Surface Singularities in Eddington-Inspired Born-Infeld Gravity

Paolo Pani1 and Thomas P. Sotiriou2

1CENTRA, Departamento de Fısica, Instituto Superior Tecnico, Universidade Tecnica de Lisboa—UTL,Avenida Rovisco Pais 1, 1049 Lisboa, Portugal

2SISSA, Via Bonomea 265, 34136 Trieste, Italy and INFN, Sezione di Trieste, Italy(Received 21 September 2012; published 18 December 2012)

Eddington-inspired Born-Infeld gravity was recently proposed as an alternative to general relativity that

offers a resolution of spacetime singularities. The theory differs from Einstein’s gravity only inside matter

due to nondynamical degrees of freedom, and it is compatible with all current observations. We show that

the theory is reminiscent of Palatini fðRÞ gravity and that it shares the same pathologies, such as

curvature singularities at the surface of polytropic stars and unacceptable Newtonian limit. This casts

serious doubt on its viability.

DOI: 10.1103/PhysRevLett.109.251102 PACS numbers: 04.50.�h, 98.80.�k

Recent years have witnessed a proliferation of alterna-tive theories of gravity, motivated by long-standing puzzlesin general relativity [1]. It is pertinent to develop theoreti-cal benchmarks that will allow us to select the physicallyrelevant candidates, and to subsequently use observationalconstraints in order to single out the few theories that areactually viable alternatives.

One of the major riddles of general relativity is that itpredicts the appearance of spacetime singularities origi-nating from regular initial data, e.g., in the gravitationalcollapse of massive stars and in the early Universe. Inorder to resolve these singularities, an appealing proposalfor a modified theory of gravity, the so-called Eddington-inspired Born-Infeld (EiBI) theory, was recently put for-ward in Ref. [2] and it has been subject to scrutiny in anumber of works [3–13]. EiBI gravity is equivalent togeneral relativity in vacuum and does not propagate anydegree of freedom other than a massless graviton. On theother hand, the theory introduces nonlinear couplings tothe matter fields [6,7], which resolve at least some of thesingularities appearing in Einstein’s theory.

The big-bang singularity in early cosmology is replacedby a freezing or a bouncing behavior of the cosmologicalscale factor, depending on the extra EiBI parameter [2].The gravitational collapse of noninteracting particles doesnot lead to singular states in the nonrelativistic limit [3,6].A tensor instability of the homogeneous and isotropicuniverse was found in Ref. [9] and EiBI gravity has alsobeen studied as an alternative to the inflation paradigm[10]. Possible constraints on the theory have been consid-ered using solar models [4] and cosmological observations[5,13] (see also Ref. [14]). However, a degeneracy betweenEiBI corrections and different matter configurations [7]makes it difficult to put observational constraints withoutindependent knowledge of the matter content of the theory.

Previous literature on EiBI gravity mostly focused onphenomenological aspects of the theory; a more detailedstudy on its dynamics and on the structure of its field

equations has not been performed yet. Here, we arguethat the field equations of EiBI gravity have a peculiardifferential structure which is similar to that of PalatinifðRÞ gravity [15] and, as a result, they exhibit the samepathologies with the latter (see, e.g., Refs. [16–19] and thereview [15]) and with theories where matter is coupled tothe Ricci scalar, which also have similar characteristics[20]. Although these theories provide an appealing (and infact similar [21]) early-time cosmology, such pathologiescast serious doubts on their viability.EiBI gravity is described by the following action [2]:

S ¼ 1

4�G�

Zd4x

� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij detðgab þ �RðabÞÞj

q� ð1þ ��Þ ffiffiffi

gp �

þ SM½gab;�M�; (1)

where SM½gab;�M� is the matter action, �M genericallydenotes any matter field,Rab is the Ricci tensor built fromthe connection �c

ab, g ¼ j detðgabÞj, � turns out to be the

cosmological constant, and � is the extra EiBI parameterwhich has dimensions of length squared. Parentheses(square brackets) denote (anti-)symmetrization.In the metric approach the field equations contain

ghosts, which must be eliminated by adding extra termsto the action [22,23]. Thus, EiBI gravity is naturally basedon the Palatini formulation; i.e., the connection �c

ab is

considered as an independent field. The original EiBIproposal is based on some crucial assumptions: namely,(i) the matter action is independent from �c

ab; (ii) the

connection is symmetric, �cab ¼ �c

ba; (iii) only RðabÞappears in Eq. (1) and not Rab. This last assumption isoften implicit. That is, it is common in the literature to haveRab appearing in Eq. (1), even though any subsequentcalculation is based on the implicit assumption R½ab� ¼0(or on the nonstandard definition Rab � RðabÞ). This

would be an extra constraint as, even for a symmetricconnection, R½ab� ¼ �@½b�l

a�l and it does not vanish ge-

nerically. This extra constraint is not required provided that

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only the symmetric part of the Ricci is used in the action.See Ref. [24] for a similar discussion for generalizedPalatini gravity.

In principle, assumptions (i), (ii), and (iii) are notrequired and, together with the action (1), they define aparticular version of EiBI gravity. Relaxing (i)–(iii) wouldlead to a metric-affine version of the theory, similar in spiritto the case of metric-affine fðRÞ theories [25], but verydifferent from the original theory. We will explore thispossibility in a separate publication. Here we shall relyon the assumptions above, as in the original proposal [2].

We start by expanding the action (1) at second order in�. This yields

S ¼ 1

8�G

Zd4x

ffiffiffig

p �R� 2�þ �

4ðR2 � 2RðabÞRðabÞÞ

�

þ SM½gab;�M� þOð�2Þ; (2)

where R ¼ gabRab. For simplicity, we shall use unitssuch that 8�G ¼ 1. When � ¼ 0, EiBI gravity reducesto the Palatini formulation of general relativity with acosmological constant. As is well known, in this case thefield equations impose that the connection is the Levi-Civita one and the theory reduces to Einstein’s gravity.However, at first order in �, quadratic corrections in thecurvature tensor built from the independent connectionappear in the action (2). The Palatini formulation guaran-tees that, despite these extra terms, no higher derivativesof the metric field would appear in the field equations.Note also that, when expanded order by order in �, theaction (1) takes the form of a specific Palatini fðR;RabÞtheory [24,26].

We now return to action (1). Independent variation withrespect to the metric and the connection yields

ffiffiffiq

pqab ¼ ffiffiffi

gp ½ð1þ ��Þgab � �Tab�; (3)

~r c½ ffiffiffiq

pqðabÞ� � ~rl½ ffiffiffi

qp

qðal��bÞc ¼ 0; (4)

where we have defined qab � gab þ �RðabÞ and ~ra is

the covariant derivative defined with �cab, Tab �

ðgÞ�1=2�SM=�gab is the standard stress-energy tensor,whose indices are raised and lowered by gab, whereasqab is the inverse of qab. After some manipulations,Eq. (4) takes the form

�cab ¼ 1

2qcdð@aqbd þ @bqad � @dqabÞ: (5)

On the other hand, using Eq. (3) we obtain

qab ¼ ð1þ ��Þgab � �Tab

ffiffiffig

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidet½ð1þ ��Þgab � �Tab�p ; (6)

which can be rewritten as

�RðabÞ ¼ ffiffiffig

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidet½ð1þ ��Þgab � �Tab�

q� ½ð1þ ��Þgab � �Tab��1 � gab: (7)

Equation (6) determines qab algebraically in terms of gaband Tab, whereas Eq. (5) determines �c

ab as the Levi-Civita

connection of qab. Hence, one can use these equations toeliminate �c

ab from Eq. (7). Then, the latter is the equation

that has to be used to determine gab.It is straightforward to see that, after eliminating �c

ab,

Eq. (7) becomes a second-order partial differential equa-tion in gab. However, it is equally straightforward to seethat it also contains second derivatives of Tab. This is truein the full theory, but it becomes more explicit if we expandEq. (7) at first order in �. The expansion can be foundeasily by noting that

qab ¼ gab � ��ab þOð�2Þ;where �ab � Tab � 1

2gabT þ�gab. Using the expression

above, we get a single equation for the metric gab only:

Rab ¼ �gab þ Tab � 1

2Tgab þ �

�Sab � 1

4Sgab

�

þ �

2½rarb�� 2rcrða�cbÞ þh�ab� þOð�2Þ; (8)

where Sab ¼ TcaTcb � 1

2TTab and we have used the fact that

�ab is symmetric. Note that now Rab is built solely from theLevi-Civita connection of gab and ra is the covariantderivative associated with gab. Therefore, � inside �abdoes not contribute to the second line of Eq. (8).General relativity corresponds to � ¼ 0, and for � � 0

Eq. (8) contains second derivatives of Tab, i.e., at least thirdderivatives of the matter fields (unless we consider a fluidapproximation of matter). This is in contrast to Einstein’stheory, where usually only first derivatives of the matterfields appear on the right-hand side of Einstein’s equations.This different structure is also evident in the Newtonianlimit of the theory [2], which can be straightforwardlyobtained from Eq. (8), but the result holds for any valueof � and does not hinge on the small � expansion. Themodified Poisson equation is r2� ¼ �

2 þ �4r2�, whose

solution reads

� ¼ �N þ �

4�; (9)

where � is the matter density and �N is the standardNewtonian potential. Although the modified Newtonianregime has been studied in some detail and it providesinteresting phenomenology [4,6], Eq. (9) shows that thegravitational potential � is algebraically related to �.This demonstrates that gravity is noncommulative: unlikein Einstein’s theory, the metric in EiBI gravity is not justan integral over the sources, but it receives an algebraiccontribution from the matter fields and their derivatives.Any matter configuration which is discontinuous or just notsmooth enough will produce discontinuities in the metric

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and singularities in the curvature invariants (which dependon the second derivatives of �), leading to unacceptablephenomenology. Clearly, this behavior persists in the post-Newtonian limit [cf. Ref. [15] for a discussion in PalatinifðRÞ gravity]. These problems have been overlooked inthe literature of EiBI theory.

As mentioned earlier, the same pathologies arise inPalatini fðRÞ gravity [15] and in theories with mattercoupled to the Ricci scalar [20]. Indeed, the structure ofEq. (8) is the same as in Palatini fðRÞ theory [cf. Eq. (28)in Ref. [15]]. A qualitative difference is that our Eq. (8)contains derivatives of the full stress-energy tensor,whereas the field equations in Palatini fðRÞ gravity onlycontain derivatives of the trace T ¼ gabTab. Thus, theproblems we are discussing are already manifest whenEiBI gravity is simply coupled to a Maxwell field, whosestress-energy tensor is traceless, T � 0.

Let us now discuss some problems in constructingperfect-fluid equilibrium structures in EiBI gravity, whichare related to its peculiar differential structure. We shallbuild on previous works which discuss similar pathologiesin other theories [15,20]. Static and spherically symmetricperfect-fluid stars were discussed in detail in Refs. [3,6].Without loss of generality, a convenient Ansatz for themetrics reads

qabdxadxb ¼ �pðrÞdt2 þ hðrÞdr2 þ r2d�2; (10)

gabdxadxb ¼ �FðrÞdt2 þ BðrÞdr2 þ AðrÞr2d�2; (11)

where we have used the gauge freedom to fix the functionin front of the spherical part of the auxiliary qab metric. Weconsider perfect-fluid stars whose stress-energy tensor reads

Tab ¼ ½�þ P�uaub þ gabP, where ua ¼ ð1= ffiffiffiffiF

p; 0; 0; 0Þ

and �ðrÞ and PðrÞ denote the energy density and thepressure, respectively.

Notice that the field equations (3) are simply algebraicrelations between qab and gab. Inserting the Ansatze aboveinto Eq. (3), one can solve for the coefficients of gab interms of qab and Tab [6]. Then, the dynamical equations (7)do not explicitly contain second derivatives of the matterfields and can be solved for p, h, and P, assuming anequation of state of the form P ¼ Pð�Þ. Interestingly, inthis formulation the field equations are equivalent to thestandard Tolman-Oppenheimer-Volkoff equations for themetric qab, with an effective equation of state [7]. Sincematter is covariantly coupled to the gab metric, the stan-dard conservation of the stress-energy tensor follows,raT

ab ¼ 0. Finally, the interior solution is matched tothe (unique) exterior Schwarzschild metric through appro-priate junction conditions at the stellar surface [3,6].However, in this formulation the physical gab metric isnot a dynamical quantity, and a successful numerical inte-gration does not necessarily mean that geometric invari-ants, which involve derivatives of gab, are regular. Indeed,we show here that the Ricci curvature of the physical gab

metric is divergent at the surface. This singularity has beenmissed in previous literature on EiBI gravity.For simplicity, we consider the asymptotically flat case,

� ¼ 0. Using Eqs. (10) and (11), the Ricci curvature reads

Rg ¼ ½2r2A2B2F2��1fr2BF2A02 þ 2AF½2B2Fþ r2FA0B0

� rBðA0ð6Fþ rF0Þ þ 2rFA00Þ� þA2½rFB0ð4Fþ rF0ÞþBð� 4F2 þ r2F02 � 2rFð2F0 þ rF00ÞÞ�g; (12)

which involves first and second derivatives of the gabmetric coefficients F, B, and A. By using the algebraicrelations, we can write Rg only in terms of the qab-metric

coefficients p and h and the matter fields P and �. The finalexpression can be schematically written as

Rg ¼ Rgðp; p0; p00; h; h0; P; P0; P00Þ; (13)

where we have used the equation of state to eliminate � andits derivatives. In general relativity the Ricci curvaturesimply reads Rg ¼ �T ¼ �� 3P; i.e., no derivatives of

the matter fields appear. This has profound implications.For example, if the function PðrÞ is continuous but notdifferentiable at the stellar surface, then P0 would bediscontinuous at the radius and P00 would introduce anunacceptable Dirac delta contribution to the curvature.However, the differentiability of PðrÞ at the surface ishard to judge before having solved the field equations.In the specific case of polytropic equations of state,

however, where P ¼ K��0 , with �0 being the rest-mass

density and K and � constants, one can determine thebehavior of Rg at the surface without actually having to

solve the equations fully. The energy density can then be

written explicitly as �ðPÞ ¼ ½P=K�1=� þ P=ð�� 1Þ. Wecan use the field equations [6] to eliminate the derivativesin Eq. (13). Evaluating Rg at the stellar surface, i.e., as

r ! RS and P ! 0, for any � � 0 we get

RgðP ! 0Þ �

8>>><>>>:�� 0< � � 3=2

��P�2þ3=� 3=2< �< 2

��P�1=� � � 2:

(14)

In the equation above, �� ¼ 0 if 0< �< 3=2, whereas

�� ¼

8>>>>><>>>>>:

�ð2��Þ2�K3=��2 3=2 � �< 2

�8�2

½8þ�=K�3�K5=2 � ¼ 2

8ð1��ÞK1=�

�� �> 2;

(15)

with � ¼ R3SðRS � 2MÞ=M2 and M is the total mass

defined by hðRSÞ ¼ ½1� 2M=RS��1 [6,12]. Therefore,for any �> 3=2 the scalar curvature diverges at the sur-face. The diverging terms originate from the derivatives ofthe matter fields in Eq. (13) and more specifically from the

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terms ���2�PP, ���P, and �PP=ð��2PÞ, where the sub-

scripts denote partial derivatives with respect to P.A similar result, i.e., the divergence of the Ricci scalar

at the surface of polytropic matter configurations, wasobtained in Palatini fðRÞ gravity and the consequencesare discussed in detail Refs. [17–19]. In that case, thedivergence occurs only for 3=2< �< 2. In EiBI gravityone has �� � 1=� for �> 2, so that this singular solutiondoes not appear in a small � expansion in which EiBIgravity resembles Palatini fðR;RabÞ.

What we have established is that EiBI gravity does notadmit any regular solution for polytropic spheres with�> 3=2, even for arbitrarily small values of �. At leasttwo physical matter configurations are exactly describedby a polytropic equation of state with �> 3=2: a degener-ate gas of nonrelativistic electrons and a monoatomicisentropic gas, both having � ¼ 5=3. These perfectly rea-sonable systems, which can even be described withinNewtonian theory, have no description in EiBI theory.This renders the theory at best incomplete.

It is worth stressing that our analysis relies only on thefield equations and on the form of the equation of stateclose to the stellar surface. Anymatter configuration whosebehavior resembles a polytrope (as an effective descrip-tion) with adequate accuracy in the immediate vicinity ofthe surface will be singular, regardless of any complicatedmicrophysics describing the stellar interior. There aremany known examples of stars that satisfy this property.For instance, the atmosphere of a white dwarf is wellapproximated by a polytrope with � ¼ 10=3 (see, e.g.,Refs. [27,28]).

At this stage one might claim that the polytropic,perfect-fluid description will break down at very smalldensities, and this might be a potential way out.Nonetheless, this would imply that no solutions for whitedwarfs (and for many other systems) are allowed in EiBIgravity without precise knowledge of the microphysics ofthe matter near the surface. Even after introducing somemicrophysical description, there are no guarantees that thesolution would be regular. Indeed, abandoning the fluidapproximation would just increase the differential order ofthe field equations in the matter sector, making the curva-ture even more sensitive to sharp variations in the matterfields.

For the sake of the argument, though, let us suppose thata polytropic equation of state provides a reliable descrip-tion close to the surface down to, say, �� 10�n kg=m3.Strong deviations from general relativity are expectedwhen REiBI

g � RGRg ¼ �� 3P at some radius very near

the surface. For example, surface singularities would giverise to divergent tidal forces, which can be orders ofmagnitude larger than in Einstein’s theory [18]. Let usthen require that REiBI

g & RGRg . This yields

� * 4� 1024þ2n m5kg�1 s�2; (16)

where we have assumed M� 1:4M, RS � 10�2R, and� ¼ 10=3. For n ¼ 10, the absence of strong near-surfacecurvature effects would imply � * 4� 1044 m5 kg�1 s�2.This bound, though admittedly simplistic, is nonethelessabout 40 orders of magnitude larger than other currentconstraints [3–5,11,14]. Additionally, Eq. (16) is a lowerbound whereas all other constraints (including those onecould derive by applying similar arguments to matterconfigurations well described by polytropes with 3=2<�< 2) are upper bounds that are totally incompatible withEq. (16).We emphasize that the surface singularities found above

are not a prerogative of some polytropic fluid descriptionof matter near the surface. Polytropes just allow for aconvenient analytical treatment of the problem. The keyissue is that higher-order derivatives of matter fields, whichappear in the EiBI field equations as a result of integratingout nondynamical degrees of freedom, make the geometryunacceptably sensitive to sharp variations in the matterconfiguration. Surface singularities in polytropic spheresare just one possible manifestation of this sensitivity, butsimilar phenomena will be present in real world systems,where sharp density variations are common. This short-coming, the rest of the pathologies related to the presenceof nondynamical fields, and a thorough discussion on thelimitations of the polytropic fluid approximation can beenfound in Ref. [19] for Palatini fðRÞ gravity (see alsoRefs. [29,30]). We will avoid repeating this discussionhere for EiBI gravity, as it would be nearly identical.In summary, we have shown that EiBI gravity is plagued

by serious pathologies, whose root is the fact that thetheory contains an auxiliary connection. The latter can beeliminated in order to obtain second-order dynamical equa-tions for the metric only. However, the differential struc-ture of these equations is profoundly different from generalrelativity, as higher derivatives of the matter fields appear.This makes gravity noncommulative and spacetime ge-ometry particularly sensitive to sharp changes in the matterconfiguration. In particular, a discontinuity in the matterdensity or even only in its derivatives is enough to producecurvature singularities, leading to unacceptable phenome-nology. This fact has been missed in the recent literature onEiBI gravity but it has profound consequences for theviability of the theory, similarly to the case of PalatinifðRÞ gravities [17–19] and to theories where matter iscoupled to the Ricci scalar [20].These problems appear to be a generic prerogative of

gravitational theories which do not propagate any degree offreedom other than the massless spin-2 field but insteadcontain auxiliary fields that are algebraically related to themetric and to matter. Although these theories are equiva-lent to general relativity in vacuum, once the auxiliaryfields are eliminated using some algebraic relation, thedynamical gravitational equations contain higher deriva-tives of the matter fields. This is the case discussed here,

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where the auxiliary field is the connection or, equivalently,the qab metric.

In principle, this pathology could be alleviated by intro-ducing higher-order derivatives of the metric as well (thisis, for example, the case in which matter is coupled to somenonlinear function of the Ricci scalar [31]). However, thisusually produces different problems, such as unitarity loss,instabilities, etc.

Finally, in this work we followed the original proposal,assuming the independent connection to be symmetric, thematter action to be independent from it, and that onlyRðabÞappears in the action. Relaxing these conditions would leadto the most general, metric-affine [25] version of EiBIgravity, which presumably has a richer phenomenology.Understanding whether this more general theory mayevade the pathologies discussed here would be an interest-ing extension of the present work.

We wish to thank T. Delsate, J. Steinhoff, V. Vitagliano,and especially E. Barausse for useful discussions. P. P. issupported by FCT-Portugal through Projects No. PTDC/FIS/098025/2008, No. PTDC/FIS/098032/2008, andNo. CERN/FP/123593/2011, and by the EuropeanCommunity through the Intra-European Marie CurieContract No. aStronGR-2011-298297. T. P. S. acknowl-edges partial financial support provided under the MarieCurie Career Integration Grant No. LIMITSOFGR-2011-TPS, the ‘‘Young SISSA Scientists Research Project’’scheme, and the European Union’s FP7 ERC GrantAgreement No 306425.

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