# Physical Review D 99, 124027 (2019)

## Transcript Of Physical Review D 99, 124027 (2019)

PHYSICAL REVIEW D 99, 124027 (2019)

Palatini f (R;Lm;RμνTμν) gravity and its Born-Infeld semblance

Matthew S. Fox*

Department of Physics, Harvey Mudd College, Claremont, California 91711, USA

(Received 2 December 2018; published 19 June 2019)

We investigate Palatini fðR; Lm; RμνTμνÞ modified theories of gravity. As such, the metric and affine connection are treated as independent dynamical fields, and the gravitational Lagrangian is made a function of the Ricci scalar R, the matter Lagrangian density Lm, and a “matter-curvature scalar” RμνTμν. The field equations and the equations of motion for massive test particles are derived, and we find that the independent connection can be expressed as the Levi-Civita connection of an auxiliary, energy momentum–dependent metric that is related to the physical metric by a matrix transformation. Similar to metric fðR; T; RμνTμνÞ gravity, the field equations impose the nonconservation of the energymomentum tensor, leading to the appearance of an extra force on massive test particles. We obtain the explicit form of the field equations for massive test particles in the case of a perfect fluid and an expression for the extra force. The nontrivial modifications to scalar fields and both linear and nonlinear electrodynamics are also considered. Finally, we detail the conditions under which the present theory is equivalent to the Eddington-inspired Born-Infeld theory of gravity.

DOI: 10.1103/PhysRevD.99.124027

I. INTRODUCTION

Observations of the cosmic microwave background (CMB) [1] and direct measurements of the light curves from several hundred type Ia supernovae [2] suggest that the Universe is currently undergoing a phase of late-time, accelerated expansion. While the physics underlying this phenomenon remain unsettled, at least one thing is certain: the acceleration is either a trait of the gravitational interaction itself, or it is a gravitational manifestation of something else (dark energy). By and large, the copious models of the former type derive from revisions to the Einstein-Hilbert action

Z

1 d4xpﬃ−ﬃﬃﬃgﬃﬃR;

ð1Þ

SEH ¼ 2κE

where κE is the Einstein constant, R is the Ricci scalar, and g is the determinant of the spacetime metric gμν.

Among the most straightforward generalizations of SEH are the fðRÞ models. These constitute a class of higherorder gravity theories in which SEH is restyled with terms of higher degree in the scalar curvature. Indeed, the mystery of

cosmic expansion can be unraveled in this approach [3].

Some models [4] even appear to avoid the fatal instabilities

and acute weak-field constraints that bar many other proposals [5]. Incidentally, the lure of fðRÞ gravity is broader in application than to just cosmic speed-up. For

instance, theories with higher-order curvature invariants

*[email protected]

show promise as effective first-order approximations to

quantum gravity and can encourage quantum and gravita-

tional fields to be well behaved in the ultraviolet regions neighboring curvature singularities [6]. Further fðRÞ phenomenology has been extensively surveyed in the

literature [7]. Interesting extensions of the fðRÞ models are those

theories which include in the action an explicit nonminimal

coupling between matter and curvature invariants. A notable subset of these models is so-called fðR; LmÞ gravity (Lm being the matter Lagrangian density) introduced by Bertolami et al. in Ref. [8]. Their model was

linear in the nonminimal coupling, which prompted the author of Ref. [9] to study the maximal extension of SEH in which R and Lm are coupled arbitrarily. Cosmological and astrophysical phenomena have been studied in various fðR; LmÞ frameworks [10], in addition to more general studies into the properties of the theory itself [11].

Generally speaking, nonminimal theories such as fðR; LmÞ gravity do not admit chart transitions that locally transform away the influence of a gravitational field on

matter [12]. In turn, the covariant divergence of the energy-

momentum tensor is generally nonzero, the motion of test

particles is generally nongeodesic (due to the presence of

an extra force orthogonal to the 4-velocity [8]), and thus the

equivalence principle (EP) is generally violated. Hence,

these theories are stringently constrained by tests of the EP.

It is important to note, however, that a violation of the EP

does not in principle disqualify the specific theory [13]. A set of models related to fðR; LmÞ gravity derives from

the case in which the functional dependence on Lm

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© 2019 American Physical Society

MATTHEW S. FOX

manifests via a dependence on the trace T of the energymomentum tensor. These so-called fðR; TÞ models have drawn significant attention and were explicitly introduced by Harko et al. in Ref. [14]. However, Poplawski [15] was first to consider a model in which the cosmological constant is a function of T, which is considered a relativistically covariant model for interacting dark energy and which is evidently a subset of the fðR; TÞ theory. We note that explicit dependences on T may be induced by quantum effects (e.g., conformal anomalies) or exotic imperfect fluids. The reader is referred to the review [16] for additional fðR; LmÞ and fðR; TÞ phenomenology.

Further extensions to the fðR; LmÞ and fðR; TÞ theories were proposed in Refs. [17,18], in which terms of the form RμνTμν, where Rμν is the Ricci tensor, were incorporated into the fðR; TÞ Lagrangian. Instances of this coupling are known to arise in Born-Infeld models of gravity [19] when one Taylor expands the Lagrangian. The cosmological implications of these so-called fðR; T; RμνTμνÞ gravity theories were surveyed in Refs. [17,18,20], and the criterion to circumvent the Dolgov-Kawasaki instability [21] can be found in Ref. [18]. Moreover, energy conditions and thermodynamic laws in fðR; T; RμνTμνÞ gravity were considered in Ref. [22]. Finally, it is known that metric fðR; T; RμνTμνÞ gravity acquires ghostlike instabilities due to the additional RμνTμν coupling [23] and that these instabilities can be avoided with a Palatini or metric-affine variation [24].

The appearance of the RμνTμν coupling in Born-Infeld gravity is the chief motivation for our study. The BornInfeld models themselves, akin to Born-Infeld electromagnetism, modify the determinantal structure of SEH. Among the many Born-Infeld models, a prominent one is the Eddington-inspired Born-Infeld (EiBI) theory proposed in Ref. [25]. Whereas many fðRÞ models differ from general relativity (GR) even in vacuum, EiBI does not. Yet in ultraviolet regions, such as near cosmological singularities, EiBI gravity is characterized by curing the geometrical divergences plaguing GR [25]. See Ref. [26] for a recent review on Born-Infeld modifications to gravity.

Importantly, in all of these theories, independent of the details of the modification, one must ultimately choose between two ostensibly similar methods for varying the action: either one treats the metric as the sole dynamical entity and fixes a priori the connection to be the LeviCivita connection of gμν (the metric formalism), or one regards the metric and affine connections as independent dynamical structures (the metric-affine or Palatini formalisms, depending on whether matter couples to the connection or not, respectively). In GR, the distinction is superfluous as they both lead to the same physics. However, in general, nearly all the aforementioned theories forecast different physics depending on whether the metric and affine structures are handled independently or not. In fact, in some theories, such as EiBI gravity [25] and

PHYS. REV. D 99, 124027 (2019)

(already mentioned) fðR; T; RμνTμνÞ gravity [24], the

Palatini formalism will remove ghostlike instabilities that

otherwise afflict their metric counterparts. Whether the

affine connection is determined by the metric degrees of

freedom (d.o.f.) or not is a truly fundamental question and

demands experimental investigation.

Though matter couplings to the connection may arise

due to quantum gravitational corrections, we shall ignore

that possibility here, and so we exercise the Palatini

formalism. Studies of Palatini fðRÞ and fðR; TÞ models

can be found in Refs. [27] and [28,29], respectively, and

more general actions varied a` la Palatini and metric-affine,

including the role of torsion, can be found in Ref. [24]. To

the best of our knowledge, no studies of pure Palatini fðR; T; RμνTμνÞ or Palatini fðR; Lm; RμνTμνÞ gravity

have yet been completed, though indirect pursuits exist

(see, e.g., Ref. [24]). In this paper, we shall investigate Palatini fðR; Lm; RμνTμνÞ gravity, from which Palatini fðR; T; RμνTμνÞ gravity follows after a simple modifica-

tion to the field equations. In addition to studying fðR; Lm; RμνTμνÞ gravity on its own, we ultimately seek

the conditions under which our theory corresponds to EiBI

gravity.

The present paper is structured as follows. In Sec. II,

we vary the fðR; Lm; RμνTμνÞ action a` la Palatini and derive the theory’s equations of motion and an explicit

form for the independent connection. In Sec. III, we survey the bimetric structure of fðR; Lm; RμνTμνÞ gravity in

addition to the nonminimal structure of the field equa-

tions. In Sec. IV, we explore various properties of the fðR; Lm; RμνTμνÞ field equations, including their non-

conservation equation, the nongeodesic motion of test

particles, the nature of the extra force, the weak-field limit,

and the modified Poisson equation. In Sec. V, we derive the fðR; Lm; RμνTμνÞ field equations for the cases of linear

and nonlinear electromagnetic fields as well as canonical

scalar fields. Finally, in Sec. VI, we derive the conditions under which the fðR; Lm; RμνTμνÞ model responds iden-

tically to the EiBI theory for specific matter sectors.

In this paper, we shall operate in a four-dimensional

spacetime

ðM

;

gμν

;

Γ

α μν

Þ

in

which

the

metric

gμν

and

connection Γαμν will be treated as independent dynamical

fields. We shall utilize the metric signature ð−; þ; þ; þÞ

and, where appropriate, adopt the natural system of units in

which c ¼ 8πG ¼ 1.

II. FIELD EQUATIONS OF f (R;Lm;RμνTμν) GRAVITY

The Ricci tensor can be defined solely in terms of the affine connection, and this underpins the Palatini and metric-affine formalisms. Explicitly, the Ricci tensor follows from the Riemann curvature tensor

Rαβμν ¼ ∂μΓανβ − ∂νΓαμβ þ ΓαμλΓλνβ − ΓανλΓλμβ ð2Þ

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PALATINI fðR; Lm; RμνTμνÞ GRAVITY AND ITS …

via the contraction RμνðΓÞ ≡ RαμανðΓÞ. Only now, one needs to invoke the metric to define the Ricci scalar Rðg; ΓÞ ≡ gμνRμνðΓÞ and a matter-curvature scalar Vðg; Γ; ΨÞ ≡ RμνðΓÞTμνðg; ΨÞ, where Tμν is the symmetric (Hilbert) energy-momentum tensor. As we shall see below

in Eq. (5), the energy-momentum tensor is constructed a` la

Palatini so that Tμν depends only on the metric and a set of matter fields Ψ. We note that the symmetry of gμν and Tμν imposes that only the symmetric part of the Ricci tensor enters into this theory’s action. This considerably simplifies the role of torsion in the theory and renders a separate

consideration for fermionic matter immaterial [24].

With all this in mind, the action considered in this work

bears the form

Z 1 d4xpﬃ−ﬃﬃﬃgﬃﬃfðR; L ; VÞ þ S ½g; Ψ; ð3Þ

S½g; Γ; Ψ ¼ 2κ

m

m

where κ is a coupling constant with suitable dimensions. Here, the matter Lagrangian density Lm, encoded in both the function fðR; Lm; VÞ ¼ fðR; Lm; RμνTμνÞ and the matter action

Z pﬃﬃﬃﬃﬃﬃ

−gL ½g; Ψ;

ð4Þ

Sm½g; Ψ ¼ d4x

m

is assumed to capture all matter fields Ψ present in M. Moreover, Lm determines the manifestly symmetric energy-momentum tensor

2 δðpﬃ−ﬃﬃﬃgﬃﬃLmÞ

Tμν ≡ − pﬃ−ﬃﬃﬃgﬃﬃ δgμν ;

ð5Þ

which again is independent of the affine connection in the

Palatini formulation. If we denote by δSg and δSΓ the variation of Eq. (3) with

respect to the metric and connection, respectively, then δS ¼ δSg þ δSΓ with

Z

δSg ¼ 1 d4xpﬃ−ﬃﬃﬃgﬃﬃ − 1 fgμν þ fRRμν þ fLΞμν

2κ

2

þ fV Πμν − κTμν δgμν

ð6Þ

and

1 Z pﬃﬃﬃﬃﬃﬃ

δRμν

δSΓ ¼ 2κ d4x −g ðfRgμν þ fV TμνÞ δΓλ δΓλαβ: ð7Þ

αβ

Here, we have introduced the definitions fR ≡ ∂Rf, fV ≡ ∂Vf; fL ≡ ∂Lmf as well as the manifestly symmetric matter and matter-curvature tensors

PHYS. REV. D 99, 124027 (2019)

Ξμν ≡ ∂∂Lgμmν ; ð8Þ

Πμν ≡ Rαβ δδTgμανβ ; ð9Þ

respectively. Since gμν and Γαμν are independent fields, δS ¼ 0 if and only if δSg and δSΓ vanish separately. In the case of the metric variation (6), δSg ¼ 0 implies

1 fRRμν − 2 fgμν ¼ κTμν − fLΞμν − fV Πμν: ð10Þ

This is the fðR; Lm; VÞ generalization of Einstein’s equation. Its properties shall be explored in the coming sections. We note here, however, that as a consequence of the nonminimal coupling there appear in Eq. (10) strict couplings between matter fields and curvature terms. This is very much unlike GR and other minimally coupled theories in which matter fields are wholly separable from curvature terms such that the field equations may be written in a “curvature = matter” type representation. Ultimately, however, writing the field equations in this way is more for physical tidiness and less for mathematical substance. Hence, the mathematical representation of these equations may as well be chosen such that it facilitates later computation. To this end, we define a curvature-dependent effective energy-momentum tensor by

Σμν ≡ Tμν − fκL Ξμν − fκV Πμν; ð11Þ

which refashions the field equations (10) into a form similar to those in Palatini fðRÞ gravity:

1

fRRμν − 2 fgμν ¼ κΣμν:

ð12Þ

The variation with respect to the connection takes more

care. We refer the reader to Ref. [24], in which a nearly complete derivation is given. One shall find that δSΓ ¼ 0 only if

∇ðσpÞ½pﬃ−ﬃﬃﬃgﬃﬃðfRgμν þ fVTμνÞ ¼ 0;

ð13Þ

where ∇ðpÞ is the derivative operator associated with the independent connection and which is manifestly distinct from ∇ðgÞ, the covariant derivative compatible with the spacetime metric gμν. The resemblance of Eq. (13) to the companion EiBI field equation will be studied in Sec. VI.

We note that Eq. (13) holds well even in the presence of torsion. This follows from this theory’s insensitivity to the projective d.o.f. in projective transformations of the independent connection, which ultimately derives from only the symmetric part of the Ricci tensor entering into the action (3). See Ref. [24] for details.

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MATTHEW S. FOX

Together, Eqs. (10) and (13) comprise the field equations for fðR; Lm; VÞ gravity. We see in Eq. (13) a natural auxiliary metric ingrained into this theory’s mathematical

structure, namely, a metric pμν of which the inverse, denoted pμν [30], satisfies

pﬃ−ﬃﬃﬃpﬃﬃpμν ¼ pﬃ−ﬃﬃﬃgﬃﬃðfRgμν þ fVTμνÞ;

ð14Þ

where p ≡ detðpμνÞ. Evidently, the symmetry of gμν and Tμν forces pμν (and hence pμν) to be symmetric. Moreover, pμν satisfies ∇ðσpÞðpﬃ−ﬃﬃﬃpﬃﬃpμνÞ ¼ 0 by construction, so pμν is compatible with ∇ðpÞ, provided the coefficients of the

independent connection are the Christoffel symbols in pμν,

Γαμν ¼ 1 pασð∂μpσν þ ∂νpμσ − ∂σpμνÞ: 2

ð15Þ

Consequently, the independent connection is the Levi-

Civita connection in the auxiliary metric pμν. Note also that the determinant p can be computed explicitly with Eq. (14) and the relation p ¼ det−1ðpμνÞ. One finds

p ¼ g2 det ðfRgμν þ fV TμνÞ:

ð16Þ

We shall apply these formulas to various physical phenomena in the coming sections. But first, we briefly comment on some general characteristics of the field equations.

III. REMARKS ON THE f ðR;Lm;VÞ FIELD EQUATIONS

As noted above, for theories in which couplings are minimal, the matter fields can in general be placed on one side of the theory’s field equation, and the symmetric part of the Ricci tensor will be given solely in terms of gμν. But for nonminimal theories, the matter fields are generally inseparable from the geometry terms, and the symmetric part of the Ricci tensor need not be given solely in terms of gμν. Such is the case for fðR; Lm; VÞ gravity, as made evident by the field equations (10) and (13). Other aspects of the present theory’s nonminimal character are addressed in this section.

A. Matter and matter-curvature tensors

The matter-curvature tensor Πμν is a hallmark of the present theory’s nonminimal coupling. For the sake of computation, it is of interest to write this tensor in a form entirely in terms of the matter Lagrangian and Ricci tensor. To this end, assuming the matter Lagrangian is independent of derivatives of the metric, one can show that Eq. (5) is equivalent to

Tμν ¼ L gμν þ 2 ∂Lm :

ð17Þ

m

∂ gμν

PHYS. REV. D 99, 124027 (2019)

Incidentally, this equation has the matter tensor Ξμν implicitly built into it,

1

Ξμν ¼ 2 ðLmgμν − TμνÞ;

ð18Þ

which we shall find useful later on. Moreover, Eq. (17) facilitates calculating the functional derivative

δTαβ ¼ gαβ ∂Lm þ 2 ∂2Lm þ L δðαβÞ ;

δgμν

∂gμν ∂gμν∂gαβ m μν

ð19Þ

where

δðαβÞμν

¼

1 2

ðδα μ δβ ν

þ

δβ μ δα ν Þ

is

the

upper

symmetric

part of the generalized Kronecker symbol (we herein

denote symmetrization by parentheses). Combining this

result with the definition of the matter-curvature tensor in

Eq. (9) implies

Π ¼ R ∂Lm þ 2R ∂2Lm þ R L :

μν

∂ gμν

αβ ∂gμν∂gαβ

μν m

ð20Þ

Another useful identity is

Πμν ¼ 2RλðμTλνÞ þ Rαβ δδTgμανβ ;

ð21Þ

which follows from substituting Tμν ¼ gμαgνβTαβ into the definition (9) of the matter-curvature tensor.

A notable matter sector is that of a perfect fluid (PF), for which we shall take Lm ¼ P [31], where P is the isotropic pressure of the fluid. The corresponding energy-

momentum tensor is

TðμPνFÞ ¼ ðρ þ PÞuμuν þ Pgμν;

ð22Þ

where ρ is the energy density of the fluid and the fluid’s 4-velocity uμ satisfies the condition uμuμ ¼ −1. One can

show the pressure P satisfies

δP ¼ − 1 ðρ þ PÞuμuνδgμν

ð23Þ

2

by using Eq. (17) with Lm ¼ P and comparing to Eq. (22). Moreover, one has [33]

δρ ¼ 1 ρðgμν − uμuνÞδgμν;

ð24Þ

2

which ultimately follows from the conservation of the matter fluid current, ∇ðμgÞðρuμÞ ¼ 0. Using these formulas appropriately, one shall find

ΞðμPνFÞ ¼ − 1 ðρ þ PÞuμuν

ð25Þ

2

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PALATINI fðR; Lm; RμνTμνÞ GRAVITY AND ITS …

and,

from

Eq.

(21)

and

the

identity

δuα δgμν

¼

−

1 2

gαðμ

uνÞ,

ΠðμPνFÞ ¼ RλðμTðνPÞλFÞ þ 12 ρRαβuαuβgμν − 1 ½ð2ρ þ PÞRαβuαuβ þ ðρ þ PÞRuμuν: 2

ð26Þ

The effective energy-momentum tensor for a perfect fluid then follows from its definition (11). The rather exotic coupling of matter and the 4-velocity to the Ricci tensor in Eq. (26) suggests that the matter-curvature tensor will play a significant role in the field equations in regions of high density, such as within a black hole or in the very early Universe. This is quantitatively similar to EiBI gravity, which has in its field equations a similar RμνTμν coupling that also gives rise to couplings between the Ricci tensor and the 4-velocity of perfect fluids (see Ref. [26] or Sec. VI of this paper). It is natural to hypothesize, then, that fðR; Lm; VÞ gravity may be fashionable such that it corresponds to GR in the weak-field regime but then cures the curvature singularities of GR in high density regions— behavior that mimics the preeminent characteristics of EiBI gravity.

B. Auxiliary metric

The introduction of the “natural” auxiliary metric pμν into the present theory affords a specific bimetric structure to the fðR; Lm; VÞ model. In addition to the physical spacetime metric gμν, through which the gravitational observables manifest, there is the auxiliary metric upon which the mathematical edifice of the theory is best supported. This structure is analogous to EiBI gravity wherein there also exists a natural bimetric arrangement [25,26]. In the present theory, however, unlike the minimal nature of EiBI theory, the gravitational Lagrangian has built into it a direct coupling between the matter fields and the auxiliary metric via the scalar curvature R and the matter-curvature scalar V≡ RμνTμν. This coupling appears through the explicit dependence of the independent connection (15) on pμν and its inverse. Such a coupling suggests that there is some physical nature tied to the auxiliary metric. But because all physical observables manifest via the spacetime metric, the nonminimal coupling suggest a general link between the auxiliary and spacetime metrics. Obviously, the particulars of this link cannot be properly realized until the details of the nonminimal coupling are known (which necessitates specifying a particular function f). However, a general relationship can be drawn.

A natural link to proffer is that of a conformal relationship, in which pμν ¼ Θ2gμν for some real-valued, smooth function Θ defined on M. This approach, however, is consistent only for specific matter sectors [34]. Thus, conformality between pμν and gμν fails to capture the general framework we seek. A more general approach,

PHYS. REV. D 99, 124027 (2019)

again analogous to EiBI gravity, is to introduce a differentiable deformation matrix Ωμν satisfying

pμν ¼ gμλΩλν:

ð27Þ

In matrix notation, this reads p ¼ gΩ so that p−1 ¼ Ω−1g−1. Direct comparison to Eq. (14) reveals that

Ω−1 ¼ p1ﬃﬃﬃﬃ ðfRI þ fVg−1TÞ;

ð28Þ

Ω

where I is the identity matrix and Ω ≡ detðΩÞ follows from Eq. (16). It is now an algebraic problem to solve for Ω and hence pμν, explicitly, following the specification of the matter Lagrangian and the fðR; Lm; VÞ model of interest. One subsequently obtains the form of the connection and

related curvature terms for the specific theory, and all that

remains to resolve a given problem is the differential

equations (10). An example of this procedure, in the

context of EiBI gravity, can be found in Ref. [26].

C. Likeness to other f theories

The Palatini fðR; Lm; VÞ formalism contains as special cases the Palatini fðRÞ and fðR; LmÞ theories but not in general the Palatini fðR; TÞ and fðR; T; VÞ theories.

Evidently, Palatini fðR; Lm; VÞ and fðR; T; VÞ gravity correspond only when Lm ¼ T, which is a hefty constraint

by which most matter fields do not abide [35]. That said, for

matter fields with a vanishing energy-momentum trace

(such as electromagnetic fields), the fðR; Lm; VÞ model clearly contains the fðR; T; VÞ model. We say that Palatini

fðR; Lm; VÞ and fðR; T; VÞ are circumstantially equivalent

theories of gravity since their equivalence is such that it holds

only for specific matter fields (this notion is made more

precise in Sec. VI). There is, however, a simple procedure to

obtain Palatini fðR; T; VÞ gravity from the fðR; Lm; VÞ theory for arbitrary matter sectors: merely replace the fLΞμν

term in the field equations (10) by fT ∂∂gTμν, and continue on that way. Since in the Palatini formalism the trace T ≡ Tμμ is

independent of the independent connection, its incorporation

into the function f will not affect Eq. (13). In this respect,

most results derived herein afford similar mathematical

structure to Palatini fðR; T; VÞ gravity, up to the replace-

ment

of

all

fLΞμν

terms

with

fT

∂T ∂ gμν

terms

and

the

subsequent manipulations of those terms. Evidently, the

exception to this rule is those results which utilize, in a

nontrivial manner, the full entourage of dependencies in the

fðR; Lm; VÞ model, such as the present theory’s circum-

stantial equivalence to EiBI gravity (see Sec. VI).

IV. PROPERTIES OF THE f (R;Lm;V) FIELD EQUATIONS

Here, we shall consider various properties of the fðR; Lm; VÞ field equations, including their conservation

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MATTHEW S. FOX

PHYS. REV. D 99, 124027 (2019)

equation, their effect on the motion of massive test particles, and their weak-field limit.

A. Conservation equation

In fðR; Lm; VÞ gravity, matter is nonminimally coupled to curvature. Hence, the covariant divergence of the energymomentum tensor is not necessarily zero. In this section, we derive an explicit expression for such nonconservation of the energy-momentum tensor. In what follows, we use tildes to decorate tensors which have been transvected by the auxiliary metric pμν.

We begin with the field equations (12) in the form

G˜ μν ¼ f1R κΣ˜ μν þ 12 fðΩ−1Þμν − 12 δμνR˜ ; ð29Þ

where G˜ μν is the Einstein tensor raised by pμν and ðΩ−1Þμν is Ω−1 in index notation. Using the definition (14), one finds

rﬃﬃﬃ Σ˜ μν ¼ gðfRΣμν þ fV TμλΣλνÞ;

p

ð30aÞ

rﬃﬃﬃ

R˜ ¼

g ðRfR þ VfVÞ:

p

ð30bÞ

The condition we seek follows from the contracted Bianchi identities, ∇ðμpÞG˜ μν ¼ 0. All that remains is a straightforward problem in algebra: expand Σμν in

Eq. (30a) using its definition (11), then isolate the covariant divergence of Tμν. We find

rﬃﬃﬃ rﬃﬃﬃ

κ∇ðμpÞTμν ¼

p ∇ðμpÞ g

pg fLΞμν þ fV Πμν − ffRV TμλΣλν − 2fffRV Tμν

rﬃﬃﬃ

rﬃﬃﬃ

− 1 ∂ν

g ðf − RfR − VfVÞ

− κTμν∂μ

g :

ð31Þ

2

p

p

Alternatively, this nonconservation can be expressed in terms of the connection ∇ðgÞ compatible with the

spacetime metric gμν. The relationship between the covariant derivatives ∇ðpÞ (that defined with the independent connection of the auxiliary metric) and ∇ðgÞ is

the following:

∇ðμpÞTμν ¼ ∇ðμgÞTμν þ CμμλTλμ − CλμνTμλ;

ð32Þ

where

Cαμν ¼ 1 pασð∇ðμgÞpσν þ ∇ðνgÞpμσ − ∇ðσgÞpμνÞ: 2

ð33Þ

The metric/auxiliary metric relationship (27), the compatibility of gμν with ∇ðgÞ, and the symmetry property of the auxiliary metric imply the coefficients can be expressed

in a form that is independent of pμν:

Cαμν ¼ 12 ðΩ−1Þασð∇ðμgÞΩσν þ ∇ðνgÞΩσμÞ

− 1 gμλðΩ−1Þασ∇σ Ωλν:

2

ðgÞ

ð34Þ

We note that any covariant derivative with respect to Ωμν can be replaced by a derivative with respect to ðΩ−1Þμν, as their inverse relationship implies

ðΩ−1Þλν∇ðσgÞΩμλ þ Ωμλ∇ðσgÞðΩ−1Þλν ¼ 0: ð35Þ

With this in mind, the coefficients (34) become

Cαμν ¼ − 12 Ωλν∇ðμgÞðΩ−1Þαλ − 12 Ωλμ∇ðνgÞðΩ−1Þαλ

þ 1 gμλðΩ−1ÞασΩγνΩλϵ∇σ ðΩ−1Þϵλ;

2

ðgÞ

ð36Þ

and the nonconservation of the energy-momentum tensor turns out to be

rﬃﬃﬃ rﬃﬃﬃ

∇ðμgÞTμν ¼ 1 p ∇ðμpÞ g fLΞμν þ fV Πμν − fV TμλΣλν − ffV Tμν

κg

p

rﬃﬃﬃ

fR 2fR

− 1 ∂ν

g ½f − RfR − VfV

þ CλμνTμλ:

ð37Þ

2

p

Clearly, the curvature and energy-momentum depend-

ences in Eq. (37) and the energy-momentum dependence of ðΩ−1Þμν restrict these formulas from simplifying much beyond what is given here. We emphasize, therefore, that

∇ðμgÞTμν does not in general vanish. Hence, the energymomentum tensor in Palatini fðR; Lm; VÞ gravity is in general not conserved. On the other hand, in Sec. VI, we

shall indirectly derive two nontrivial functions of

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PHYS. REV. D 99, 124027 (2019)

fðR; Lm; VÞ for which the covariant divergence of specific but nontrivial Tμν necessarily vanish, hence conserving the energy-momentum tensor. That said, this conservation will

not be obvious at the level of Eq. (37), though it will

nevertheless be true. Finally, we note that for the EinsteinHilbert model fðR; Lm; VÞ ¼ R − 2Λ the conservation of Tμν is restored, as desired.

Here, the left-hand side coincides with the well-known identity

uμ∇ðμgÞuν ¼ d2xν þ Γν

dxα dxβ :

ð39Þ

ds2 αβ ds ds

B. Motion of test particles

For clarity, we denote by Δν the right-hand side of Eq. (37). Then, ∇ðμgÞTμν ¼ Δν. For the case of a perfect fluid, for which Tμν ¼ ðρ þ PÞuμuν þ Pgμν, it is straightforward to show, using the constraint from the conservation of the matter fluid current, ∇ðμgÞðρuμÞ ¼ 0, that

uμ∇ðμgÞuν ¼ Δν − uν∇ðμgÞðPuμÞ − ∂νP :

ð38Þ

Pþρ

Therefore, Eq. (38) is the equation of motion for particles in the presence of an isotropic pressure P. Absent this pressure, the equation reduces to

d2xν þ Γν dxα dxβ ¼ fν;

ð40Þ

ds2 αβ ds ds

where the extra force fν ¼ ρ−1ΔνðP¼0Þ with

rﬃﬃﬃ rﬃﬃﬃ

rﬃﬃﬃ

ΔðνP¼0Þ ¼ 1 p ∇ðμpÞ − g ρ fLuμuν þ fV ρ gð½Rσμuν þ Rσνuμuσ þ Rαβuαuβ½δμv − 2uμuν − RuμuνÞ

κg

p2

2p

rﬃﬃﬃ gf

ρ2

f

þ p κVf

½κ − fLuμuν þ

V

2

½Ruμuν

−

uμRανuα

þ

4Rαβ uα uβ uμ uν

rﬃﬃﬃ R

rﬃﬃﬃ

− pg 2fffRV ρuμuν − 12 ∂ν pg½f − RfR − VfV þ ρCαβνuβuα ð41Þ

[see Eqs. (25) and (26) to derive this]. Since ΔνðP¼0Þ is in general nonzero, the extra force fν is in general nonzero. Hence, test particles in fðR; Lm; VÞ gravity do not in general obey the geodesic equation. In other words, test particles traverse geodesics of gμν if and only if ΔμðP¼0Þ ¼ 0.

C. Newtonian limit

In the weak-field regime, we consider the gravitational effect of nonrelativistic dust, for which Tμν ¼ ρuμuν where uμ ¼ ð∂0Þμ is the rest frame 4-velocity and ρ is the dust’s energy density [38]. We shall linearize the fðR; Lm; VÞ equations by keeping terms linear in ρ and in the perturbations introduced below. To facilitate the coming analysis, we adopt the following notation.

Let γμν and γˆμν be smooth 2-forms (soon to be perturbations). Further, let A and B be mathematical objects composed, in some acceptable fashion, of the objects ρ; γμν, and γˆμν. Then, by A ≪ B, we shall mean A is first order (linear) in at least one of ρ, γμν, or γˆμν, while B is zeroth order in all. Moreover, by A ≅ B, we shall mean A ¼ B up to at least linear corrections in all ρ, γμν, and γˆμν. Finally, by A ∼ B, we shall mean A and B are of the same order in ρ, γμν, or γˆμν, but not necessarily equal (thus, A ≅ B implies A ∼ B).

Consider the metric/auxiliary metric relation posited in Eq. (27). This establishes that any perturbation δpμν upon pμν satisfies

δpμν ¼ gμλδΩλν þ Ωλνδgμλ:

ð42Þ

Specifically, we shall consider perturbations δpμν ¼ γˆμν

and δgμν ¼ γμν upon a Minkowski background ημν.

Then, pμν ¼ ημν þ γˆμν and gμν ¼ ημν þ γμν such that

γˆμν; γμν ≪ ημν. Here, pμν and gμν are related by Eq. (27),

and furthermore, the perturbations γˆμν and γμν satisfy Eq. (42). Together, these imply ημλΩλν − ημν ≪ ημν, which is possible if and only if Ωμν ≅ δμν þ kμν, where kμν ≪ δμν.

In deriving this, one must also assume that the deformation

matrix reacts smoothly to slight perturbations upon the

Minkowski background, i.e., that gμλδΩλν ≪ ημν, which necessitates ημλδΩλν ∼ γμν and γμλδΩλν ≅ 0.

Transcribed to matrix notation, we have Ω ≅ I þ k.

Thus, Ω−1 ≅ I − k, to which we shall directly compare

Eq. (28). Since the tensor Tμν → g−1T is in general

pdifﬃﬃfﬃﬃerent from the idpenﬃﬃﬃtﬃity matrix,piﬃtﬃﬃﬃmust be that fR ≅

Ω and fVg−1T ≅

Ωk, where

Ω

≅

1

þ

1 2

Tr

ðkÞ

.

But

since TrðkÞk ≅ 0, with 0 the zero matrix, we simply have

k ≅ fVg−1T. Together, these results yield both pμν and pμν

to the desired first-order precision:

124027-7

MATTHEW S. FOX

pμν ≅ ημν þ γμν þ fV Tμν;

ð43aÞ

pμν ≅ ημν − γμν − fV Tμν;

ð43bÞ

where Tμν ¼ ημαηνβTαβ and γμν ¼ ημαηνβγαβ. We see from Eq. (43a) that

γˆμν ≅ γμν þ fV Tμν

ð44Þ

and similarly from Eq. (43b) that γˆμν ≅ γμν þ fVTμν. Hence, the connection coefficients (15) are, to linear order in γˆμν,

Γαμν ≅ 1 ηασð∂μγˆσν þ ∂νγˆμσ − ∂σγˆμνÞ;

ð45Þ

2

which yields the Ricci tensor

Rμν ≅ ∂σ∂ðνγˆμÞσ − 1 ∂σ∂σγˆμν − 1 ∂μ∂νγˆ; ð46Þ

2

2

where γˆ ≡ γˆμμ. Note that Eq. (46) is entirely of linear order in γˆμν. Hence, V ¼ RμνTμν ≅ 0 since Tμν is linear in ρ. Moreover, fVΠμν ≅ 0 since δTαβ=δgμν ∼ ρ for dust [see Eq. (24)], and Rμν ∼ γˆμν. We shall impose the Lorenz gauge ∂σγˆμσ ¼ 0 so that the fðR; Lm; VÞ field equations, to linear order in γˆμν and ρ, bear the form

− 1 ∂σ∂σγˆμν − 1 ∂μ∂νγˆ − 1 fgμν ≅ κTμν − fLΞμν: ð47Þ

2

2

2

To obtain the matter tensor, we necessarily take Lm ¼ −ρ

for the matter Lagrangian of the pressureless dust. Hence,

from

Eq.

(24),

Ξμν

¼

1 2

ρðuμuν

−

ημν

−

γμνÞ.

Here,

ρ

is

the

leading-order correction from the matter sector; thus, the

product ργμν must be regarded as a second-order correction.

This

implies

that,

to

first

order,

Ξμν

≅

1 2

ρðuμuν

−

ημνÞ

and

hence

Ξ

≡

Ξμμ

≅

−

5 2

ρ.

It

follows

that

the

trace

of

the

field

equations (10) is, to first order,

5

fRR − 2f ≅ ρ 2 fL − κ :

ð48Þ

Note that Πμμ ≅ 0 since Πμν ≅ 0. Thus, Πμμ is absent from Eq. (48) in this approximation. Note also that the already first-order corrections ρ and R force f to be at least a firstorder correction; hence, we can rewrite Eq. (47) as

− 1 ∂σ∂σγˆμν − 1 ∂μ∂νγˆ − 1 fημν

2

2

2

1

≅ κρuμuν − 2 ρfLðuμuν − ημνÞ:

ð49Þ

The 00 component of this equation encodes the weak-field dynamics in which we are interested. Since the spacetime is

PHYS. REV. D 99, 124027 (2019)

assumed static, the time derivatives vanish, leaving the expression

1

1

− 2 Δγˆ00 ≅ κρ − 2 f − ρfL;

ð50Þ

where Δ ≡ ∇2 is the Laplacian operator. Using Eq. (44)

and

the

definition

of

the

Newtonian

potential,

Φ

≡

−

γ00 4

,

we

obtain the modified Poisson equation in fðR; Lm; VÞ

gravity:

1 11

1

ΔΦ ≅ 2 κρ − 4 f − 2 ρfL þ 4 ΔðfVρÞ: ð51Þ

Since f is implicitly a function of Tμν, and hence of ρ, the

quantity

1 2

κρ

−

1 4

f

−

1 2

ρfL

acts

as

a

sort

of

effective

density

1 2

κρ¯ .

In

these

terms,

Eq.

(51)

reads

11

ΔΦ ≅ 2 κρ¯ þ 4 ΔðfVρÞ:

ð52Þ

This modification to Poisson’s equation is formally identical to those in both EiBI and Palatini fðR; TÞ gravity (see Refs. [26,28], respectively). Consequently, we expect all these theories to afford similar nonrelativistic phenomenology.

V. SOME APPLICATIONS

The weak-field equations considered above disclosed a relationship between fðR; Lm; VÞ and other theories of gravity in the Newtonian regime. In this section, we derive the field equations governing the response of fðR; Lm; VÞ gravity in other regimes, in particular the electromagnetic and scalar field sectors.

A. Electromagnetic fields

Consider next the traditional linear electrodynamics

(LED) of Maxwell for which the matter Lagrangian is

LðLEDÞ

¼

−

1 16π

FμνFμν,

where

Fμν ≡ ∂μAν − ∂νAμ

is

the

Faraday tensor. The LED matter tensor follows quickly

from

its

definition

(8),

ΞðμLνEDÞ

¼

−

1 8π

FμλFλν.

Alternatively,

using Eq. (18),

ΞðμLνEDÞ ¼ 1 ðLðLEDÞgμν − TðμLνEDÞÞ: 2

ð53Þ

Similarly, from the symmetry of the Ricci tensor and Eqs. (21) and (53), the LED matter-curvature tensor turns out to be

ΠðμLνEDÞ ¼ 2RλðμTðνLÞλEDÞ − RμνLðLEDÞ − 81π RFμλFλν þ 1 RαβFμβFαν: 4π

ð54Þ

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PHYS. REV. D 99, 124027 (2019)

The LED effective energy-momentum tensor, ΣðμLνEDÞ, then follows trivially from its definition (11).

The matter-curvature couplings in Eq. (54) are very much unlike, for instance, Palatini fðR; TÞ theory, in which TðLEDÞ ¼ gμνTðμLνEDÞ ¼ 0. Hence, the fðR; TÞ models (whether Palatini or not) respond to linear electromagnetic fields as an fðRÞ model. This is evidently not the case for the present theory, in which there are nontrivial couplings between curvature and matter terms, all of which have the potential to invite new gravitational electrodynamic behavior. We note that in the Palatini fðR; T; VÞ model all the fV couplings in Eq. (54) persist; therefore, even with a vanishing trace [making the gravitational response a Palatini fðR; VÞ theory], there remain new and nontrivial corrections to the linear electrodynamics.

However, it is well known that the linear electrodynamics in vacuo are only an approximation to the full electrodynamic theory. General relativity, for example, demands a gravitational coupling between electromagnetic fields, which affords nonlinear electrodynamic behavior. That said, more considerable nonlinearity arises from quantum field effects, such as vacuum polarization [39]. It is therefore of interest to also derive the fðR; Lm; VÞ field

equations associated with a general set of nonlinear electrodynamic (NED) theories. To this end, we set the matter sector action to

Z

SðNEDÞ ¼ 1 d4xpﬃ−ﬃﬃﬃgﬃﬃχðI; JÞ;

ð55Þ

8π

where χ is a well-behaved function of the algebraic

invariants I ≡ 12FμνFμν and J ≡ Fμνð⋆FÞμν. Here, ð⋆FÞμν ¼

1 2

ð−gÞ−12

ϵμναβ

F

αβ

is

the

Hodge

dual

of

the

Faraday

tensor,

with ϵμναβ denoting the Levi-Civita symbol. We note that I

and J are the unique algebraic invariants constructible from

Fμν and gμν [40,41] and also that the choice χðI; JÞ ¼ −I

corresponds to the LED theory considered above.

With

1 8π

χ

ðI

;

J

Þ

as

the

NED

matter

Lagrangian,

and

defining

χI

≡

∂χ ∂I

and

χJ

≡

∂∂Jχ,

we

find

ΞðμNνEDÞ ¼ 1 χIFμλFλν þ 1 χJJgμν :

8π

2

ð56Þ

It then follows from Eq. (21) and the NED equivalent of Eq. (53) that

ΠðμNνEDÞ ¼ 2RλðμTðνNÞλEDÞ þ 81π ðχJJ − χÞRμν þ 41π

1 RχI − RαβFαλFλβχII − 1 RχJIJ2

2

2

FμλFλν

− 1 RαβFαλFλβχIJJ þ 1 RχJJJ2 gμν − 1 RαβχIFαμFνβ:

8π

2

4π

ð57Þ

As before, ΣðμNνEDÞ then follows from its definition (11), and TðμNνEDÞ follows from the NED equivalent of Eq. (53). Note that, as expected, upon fixing χðI; JÞ ¼ −I, Eq. (57) reduces to Eq. (54). As in the LED case, these field

equations have in them nontrivial matter-curvature cou-

plings which again bear new possibilities for NED gravi-

tational dynamics, such as in studies of nonsingular black

holes. We also note that these equations again differ

drastically in their matter-curvature couplings from the field equations for NED in fðR; TÞ gravity (see, e.g., Ref. [28]). This much is evident from the fV coupling terms, which persist only in the fðR; Lm; VÞ framework.

B. Canonical scalar fields

Scalar fields comprise another set of generic matter fields

for which fðR; Lm; VÞ gravity admits new and nontrivial

dynamics. Here, we shall consider the effect of a real-

valued scalar field ϕ in a potential UðϕÞ, the Lagrangian

density

of

which

bears

the

form

LðϕÞ

¼

−

1 2

∂λϕ∂λϕ

−

UðϕÞ.

One

shall

find

ΞðμϕνÞ

¼

−

1 2

∂μϕ∂νϕ

and,

from

Eq. (20),

ΠðμϕνÞ ¼ − 1 R∂μϕ∂νϕ þ ∂λϕ∂ðμϕRλνÞ þ RμνLðϕÞ: 2

ð58Þ

Hence,

κΣðμϕνÞ ¼ κTðμϕνÞ þ 1 ðfL þ RfV Þ∂μϕ∂νϕ − fV LðϕÞRμν 2

− fV ∂λϕ∂ðμϕRλνÞ:

ð59Þ

As with the electromagnetic field, specifying particular fðR; Lm; VÞ functions and solving the associated field equations will conceivably yield new nonminimal correc-

tions to ordinary GR problems, which brings about new possibilities. For example, as posited for Palatini fðR; TÞ gravity [28], free [UðϕÞ ¼ 0] geonic solutions of the kind in EiBI gravity [42] are conceivable in the present theory.

VI. COMPATIBILITY WITH EiBI GRAVITY

In this section, we shall investigate the conditions under which the fðR; Lm; VÞ paradigm encapsulates the EiBI theory. We shall denote by fBI any fðR; Lm; VÞ function that does this. To begin, it is imperative that we be precise

124027-9

MATTHEW S. FOX

with the meaning of “one gravitational theory corresponding to another.”

Let A and B be two Palatini theories of gravity defined on a world manifold M, and let Ψ be a matter field on M. Further, let gðμAν Þ and gðμBνÞ be the solutions generated from A and B, respectively, in response to Ψ, and ∇ðAÞ and ∇ðBÞ be the derivative operators of A and B, respectively, defined on M. On one hand, we say A and B are equivalent if, for all Ψ, (i) ∇ðσAÞξμ ¼ ∇ðσBÞξμ for all vectors ξμ defined on some tangent space in the tangent bundle of M and (ii) gðμAν Þ ¼ Θ2gðμBνÞ for some real-valued, smooth conformal factor Θ defined on M. Evidently, condition i ensures that both theories measure the same intrinsic curvature of M,

that both have the same notion of transport, and so forth,

while condition ii establishes that the gravitational dynam-

ics of the two theories are the same (since they afford the

same solution, up to a conformal factor, for a given matter sector Ψ). On the other hand, we say A and B are

circumstantially equivalent if conditions i and ii hold only for particular Ψ. Indeed, we shall prove in this section that EiBI and fðR; Lm; VÞ are circumstantially equivalent theories of gravity; in particular, that condition i shall hold well for all Ψ but that condition ii shall hold well only for specific Ψ.

The (Palatini) EiBI action bears the form [26]

1 Z hqﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃi SBI½g; Γ; Ψ ¼ 2κϵ d4x jgμν þ ϵRμνðΓÞj − λ −g

þ Sm½g; Ψ;

ð60Þ

where ϵ is a coupling parameter, λ is related to the cosmological constant Λ by λ ¼ 1 þ ϵΛ, and the vertical bars denote the absolute value of the determinant. The reader is referred to Refs. [25,26] for details on the variation. The field equations are

qμν ¼ gμν þ ϵRμν; pﬃ−ﬃﬃﬃqﬃﬃqμν ¼ pﬃ−ﬃﬃﬃgﬃﬃðλgμν − κϵTμνÞ;

ð61aÞ ð61bÞ

where q is the determinant of the auxiliary metric qμν and qμν satisfies both qμλqλν ¼ δμν and ∇ðσBIÞðpﬃ−ﬃﬃﬃqﬃﬃqμνÞ ¼ 0, where ∇ðBIÞ is the derivative operator associated with the Palatini EiBI theory. Hence, ∇ðσBIÞqμν ¼ 0.

We note that λ ≠ 0 (equivalently pΛﬃﬃ≠ﬃﬃﬃﬃ −ϵ−1), for otherwise Eq. (61b) implies that in vacuo −qqμν ¼ 0, which is nonsense. Moreover, with Tμν ¼ 0 and λ ≠ 1, the solutions from the two theories do not coincide; EiBI affords a de Sitter or anti-de Sitter universe, while fðR; Lm; VÞ outputs Minkowski space. In speaking of a possible equivalence

between the theories, it is natural to demand that at least

the vacuum solutions correspond. To this end, we shall hereafter fix λ ¼ 1, making EiBI Minkowskian in vacuo.

PHYS. REV. D 99, 124027 (2019)

Note that there is no loss of generality in doing this. Should

one wish to append a cosmological constant to either

theory, one would simply do so via the matter sector. We have merely “tared” the two theories at the level of their

vacuum solutions. As previously defined, ∇ðpÞ is the derivative operator

associated with the Palatini fðR; Lm; VÞ theory. Thus, for an EiBI=fðR; Lm; VÞ equivalence to exist, condition i demands that ∇ðσBIÞξμ ¼ ∇ðσpÞξμ for all smooth vectors ξμ. This implies, in particular, that

∇ðσBIÞqμν ¼ ∇ðσpÞqμν ¼ ∇ðσpÞpμν ¼ 0:

ð62Þ

The connections of both fðR; Lm; VÞ and EiBI gravity are torsion free. Hence, as required by the fundamental theorem of Riemannian geometry, Eq. (62) holds well if apnﬃdﬃﬃﬃﬃﬃonly ipf ﬃﬃpﬃﬃﬃμﬃν ¼ qμν, which is true if and only if

−qqμν ¼ −ppμν. Therefore, the definitions (14) and

(61), together with condition ii, i.e., the requisite conformal relationship gðμfνÞ ¼ Θ2gðμBνIÞ [gðμfνÞ being the solution from the fðR; Lm; VÞ theory], imply (with λ ¼ 1)

ð1 − Θ2fRÞgμðBνIÞ − ðκϵTμðBνIÞ þ Θ4fV TμðfνÞÞ ¼ 0; ð63Þ

where TμðBνIÞ and TμðfνÞ are the energy-momentum tensors of the EiBI and fðR; Lm; VÞ theories, respectively, each raised by their respective metric. We cannot impose a priori that these energy-momentum tensors be the same since they are functions of their respective metrics. We can impose, however, that the two parenthetical terms in Eq. (63) vanish separately. This is necessarily the case if we seek generality in the matter sector, as, for instance, Eq. (63) holds in vacuo if and only if the two parenthetical terms vanish separately. Consequently, Θ2fR ¼ 1, and κϵTμðBνIÞ ¼ Θ4fVTμðfνÞ. Differentiating the former with respect to R demands that fR is constant and hence that the conformal factor Θ is constant. The same is true for the latter, where differentiation upon V implies fV is constant.

These results indicate two things. First, TBμνI ∝ TðμfνÞ. For equivalence between the two theories to hold, this constant proportionality must hold in general, for arbitrary choices of the matter sector. But since the conformal transformation properties of the energy-momentum tensor depend on the matter sector, constant proportionality is guaranteed only with exact equality between the metrics, i.e., with Θ2 ¼ 1 and hence fV ¼ −κϵ. Second, the vanishing of the second derivatives fRR and fVV implies that the EiBI= fðR; Lm; VÞ function fBI is of the form fBIðR; Lm; VÞ ¼ f1ðRÞ þ hðLmÞ þ f2ðVÞ for well-behaved functions f1, h, and f2. In fact, with the conformal factor set at unity and the energy-momentum tensors identical, we simply have from Eq. (63) that f1ðRÞ ¼ R and f2ðVÞ ¼ −κϵV. Hence, from Eq. (10), the fBI field equations bear the form

124027-10

Palatini f (R;Lm;RμνTμν) gravity and its Born-Infeld semblance

Matthew S. Fox*

Department of Physics, Harvey Mudd College, Claremont, California 91711, USA

(Received 2 December 2018; published 19 June 2019)

We investigate Palatini fðR; Lm; RμνTμνÞ modified theories of gravity. As such, the metric and affine connection are treated as independent dynamical fields, and the gravitational Lagrangian is made a function of the Ricci scalar R, the matter Lagrangian density Lm, and a “matter-curvature scalar” RμνTμν. The field equations and the equations of motion for massive test particles are derived, and we find that the independent connection can be expressed as the Levi-Civita connection of an auxiliary, energy momentum–dependent metric that is related to the physical metric by a matrix transformation. Similar to metric fðR; T; RμνTμνÞ gravity, the field equations impose the nonconservation of the energymomentum tensor, leading to the appearance of an extra force on massive test particles. We obtain the explicit form of the field equations for massive test particles in the case of a perfect fluid and an expression for the extra force. The nontrivial modifications to scalar fields and both linear and nonlinear electrodynamics are also considered. Finally, we detail the conditions under which the present theory is equivalent to the Eddington-inspired Born-Infeld theory of gravity.

DOI: 10.1103/PhysRevD.99.124027

I. INTRODUCTION

Observations of the cosmic microwave background (CMB) [1] and direct measurements of the light curves from several hundred type Ia supernovae [2] suggest that the Universe is currently undergoing a phase of late-time, accelerated expansion. While the physics underlying this phenomenon remain unsettled, at least one thing is certain: the acceleration is either a trait of the gravitational interaction itself, or it is a gravitational manifestation of something else (dark energy). By and large, the copious models of the former type derive from revisions to the Einstein-Hilbert action

Z

1 d4xpﬃ−ﬃﬃﬃgﬃﬃR;

ð1Þ

SEH ¼ 2κE

where κE is the Einstein constant, R is the Ricci scalar, and g is the determinant of the spacetime metric gμν.

Among the most straightforward generalizations of SEH are the fðRÞ models. These constitute a class of higherorder gravity theories in which SEH is restyled with terms of higher degree in the scalar curvature. Indeed, the mystery of

cosmic expansion can be unraveled in this approach [3].

Some models [4] even appear to avoid the fatal instabilities

and acute weak-field constraints that bar many other proposals [5]. Incidentally, the lure of fðRÞ gravity is broader in application than to just cosmic speed-up. For

instance, theories with higher-order curvature invariants

*[email protected]

show promise as effective first-order approximations to

quantum gravity and can encourage quantum and gravita-

tional fields to be well behaved in the ultraviolet regions neighboring curvature singularities [6]. Further fðRÞ phenomenology has been extensively surveyed in the

literature [7]. Interesting extensions of the fðRÞ models are those

theories which include in the action an explicit nonminimal

coupling between matter and curvature invariants. A notable subset of these models is so-called fðR; LmÞ gravity (Lm being the matter Lagrangian density) introduced by Bertolami et al. in Ref. [8]. Their model was

linear in the nonminimal coupling, which prompted the author of Ref. [9] to study the maximal extension of SEH in which R and Lm are coupled arbitrarily. Cosmological and astrophysical phenomena have been studied in various fðR; LmÞ frameworks [10], in addition to more general studies into the properties of the theory itself [11].

Generally speaking, nonminimal theories such as fðR; LmÞ gravity do not admit chart transitions that locally transform away the influence of a gravitational field on

matter [12]. In turn, the covariant divergence of the energy-

momentum tensor is generally nonzero, the motion of test

particles is generally nongeodesic (due to the presence of

an extra force orthogonal to the 4-velocity [8]), and thus the

equivalence principle (EP) is generally violated. Hence,

these theories are stringently constrained by tests of the EP.

It is important to note, however, that a violation of the EP

does not in principle disqualify the specific theory [13]. A set of models related to fðR; LmÞ gravity derives from

the case in which the functional dependence on Lm

2470-0010=2019=99(12)=124027(13)

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© 2019 American Physical Society

MATTHEW S. FOX

manifests via a dependence on the trace T of the energymomentum tensor. These so-called fðR; TÞ models have drawn significant attention and were explicitly introduced by Harko et al. in Ref. [14]. However, Poplawski [15] was first to consider a model in which the cosmological constant is a function of T, which is considered a relativistically covariant model for interacting dark energy and which is evidently a subset of the fðR; TÞ theory. We note that explicit dependences on T may be induced by quantum effects (e.g., conformal anomalies) or exotic imperfect fluids. The reader is referred to the review [16] for additional fðR; LmÞ and fðR; TÞ phenomenology.

Further extensions to the fðR; LmÞ and fðR; TÞ theories were proposed in Refs. [17,18], in which terms of the form RμνTμν, where Rμν is the Ricci tensor, were incorporated into the fðR; TÞ Lagrangian. Instances of this coupling are known to arise in Born-Infeld models of gravity [19] when one Taylor expands the Lagrangian. The cosmological implications of these so-called fðR; T; RμνTμνÞ gravity theories were surveyed in Refs. [17,18,20], and the criterion to circumvent the Dolgov-Kawasaki instability [21] can be found in Ref. [18]. Moreover, energy conditions and thermodynamic laws in fðR; T; RμνTμνÞ gravity were considered in Ref. [22]. Finally, it is known that metric fðR; T; RμνTμνÞ gravity acquires ghostlike instabilities due to the additional RμνTμν coupling [23] and that these instabilities can be avoided with a Palatini or metric-affine variation [24].

The appearance of the RμνTμν coupling in Born-Infeld gravity is the chief motivation for our study. The BornInfeld models themselves, akin to Born-Infeld electromagnetism, modify the determinantal structure of SEH. Among the many Born-Infeld models, a prominent one is the Eddington-inspired Born-Infeld (EiBI) theory proposed in Ref. [25]. Whereas many fðRÞ models differ from general relativity (GR) even in vacuum, EiBI does not. Yet in ultraviolet regions, such as near cosmological singularities, EiBI gravity is characterized by curing the geometrical divergences plaguing GR [25]. See Ref. [26] for a recent review on Born-Infeld modifications to gravity.

Importantly, in all of these theories, independent of the details of the modification, one must ultimately choose between two ostensibly similar methods for varying the action: either one treats the metric as the sole dynamical entity and fixes a priori the connection to be the LeviCivita connection of gμν (the metric formalism), or one regards the metric and affine connections as independent dynamical structures (the metric-affine or Palatini formalisms, depending on whether matter couples to the connection or not, respectively). In GR, the distinction is superfluous as they both lead to the same physics. However, in general, nearly all the aforementioned theories forecast different physics depending on whether the metric and affine structures are handled independently or not. In fact, in some theories, such as EiBI gravity [25] and

PHYS. REV. D 99, 124027 (2019)

(already mentioned) fðR; T; RμνTμνÞ gravity [24], the

Palatini formalism will remove ghostlike instabilities that

otherwise afflict their metric counterparts. Whether the

affine connection is determined by the metric degrees of

freedom (d.o.f.) or not is a truly fundamental question and

demands experimental investigation.

Though matter couplings to the connection may arise

due to quantum gravitational corrections, we shall ignore

that possibility here, and so we exercise the Palatini

formalism. Studies of Palatini fðRÞ and fðR; TÞ models

can be found in Refs. [27] and [28,29], respectively, and

more general actions varied a` la Palatini and metric-affine,

including the role of torsion, can be found in Ref. [24]. To

the best of our knowledge, no studies of pure Palatini fðR; T; RμνTμνÞ or Palatini fðR; Lm; RμνTμνÞ gravity

have yet been completed, though indirect pursuits exist

(see, e.g., Ref. [24]). In this paper, we shall investigate Palatini fðR; Lm; RμνTμνÞ gravity, from which Palatini fðR; T; RμνTμνÞ gravity follows after a simple modifica-

tion to the field equations. In addition to studying fðR; Lm; RμνTμνÞ gravity on its own, we ultimately seek

the conditions under which our theory corresponds to EiBI

gravity.

The present paper is structured as follows. In Sec. II,

we vary the fðR; Lm; RμνTμνÞ action a` la Palatini and derive the theory’s equations of motion and an explicit

form for the independent connection. In Sec. III, we survey the bimetric structure of fðR; Lm; RμνTμνÞ gravity in

addition to the nonminimal structure of the field equa-

tions. In Sec. IV, we explore various properties of the fðR; Lm; RμνTμνÞ field equations, including their non-

conservation equation, the nongeodesic motion of test

particles, the nature of the extra force, the weak-field limit,

and the modified Poisson equation. In Sec. V, we derive the fðR; Lm; RμνTμνÞ field equations for the cases of linear

and nonlinear electromagnetic fields as well as canonical

scalar fields. Finally, in Sec. VI, we derive the conditions under which the fðR; Lm; RμνTμνÞ model responds iden-

tically to the EiBI theory for specific matter sectors.

In this paper, we shall operate in a four-dimensional

spacetime

ðM

;

gμν

;

Γ

α μν

Þ

in

which

the

metric

gμν

and

connection Γαμν will be treated as independent dynamical

fields. We shall utilize the metric signature ð−; þ; þ; þÞ

and, where appropriate, adopt the natural system of units in

which c ¼ 8πG ¼ 1.

II. FIELD EQUATIONS OF f (R;Lm;RμνTμν) GRAVITY

The Ricci tensor can be defined solely in terms of the affine connection, and this underpins the Palatini and metric-affine formalisms. Explicitly, the Ricci tensor follows from the Riemann curvature tensor

Rαβμν ¼ ∂μΓανβ − ∂νΓαμβ þ ΓαμλΓλνβ − ΓανλΓλμβ ð2Þ

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PALATINI fðR; Lm; RμνTμνÞ GRAVITY AND ITS …

via the contraction RμνðΓÞ ≡ RαμανðΓÞ. Only now, one needs to invoke the metric to define the Ricci scalar Rðg; ΓÞ ≡ gμνRμνðΓÞ and a matter-curvature scalar Vðg; Γ; ΨÞ ≡ RμνðΓÞTμνðg; ΨÞ, where Tμν is the symmetric (Hilbert) energy-momentum tensor. As we shall see below

in Eq. (5), the energy-momentum tensor is constructed a` la

Palatini so that Tμν depends only on the metric and a set of matter fields Ψ. We note that the symmetry of gμν and Tμν imposes that only the symmetric part of the Ricci tensor enters into this theory’s action. This considerably simplifies the role of torsion in the theory and renders a separate

consideration for fermionic matter immaterial [24].

With all this in mind, the action considered in this work

bears the form

Z 1 d4xpﬃ−ﬃﬃﬃgﬃﬃfðR; L ; VÞ þ S ½g; Ψ; ð3Þ

S½g; Γ; Ψ ¼ 2κ

m

m

where κ is a coupling constant with suitable dimensions. Here, the matter Lagrangian density Lm, encoded in both the function fðR; Lm; VÞ ¼ fðR; Lm; RμνTμνÞ and the matter action

Z pﬃﬃﬃﬃﬃﬃ

−gL ½g; Ψ;

ð4Þ

Sm½g; Ψ ¼ d4x

m

is assumed to capture all matter fields Ψ present in M. Moreover, Lm determines the manifestly symmetric energy-momentum tensor

2 δðpﬃ−ﬃﬃﬃgﬃﬃLmÞ

Tμν ≡ − pﬃ−ﬃﬃﬃgﬃﬃ δgμν ;

ð5Þ

which again is independent of the affine connection in the

Palatini formulation. If we denote by δSg and δSΓ the variation of Eq. (3) with

respect to the metric and connection, respectively, then δS ¼ δSg þ δSΓ with

Z

δSg ¼ 1 d4xpﬃ−ﬃﬃﬃgﬃﬃ − 1 fgμν þ fRRμν þ fLΞμν

2κ

2

þ fV Πμν − κTμν δgμν

ð6Þ

and

1 Z pﬃﬃﬃﬃﬃﬃ

δRμν

δSΓ ¼ 2κ d4x −g ðfRgμν þ fV TμνÞ δΓλ δΓλαβ: ð7Þ

αβ

Here, we have introduced the definitions fR ≡ ∂Rf, fV ≡ ∂Vf; fL ≡ ∂Lmf as well as the manifestly symmetric matter and matter-curvature tensors

PHYS. REV. D 99, 124027 (2019)

Ξμν ≡ ∂∂Lgμmν ; ð8Þ

Πμν ≡ Rαβ δδTgμανβ ; ð9Þ

respectively. Since gμν and Γαμν are independent fields, δS ¼ 0 if and only if δSg and δSΓ vanish separately. In the case of the metric variation (6), δSg ¼ 0 implies

1 fRRμν − 2 fgμν ¼ κTμν − fLΞμν − fV Πμν: ð10Þ

This is the fðR; Lm; VÞ generalization of Einstein’s equation. Its properties shall be explored in the coming sections. We note here, however, that as a consequence of the nonminimal coupling there appear in Eq. (10) strict couplings between matter fields and curvature terms. This is very much unlike GR and other minimally coupled theories in which matter fields are wholly separable from curvature terms such that the field equations may be written in a “curvature = matter” type representation. Ultimately, however, writing the field equations in this way is more for physical tidiness and less for mathematical substance. Hence, the mathematical representation of these equations may as well be chosen such that it facilitates later computation. To this end, we define a curvature-dependent effective energy-momentum tensor by

Σμν ≡ Tμν − fκL Ξμν − fκV Πμν; ð11Þ

which refashions the field equations (10) into a form similar to those in Palatini fðRÞ gravity:

1

fRRμν − 2 fgμν ¼ κΣμν:

ð12Þ

The variation with respect to the connection takes more

care. We refer the reader to Ref. [24], in which a nearly complete derivation is given. One shall find that δSΓ ¼ 0 only if

∇ðσpÞ½pﬃ−ﬃﬃﬃgﬃﬃðfRgμν þ fVTμνÞ ¼ 0;

ð13Þ

where ∇ðpÞ is the derivative operator associated with the independent connection and which is manifestly distinct from ∇ðgÞ, the covariant derivative compatible with the spacetime metric gμν. The resemblance of Eq. (13) to the companion EiBI field equation will be studied in Sec. VI.

We note that Eq. (13) holds well even in the presence of torsion. This follows from this theory’s insensitivity to the projective d.o.f. in projective transformations of the independent connection, which ultimately derives from only the symmetric part of the Ricci tensor entering into the action (3). See Ref. [24] for details.

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MATTHEW S. FOX

Together, Eqs. (10) and (13) comprise the field equations for fðR; Lm; VÞ gravity. We see in Eq. (13) a natural auxiliary metric ingrained into this theory’s mathematical

structure, namely, a metric pμν of which the inverse, denoted pμν [30], satisfies

pﬃ−ﬃﬃﬃpﬃﬃpμν ¼ pﬃ−ﬃﬃﬃgﬃﬃðfRgμν þ fVTμνÞ;

ð14Þ

where p ≡ detðpμνÞ. Evidently, the symmetry of gμν and Tμν forces pμν (and hence pμν) to be symmetric. Moreover, pμν satisfies ∇ðσpÞðpﬃ−ﬃﬃﬃpﬃﬃpμνÞ ¼ 0 by construction, so pμν is compatible with ∇ðpÞ, provided the coefficients of the

independent connection are the Christoffel symbols in pμν,

Γαμν ¼ 1 pασð∂μpσν þ ∂νpμσ − ∂σpμνÞ: 2

ð15Þ

Consequently, the independent connection is the Levi-

Civita connection in the auxiliary metric pμν. Note also that the determinant p can be computed explicitly with Eq. (14) and the relation p ¼ det−1ðpμνÞ. One finds

p ¼ g2 det ðfRgμν þ fV TμνÞ:

ð16Þ

We shall apply these formulas to various physical phenomena in the coming sections. But first, we briefly comment on some general characteristics of the field equations.

III. REMARKS ON THE f ðR;Lm;VÞ FIELD EQUATIONS

As noted above, for theories in which couplings are minimal, the matter fields can in general be placed on one side of the theory’s field equation, and the symmetric part of the Ricci tensor will be given solely in terms of gμν. But for nonminimal theories, the matter fields are generally inseparable from the geometry terms, and the symmetric part of the Ricci tensor need not be given solely in terms of gμν. Such is the case for fðR; Lm; VÞ gravity, as made evident by the field equations (10) and (13). Other aspects of the present theory’s nonminimal character are addressed in this section.

A. Matter and matter-curvature tensors

The matter-curvature tensor Πμν is a hallmark of the present theory’s nonminimal coupling. For the sake of computation, it is of interest to write this tensor in a form entirely in terms of the matter Lagrangian and Ricci tensor. To this end, assuming the matter Lagrangian is independent of derivatives of the metric, one can show that Eq. (5) is equivalent to

Tμν ¼ L gμν þ 2 ∂Lm :

ð17Þ

m

∂ gμν

PHYS. REV. D 99, 124027 (2019)

Incidentally, this equation has the matter tensor Ξμν implicitly built into it,

1

Ξμν ¼ 2 ðLmgμν − TμνÞ;

ð18Þ

which we shall find useful later on. Moreover, Eq. (17) facilitates calculating the functional derivative

δTαβ ¼ gαβ ∂Lm þ 2 ∂2Lm þ L δðαβÞ ;

δgμν

∂gμν ∂gμν∂gαβ m μν

ð19Þ

where

δðαβÞμν

¼

1 2

ðδα μ δβ ν

þ

δβ μ δα ν Þ

is

the

upper

symmetric

part of the generalized Kronecker symbol (we herein

denote symmetrization by parentheses). Combining this

result with the definition of the matter-curvature tensor in

Eq. (9) implies

Π ¼ R ∂Lm þ 2R ∂2Lm þ R L :

μν

∂ gμν

αβ ∂gμν∂gαβ

μν m

ð20Þ

Another useful identity is

Πμν ¼ 2RλðμTλνÞ þ Rαβ δδTgμανβ ;

ð21Þ

which follows from substituting Tμν ¼ gμαgνβTαβ into the definition (9) of the matter-curvature tensor.

A notable matter sector is that of a perfect fluid (PF), for which we shall take Lm ¼ P [31], where P is the isotropic pressure of the fluid. The corresponding energy-

momentum tensor is

TðμPνFÞ ¼ ðρ þ PÞuμuν þ Pgμν;

ð22Þ

where ρ is the energy density of the fluid and the fluid’s 4-velocity uμ satisfies the condition uμuμ ¼ −1. One can

show the pressure P satisfies

δP ¼ − 1 ðρ þ PÞuμuνδgμν

ð23Þ

2

by using Eq. (17) with Lm ¼ P and comparing to Eq. (22). Moreover, one has [33]

δρ ¼ 1 ρðgμν − uμuνÞδgμν;

ð24Þ

2

which ultimately follows from the conservation of the matter fluid current, ∇ðμgÞðρuμÞ ¼ 0. Using these formulas appropriately, one shall find

ΞðμPνFÞ ¼ − 1 ðρ þ PÞuμuν

ð25Þ

2

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PALATINI fðR; Lm; RμνTμνÞ GRAVITY AND ITS …

and,

from

Eq.

(21)

and

the

identity

δuα δgμν

¼

−

1 2

gαðμ

uνÞ,

ΠðμPνFÞ ¼ RλðμTðνPÞλFÞ þ 12 ρRαβuαuβgμν − 1 ½ð2ρ þ PÞRαβuαuβ þ ðρ þ PÞRuμuν: 2

ð26Þ

The effective energy-momentum tensor for a perfect fluid then follows from its definition (11). The rather exotic coupling of matter and the 4-velocity to the Ricci tensor in Eq. (26) suggests that the matter-curvature tensor will play a significant role in the field equations in regions of high density, such as within a black hole or in the very early Universe. This is quantitatively similar to EiBI gravity, which has in its field equations a similar RμνTμν coupling that also gives rise to couplings between the Ricci tensor and the 4-velocity of perfect fluids (see Ref. [26] or Sec. VI of this paper). It is natural to hypothesize, then, that fðR; Lm; VÞ gravity may be fashionable such that it corresponds to GR in the weak-field regime but then cures the curvature singularities of GR in high density regions— behavior that mimics the preeminent characteristics of EiBI gravity.

B. Auxiliary metric

The introduction of the “natural” auxiliary metric pμν into the present theory affords a specific bimetric structure to the fðR; Lm; VÞ model. In addition to the physical spacetime metric gμν, through which the gravitational observables manifest, there is the auxiliary metric upon which the mathematical edifice of the theory is best supported. This structure is analogous to EiBI gravity wherein there also exists a natural bimetric arrangement [25,26]. In the present theory, however, unlike the minimal nature of EiBI theory, the gravitational Lagrangian has built into it a direct coupling between the matter fields and the auxiliary metric via the scalar curvature R and the matter-curvature scalar V≡ RμνTμν. This coupling appears through the explicit dependence of the independent connection (15) on pμν and its inverse. Such a coupling suggests that there is some physical nature tied to the auxiliary metric. But because all physical observables manifest via the spacetime metric, the nonminimal coupling suggest a general link between the auxiliary and spacetime metrics. Obviously, the particulars of this link cannot be properly realized until the details of the nonminimal coupling are known (which necessitates specifying a particular function f). However, a general relationship can be drawn.

A natural link to proffer is that of a conformal relationship, in which pμν ¼ Θ2gμν for some real-valued, smooth function Θ defined on M. This approach, however, is consistent only for specific matter sectors [34]. Thus, conformality between pμν and gμν fails to capture the general framework we seek. A more general approach,

PHYS. REV. D 99, 124027 (2019)

again analogous to EiBI gravity, is to introduce a differentiable deformation matrix Ωμν satisfying

pμν ¼ gμλΩλν:

ð27Þ

In matrix notation, this reads p ¼ gΩ so that p−1 ¼ Ω−1g−1. Direct comparison to Eq. (14) reveals that

Ω−1 ¼ p1ﬃﬃﬃﬃ ðfRI þ fVg−1TÞ;

ð28Þ

Ω

where I is the identity matrix and Ω ≡ detðΩÞ follows from Eq. (16). It is now an algebraic problem to solve for Ω and hence pμν, explicitly, following the specification of the matter Lagrangian and the fðR; Lm; VÞ model of interest. One subsequently obtains the form of the connection and

related curvature terms for the specific theory, and all that

remains to resolve a given problem is the differential

equations (10). An example of this procedure, in the

context of EiBI gravity, can be found in Ref. [26].

C. Likeness to other f theories

The Palatini fðR; Lm; VÞ formalism contains as special cases the Palatini fðRÞ and fðR; LmÞ theories but not in general the Palatini fðR; TÞ and fðR; T; VÞ theories.

Evidently, Palatini fðR; Lm; VÞ and fðR; T; VÞ gravity correspond only when Lm ¼ T, which is a hefty constraint

by which most matter fields do not abide [35]. That said, for

matter fields with a vanishing energy-momentum trace

(such as electromagnetic fields), the fðR; Lm; VÞ model clearly contains the fðR; T; VÞ model. We say that Palatini

fðR; Lm; VÞ and fðR; T; VÞ are circumstantially equivalent

theories of gravity since their equivalence is such that it holds

only for specific matter fields (this notion is made more

precise in Sec. VI). There is, however, a simple procedure to

obtain Palatini fðR; T; VÞ gravity from the fðR; Lm; VÞ theory for arbitrary matter sectors: merely replace the fLΞμν

term in the field equations (10) by fT ∂∂gTμν, and continue on that way. Since in the Palatini formalism the trace T ≡ Tμμ is

independent of the independent connection, its incorporation

into the function f will not affect Eq. (13). In this respect,

most results derived herein afford similar mathematical

structure to Palatini fðR; T; VÞ gravity, up to the replace-

ment

of

all

fLΞμν

terms

with

fT

∂T ∂ gμν

terms

and

the

subsequent manipulations of those terms. Evidently, the

exception to this rule is those results which utilize, in a

nontrivial manner, the full entourage of dependencies in the

fðR; Lm; VÞ model, such as the present theory’s circum-

stantial equivalence to EiBI gravity (see Sec. VI).

IV. PROPERTIES OF THE f (R;Lm;V) FIELD EQUATIONS

Here, we shall consider various properties of the fðR; Lm; VÞ field equations, including their conservation

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MATTHEW S. FOX

PHYS. REV. D 99, 124027 (2019)

equation, their effect on the motion of massive test particles, and their weak-field limit.

A. Conservation equation

In fðR; Lm; VÞ gravity, matter is nonminimally coupled to curvature. Hence, the covariant divergence of the energymomentum tensor is not necessarily zero. In this section, we derive an explicit expression for such nonconservation of the energy-momentum tensor. In what follows, we use tildes to decorate tensors which have been transvected by the auxiliary metric pμν.

We begin with the field equations (12) in the form

G˜ μν ¼ f1R κΣ˜ μν þ 12 fðΩ−1Þμν − 12 δμνR˜ ; ð29Þ

where G˜ μν is the Einstein tensor raised by pμν and ðΩ−1Þμν is Ω−1 in index notation. Using the definition (14), one finds

rﬃﬃﬃ Σ˜ μν ¼ gðfRΣμν þ fV TμλΣλνÞ;

p

ð30aÞ

rﬃﬃﬃ

R˜ ¼

g ðRfR þ VfVÞ:

p

ð30bÞ

The condition we seek follows from the contracted Bianchi identities, ∇ðμpÞG˜ μν ¼ 0. All that remains is a straightforward problem in algebra: expand Σμν in

Eq. (30a) using its definition (11), then isolate the covariant divergence of Tμν. We find

rﬃﬃﬃ rﬃﬃﬃ

κ∇ðμpÞTμν ¼

p ∇ðμpÞ g

pg fLΞμν þ fV Πμν − ffRV TμλΣλν − 2fffRV Tμν

rﬃﬃﬃ

rﬃﬃﬃ

− 1 ∂ν

g ðf − RfR − VfVÞ

− κTμν∂μ

g :

ð31Þ

2

p

p

Alternatively, this nonconservation can be expressed in terms of the connection ∇ðgÞ compatible with the

spacetime metric gμν. The relationship between the covariant derivatives ∇ðpÞ (that defined with the independent connection of the auxiliary metric) and ∇ðgÞ is

the following:

∇ðμpÞTμν ¼ ∇ðμgÞTμν þ CμμλTλμ − CλμνTμλ;

ð32Þ

where

Cαμν ¼ 1 pασð∇ðμgÞpσν þ ∇ðνgÞpμσ − ∇ðσgÞpμνÞ: 2

ð33Þ

The metric/auxiliary metric relationship (27), the compatibility of gμν with ∇ðgÞ, and the symmetry property of the auxiliary metric imply the coefficients can be expressed

in a form that is independent of pμν:

Cαμν ¼ 12 ðΩ−1Þασð∇ðμgÞΩσν þ ∇ðνgÞΩσμÞ

− 1 gμλðΩ−1Þασ∇σ Ωλν:

2

ðgÞ

ð34Þ

We note that any covariant derivative with respect to Ωμν can be replaced by a derivative with respect to ðΩ−1Þμν, as their inverse relationship implies

ðΩ−1Þλν∇ðσgÞΩμλ þ Ωμλ∇ðσgÞðΩ−1Þλν ¼ 0: ð35Þ

With this in mind, the coefficients (34) become

Cαμν ¼ − 12 Ωλν∇ðμgÞðΩ−1Þαλ − 12 Ωλμ∇ðνgÞðΩ−1Þαλ

þ 1 gμλðΩ−1ÞασΩγνΩλϵ∇σ ðΩ−1Þϵλ;

2

ðgÞ

ð36Þ

and the nonconservation of the energy-momentum tensor turns out to be

rﬃﬃﬃ rﬃﬃﬃ

∇ðμgÞTμν ¼ 1 p ∇ðμpÞ g fLΞμν þ fV Πμν − fV TμλΣλν − ffV Tμν

κg

p

rﬃﬃﬃ

fR 2fR

− 1 ∂ν

g ½f − RfR − VfV

þ CλμνTμλ:

ð37Þ

2

p

Clearly, the curvature and energy-momentum depend-

ences in Eq. (37) and the energy-momentum dependence of ðΩ−1Þμν restrict these formulas from simplifying much beyond what is given here. We emphasize, therefore, that

∇ðμgÞTμν does not in general vanish. Hence, the energymomentum tensor in Palatini fðR; Lm; VÞ gravity is in general not conserved. On the other hand, in Sec. VI, we

shall indirectly derive two nontrivial functions of

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PALATINI fðR; Lm; RμνTμνÞ GRAVITY AND ITS …

PHYS. REV. D 99, 124027 (2019)

fðR; Lm; VÞ for which the covariant divergence of specific but nontrivial Tμν necessarily vanish, hence conserving the energy-momentum tensor. That said, this conservation will

not be obvious at the level of Eq. (37), though it will

nevertheless be true. Finally, we note that for the EinsteinHilbert model fðR; Lm; VÞ ¼ R − 2Λ the conservation of Tμν is restored, as desired.

Here, the left-hand side coincides with the well-known identity

uμ∇ðμgÞuν ¼ d2xν þ Γν

dxα dxβ :

ð39Þ

ds2 αβ ds ds

B. Motion of test particles

For clarity, we denote by Δν the right-hand side of Eq. (37). Then, ∇ðμgÞTμν ¼ Δν. For the case of a perfect fluid, for which Tμν ¼ ðρ þ PÞuμuν þ Pgμν, it is straightforward to show, using the constraint from the conservation of the matter fluid current, ∇ðμgÞðρuμÞ ¼ 0, that

uμ∇ðμgÞuν ¼ Δν − uν∇ðμgÞðPuμÞ − ∂νP :

ð38Þ

Pþρ

Therefore, Eq. (38) is the equation of motion for particles in the presence of an isotropic pressure P. Absent this pressure, the equation reduces to

d2xν þ Γν dxα dxβ ¼ fν;

ð40Þ

ds2 αβ ds ds

where the extra force fν ¼ ρ−1ΔνðP¼0Þ with

rﬃﬃﬃ rﬃﬃﬃ

rﬃﬃﬃ

ΔðνP¼0Þ ¼ 1 p ∇ðμpÞ − g ρ fLuμuν þ fV ρ gð½Rσμuν þ Rσνuμuσ þ Rαβuαuβ½δμv − 2uμuν − RuμuνÞ

κg

p2

2p

rﬃﬃﬃ gf

ρ2

f

þ p κVf

½κ − fLuμuν þ

V

2

½Ruμuν

−

uμRανuα

þ

4Rαβ uα uβ uμ uν

rﬃﬃﬃ R

rﬃﬃﬃ

− pg 2fffRV ρuμuν − 12 ∂ν pg½f − RfR − VfV þ ρCαβνuβuα ð41Þ

[see Eqs. (25) and (26) to derive this]. Since ΔνðP¼0Þ is in general nonzero, the extra force fν is in general nonzero. Hence, test particles in fðR; Lm; VÞ gravity do not in general obey the geodesic equation. In other words, test particles traverse geodesics of gμν if and only if ΔμðP¼0Þ ¼ 0.

C. Newtonian limit

In the weak-field regime, we consider the gravitational effect of nonrelativistic dust, for which Tμν ¼ ρuμuν where uμ ¼ ð∂0Þμ is the rest frame 4-velocity and ρ is the dust’s energy density [38]. We shall linearize the fðR; Lm; VÞ equations by keeping terms linear in ρ and in the perturbations introduced below. To facilitate the coming analysis, we adopt the following notation.

Let γμν and γˆμν be smooth 2-forms (soon to be perturbations). Further, let A and B be mathematical objects composed, in some acceptable fashion, of the objects ρ; γμν, and γˆμν. Then, by A ≪ B, we shall mean A is first order (linear) in at least one of ρ, γμν, or γˆμν, while B is zeroth order in all. Moreover, by A ≅ B, we shall mean A ¼ B up to at least linear corrections in all ρ, γμν, and γˆμν. Finally, by A ∼ B, we shall mean A and B are of the same order in ρ, γμν, or γˆμν, but not necessarily equal (thus, A ≅ B implies A ∼ B).

Consider the metric/auxiliary metric relation posited in Eq. (27). This establishes that any perturbation δpμν upon pμν satisfies

δpμν ¼ gμλδΩλν þ Ωλνδgμλ:

ð42Þ

Specifically, we shall consider perturbations δpμν ¼ γˆμν

and δgμν ¼ γμν upon a Minkowski background ημν.

Then, pμν ¼ ημν þ γˆμν and gμν ¼ ημν þ γμν such that

γˆμν; γμν ≪ ημν. Here, pμν and gμν are related by Eq. (27),

and furthermore, the perturbations γˆμν and γμν satisfy Eq. (42). Together, these imply ημλΩλν − ημν ≪ ημν, which is possible if and only if Ωμν ≅ δμν þ kμν, where kμν ≪ δμν.

In deriving this, one must also assume that the deformation

matrix reacts smoothly to slight perturbations upon the

Minkowski background, i.e., that gμλδΩλν ≪ ημν, which necessitates ημλδΩλν ∼ γμν and γμλδΩλν ≅ 0.

Transcribed to matrix notation, we have Ω ≅ I þ k.

Thus, Ω−1 ≅ I − k, to which we shall directly compare

Eq. (28). Since the tensor Tμν → g−1T is in general

pdifﬃﬃfﬃﬃerent from the idpenﬃﬃﬃtﬃity matrix,piﬃtﬃﬃﬃmust be that fR ≅

Ω and fVg−1T ≅

Ωk, where

Ω

≅

1

þ

1 2

Tr

ðkÞ

.

But

since TrðkÞk ≅ 0, with 0 the zero matrix, we simply have

k ≅ fVg−1T. Together, these results yield both pμν and pμν

to the desired first-order precision:

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MATTHEW S. FOX

pμν ≅ ημν þ γμν þ fV Tμν;

ð43aÞ

pμν ≅ ημν − γμν − fV Tμν;

ð43bÞ

where Tμν ¼ ημαηνβTαβ and γμν ¼ ημαηνβγαβ. We see from Eq. (43a) that

γˆμν ≅ γμν þ fV Tμν

ð44Þ

and similarly from Eq. (43b) that γˆμν ≅ γμν þ fVTμν. Hence, the connection coefficients (15) are, to linear order in γˆμν,

Γαμν ≅ 1 ηασð∂μγˆσν þ ∂νγˆμσ − ∂σγˆμνÞ;

ð45Þ

2

which yields the Ricci tensor

Rμν ≅ ∂σ∂ðνγˆμÞσ − 1 ∂σ∂σγˆμν − 1 ∂μ∂νγˆ; ð46Þ

2

2

where γˆ ≡ γˆμμ. Note that Eq. (46) is entirely of linear order in γˆμν. Hence, V ¼ RμνTμν ≅ 0 since Tμν is linear in ρ. Moreover, fVΠμν ≅ 0 since δTαβ=δgμν ∼ ρ for dust [see Eq. (24)], and Rμν ∼ γˆμν. We shall impose the Lorenz gauge ∂σγˆμσ ¼ 0 so that the fðR; Lm; VÞ field equations, to linear order in γˆμν and ρ, bear the form

− 1 ∂σ∂σγˆμν − 1 ∂μ∂νγˆ − 1 fgμν ≅ κTμν − fLΞμν: ð47Þ

2

2

2

To obtain the matter tensor, we necessarily take Lm ¼ −ρ

for the matter Lagrangian of the pressureless dust. Hence,

from

Eq.

(24),

Ξμν

¼

1 2

ρðuμuν

−

ημν

−

γμνÞ.

Here,

ρ

is

the

leading-order correction from the matter sector; thus, the

product ργμν must be regarded as a second-order correction.

This

implies

that,

to

first

order,

Ξμν

≅

1 2

ρðuμuν

−

ημνÞ

and

hence

Ξ

≡

Ξμμ

≅

−

5 2

ρ.

It

follows

that

the

trace

of

the

field

equations (10) is, to first order,

5

fRR − 2f ≅ ρ 2 fL − κ :

ð48Þ

Note that Πμμ ≅ 0 since Πμν ≅ 0. Thus, Πμμ is absent from Eq. (48) in this approximation. Note also that the already first-order corrections ρ and R force f to be at least a firstorder correction; hence, we can rewrite Eq. (47) as

− 1 ∂σ∂σγˆμν − 1 ∂μ∂νγˆ − 1 fημν

2

2

2

1

≅ κρuμuν − 2 ρfLðuμuν − ημνÞ:

ð49Þ

The 00 component of this equation encodes the weak-field dynamics in which we are interested. Since the spacetime is

PHYS. REV. D 99, 124027 (2019)

assumed static, the time derivatives vanish, leaving the expression

1

1

− 2 Δγˆ00 ≅ κρ − 2 f − ρfL;

ð50Þ

where Δ ≡ ∇2 is the Laplacian operator. Using Eq. (44)

and

the

definition

of

the

Newtonian

potential,

Φ

≡

−

γ00 4

,

we

obtain the modified Poisson equation in fðR; Lm; VÞ

gravity:

1 11

1

ΔΦ ≅ 2 κρ − 4 f − 2 ρfL þ 4 ΔðfVρÞ: ð51Þ

Since f is implicitly a function of Tμν, and hence of ρ, the

quantity

1 2

κρ

−

1 4

f

−

1 2

ρfL

acts

as

a

sort

of

effective

density

1 2

κρ¯ .

In

these

terms,

Eq.

(51)

reads

11

ΔΦ ≅ 2 κρ¯ þ 4 ΔðfVρÞ:

ð52Þ

This modification to Poisson’s equation is formally identical to those in both EiBI and Palatini fðR; TÞ gravity (see Refs. [26,28], respectively). Consequently, we expect all these theories to afford similar nonrelativistic phenomenology.

V. SOME APPLICATIONS

The weak-field equations considered above disclosed a relationship between fðR; Lm; VÞ and other theories of gravity in the Newtonian regime. In this section, we derive the field equations governing the response of fðR; Lm; VÞ gravity in other regimes, in particular the electromagnetic and scalar field sectors.

A. Electromagnetic fields

Consider next the traditional linear electrodynamics

(LED) of Maxwell for which the matter Lagrangian is

LðLEDÞ

¼

−

1 16π

FμνFμν,

where

Fμν ≡ ∂μAν − ∂νAμ

is

the

Faraday tensor. The LED matter tensor follows quickly

from

its

definition

(8),

ΞðμLνEDÞ

¼

−

1 8π

FμλFλν.

Alternatively,

using Eq. (18),

ΞðμLνEDÞ ¼ 1 ðLðLEDÞgμν − TðμLνEDÞÞ: 2

ð53Þ

Similarly, from the symmetry of the Ricci tensor and Eqs. (21) and (53), the LED matter-curvature tensor turns out to be

ΠðμLνEDÞ ¼ 2RλðμTðνLÞλEDÞ − RμνLðLEDÞ − 81π RFμλFλν þ 1 RαβFμβFαν: 4π

ð54Þ

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PALATINI fðR; Lm; RμνTμνÞ GRAVITY AND ITS …

PHYS. REV. D 99, 124027 (2019)

The LED effective energy-momentum tensor, ΣðμLνEDÞ, then follows trivially from its definition (11).

The matter-curvature couplings in Eq. (54) are very much unlike, for instance, Palatini fðR; TÞ theory, in which TðLEDÞ ¼ gμνTðμLνEDÞ ¼ 0. Hence, the fðR; TÞ models (whether Palatini or not) respond to linear electromagnetic fields as an fðRÞ model. This is evidently not the case for the present theory, in which there are nontrivial couplings between curvature and matter terms, all of which have the potential to invite new gravitational electrodynamic behavior. We note that in the Palatini fðR; T; VÞ model all the fV couplings in Eq. (54) persist; therefore, even with a vanishing trace [making the gravitational response a Palatini fðR; VÞ theory], there remain new and nontrivial corrections to the linear electrodynamics.

However, it is well known that the linear electrodynamics in vacuo are only an approximation to the full electrodynamic theory. General relativity, for example, demands a gravitational coupling between electromagnetic fields, which affords nonlinear electrodynamic behavior. That said, more considerable nonlinearity arises from quantum field effects, such as vacuum polarization [39]. It is therefore of interest to also derive the fðR; Lm; VÞ field

equations associated with a general set of nonlinear electrodynamic (NED) theories. To this end, we set the matter sector action to

Z

SðNEDÞ ¼ 1 d4xpﬃ−ﬃﬃﬃgﬃﬃχðI; JÞ;

ð55Þ

8π

where χ is a well-behaved function of the algebraic

invariants I ≡ 12FμνFμν and J ≡ Fμνð⋆FÞμν. Here, ð⋆FÞμν ¼

1 2

ð−gÞ−12

ϵμναβ

F

αβ

is

the

Hodge

dual

of

the

Faraday

tensor,

with ϵμναβ denoting the Levi-Civita symbol. We note that I

and J are the unique algebraic invariants constructible from

Fμν and gμν [40,41] and also that the choice χðI; JÞ ¼ −I

corresponds to the LED theory considered above.

With

1 8π

χ

ðI

;

J

Þ

as

the

NED

matter

Lagrangian,

and

defining

χI

≡

∂χ ∂I

and

χJ

≡

∂∂Jχ,

we

find

ΞðμNνEDÞ ¼ 1 χIFμλFλν þ 1 χJJgμν :

8π

2

ð56Þ

It then follows from Eq. (21) and the NED equivalent of Eq. (53) that

ΠðμNνEDÞ ¼ 2RλðμTðνNÞλEDÞ þ 81π ðχJJ − χÞRμν þ 41π

1 RχI − RαβFαλFλβχII − 1 RχJIJ2

2

2

FμλFλν

− 1 RαβFαλFλβχIJJ þ 1 RχJJJ2 gμν − 1 RαβχIFαμFνβ:

8π

2

4π

ð57Þ

As before, ΣðμNνEDÞ then follows from its definition (11), and TðμNνEDÞ follows from the NED equivalent of Eq. (53). Note that, as expected, upon fixing χðI; JÞ ¼ −I, Eq. (57) reduces to Eq. (54). As in the LED case, these field

equations have in them nontrivial matter-curvature cou-

plings which again bear new possibilities for NED gravi-

tational dynamics, such as in studies of nonsingular black

holes. We also note that these equations again differ

drastically in their matter-curvature couplings from the field equations for NED in fðR; TÞ gravity (see, e.g., Ref. [28]). This much is evident from the fV coupling terms, which persist only in the fðR; Lm; VÞ framework.

B. Canonical scalar fields

Scalar fields comprise another set of generic matter fields

for which fðR; Lm; VÞ gravity admits new and nontrivial

dynamics. Here, we shall consider the effect of a real-

valued scalar field ϕ in a potential UðϕÞ, the Lagrangian

density

of

which

bears

the

form

LðϕÞ

¼

−

1 2

∂λϕ∂λϕ

−

UðϕÞ.

One

shall

find

ΞðμϕνÞ

¼

−

1 2

∂μϕ∂νϕ

and,

from

Eq. (20),

ΠðμϕνÞ ¼ − 1 R∂μϕ∂νϕ þ ∂λϕ∂ðμϕRλνÞ þ RμνLðϕÞ: 2

ð58Þ

Hence,

κΣðμϕνÞ ¼ κTðμϕνÞ þ 1 ðfL þ RfV Þ∂μϕ∂νϕ − fV LðϕÞRμν 2

− fV ∂λϕ∂ðμϕRλνÞ:

ð59Þ

As with the electromagnetic field, specifying particular fðR; Lm; VÞ functions and solving the associated field equations will conceivably yield new nonminimal correc-

tions to ordinary GR problems, which brings about new possibilities. For example, as posited for Palatini fðR; TÞ gravity [28], free [UðϕÞ ¼ 0] geonic solutions of the kind in EiBI gravity [42] are conceivable in the present theory.

VI. COMPATIBILITY WITH EiBI GRAVITY

In this section, we shall investigate the conditions under which the fðR; Lm; VÞ paradigm encapsulates the EiBI theory. We shall denote by fBI any fðR; Lm; VÞ function that does this. To begin, it is imperative that we be precise

124027-9

MATTHEW S. FOX

with the meaning of “one gravitational theory corresponding to another.”

Let A and B be two Palatini theories of gravity defined on a world manifold M, and let Ψ be a matter field on M. Further, let gðμAν Þ and gðμBνÞ be the solutions generated from A and B, respectively, in response to Ψ, and ∇ðAÞ and ∇ðBÞ be the derivative operators of A and B, respectively, defined on M. On one hand, we say A and B are equivalent if, for all Ψ, (i) ∇ðσAÞξμ ¼ ∇ðσBÞξμ for all vectors ξμ defined on some tangent space in the tangent bundle of M and (ii) gðμAν Þ ¼ Θ2gðμBνÞ for some real-valued, smooth conformal factor Θ defined on M. Evidently, condition i ensures that both theories measure the same intrinsic curvature of M,

that both have the same notion of transport, and so forth,

while condition ii establishes that the gravitational dynam-

ics of the two theories are the same (since they afford the

same solution, up to a conformal factor, for a given matter sector Ψ). On the other hand, we say A and B are

circumstantially equivalent if conditions i and ii hold only for particular Ψ. Indeed, we shall prove in this section that EiBI and fðR; Lm; VÞ are circumstantially equivalent theories of gravity; in particular, that condition i shall hold well for all Ψ but that condition ii shall hold well only for specific Ψ.

The (Palatini) EiBI action bears the form [26]

1 Z hqﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃi SBI½g; Γ; Ψ ¼ 2κϵ d4x jgμν þ ϵRμνðΓÞj − λ −g

þ Sm½g; Ψ;

ð60Þ

where ϵ is a coupling parameter, λ is related to the cosmological constant Λ by λ ¼ 1 þ ϵΛ, and the vertical bars denote the absolute value of the determinant. The reader is referred to Refs. [25,26] for details on the variation. The field equations are

qμν ¼ gμν þ ϵRμν; pﬃ−ﬃﬃﬃqﬃﬃqμν ¼ pﬃ−ﬃﬃﬃgﬃﬃðλgμν − κϵTμνÞ;

ð61aÞ ð61bÞ

where q is the determinant of the auxiliary metric qμν and qμν satisfies both qμλqλν ¼ δμν and ∇ðσBIÞðpﬃ−ﬃﬃﬃqﬃﬃqμνÞ ¼ 0, where ∇ðBIÞ is the derivative operator associated with the Palatini EiBI theory. Hence, ∇ðσBIÞqμν ¼ 0.

We note that λ ≠ 0 (equivalently pΛﬃﬃ≠ﬃﬃﬃﬃ −ϵ−1), for otherwise Eq. (61b) implies that in vacuo −qqμν ¼ 0, which is nonsense. Moreover, with Tμν ¼ 0 and λ ≠ 1, the solutions from the two theories do not coincide; EiBI affords a de Sitter or anti-de Sitter universe, while fðR; Lm; VÞ outputs Minkowski space. In speaking of a possible equivalence

between the theories, it is natural to demand that at least

the vacuum solutions correspond. To this end, we shall hereafter fix λ ¼ 1, making EiBI Minkowskian in vacuo.

PHYS. REV. D 99, 124027 (2019)

Note that there is no loss of generality in doing this. Should

one wish to append a cosmological constant to either

theory, one would simply do so via the matter sector. We have merely “tared” the two theories at the level of their

vacuum solutions. As previously defined, ∇ðpÞ is the derivative operator

associated with the Palatini fðR; Lm; VÞ theory. Thus, for an EiBI=fðR; Lm; VÞ equivalence to exist, condition i demands that ∇ðσBIÞξμ ¼ ∇ðσpÞξμ for all smooth vectors ξμ. This implies, in particular, that

∇ðσBIÞqμν ¼ ∇ðσpÞqμν ¼ ∇ðσpÞpμν ¼ 0:

ð62Þ

The connections of both fðR; Lm; VÞ and EiBI gravity are torsion free. Hence, as required by the fundamental theorem of Riemannian geometry, Eq. (62) holds well if apnﬃdﬃﬃﬃﬃﬃonly ipf ﬃﬃpﬃﬃﬃμﬃν ¼ qμν, which is true if and only if

−qqμν ¼ −ppμν. Therefore, the definitions (14) and

(61), together with condition ii, i.e., the requisite conformal relationship gðμfνÞ ¼ Θ2gðμBνIÞ [gðμfνÞ being the solution from the fðR; Lm; VÞ theory], imply (with λ ¼ 1)

ð1 − Θ2fRÞgμðBνIÞ − ðκϵTμðBνIÞ þ Θ4fV TμðfνÞÞ ¼ 0; ð63Þ

where TμðBνIÞ and TμðfνÞ are the energy-momentum tensors of the EiBI and fðR; Lm; VÞ theories, respectively, each raised by their respective metric. We cannot impose a priori that these energy-momentum tensors be the same since they are functions of their respective metrics. We can impose, however, that the two parenthetical terms in Eq. (63) vanish separately. This is necessarily the case if we seek generality in the matter sector, as, for instance, Eq. (63) holds in vacuo if and only if the two parenthetical terms vanish separately. Consequently, Θ2fR ¼ 1, and κϵTμðBνIÞ ¼ Θ4fVTμðfνÞ. Differentiating the former with respect to R demands that fR is constant and hence that the conformal factor Θ is constant. The same is true for the latter, where differentiation upon V implies fV is constant.

These results indicate two things. First, TBμνI ∝ TðμfνÞ. For equivalence between the two theories to hold, this constant proportionality must hold in general, for arbitrary choices of the matter sector. But since the conformal transformation properties of the energy-momentum tensor depend on the matter sector, constant proportionality is guaranteed only with exact equality between the metrics, i.e., with Θ2 ¼ 1 and hence fV ¼ −κϵ. Second, the vanishing of the second derivatives fRR and fVV implies that the EiBI= fðR; Lm; VÞ function fBI is of the form fBIðR; Lm; VÞ ¼ f1ðRÞ þ hðLmÞ þ f2ðVÞ for well-behaved functions f1, h, and f2. In fact, with the conformal factor set at unity and the energy-momentum tensors identical, we simply have from Eq. (63) that f1ðRÞ ¼ R and f2ðVÞ ¼ −κϵV. Hence, from Eq. (10), the fBI field equations bear the form

124027-10