. (42)

Thus we have a quadratic potential forφ, which originatesfrom R + αR2 gravity. Matching the potential to thatderived in the Einstein frame, given by Eq. (33), we obtainthe following relation: � = −9/(128α).

• Let us now consider a more general f (R) gravity modelfor which f (R) = R + αR2 + βR4, where α and β aredimensionful constant coefficients, with values such thatthe model becomes ghost free. Then from Eq. (39) weobtain R = (

√3κ5φ)/(4α), such that the potential turns

out to be

V (φ) = αR2 + 3βR4

2κ25

= 3

32αφ2 + 1

2

33βκ25

44α4 φ4, (43)

2 The quadratic correction is well known in the literature; see for exam-ple [69–73].

where we have assumed α � β � α2, consistent withthe ghost free criterion for this f (R) model. Compar-ing this with the potential obtained by solving bulk fieldequations in the Einstein frame, presented in Eq. (36) weimmediately obtain

a =√

−3r2c�

2; b = −4aκ2

5

3±

√16a2κ4

5

9+ 3r2

c κ45

16α;

β = −45b2α4

34r2c κ4

5

. (44)

This completes the connection between Einstein and Jordanframes. The potentials obtained in the Einstein frame getmapped to respective f (R) theories.

Solutions in Jordan frameWe have now reached the final stepof our flowchart, viz., a solution to the bulk field equations inthe Jordan frame. For this purpose we can use the connectionwith the Einstein frame derived earlier.

• For f (R) = R + αR2, the corresponding scalar fieldpotential in the Einstein frame is quadratic with the map-ping being given by Eq. (42). Thus, the conformal factorturns out to be � = (1+2αR)1/3 = [1+(

√3κ5φ/2)]1/3.

Hence the bulk solution in the Jordan frame correspondsto

ds2 =[

1 +√

3κ5φ(y)

2

]−2/3

×{e−2A(y)ημνdxμdxν + r2

c dy2}

, (45)

φ(y) = φ0 + c

κ25

y; A(y) = A0 + cφ0

3y + c2

6κ25

y2,

where c and φ0 are arbitrary constants of integration.Thus for the quadratic f (R) model under consideration,one can map it to the Einstein frame and obtain the respec-tive potential. For this particular case, the field equationsin the Einstein frame become exactly solvable and henceby a conformal transformation one can obtain the cor-responding solution in the Jordan frame. Further, fromEq. (45) it turns out that the warp factor is governed bythe factors cφ0 and c/κ5. Hence in order to have propersuppression of the Planck scale on the visible brane onemust have the conditions c < κ5 and cφ0 ∼ 36. Henceone arrives at φ0 > κ−1

5 . Further, in this model the radionfield varies with extra dimension y as (a+by)2/3, wherea and b depend on c, κ5 and φ0.

• For the other model, i.e., f (R) = R + αR2 + βR4,the scalar field potential in the Einstein frame is quar-tic. From this one can relate the parameters α, β withthe respective ones in the Einstein frame. Following the

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Eur. Phys. J. C (2016) 76 :552 Page 9 of 13 552

same strategy as above, the conformal factor turns out toyield � = (1 + 2αR + 4βR3)1/3 = [1 + (

√3κ5φ/2) +

(3√

3βκ35φ3/16α3)]1/3. Using this the solution in the Jor-

dan frame becomes

ds2 =[

1 +√

3κ5φ(y)

2+ 3

√3βκ3

5φ(y)3

16α3

]−2/3

×{e−2A(y)ημνdxμdxν + r2

c dy2}

, (46)

φ(y) = φ0 exp

(− b

κ25

y

);

A(y) = A0 +√

−�r2c

6y + κ2

5

6φ2

0 exp

(−2

b

κ25

y

).

This demonstrates another f (R) theory for which thecorresponding potential in the Einstein frame leads to anexact solution. Using this and the mapping between Ein-stein and Jordan frame one obtains the respective solutionin the Jordan frame. However, in contrast with the previ-ous situation, in this case the warp factor behaves exactlylike the Randall–Sundrum scenario, since all the correc-tions are exponentially suppressed (see Eq. (46)). Theradion field is almost constant due to identical exponen-tial suppression. Hence the f (R) model with quartic cor-rection is more favored in the extra-dimensional physicsthan the earlier one.

Aside: comment on domain of applicability After illustratingtwo examples on how to obtain solutions to higher orderfield equations, by using the Einstein frame judiciously, wewould like to comment on the domain of applicability of thisapproach.

We should emphasize that we are working in a mesoscopicenergy scale, i.e., the energy scale is larger compared to gen-eral relativity, such that the effect of higher order terms, e.g.,αR2 cannot be ignored. On the other hand, the energy scaleis much smaller compared to the Planck scale so that theadditional contributions are still sub-dominant, i.e., αR < 1.This is important, in particular when one obtains the scalarfield in the Einstein frame in terms of the curvature. In orderto obtain a closed form expression one has to expand thepotential near its minimum, and this in turn requires one toneglect higher order curvature corrections, e.g., one mightneglect α2R2 in comparison with αR. In a nutshell, we areworking in a high curvature regime such that the effect off (R) gravity can be felt, but it is not high enough that theEinstein–Hilbert action becomes sub-dominant.

Another point that requires clarification is the approxima-tions involved in general scenarios. The conformal transfor-mation and hence the conversion of a potential to a corre-sponding f (R) model is not at all straightforward. In most

of the cases the relations turn out to be non-invertible, andone needs to resort to approximations. As explained earlier,on physical grounds, one can assume that the higher orderterms are sub-leading and hence one can keep only linearorder terms. While dealing with complicated potentials, mostoften one needs to resort to these approximations, justifiedby physical intuitions. However, at the Planck scale theseapproximations break down, since the assumption that higherorders terms are sub-leading cannot be trusted.

Having discussed two possible scenarios in the context ofbulk physics let us now consider brane dynamics. In partic-ular, we will be interested in one spherically symmetric andone cosmological application.

6 Jordan to Einstein frame in the brane: explicitexamples

Both examples depicted above are related to bulk spacetimes.To complete the discussion we will also derive the metric inthe Einstein frame starting from the Jordan frame, but in thebrane spacetime. This is to explicitly demonstrate that thetechnique works both ways – whenever it is convenient tosolve in the Einstein frame, we can solve it and transformback to the Jordan frame, while if the solution is simpler inthe Jordan frame it can give insight into what happens inscalar coupled gravity, viz., the Einstein frame.

• In this example, we will start with a particular f (R)

model on the brane, solve the effective field equations,and obtain a cosmological solution. Then using a con-formal transformation the corresponding solution in theEinstein frame can be obtained.

Solution in the Jordan frame Let us start with the f (R)

model given by f (R) = f0(R − R0)α , where f0 and R0

are constants and α �= 1. The corresponding solution forthe scale factor on the brane can be obtained by solvingthe effective field equations derived in [48,74]. This leadsto the power law solution a(t) ∼ tn , where n is relatedto α and the matter fields present on the brane.

Converting back to Einstein frame We need to convert itback to the Einstein frame and hence obtain the corre-sponding solution in the scalar coupled gravity. For thischoice for f (R), we obtain for the scalar field

κ5φ= 2√3

ln ( f0α)+ 2√3

(α − 1) ln (R − R0) , (47)

which in turn can be inverted, leading to, R − R0 =exp[aκ5(φ−φ0)], where a = √

3/(2(α−1)) and κ5φ0 =(2/

√3) ln ( f0α). Then the potential can be determined

readily, using Eq. (5), leading to

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552 Page 10 of 13 Eur. Phys. J. C (2016) 76 :552

V (φ) = α − 1

2κ25 f 2/3

0 α5/3exp

[κ5a

(5

3− 2α

3

)(φ − φ0)

].

(48)

Hence the power law behavior of the f (R) theory trans-forms back to an exponential potential.

Solution in Einstein frame The corresponding cosmolog-ical solution in the Einstein frame could be obtainedby transforming the Jordan frame solution using theconformal factor. In this particular class of f (R) mod-els, the conformal factor becomes � = ( f0α)1/3(R −R0)

(α−1)/3 ∼ t−2(α−1)/3. Thus the corresponding solu-tion in the Einstein frame is again cosmological with anew scale factor: a(t) ∼ t [3n−4(α−1)]/3. Hence the cos-mological solution in the Einstein frame with an expo-nential potential is still a power law.

• So far we have been dealing with power law f (R) theo-ries. However, to show the applicability of our method tomore general scenarios, we will consider the followingf (R) model: R−(α/R) on the brane. This f (R) model infour-dimensional spacetime has been discussed in detailin [75], however, no such solution in the context of effec-tive field equations exists, which by itself would be aninteresting future work. However, in this work we willcontent ourselves by providing basic ingredients regard-ing this model. Furthermore, given a solution in the Jor-dan frame, use of a conformal transformation will leadto the corresponding solution in the Einstein frame.

Solution in the Jordan frame Let us start with the abovementioned f (R) model. The corresponding solution forthe scale factor on the brane can be obtained by solvingthe effective field equations and can be taken to be a(t) ∼tn , where n should be related to α and the matter fieldspresent on the brane.

Converting back to Einstein frame We need to convert itback to Einstein frame and hence obtain the correspond-ing solution in the scalar coupled gravity. For this choicefor f (R), we obtain for the scalar field

κ5φ = 2√3

ln(

1 + α

R2

), (49)

which in turn can be inverted, leading to R = √α

(exp[(2/√

3)κ5φ] − 1)−1/2. Then the potential can bedetermined readily, using Eq. (5), leading to

V (φ) = √α exp

(− 5

2√

3κ5φ

) √exp[(2/

√3)κ5φ] − 1 .

(50)

Expanding for small φ, we obtain V (φ) =√

ακ5√

3φ/2

(1−(5κ5φ/2√

3)). Hence, the negative power law behav-

ior of the f (R) theory transforms back to a potential with√φ as the leading order contribution.

Solution in Einstein frame The corresponding cosmolog-ical solution in the Einstein frame could be obtained bytransforming the Jordan frame solution using the confor-mal factor. In this particular class of f (R) model, theconformal factor becomes � = [1 + (α/R2)]1/3 ∼ [1 +αt4]1/3, since R ∼ t−2. Thus for late times, � ∼ t4/3.Thus the corresponding late time solution in the Ein-stein frame is again cosmological with a new scale fac-tor: a(t) ∼ tn+(8/3), again a power law behavior. Hencethe cosmological solution in the Einstein frame with

√φ

potential is still a power law.• Another explicit spherically symmetric solution on the

brane in the Jordan frame has been constructed in [48]by decomposing the electric part of the Weyl tensor into adark radiation term U (r) and a dark pressure term P(r).

Solution in Jordan frame The dark radiation U (r) anddark pressure P(r) act as auxiliary sources to the effec-tive gravitational field equations [76]. A possible solutioncan be obtained when an “equation of state” betweenU (r) and P(r) is specified. For the particular choice2U (r) + P(r) = 0, we immediately obtain the corre-sponding spherically symmetric solution [48]

ds2 = − f (r)dt2 + dr2

f (r)+ r2d�2;

f (r) = 1 − 2GM + Q0

r− 3κP0

2r2 + F(R) − �4

3r2 .

(51)

Here Q0 and P0 correspond to constants of integration, κcaptures the effect of bulk spacetime, i.e., it depends onthe bulk gravitational constant, and F(R), evaluated at thebrane location, can be constructed from the original f (R)

theory by taking an appropriate derivative. Assuming thatthe bulk scalar depends only on the bulk coordinates andfor a f (R) theory of the form f (R) = R + αR2 + βR4,the leading order behavior of F(R) is like an effectivefour-dimensional cosmological constant �4.

Converting back to Einstein frame The correspondingscalar–tensor solution can be obtained by transform-ing the metric in Eq. (51) using the appropriate con-formal factor: � = (1 + 2αR + 4βR3)1/3 = [1 +(3κ5φ/2) + (27βκ3

5 φ3/64α3)]1/3. Since the scalar fielddepends on the extra coordinate only, the conformalfactor evaluated on the location of the brane is just aconstant.

Solution in Einstein frame Since the corresponding con-formal factor is just a constant it will scale the metric,which can be absorbed by rescaling of the time and radial

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Eur. Phys. J. C (2016) 76 :552 Page 11 of 13 552

coordinate by the conformal factor. Hence the solution inthe Einstein frame would remain the same.

Thus even in the context of brane models solving effectivegravitational field equations in one frame and obtaining thesolution in the other often requires approximations. One hasto keep in mind that we are working in the mesoscopic scale,where neither the higher curvature terms are dominant norare they negligible. This allows one to invert various relationsconnecting the Einstein frame scalar with Jordan frame cur-vature, a key aspect while converting solutions in one frameto another.

7 Conclusions

A technique for solving field equations of higher curvaturegravity theories has been proposed. The technique essentiallyhinges on the mathematical equivalence of higher curvaturegravity theory, e.g., f (R) theories of gravity with scalar–tensor representation. Earlier this equivalence was knownonly at the level of the action principle. It was not clear a pri-ori whether the field equations derived from either the f (R)

representation or the scalar–tensor representation would alsobe equivalent. In this work starting from a five-dimensionaltheory we have explicitly demonstrated – (a) the bulk gravita-tional field equations derived from Jordan and Einstein frameare equivalent and (b) the effective field equations on thebrane in these two approaches are also equivalent. Using thisequivalence we have argued that, if one can solve for the fieldequations in one frame, the solution in the other frame caneasily be obtained. Even though for simple models one canperform the above operation exactly, often it requires suitableapproximations. The approximations essentially requires oneto work in mesoscopic energy scales, viz., higher than theweak scale but less than the Planck scale. For practical appli-cation of the technique, we have illustrated it in two relatedsituations:

• The bulk field equations in the Einstein frame have beensolved in the context of warped geometry models for twochoices of the potential – quadratic and quartic. Follow-ing expectation, the warp factor behaves differently inthese two scenarios, but leads to the desired exponentialwarping. These potential through conformal transforma-tions are related to two f (R) models – (a) R + αR2 and(b) R + αR2 + βR4, respectively. From the solution inthe Einstein frame we have obtained the solution in thef (R) representation as well, having a different warp fac-tor behavior and an extra dimension dependent radionfield.

• Second, using the known solutions to the effective fieldequations in the f (R) representations we have obtained

the corresponding solutions in the scalar–tensor repre-sentation. In the cosmological context, the scale factorstill exhibits a power law behavior, but with a differ-ent power. The spherically symmetric solution results inmere rescaling of the coordinates.

In the two examples depicted in this work we have exploreda practical illustration of the technique for both the bulk andthe brane spacetimes. Further it turns out that, even though inthe Einstein frame the brane separation has been fixed at thestabilized value, in the Jordan frame it starts depending onthe extra dimension. Interestingly, the warp factors in the twoframes are different, leading to different suppression of thePlanck scale on the visible brane. This might lead to potentialobservables, distinguishing the two frames in the context ofrecent LHC experiments.

Acknowledgments Research of S.C. is funded by a SPM fellowshipfrom CSIR, Government of India. He also thanks IACS, India for warmhospitality; a part of this work was completed there during a visit.

Open Access This article is distributed under the terms of the CreativeCommons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,and reproduction in any medium, provided you give appropriate creditto the original author(s) and the source, provide a link to the CreativeCommons license, and indicate if changes were made.Funded by SCOAP3.

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