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On the equivalence between f(R)theories and Einstein gravity

Soham Bhattacharyya1, ∗

1Department of Physics, IIT Madras, Chennai, India

In this brief note we present a somewhat trivial result. Namely, we show that perturbative oﬀ-shell

f(R)-theory is equivalent to Einstein gravity, as well as to the Brans-Dicke theory and the Einstein

scalar ﬁeld model. We also discuss possible generalisation of this result to higher-order gravitational

ﬁeld models.

I. INTRODUCTION

Signiﬁcant strides were made in the early parts of

the twentieth century in the formulation of gravity as

a classical ﬁeld theory. Out of that eﬀort came forth

General Relativity (GR), which passed almost all the

tests thrown at it, starting from the short, terrestrial (or

weak gravity region) length scales [1], to solar system

tests [2], to binary pulsar (strong gravity region) tests

[2], and ﬁnally constraints from Gravitational Wave

(GW) (extreme gravity region) observations in [3].

GR is a theory with two massless degrees of freedom.

The degrees of freedom manifest at asymptotic inﬁnity as

the so called plus and cross polarizations in gravitational

wave detectors. Eﬀorts have been made to ascertain

whether there are more than two degrees of freedom in

the observed gravitational wave signal, as in [4–6], and

others. However, so far there is no conclusive proof of

the presence of more than two degrees of freedom. On

the other hand, there are more than enough theoretical

and observational considerations that GR cannot be

a complete theory of gravity. In the theoretical side,

prediction of singularities in extreme situations like

black holes or big bang render the theory useless in

such scenarios. Similarly, in the observational side, the

presence of ﬂat velocity proﬁle of galaxies instead of

decaying ones, or the observational challenges faced by

the ΛCDM model provide much needed motivation to

come up with modiﬁed theories of gravity.

However, modifying gravity has its costs. Most

modiﬁcations to gravity leads to certain observational

features that have so far eluded experiments. Generally

speaking, modiﬁcations to GR leads to either increase

in degrees of freedom or violations of one or more of the

postulates of GR. On the increase in degrees of freedom

aspect, it is possible for gravity to have six degrees of

freedom, given that we live in a four dimensional space-

time. In the most naive sense, one can make coeﬃcients

of the modiﬁcation to GR go to very small values in

the action level and claim that the modiﬁcations are

un-excitable. However, that theory loses any predictive

power. However, there are other ways in which one can

∗xeonese@gmail.com

show modiﬁcations to gravity can be “pushed” to sub-

dominant regimes. For example, in some special cases it

can be shown that the extra degrees of freedom turn out

to be non-propagating (hanging around perturbed black

holes as some massive scalar cloud), like for the case of

Ricci ﬂat solutions of f(R) gravity (see Chapter 6 of the

thesis [7]), or it is shielded by means of the so called

Chameleon mechanism [8]. Such a mechanisms dictate

that extra degrees of freedom become highly massive

(that is they cannot be excited by standard solar system

excitations) in dense environments and light in very low

density scenarios, leading to expansion of the universe

as a whole. Similarly, in the violations of postulates

aspect, for example, in Chern-Simons theory of gravity,

a pseudo-scalar ﬁelds breaks the parity invariance of GR

and leads to the birefringence of vacuum, as in [9].

There exists another little known method to “push”

leading order modiﬁcations to GR to sub-leading orders.

This is the case of the ﬁeld redeﬁnition. In such cases,

the metric is redeﬁned in a particular manner such that

leading order eﬀects (such as quadratic modiﬁcations

to GR) can be pushed via conformal transformations

to cubic orders or higher. For example, in [10], it was

shown that under ﬁeld redeﬁnition of the metric tensor,

certain combinations of the curvature scalars appearing

as modiﬁcations to the Einstein-Hilbert action can be

pushed to higher orders. That is, any observer whose

eﬀective metric is the redeﬁned metric will no longer

see any quadratic modiﬁcation, as well as some cubic

modiﬁcations. Since the energy scales at which quadratic

modiﬁcations appear are lower than the energy scales

of the cubic modiﬁcations, it becomes harder for any

astrophysical source to perturb or excite the additional

degrees of freedom in the redeﬁned metric scenario. In

the language of [10], dimension four terms have been

pushed to dimension eight under the ﬁeld redeﬁni-

tion. In this article, we generalize the quadratic order

calculations of [10] to arbitrary powers of the Ricci scalar.

At the low energy limit, string theoretic/loop quantum

gravity or in other words, eﬀective ﬁeld theory actions

can lead to a Lagrangian of the following form

S=1

16 πZd4xR+R2+R3+OR4+Lm,

(1)

where Rnare general curvature invariants of dimension

2

2nconsisting of the Ricci scalar, Ricci tensor, and the

Riemann tensor. Lmis the matter Lagrangian with a

minimal coupling to the curvature terms, and is also a

functional of the metric tensor. If in Eq. (1), one only

considers till curvature squared terms of the most general

form like that of the Gauss-Bonnet type,

R2=a R2+b Rµν Rµν +c Rµρνσ Rµρν σ,(2)

then it was shown by [11] that such a model is renor-

malizable.

Authors like [12] decided to approach the inverse prob-

lem while trying to perturbatively renormalize gravity.

They started with GR, and under ﬁeld redeﬁnition was

able to obtain the most general quadratic theory, that is

Eq. (1) truncated till R2.

In this article we do not pursue all possible type of

independent scalar invariants at each dimension, but re-

strict ourselves to only vacuum polynomial f(R) theories

of the form

S=1

16πZd4x√−g[R+f(R)] ,(3)

where Ris the Ricci scalar and the form of f(R) is

given in Eq. (11). Whereas GR has only two massless

spin-2 (or purely tensorial) degrees of freedom (dof),

f(R) theories of gravity on the other hand contain, in

addition to the massless dof of GR, an extra massive

scalar degree of freedom. This is best understood by

going to the so called ‘Einstein frame’ by means of

a conformal transformation of the original frame (or

‘Jordan frame’) metric of the f(R) theory which frames

the theory eﬀectively as GR plus scalar ﬁeld with a

potential (the form of which is ﬁxed by the particular

form of the f(R) theory at hand). Examples of such

can be found in [13–18].

We however ﬁnd that in perturbative f(R) theories

of gravity, with f(R) of the form of Eq. (11), under a

suitable ﬁeld redeﬁnition, can be mapped back to GR ad

inﬁnitum. We ﬁnd that the ﬁeld redeﬁnition responsible

for mapping the theory back to GR is a conformal

transformation on the metric.

Our paper is organized as follows. In Sec. II we

brieﬂy review the Einstein and f(R)-gravity theories.

In Sec. III we review the ﬁeld redeﬁnition method

to excise quadratic (and some cubic) modiﬁcations

to GR. In Sec. IV we prove the perturbative oﬀ-shell

equivalence of these theories. Section V contains a

brief review of Brans-Dicke theory and describe its

oﬀ-shell equivalence with f(R)-theory. In Sec. VI we

review oﬀ-shell equivalence between f(R)-gravity and

the Einstein-scalar ﬁeld model. In Sec. VII we establish

perturbative oﬀ-shell equivalence between f(R)-gravity,

Brans-Dicke theory, and the Einstein-scalar ﬁeld model

and illustrate it via equivalence maps. Finally, in

Sec. VIII we conjecture on the potential eﬀects of the

conformal transformation when matter ﬁelds are present.

In this paper we shall use the geometrized system of

units, setting c=G= 1. We use space-time signature

+2 and conventions adopted in the book [19], unless ex-

plicitly stated otherwise.

II. EINSTEIN AND f(R)-GRAVITY MODELS

The famous 4-dimensional Einstein-Hilbert action in

the presence of matter is

SEH =1

16πZd4x√−gR +Sm(gαβ , ψ),(4)

where gis determinant of space-time metric gαβ in the

chosen coordinates xα,Ris the Ricci scalar, and the in-

tegral is taken over the space-time manifold. The matter

action Smcontains matter ﬁelds ψ. One of the simplest

generalization of the Einstein-Hilbert action is the so-

called f(R) model,

S=1

16πZd4x√−g[R+f(R) + Lm],(5)

where f(R) is an arbitrary function of the Ricci scalar.

f(R) theories shot to prominence with the quadratic

version of Eq. (5), or the ‘Starobinsky model’ given

by R+α R2([20]), which was shown to demonstrate

inﬂation, and remains a viable alternative to scalar ﬁeld

inﬂationary models.

Equations of motion or ﬁeld equations which have

higher than second order derivatives suﬀer from an

instability where their total energy or the Hamiltonian

is unbounded from below. Implying that in general

these theories do not have a stable vacuum state.

This is referred to as the Ostrogradski instability in

literature. f(R) theories of gravity however do not

suﬀer from the Ostrogradski instability, as was found

in [21]. This is in contrast to more generalized theories

of gravity containing scalar invariants like Ricci tensor

and Riemann contractions whose ﬁeld equations are

also fourth order in nature, and they suﬀer from the so

called ‘ghost’ degrees of freedom [11] whose energy is

unbounded from below.

There are two diﬀerent formalisms of f(R) theories

of gravity, the metric and the Palatini (or metric-aﬃne)

formalism. While the metric formalism is the standard

that assumes the metric to be the dynamical ﬁeld

in the space-time, the Palatini formalism takes the

connection to be an independent variable as well in

addition to the metric tensor (a review in [22]), lead-

ing to some interesting features, like the suppression of

the massive scalar degree of freedom as was found in [23].

3

It has been noticed by more and more accurate ex-

periments that there exist in nature, an equivalence (or

a strict equality) between the gravitational ‘charge’ (or

gravitational mass) and the inertial mass. GR as a theory

satisﬁes the equivalence principle. The fate of the equiva-

lence principle in f(R) theories of gravity is still debated

in the gravity community. On one hand, various authors

argue that the principle is violated, for example in the

Palatini formalism in [24]. Also, since f(R) theories can

be mapped to scalar-tensor theories of a particular form

(as in [25]), it was shown in [26] that equivalence prin-

ciple is violated in such theories. On the other hand,

it was shown in [27] that for transition from dense re-

gions of space-time to vacuum, the exterior solution can

be represented by a Schwarzschild-like metric, with the

extra Yukawa type ﬁeld (arising due to f(R) modiﬁca-

tion) only appearing as a very thin shell around the dense

object. In such a case, the weak equivalence principle is

shown to hold to a very high accuracy, like for solar sys-

tem tests of the equivalence principle. Constraints on

ﬁfth force parameters can also be found in [27]

III. THE FIELD REDEFINITION METHOD

Following the notations of [10] only for this section, we

brieﬂy review the ﬁeld redeﬁnition method here. For a

Lagrangian of the following form [28] in vacuum

L=√−gM2

P l

2R+LD4+LD6+ORiemann4

M4,

(6)

where the ﬁrst term is the Einstein-Hilbert action and

the higher dimensional operators are

LD4=√−gcR2R2+cW2W2

µανβ +cGB R2

GB,(7)

LD6=√−g

M2d1R□R+d2Rµν □Rµν +d3R3

d4RR2

µν +d5RR2

µναβ +d6R3

µν +d7Rµν RαβRµναβ

d8Rµν Rµαβγ Rαβγ

ν+d9Rαβ

µν Rγσ

αβ Rµν

γσ

+d10Rα β

µ ν Rγ σ

α β Rµ ν

γ σ i,(8)

where Wµανβ is the Weyl tensor and R2

GB is the Gauss-

Bonnet topological term which can be ignored since in

this article we restrict ourselves to four dimensions. One

can perform a perturbative ﬁeld redeﬁnition of the metric

as follows

gµν →gµν −2

M2

P l −2cW2Rµν +cR2+1

3cW2gµν R

−2

M2

P l

1

M2−d2□Rµν −d4RRµν −d6Rα

µRνα

−d8Rαβγ

µRναβγ +gµµ d1+d2

2□R+ (d3

+d4

2R2+d6+d7

2R2

αβ +d5+d8

2R2

αβγσ,

(9)

which leads to the Lagrangian (6) becoming as follows

L=√−g"M2

pl

2R+1

M2d9Rαβ

µν Rγσ

αβ Rµν

γσ

+d10Rα β

µ ν Rγ σ

α β Rµ ν

γ σ i+O1

M4.(10)

One can see that due to the ﬁeld redeﬁnition entirety

of the quadratic and some of the cubic terms, except

pure Weyl cube terms which cannot be ﬁeld redeﬁned

away, have been pushed to higher orders. In the follow-

ing section, ignoring Ricci tensor and Riemann tensor

terms in the Lagrangian, we generalize the ﬁeld redeﬁni-

tion method to higher powers of the Ricci scalar.

IV. EQUIVALENCE WITH EINSTEIN

GRAVITY

Here we shall consider perturbative approach to f(R)

gravity. Namely, we introduce a parameter λwhich char-

acterizes strength of perturbation, such that λR ≪1.

This parameter can be considered as squared character-

istic length of the model (measured in units of mP). In

the string eﬀective action it corresponds to the string

slope parameter α′, which in natural units is equal to the

squared string length. Hence, the dimension of λis length

squared. The perturbative f(R) takes the following form

(see, e.g. [10, 29]):

f(R) = X

k≥1

ckλkRk+1 ,(11)

where ck’s are dimensionless expansion coeﬃcients [30].

Our goal is to establish perturbative equivalence between

the f(R) theory (5) with f(R) given by (11) and the Ein-

stein gravity (4). Such an equivalence was established be-

tween Lagrangian density with quadratic and cubic terms

in R,Rαβ, and the Riemann tensor components in ear-

lier works by means of ﬁeld redeﬁnition (see, for example,

[17, 28, 29, 31]). In the case of Rterms only, the ﬁeld

redeﬁnition is equivalent to conformal transformation of

space-time metric. Here we shall use the same approach,

which can also be suggested by the Scherk-Schwarz for-

malism for performing dimensional reduction, which we

do not apply here [32].

4

The conformal metric transformation

¯gαβ = Ω2gαβ ,(12)

induces the following Ricci scalar conformal transforma-

tion in 4-dimensional space-time (see e.g. [33]):

¯

R= Ω−2R−6 Ω−1∇2Ω,(13)

where ∇is the covariant derivative operator associated

with the space-time metric gαβ,∇αgβγ = 0. Here and in

what follows, we shall use the notations ∇2=gαβ∇α∇β

and (∇Ω)2=gαβ(∇αΩ)(∇βΩ). It is to be noted that

under the conformal transformation (12), the coupling of

matter with the space-time curvature changes. This is

because of the fact that the matter Lagrangian density

Lmexplicitly contains the metric, and any redeﬁnition

of the metric will correspondingly feature in Lmas well.

The point has been illustrated in [34]. However, for sim-

plicity, and to illustrate our point, we will only consider

the vacuum case.

Now we require that for a certain conformal factor the

f(R) theory (5) is equivalent to the Einstein gravity (4),

that is

√−g[R+f(R)] ⊜√−¯g¯

R , (14)

where ¯gis determinant of ¯gαβ and the symbol ⊜stands

for equality modulo total derivatives, which give vanish-

ing within our theory boundary terms. Let us remark

that expression with ⊜is always understood under the

integral sign, i.e. we can multiply the expression like (14)

by a constant, but not by a scalar function, except for

a function of the metric gαβ and its determinant. Using

the expression (13) and the relation √−¯g= Ω4√−gwe

present (14) in the following form:

6Ω∇2Ω−(Ω2−1)R+f(R)⊜0.(15)

Let us observe ﬁrst that for f(R) = 0 the expression (5)

reduces to the expression (4), which implies that Ω = 1,

which is a trivial solution to (15) with f(R) = 0. Be-

cause f(R) is given in terms of series (11), we shall look

for a solution to (15) in the perturbative form of the cor-

responding to f(R) order,

Ω = 1 + X

k≥1

λkPk[R],(16)

where Pk[R] = Pk(R, ∇2R, ...) are sought polynomials

composed of the Ricci scalar and its covariant derivatives.

Then, the conformal factor Ω2can be computed from (16)

for a given λ-order nof f(R) as follows:

Ω2= 1 +

n

X

k=1

λk 2Pk+

k−1

X

l=1

Pk−lPl!,(17)

where for brevity we dropped the [R] notation in the

Pk’s terms. Substituting (16) into (15) and using (11)

we derive the following λnorder term:

n−1

X

k=1

(6Pn−k∇2Pk−R Pn−kPk)−2R Pn+cnRn+1 ,(18)

where n≥1. Each λ-order expression should vanish

independently modulo the boundary term. As a result,

we can ﬁnd the following nonlinear recurrence relation of

order n−2 for Pn:

Pn=cn

2Rn+

n−1

X

k=1 3R−1Pn−k∇2Pk−1

2PkPn−k.

(19)

As an example, for f(R) of λ-order 2 terms are

P1=c1

2R , P2=c2

2R2+c2

1

43∇2R−1

2R2.(20)

They deﬁne the following conformal factor:

Ω2= 1 + λ1c1R+λ2c2R2+3

2c2

1∇2R.(21)

The recurrence relation (19) together with (17) solves

equation (15) and proves by induction that the perturba-

tive f(R) theory (11) is equivalent to the Einstein gravity

(4).

V. EQUIVALENCE WITH BRANS-DICKE

THEORY WITH ω0= 0

The Brans-Dicke theory with ω0= 0 parameter with-

out matter ﬁelds reads (see, e.g. [14, 18, 35])

S=1

16πZd4x√−g[ϕ R −V(ϕ)] ,(22)

where ϕis the Brans-Dicke scalar ﬁeld and V(ϕ) is its

potential. The corresponding ﬁeld equations are

Gαβ =1

ϕ(∇α∇βϕ−gαβ∇2ϕ)−gαβ

2ϕV(ϕ),(23)

R=V′(ϕ),(24)

where the prime stands for the derivative with respect to

ϕ. Assume now that the scalar ﬁeld depends perturba-

tively on Rand consider the following expansion:

ϕ(R) = X

k≥0

akλkRk,(25)

where λ≪1 as before and ak’s are constant expansion

coeﬃcients. Then, integrating the expression (24) and

imposing the condition V(0) = 0 we derive

V(R) = X

k≥0

akk

k+ 1λkRk+1 .(26)

Thus we see that taking a0= 1 and ak=ck(k+ 1) for

k≥1 the Brans-Dicke action (22) is equivalent to the

f(R) theory (5) and, as it follows from Sec. II, can be

transformed to the Einstein gravity (4).

5

VI. EQUIVALENCE WITH EINSTEIN-SCALAR

FIELD MODEL

The Brans-Dicke model (22) can be viewed as Einstein-

scalar ﬁeld model in the Jordan frame. Here we present

it in the Einstein frame. Let us ﬁrst rewrite the action

(22) in the “barred” form

S=1

16πZd4x√−¯gϕ¯

R−V(ϕ).(27)

Then, we apply the conformal transformations (12), (13)

with the conformal factor

Ω2=ϕ−1.(28)

and present the kinetic term modulo the boundary term

as follows: 6ϕ1/2∇2ϕ−1/2⊜−6(∇ϕ1/2)(∇ϕ−1/2). This

brings us to the action

S=1

16πZd4x√−gR−3

2ϕ−2(∇ϕ)2−ϕ−2V(ϕ).

(29)

The ﬁnal step is to write the kinetic term in the standard

form by deﬁning new scalar ﬁeld φ, such that

ϕ= exp φ/√3(30)

and the related potential

U(φ) = exp −2φ/√3V(φ).(31)

This results in the Einstein-scalar ﬁeld model

S=1

16πZd4x√−gR−1

2(∇φ)2−U(φ).(32)

This procedure is well known and presented in many

works (see, e.g. [14, 18]). Our goal is to establish its re-

lation with the perturbative f(R) model (5), (11), and,

as a result, with the Einstein gravity (4). We observe

that relations inverse to (28), (30), and (31) transform

the Einstein-scalar ﬁeld model (32) to the Brans-Dicke

model (22). Then, as it was illustrated in the previous

section, the Brans-Dicke model can also be transformed

perturbatively via the expressions (25) and (26) to the

Einstein gravity (4).

VII. EQUIVALENCE MAPS

In the previous sections we established oﬀ-shell equiv-

alence between perturbative f(R) model, the Einstein

gravity, the Brans-Dicke with ω0= 0 and the Einstein-

scalar ﬁeld models. Here we explore this equivalence in

detail.

It is well known that f(R) gravity is equivalent to

Brans-Dicke theories (see, e.g. [14, 18] and the equiva-

lence map presented therein.) This equivalence exists for

any f(R) function of class C2and is established by means

of the Legendre and conformal transformations. Equiva-

lence between the Einstein gravity and f(R) model can

be established by means of the conformal transforma-

tion (12) where the conformal factor solves equation (15).

This equation is a nonlinear PDE of the ﬁrst (or ignoring

boundary term second) order. We do not know whether

a general solution to this equation for arbitrary f(R) of

class C2exists and if so, how to ﬁnd it. It could be that

there is some class of transformations or generating tech-

niques allowing to ﬁnd for a given f(R) corresponding

solutions to this equation. An analysis of this problem

goes beyond the scope of this work.

We were able to ﬁnd a perturbative solution to this

equation for the corresponding form of f(R) function

(11). This equivalence implies that the perturbative f(R)

and the related to it Brans-Dicke and Einstein scalar ﬁeld

models are conformally equivalent to the Einstein gravity.

In other words, all these models lie within the Einstein

gravity and can be “perturbatively revealed” by a suit-

able conformal factor. One can say accordingly, that the

Einstein theory is rich enough to accommodate pertur-

batively via conformal transformation f(R) and related

gravity models.

VIII. SCENARIO IN THE PRESENCE OF

MATTER AND APPLICATIONS TO

COSMOLOGY

As the current work deals with vacuum scenarios, it

is important to note that the scenario maybe completely

diﬀerent in the presence of matter. While it is clear from

this work that in vacuum the extra degree of freedom is

suppressed inﬁnitely for f(R) theories (and related theo-

ries that f(R) can be mapped to), the presence of matter

will lead to complications. To be precise, the Lagrangian

for the matter ﬁelds contain the metric; any changes to

the metric, like a conformal transformation, will lead to

a change in the matter Lagrangian. In eﬀect, the confor-

mal transformation in the current work and other such

transformations will lead to the non-minimal coupling of

the matter ﬁeld with gravity, resulting in an actual de-

viation from GR. As an example, consider the conformal

transformation on the metric which takes an f(R) theory

in Jordan frame to its equivalent Einstein+scalar model

in the Einstein frame. The conformal transformation is

given as follows

˜gµν =f′(R)gµν ,(33)

where f′(R) = df(R)

dR . In the Einstein frame, due to

this particular conformal transformation, the matter La-

grangian picks up the conformal factor and hence be-

comes non-minimally coupled to gravity. This particu-

lar non-minimal coupling leads to a variable mass of the

scalar ﬁeld which depends on the ambient matter density

[36]. To be precise, the mass of the scalar ﬁeld becomes

6

heavy in regions of high mass density (leading to a sharp

fall oﬀ of the scalar ﬁeld proﬁle outside a mass distribu-

tion) like the surface of planets, stars, etc. On the other

hand when the mass density is low, the mass of the scalar

ﬁeld becomes light, which can support cosmic accelera-

tion. This eﬀect is known as the Chameleon mechanism.

Now consider the leading order f(R) theory of the form

f(R)≡R+α R2.(34)

The conformal transformation required to take this the-

ory into the Einstein frame is then given by

f′(R) = 1 + 2 α R. (35)

Compare this with the leading order conformal trans-

formation for taking the (R) theory to Einstein gravity

in Eq. (21) which goes as 1 + λ c1R. This is not a

coincidence. While the former serves to convert higher

powers of the Ricci scalar (and in essence higher powers

of derivatives of the metric) into kinetic and potential

terms of the scalar ﬁeld, the latter ‘pushes’ the quadratic

powers of the Ricci scalar into cubic and higher powers.

Both conformal transformations have the same eﬀect of

getting rid of the quadratic Ricci term. Both conformal

transformations will lead to a non-minimal coupling of

the matter ﬁelds with gravity, and in return will gener-

ate Chameleon screening of the scalar ﬁeld, which then

becomes important for cosmological scenarios.

IX. DISCUSSIONS AND FUTURE WORK

In this article we were able to show the pertubative

equivalence between polynomial f(R) theories of gravity

and GR. At the same time equivalences between Brans-

Dicke theories, scalar-tensor theories, and polynomial

f(R) were also reviewed. This exercise shows that it

is possible to generate theories of higher complexity and

ﬁeld content from GR itself through proper ﬁeld redeﬁni-

tions. Speciﬁcally, the graviton, with it’s three degrees of

freedom in f(R) theories of gravity is seen to apparently

shed one massive scalar degree of freedom through the

particular ﬁeld redeﬁnition in Eqs. (12) and (21), and

have only two massless tensor degrees of freedom of GR,

to which it was mapped to. The equivalences shown im-

ply that if a solution to the vacuum ﬁeld equations of GR

is known, one can generate the corresponding solutions

of the vacuum ﬁeld equations of the modiﬁed theories of

gravity that were discussed in this article. However, this

will require further study and will be published elsewhere.

The equivalences raise the question about the ‘physi-

cality’ of the conformal ‘frames’. That is, whether a black

hole solution in GR in the conformally transformed GR

‘frame’ is the physical solution, or whether the GR frame

is simply a convenient frame to generate solutions, and

the physical object in question (the black hole solution)

is described by the solution of the vacuum ﬁeld equations

of modiﬁed gravity frame. However, this question is be-

yond the scope of this article and will be pursued in a

following publication.

In this article, we have only pursued vacuum La-

grangian densities. However, ﬁeld redeﬁnitions of the

form (12) induce non-minimal coupling of the curvature

with the matter Lagrangian density since it is an explicit

function of the metric tensor as well. This will lead to

the comformally transformed Lagrangian having a more

complicated form in the GR ‘frame’ and require further

investigation.

For future work we will also generalize from f(R) the-

ories to the fully general Lagrangian of Eq. (1), and will

aim to come up with a ﬁeld redeﬁnition that maps it to

GR.

ACKNOWLEDGMENTS

The author would like to acknowledge Andrey Shoom

for giving signiﬁcant help and directions throughout the

duration of the study. The author would also like to

thank Alok Laddha and Ashoke Sen for giving helpful

comments. The author wishes to express his gratitude to

the MPI f¨ur Gravitationsphysik (Hannover) and ICTS-

TIFR (Bangalore) for hospitality. The author would also

like the thank the anonymous reviewers for their ques-

tions and suggestions.

Appendix A: Derivation of the master equation

From Eq. (14), one has the following

√−g[R+f(R)] −√−¯g¯

R= 0.(A1)

The conformally transformed square root of metric de-

terminant and the Ricci scalar transform as

√−¯g= Ω4√−g, (A2)

¯

R=R

Ω2−6□Ω

Ω3.(A3)

Substituting the above in Eq. (A1) leads to the following

√−g[R+f(R)] −√−gΩ2R−6 Ω □Ω= 0.(A4)

Canceling √−gon both sides one obtains

6Ω□Ω−(Ω2−1)R+f(R)=0.(A5)

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