Nonsingular Promises from Born-Infeld Gravity

Franco Fiorini*

Centro Atomico Bariloche, Comision Nacional de Energıa Atomica, R8402AGP Bariloche, Argentina(Received 27 April 2013; published 23 July 2013)

Born-Infeld determinantal gravity formulated in Weitzenbock spacetime is discussed in the context of

Friedmann-Robertson-Walker (FRW) cosmologies. It is shown how the standardmodel big bang singularity

is absent in certain spatially flat FRW spacetimes, where the high energy regime is characterized by a de

Sitter inflationary stage of geometrical character, i.e., without the presence of the inflaton field. This taming

of the initial singularity is also achieved for some spatially curved FRWmanifolds where the singularity is

replaced by a de Sitter stage or a big bounce of the scale factor depending on certain combinations of free

parameters appearing in the action. Unlike other Born-Infeld-like theories invogue, the one here presented is

also capable of deforming vacuum general relativistic solutions.

DOI: 10.1103/PhysRevLett.111.041104 PACS numbers: 04.50.Kd, 04.20.Dw

After the proposal addressed in [1], the quest for non-singular classical spacetimes was carried out several timesin the context of extended theories of gravity with Born-Infeld-like structure [2–11]. The choice of Born-Infeldactions for the gravitational field is strongly suggestedfor taming the singularities present in Einstein’s theory,especially if one remembers the early success of the theoryin the analogue framework of classical electrodynamics,i.e., in the problem concerning the infiniteness of thepointlike charge self-energy characterizing Maxwell’s the-ory [12]. Remarkable enough, the Born-Infeld electromag-netic action also plays a prominent role in string theory,being the proper action for describing the electromagneticfield in D-branes [13,14], and it was shown recently thatBorn-Infeld gravitational structures of the sort consideredin Refs. [9,10] appear naturally as counterterms in four-dimensional anti–de Sitter spacetime [15]. However, in allthe abovementioned gravitational Born-Infeld schemes,and due to the fourth order field equations for the metrictensor coming from the higher order curvature componentsin the action, no single exact solution describing a regularspacetime has ever been found.

Along this quest, another closely related nonsingularprogram for the gravitational field, the so-calledEddington inspired Born-Infeld gravity, was introducedby Banados in [16] and studied carefully very recently,mostly in cosmological and astrophysical environments[17–26]. Second order motion equations in this frameworkare assured due to the independent role played by themetric and the connection, and a number of exact cosmo-logical solutions without the big bang singularity werefound [18,24], even though the theory seems to also pos-sess singular states [27,28]. Nevertheless, Eddington-Born-Infeld gravity differs from Einstein theory only whenmatter sources are present, an unfortunate fact thatdeprives us from regular black hole states in pure vacuum.

The purpose of this work is to present a number ofregular cosmological solutions arising from the previously

featured Born-Infeld determinantal gravity [29]. In manysenses, this theory captures the most distinguished pointsof all the abovementioned Born-Infeld-like schemes, for itassures second order motion equations for the vielbein fieldea and, unlike Eddington-Born-Infeld gravity, it is alsocapable of deforming vacuum general relativistic space-times. Actually, in Ref. [29] we have shown how theconical singularity present in vacuum three-dimensionalEinstein gravity was deformed into a geodesically com-plete curved spacetime by solving exactly the gravitationalBorn-Infeld motion equations.Born-Infeld-like actions are not only good candidates in

order to deal with singularities, but they are also quitenatural. By its very nature, the Lagrangian density Lmust be a scalar density of weight one and, in aD-dimensional orientable manifold, the Lagrangian ~Lis represented by a D-form which locally looks like~L ¼ Lð�; @j�Þdx1 ^ � � � ^ dxn. Here we suppose that Ldepends on certain fields� and its derivatives up to order j.In order to construct this density we have at hand acanonical procedure: just take a linear combination ofsquared roots of determinants of second rank tensors.Hence we can write in general

Lð�; @j�Þ ¼ �k

Xk

ffiffiffiffiffiffiffiffiffiffiffiffijLðkÞ

��jq

; (1)

where j j stands for the absolute value of the determinant.Formally, each one of the tensors in the combination (1)can be decomposed according to

L�� ¼ �0g��ð�Þ þ �1Fð1Þ��ð�; @�Þ

þ � � � þ �jFðjÞ��ð�; . . . ; @j�Þ; (2)

where �i, 0 � i � j, are arbitrary couplings at this point.The splitting (2) emphasizes that L�� can be viewed for-

mally as a sum of tensors FðiÞ��ð�; . . . ; @i�Þ containing

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derivatives up to order i, where we have written Fð0Þ��ð�Þ �

g��ð�Þ. This last tensor has a prominent role in (2) because

it contains no derivatives at all.Even though we do not account with a general rule for

the number of terms we have to consider in (2), a sufficientcondition in order to obtain second order differential equa-tions for the fields � results by interrupting the sum in

Fð1Þ��ð�; @�Þ � F�� [30]. Additionally, this condition also

restricts the summands in (1) to three, otherwise we wouldfind redundant terms in the Lagrangian. So, the Lagrangiantakes the form

L¼�0

ffiffiffiffiffiffiffiffiffiffiffijg��j

qþ�1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijg��þ2��1F��j

qþ�2

ffiffiffiffiffiffiffiffiffiffiffiffijF��j

q; (3)

where we called 2��1 � �1=�0, and a redefinition of theconstants �k (k ¼ 0; 1; 2) was performed. An additionalreduction will assure the proper low field limit of thetheory; taking �0 ¼ ��1 and �2 ¼ 0 the Lagrangianwill depend on just one undetermined constant (say, �),and the action will finally acquire the form

IBIG ¼ �

16�G

ZdDx½

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijg�� þ 2��1F��j

q�

ffiffiffiffiffiffiffiffiffiffiffijg��j

q�: (4)

Note that, whatever the (at this point unspecified) tensorF�� is, the low energy limit (� ! 1) of the action (4) can

be easily obtained by factoring outffiffiffiffiffiffiffiffiffiffiffijg��j

qand using

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijIþ 2��1Fj

q¼ 1þ ��1TrðFÞ þOð��2Þ; (5)

where F � F�� and I is the identity. Hence, in the low

energy limit we get the action

I# ¼ 1

16�G

ZdDx

ffiffiffiffiffiffiffiffiffiffiffijg��j

qTrðFÞ: (6)

It only remains to find the tensors g�� and F��. Note that

F�� contains just first derivatives of the dynamical field�,

so to look for it in a Riemannian context would be futile; nosecond rank tensor built with first derivatives of the metricwill give rise through its trace to the scalar curvature Rcharacterizing the Hilbert-Einstein action of generalrelativity (GR).

To link this low energy action to the Einstein-Hilbertaction, we can evoke the teleparallel representation of GR.The linkage between the standard (Riemannian) descrip-tion of general relativity and its absolute parallelism(Weitzenbock) version is summarized in the equation

T ¼ �Rþ 2e�1@�ðeT��� Þ; (7)

where T is the so-called Weitzenbock invariant,

T � S��� T�

��; (8)

with S��� defined as

S��� � 1

4ðT��

� � T��� þ T��

�Þ þ 1

2��T

��� � 1

2��T��

� ;

(9)

and T��� are the components of the Weitzenbock torsion

Ta � dea, i.e., T��� ¼ e

�að@�ea� � @�e

a�Þ. This torsion

emerges out from Weitzenbock connection ���� ¼

e�a@�e

a� (here, e

�a is the inverse of ea�, so e

�aeb� ¼ b

a). In

the Weitzenbock representation of GR, the dynamical fieldis the vielbein ea, and the metric is a subsidiary field relatedto the vielbein in the form

g�� ¼ abea�e

b�: (10)

This last equation determines the tensor g�� in terms of the

fundamental fields � ¼ ea. This implies e � detðea�Þ ¼ffiffiffiffiffiffiffiffiffiffiffijg��j

q, which appears in (7). According to Eq. (7), the

theories described by the Lagrangian densities eT and eRare dynamically equivalent, because these two quantitiesdiffer in a total derivative. Therefore, the low energyregime governed by action (6) will be general relativityprovided TrðFÞ ¼ T. This is the sole constraint for thetensor F�� in the Born-Infeld gravitational action (4).

We are now in position to find F��. A direct inspection

shows that the more general candidate should read

F�� ¼ �S��� T��� þ �S���T��� þ �g��T; (11)

where �, �, � are dimensionless constants such as �þ�þD� ¼ 1, hence, ensuring that TrðFÞ ¼ T [31]. Withthe definitions (10) and (11), the action (4) is now deter-mined. It governs the dynamics of the vielbein field ea

and contains just first derivatives of it, thus guaranteeingsecond order differential equations. It also assures that inregions where F � � the gravitational phenomena arethose predicted by Einstein’s theory. Note that the particu-lar choice � ¼ � ¼ 0 trivializes the determinantal charac-ter of action (4), for in this case we actually have that theaction becomes

IBI0 ¼ �

16�G

ZdDx

ffiffiffiffiffiffiffiffiffiffiffijg��j

q ��1þ 2

T

D�

�D=2 � 1

�; (12)

which is an fðTÞ-type action. In fact, this constitutes thesole fðTÞ-like action that can be obtained from the deter-minantal structure (4). Finally, we should mention animportant point. The tensor (11) is not invariant under localLorentz transformations of the vielbein ea, so neither is theaction (4). However, it is mandatory to note that the break-ing of Lorentz invariance happens at a Born-Infeld scale oforder ‘2BI ¼ ��1. This is an exceedingly small length scalepossibly associated with ‘p, the Planck length, where no

fully satisfactory description of the spacetime structureseems to exist currently.In order to exhibit the power hidden in action (4), from

now on we will focus on cosmological scenarios. For this

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reason we will be conservative and fix D ¼ 4. It is ourintention to deal first with spatially flat Friedmann-Robertson-Walker (FRW)-like spacetimes. For this pur-pose we have the frame ea ¼ diagð1; aðtÞ; aðtÞ; aðtÞÞ, whichleads to the spatially flat FRW line element

ds2 ¼ dt2 � a2ðtÞ½dx2 þ dy2 þ dz2�; (13)

with aðtÞ the scale factor. Again, being very conservativewe shall consider as a source a perfect fluid withstate equation p ¼ !�, where ! is the barotropic index.In this case we can easily obtain the initial value equation,i.e., the motion equation coming from varying theaction with respect to e00. By using the comoving

frame, the energy-momentum simply reads T�� ¼

diagð�;�!�;�!�;�!�Þ, and the initial value equationresults:ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1�BH2pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�AH2

p ½1þ2BH2�3ABH4��1¼ 16�G

��; (14)

where

A ¼ 6ð�þ 2�Þ=�; B ¼ 2ð2�þ �þ 6�Þ=�: (15)

This is the modified Friedmann equation governing thedynamics of the Hubble rate H ¼ _a=a. The energy density� is linked to the scale factor by the conservation equation_�þ 3ð�þ pÞH ¼ 0. If one is dealing with a perfect fluidstate equation this conservation law assumes the form

�ðtÞ ¼ �0

�a0aðtÞ

�3ð1þ!Þ

; (16)

where �0 and a0 are two constants alluding to present-dayvalues. If we replace (16) in the right-hand side of (14), weget a first order differential equation for the scale factoraðtÞ. The other motion equations coming from the remain-ing tetrad components ea�, if non-null, are just a conse-

quence of combining the time derivative of (14) with (16),so they do not provide additional information.

It is then convenient to carefully inspect Eq. (14) for acase that, due to its simplicity, is of particular interest,namely, when B ¼ 0. In addition to B ¼ 0, the normaliza-tion condition �þ �þ 4� leads us to A ¼ 12=�, so themotion equation reduces to

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 12H2

�

q � 1 ¼ 16�G

��: (17)

Surprisingly enough [32], Eq. (17) is the same as the oneobtained previously in the Born-Infeld gravitational theorywith fðTÞ structure discussed in Refs. [33,34]. This fieldequation conduces to an exact solution with remarkableproperties which we just summarize here (see the men-tioned references for details). For every barotropic index!>�1 (radiation and dust matter lying in this interval),the scale factor describes a regular (geodesically complete)spacetime without the big bang singularity and possessing

a natural inflationary stage of geometrical character.Actually, it can be easily seen that

aðt ! �1Þ / exp

� ffiffiffiffiffiffi�

12

st

�; (18)

so there exists a maximum Hubble factor Hmax ¼ffiffiffiffiffiffiffiffiffiffiffi�=12

pas we backtrack the cosmic evolution. This maximumHubble factor assures for the early Universe a �-drivende Sitter evolution of infinite duration, and it is responsiblefor the geodesic completeness of the spacetime, for everypast directed timelike or null geodesic can be extendedto arbitrary values of the affine parameter. As a conse-quence of the presence of Hmax, it can be easily shownthat the invariants of the geometry also reach saturationvalues given by Rmax ¼ �, ðR��R��Þmax ¼ �2=4 and

ðR����R

���� Þmax ¼ �2=6 as t ! �1.

The characterization of isotropic and homogeneouscosmological manifolds would not be complete if wewould cease the analysis in the spatially flat FRW models.In order to carry on with the investigation, let us deal nowwith spatially curved cosmological models. The framesadapted for this symmetry are substantially more compli-cated than the simple diagonal frames corresponding to thespatially flat case worked above. In Ref. [35] it was showedthat a global basis of frames for spatially curved FRWspacetimes reads

e0 ¼ dt; e1 ¼ aðtÞE1; e2 ¼ aðtÞE2; e3 ¼ aðtÞE3;

(19)

where the 1-forms E1, E2, and E3 are

E1

k¼ �k cos dc þ sinðkc Þ sin cosðkc Þd

� sin2ðkc Þsin2 d�;

E2

k¼ k sin cos�dc � sin2ðkc Þ

� ½sin�� cotðkc Þ cos cos��d � sin2ðkc Þ sin ½cotðkc Þ sin�þ cos cos��d�;

E3

k¼ �k sin sin�dc � sin2ðkc Þ

� ½cos�þ cotðkc Þ cos sin��d � sin2ðkc Þ sin ½cotðkc Þ cos�� cos cos��d�:

(20)

This global basis leads to the line element

ds2 ¼ dt2 � a2ðtÞk2½dðkc Þ2� sin2ðkc Þðd 2 þ sin2 d�2Þ�; (21)

where (c , , �) are standard spherical coordinates.The parameter k appearing in Eqs. (20) and (21) takes

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the values k ¼ 1 for the spatially spherical universe andk ¼ i for the spatially hyperbolic one.

The equations of motion taking into account the fullparameter space (�, �, �) are indeed very complicated forthe spatially curved manifolds under consideration, and itis not our intention to deal with them in their full general-ity. In analogy with the study made before, we shall focuson a few cases of particular interest for our present con-cerns. The fine solution encoded in Eq. (17) has an equallynice counterpart in the curved case. Actually, the motionequation for B ¼ 0 reads

ð1� 1�a2

Þ3=2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 12H2

�

q � 1 ¼ 16�G

��; (22)

where from now on þ and � correspond to the closed andopen cases, respectively. An exact solution of this equationis rather elusive, but we can extract its most importantfeatures by writing it in the form

_y 2 þ VðyÞ ¼ 0; y ¼ aðtÞa0

(23)

after defining the effective potential VðyÞ given by

VðyÞ ¼ � �

12y2�1� ð1� k0y

�2Þ3ð1þ �0y

�3ð1þ!ÞÞ2�; (24)

where we have defined the constants k0 ¼ 1=�a20 and

�0 ¼ 16�G�0=�.If we focus on the high energy regime where the theory

supposes to make a difference with respect to the GRsingular behavior, we can expand the potential (24) in thesmall quantity y ¼ aðtÞ=a0 (from now on we will take! ¼ 1=3). Under this circumstance, Eq. (23) results in

_y 2 � �

12y2 ¼ Oðy4Þ 0; (25)

Naturally, we have then

aðtÞ exp

� ffiffiffiffiffiffi�

12

st

�; as aðtÞ=a0 ! 0: (26)

Again, as in its flat counterpart of Eq. (18), we see how theinitial singularity is removed by the presence of an infla-tionary early stage driven by the Born-Infeld constant �.Note that this result is independent of the open or closedcharacter of the spatial slices.

Some other interesting results emerge when one consid-ers the case� ¼ 0,�� 12� ¼ 0. For this particular choiceof parameters the initial value motion equation reads

1� 1�a2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1� 1�a2

� 12H2

�

q � 1 ¼ 16�G

��: (27)

This equation would be even harder to solve than the one ofthe previous case [Eq. (22)], so again we will put it in theform (23). Now the effective potential results

VðyÞ ¼ � �

12y2�1� k0y

�2 ��

1� k0y�2

1þ �0y�3ð1þ!Þ

��: (28)

Unlike in the previous case, it is easy to see that when y goesto zero we have now Vð0Þ ¼ 1=ð12a20Þ, where it is impor-

tant to take note of the sign inversion. This peculiarity leadsto different dynamics according to the open or closedcharacter of the Universe. For the closed case we canactually expand the potential in powers of y in the samefashion we did before, to obtain

_y 2 � �

12y2 � a�2

0

12 0: (29)

In this way the high energy regime for the scale factor isgiven by aðtÞ t, with an associated Hubble rateH t�1.According to this picture the closed Universe possesses asingularity in t ¼ 0, and it expands for t > 0 in an accel-erated manner due to €y > 0.A radically different picture emerges out when one

considers the open Universe. Given that the potentialgoes to a positive number when y ! 0, and that it isnegative when y ! 1, it a has a root somewhere.Actually, by inspecting Eq. (28) for the minus sign weeasily see that the root results

ymin ¼ffiffiffiffiffik0

p; ) amin ¼ 1=

ffiffiffiffi�

p: (30)

Because the ‘‘energy level’’ in Eq. (23) is null, we observe

that the Universe expands from the minimum size amin ¼��1=2. The functional form of the scale factor near thisminimum can be obtained by expanding the potential (28)in the small quantity y� ymin. In this way we find thatEq. (23) reduces to

_y 2 � �ffiffiffiffiffik0

p6

yþ �k06

0; (31)

leading to the scale factor

aðtÞ amin þffiffiffiffi�

p24

t2: (32)

This constitutes a bounce of the scale factor in the eventt ¼ 0, where H ¼ 0. Unlike the closed case consideredabove, the spacetime results now geodesically complete,and the cosmic evolution starts its accelerated expansion

from a minimum volume given by a3min ¼ �3=2 with a

maximum energy density

�max / a�4min ¼ �2: (33)

This quantity can be interpreted as the maximum energythat can be stored as a consequence of the minimumvolume existent due to the repulsive quantum effects gov-erning the very early Universe. The reason why this occursonly in the context of open models remains unclear. Thisand many other open questions, such as the existence ofregular, asymptotically flat vacuum black hole solutions,will be a matter of future works.

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I would like to thank Rafael Ferraro for sharing so manyyears of enjoyable discussions and J. Areta for a carefulreading of themanuscript. I also thank F. Canfora for makingme aware of the inclusion of a pure trace term in Eq. (11)(see Ref. [31]). This work was supported by CONICET.F. F. is a member of Carrera del Investigador Cientıfico(CONICET, Argentina).

*francof@cab.cnea.gov.ar[1] S. Deser and G.W. Gibbons, Classical Quantum Gravity

15, L35 (1998).[2] J. A. Feigenbaum, Phys. Rev. D 58, 124023 (1998).[3] J. A. Feigenbaum, P. G.O. Freund, and M. Pigli, Phys.

Rev. D 57, 4738 (1998).[4] D. Comelli, Phys. Rev. D 72, 064018 (2005).[5] D. Comelli and A. Dolgov, J. High Energy Phys. 11 (2004)

062.[6] J. A. Nieto, Phys. Rev. D 70, 044042 (2004).[7] M.N. R. Wohlfarth, Classical Quantum Gravity 21, 1927

(2004); 21, 5297 (2004).[8] R. Garcıa-Salcedo, T. Gonzalez, C. Moreno, Y. Napoles,

Y. Leyva, and I. Quiros, J. Cosmol. Astropart. Phys. 02(2010) 027.

[9] I. Gullu, T. C. Sisman, and B. Tekin, Classical QuantumGravity 27, 162001 (2010).

[10] I. Gullu, T. C. Sisman, and B. Tekin, Phys. Rev. D 81,104018 (2010).

[11] D. N. Vollick, Phys. Rev. D 72, 084026 (2005).[12] M. Born and L. Infeld, Proc. R. Soc. A 144, 425 (1934);

147, 522 (1934); 150, 141 (1935).[13] E. S. Fradkin and A.A. Tseytlin, Phys. Lett. 163B, 123

(1985).[14] A. A. Tseytlin, arXiv:hep-th/9908105.[15] D.P. Jatkar andA.Sinha,Phys.Rev.Lett.106, 171601 (2011).[16] M. Banados, Phys. Rev. D 77, 123534 (2008).

[17] M. Banados, P. G. Ferreira, and C. Skordis, Phys. Rev. D79, 063511 (2009).

[18] M. Banados and P.G. Ferreira, Phys. Rev. Lett. 105,011101 (2010).

[19] C. Escamilla-Rivera, M. Banados, and P. G. Ferreira, Phys.Rev. D 85, 087302 (2012).

[20] J. H. C. Scargill, M. Banados, and P.G. Ferreira, Phys.Rev. D 86, 03533 (2012).

[21] P. Pani, V. Cardoso, and T. Delsate, Phys. Rev. Lett. 107,031101 (2011).

[22] T. Delsate and J. Steinhoff, Phys. Rev. Lett. 109, 021101(2012).

[23] P. P. Avelino and R. Z. Ferreira, Phys. Rev. D 86, 041501(2012).

[24] P. P. Avelino, Phys. Rev. D 85, 104053 (2012).[25] P. P. Avelino, J. Cosmol. Astropart. Phys. 11 (2012) 022.[26] Y.-H. Sham, P.-T. Leung, and L.-M. Lin, Phys. Rev. D 87,

061503 (2013).[27] P. Pani and T. P. Sotiriou, Phys. Rev. Lett. 109, 251102

(2012).[28] M. Bouhmadi-Lopez, C.-Y Chen, and P. Chen,

arXiv:1302.5013.[29] R. Ferraro and F. Fiorini, Phys. Lett. B 692, 206 (2010).[30] It could be the case that specific combinations of tensors

F�� containing up to second derivatives of the fields couldlead to second order motion equations too, as, for instance,in Lovelock actions. We will not deal with those(important) subtleties in this work.

[31] In the original construction of Ref. [29] we haveconsidered the theory with � ¼ 0.

[32] It is important to be aware that the theory consideredhere and the one of Refs. [33,34] are totally different.The equivalence is valid at the level of this particularsolution.

[33] R. Ferraro and F. Fiorini, Phys. Rev. D 75, 084031 (2007).[34] R. Ferraro and F. Fiorini, Phys. Rev. D 78, 124019

(2008).[35] R. Ferraro and F. Fiorini, Phys. Lett. B 702, 75 (2011).

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