Born-Infeld inspired modifications of gravity
General Relativity has shown an outstanding observational success in the scales where it has been directly tested. However, modifications have been intensively explored in the regimes where it seems either incomplete or signals its own limit of validity. In particular, the breakdown of unitarity near the Planck scale strongly suggests that General Relativity needs to be modified at high energies and quantum gravity effects are expected to be important. This is related to the existence of spacetime singularities when the solutions of General Relativity are extrapolated to regimes where curvatures are large. In this sense, Born-Infeld inspired modifications of gravity have shown an extraordinary ability to regularise the gravitational dynamics, leading to non-singular cosmologies and regular black hole spacetimes in a very robust manner and without resorting to quantum gravity effects. This has boosted the interest in these theories in applications to stellar structure, compact objects, inflationary scenarios, cosmological singularities, and black hole and wormhole physics, among others. We review the motivations, various formulations, and main results achieved within these theories, including their observational viability, and provide an overview of current open problems and future research opportunities.
keywords:Born-Infeld gravity, Astrophysics, Black Holes, Cosmology, Early universe, Compact objects, Singularities
- 1 Preamble
2 Born-Infeld theories
- 2.1 Born-Infeld electromagnetism in a nutshell
- 2.2 The Deser-Gibbons proposal: The ghost problem of the metric formalism
- 2.3 Other proposals in the metric formalism
- 2.4 Eddington-Born-Infeld gravity
- 2.5 Field equations
- 2.6 The two frames of Born-Infeld gravity and the physical relevance of the auxiliary metric
- 2.7 Classes of Born-Infeld inspired gravity.
- 2.8 Final remarks
- 3.1 Newtonian limit and fluid approximation
- 3.2 Non-relativistic stars
- 3.3 Relativistic stars
- 3.4 Final remarks
4 Black Holes
- 4.1 Spherically symmetric solutions with matter in Born-Infeld gravities
- 4.2 Eddington-inspired Born-Infeld black hole solutions
- 4.3 Wormholes
- 4.4 Electromagnetic black holes and geons
- 4.5 Non-singular solutions
- 4.6 Higher and lower dimensional models and solutions
- 4.7 Magnetically charged solutions with cylindrical symmetry
- 4.8 Final remarks
- 5.1 Eddington-inspired Born-Infeld gravity
- 5.2 General cosmological framework for Born-Infeld inspired gravity theories
- 5.3 Elementary symmetric polynomials extension: Minimal model
- 5.4 Functional extensions of Born-Infeld gravity
- 5.5 Extensions including a Ricci scalar
- 5.6 Other extensions
- 5.7 Final remarks
- 6 Concluding remarks, open questions and prospects
1.1 Motivations and introduction
General Relativity (GR) is nowadays firmly established as the standard theory to describe the gravitational interaction with the same mathematical framework and physical principles as those used by Einstein more than one hundred years ago. After all this time, it still stands out as the most successful theory able to explain all the gravitational phenomena in a wide range of scales. Direct tests comprise from sub-milimeter to Solar System scales, where the Parameterised Post-Newtonian formalism has allowed to constrain deviations from GR in the weak field limit at the level of Will:2014kxa (). Moreover, the amazing direct observation of gravitational waves by the LIGO collaboration is also compatible with the prediction of GR for the merging of two black holes, where strong field effects are relevant Abbott:2016blz (); TheLIGOScientific:2016pea (). On the other hand, we have witnessed how the accurate measurements of the CMB anisotropies and galaxy surveys have established CDM as the standard model of cosmology, which is based on a homogeneous and isotropic Universe governed by GR as the theoretical framework for gravity. This picture requires an unobserved cold dark matter source plus a tiny cosmological constant to account for the current accelerated expansion of the Universe. Furthermore, the CDM model needs to be supplemented with the inflationary paradigm so that the primordial perturbations are generated during a short period of accelerated expansion at very early times. For a review on the current status of the CDM model, its challenges and possible alternatives, see Bull et al. Bull:2015stt (). Further observational tests, for instance via the Euclid satellite Amendola:2012ys (), will hopefully shed light on all the additional elements above and their contributions to fundamental physics.
Despite its observational success, there are strong arguments supporting and/or motivating to seek for theories beyond GR. These arguments are of two kinds. On the theoretical side, GR itself predicts the unavoidable existence of spacetime singularities, i.e., events where our ability to make predictions comes to an end Senovilla:2014gza (). Such singularities are unavoidably developed during the gravitational collapse of a fuel-exhausted star to form a black hole Joshibook (), as well as during the cosmological evolution in the early Universe. In this sense, the requirement that “nothing should cease to exist suddenly” and that “nothing should emerge out of nowhere” should be seen as basic consistency conditions for any physical theory, including GR. The existence of singularities in GR unavoidably leads to the breakdown of these conditions, and gives clear indications that we have pushed the theory beyond its regime of validity. According to the standard lore Burgess:2003jk (), GR is a good effective field theory up to a scale somewhere near the Planck mass and, therefore, those singular behaviours are regarded as manifestations that the higher order operators should be included. For this reason, quantum gravity is usually expected to regularise such singularities, although it is possible that high energy modifications of GR might allow to classically regularise some of those singularities before reaching the cut-off of the theory without invoking any quantum gravity effects.
On the phenomenological side, the unprecedented experimental precision reached by observational cosmology requires the aforementioned ad hoc extra ingredients in order to account for the observations. While the cosmological constant is fundamental part of the theory and its difficulty resides in its aesthetic value that poses naturalness problem, dark matter and inflation require the introduction of new physics and, as a consequence, a large degeneracy among all the proposed models. This degeneracy is more prominent owing to the lack of experimental signatures from laboratory experiments and particle accelerators, despite the existence of different ongoing galactic Strigari:2013iaa (); Ackermann:2015zua (), cosmic rays Aramaki:2015pii (), CMB Ade:2015rim (), collider Carpenter:2012rg () and underground laboratory Akerib:2016vxi () searches.
In view of the above situation, one may wonder if the difficulties and lack of naturalness faced in GR indicates that a new framework to describe gravity is needed, which would yield different astrophysical and cosmological observational signatures from the CDM model Joyce:2014kja (). From a conservative perspective, one may stick to the point of view that gravitation is a manifestation of the curvature of spacetime, but one that is not sufficiently well described by GR. As a matter of fact, the common factor to all the issues discussed above is the extrapolation of GR to regions where it has not been directly well tested and this may introduce significant bias in the interpretation of astrophysical and cosmological observations. The consideration of additional curvature contributions to the Einstein-Hilbert action, usually under the form of curvature invariants, has been used in the literature as a way to enlarge the phenomenology of gravity. This typically involves a number of problems such as higher-order field equations, which usually entail the presence of ghost-like instabilities Stelle:1977ry (); Stelle:1976gc (); Nunez:2004ts (); Chiba:2005nz (), or the difficulty to make these models compatible with solar system tests due to the existence of new degrees of freedom Olmo:2006eh (); Chiba:2006jp ()111Those models avoiding these shortcomings and, at the same time, being able to provide a consistent cosmological expansion which is coherent with the GR limit are usually termed as viable, see e.g. Amendola:2006kh (); Cognola:2007zu (); delaCruz-Dombriz:2015tye ().. The arbitrariness in the choice of curvature invariants also implies a strong lack of naturalness in these models. The main references regarding such models and their applications are provided by de Felice and Tsujikawa DeFelice:2010aj (), Capozziello and de Laurentis Capozziello:2011et (), and Nojiri and Odintsov Nojiri:2010wj () (see also Faraoni and Sotiriou Sotiriou:2008rp ()).
The difficulties with ghost-like instabilities in higher curvature modifications of gravity can be avoided by formulating those theories in the so-called Palatini or metric-affine formalism Olmo:2011uz (). Though this approach is sometimes viewed as a shortcut to obtain the field equations of GR (and rightly so for some specific Lagrangians), it actually represents an inequivalent formulation of gravity in which metric and affine structures are regarded as independent geometrical entities. The fact that, when formulated à la Palatini Ferraris1982 (), metric and connection are compatible in the case of GR has spread the view that such condition should always hold regardless of the form of the gravity Lagrangian. However, this is not true in general. In the metric-affine approach, the specific relation between metric and connection is determined by the field equations, not imposed a priori by mathematical conventions. In fact, whether the affine connection is determined by the metric degrees of freedom or not is as fundamental a question as the number of spacetime dimensions or the existence of supersymmetry.
The metric-affine or Palatini approach, therefore, avoids the problems with ghosts that affect extensions of GR in the usual metric formulation. In vacuum configurations, the field equations of these theories boil down to Einstein’s equations with an effective cosmological constant Ferraris:1992dx () which, apparently, supports their compatibility with orbital motion tests (see Olmo:2005zr (); Olmo:2008ye () for a discussion). Though this mathematical framework cannot solve on its own the arbitrariness in the choice of gravity Lagrangian, a novel class of extensions of GR with a solid motivation for a high-energy completion of gravity has been proposed and explored with much interest in the last few years. These models are motivated by the Born-Infeld approach to electrodynamics, where a modification of Maxwell’s Lagrangian is introduced to set an upper bound on the electromagnetic field intensity BI1934, with the result that the divergence of the self-energy of a point-like charge is regularised. This type of high-energy modification is analogous to the transformation that leads from a free particle in Newtonian mechanics to a free relativistic particle, whose maximum speed is bounded by the speed of light. The same Lagrangian structure describes the electromagnetic fields of -branes in string theories Gibbons:1997xz (); Brecher:1998su (); Callan:1997kz (). It is natural to wonder whether such an approach, now fully defined in terms of geometrical objects, could play a similar role in order to avoid divergences and spacetime singularities in the high-energy/curvature regime and, accordingly, different proposals have been considered in the literature. Indeed, a major reason for the investigation of such models is the fact that, using standard matter sources satisfying the energy conditions, they naturally lead to non-singular cosmologies, inflationary scenarios without the need for scalar fields, and black hole spacetimes without singularities, among other appealing results. Moreover, the physics of these gravity theories has been studied in numerous astrophysical, black hole and cosmological scenarios where high-energy physics is relevant.
In this work we shall refer to this kind of models, which are close to the original spirit of Born-Infeld electrodynamics, as Born-Infeld inspired modifications of gravity. They are defined by the following basic principles:
Square-root form: Some geometric object(s) appears under a square-root with a determinantal structure in the action which defines the gravitational theory, alongside with some new mass/length scale.
Consistency: No obvious pathologies are present, among which the absence of ghost-like instabilities is of utmost importance. In turn, this almost unavoidably enforces the use of a metric-affine formulation.
High-energy modification: The modifications of GR mostly occur in the ultraviolet regime, i.e., in regions of large mass/curvature or short scales. This implies that GR is recovered in the low-energy limit.
Nonetheless, as there are available proposals in the literature for these theories that run away from one (or both) of the two last requirements, for completeness of this work we shall also discuss such proposals. A more precise description and classification of such theories will be presented in section 2, alongside a criticism of each of them.
This review is intended to fill a gap in the recent literature of Born-Infeld inspired modifications of gravity by providing a comprehensible account of the many different scenarios on which these classes of theories have been considered, including the astrophysics and internal structure of compact objects, solar physics constraints, modifications on black hole structure, non-singular black holes and wormholes, early universe and bouncing solutions, inflation, and dark energy, among others. Its aim is to summarise, classify and unify the different theoretical approaches, to clarify the assumptions on which the different approaches to build the theory are formulated, to discuss the numerous theoretical and phenomenological results, to highlight the experimental constraints these theories are subjected to, to clarify some existing misunderstandings, and to provide an overview of the future research opportunities. It is designed to be useful both for pure theorists and for astrophysicists/cosmologists working on alternatives to the CDM (plus inflation) model.
For a review on modified gravity in cosmology mainly focused on infrarred modifications of gravity in connection with late-time solutions (but with little contact with Born-Infeld-inspired theories or the Palatini formalism), see instead Clifton et al. Clifton:2011jh (). For additional astrophysical and cosmological observational constraints over different modified theories of gravity deviating from GR predictions, see Berti et al. Berti:2015itd ().
The main content of this review is split in four sections, according to the context on which Born-Infeld-inspired theories of gravity have been investigated.
In section 2 we will briefly review the original Born-Infeld electrodynamics theory from which the motivation for analogue constructions within gravity emerges. After explaining the early attempts that resulted in pathological theories, we will introduce what represents the most extensively studied theory of gravity with the Born-Infeld structure. The slightly different formulations of such a theory will be discussed as well as the main equations. Along the way, we will spend some time discussing the two frames existing in these theories and clarify the physical meaning of the different geometrical objects arising in them. We will end this section with a survey on the different Born-Infeld inspired theories of gravity existing in the literature and we will provide a general mathematical framework for these theories. The general developments introduced in this section will serve as starting points for the practical applications discussed in the subsequent sections.
In section 3 some attempts to place observational constraints on the Born-Infeld theory using stellar models are reviewed. We will make special emphasis on the central role played by the energy density in the modified dynamics of this theory, which affects in a nontrivial way the mass-radius relation and maximum mass limit of compact objects, the energy transport mechanisms and oscillation frequencies of stars, the intensity of neutrino fluxes from the Sun, …providing numerous tests to confront the theory with observations. The need for a careful description of the outermost layers of compact objects is also discussed in detail, considering for this purpose some relevant examples in which the peculiarities of metric-affine theories demand additional modeling beyond the canonical approaches of GR.
In section 4 we will review the counterparts of the Schwarzschild and Reissner-Nordström black hole solutions of GR, where a coupling to a Maxwell field is considered. We will spend some time explaining the procedure for derivation of the corresponding solutions, so as to highlight some important subtleties. Then we will explain the main differences of such solutions as compared to the GR ones, in particular, regarding the modifications on the horizon structure, which bear some resemblance to that of black holes supported by Born-Infeld electrodynamics in GR. On the other hand, we will study how these black holes may affect the description of strong gravitational lensing as well as the physics regarding mass inflation. An important issue will be the existence of non-singular geometries in these theories, whose nature and properties is tested using different well-established criteria. We also review some wormhole solutions constructed out of anisotropic fluids. Finally, different extensions to higher and lower dimensions, as well as to magnetically charged solutions will be discussed.
The section 5 will be devoted to the effects of Born-Infeld inspired theories of gravity in cosmological scenarios. We will discuss the existence of homogeneous and isotropic solutions free from Big Bang singularities with standard matter sources as well as couplings of these theories to other types of fields. Anisotropic models and inhomogeneous perturbations will also be discussed. Since the Born-Infeld inspired theories are designed to modify gravity in the high curvatures regime, their natural domain of applicability is the early universe. However, there have also been studies where Born-Infeld theories are considered for late time cosmology and we will revisit them.
We will end in section 6 by giving a summary of all the material presented in the core of this review. We will discuss the most outstanding achievements and will make special emphasis on the open questions that remain as well as the prospects for future research within the field.
In this section we will review some basic ingredients of differential geometry that we will use throughout the different parts of this review. We will assume that the reader is familiarized with the concepts presented here and the main purpose of this section will be to fix the notation and the conventions for the different choices of signs and numerical factors in the definitions of relevant geometrical objects. It does not intend to be an exhaustive and rigorous exposition, but rather it should be regarded as a brief compendium of useful concepts and formulae. For a more detailed treatment we urge the reader to consult her/his favourite book on differential geometry or General Relativity or, in the lack thereof, see e.g. schouten1954ricci (); misner1973gravitation (); Waldbook (). One reference particularly useful and with numerous applications in gravitation and gauge theories is EGUCHI1980213 ().
Connection, curvature and torsion conventions
The theories that will be considered throughout the present review will be formulated either in (pseudo-)Riemannian or non-Riemannian geometries. In order to construct the necessary geometrical framework, we first introduce a 4-dimensional manifold that will eventually constitute our spacetime. In that spacetime we introduce a general connection that defines the covariant derivative of a 1-form as
This definition results in the following covariant derivative for a vector field :
These expressions can then be easily generalised to arbitrary tensors so that
In addition to objects with tensorial transformation properties under changes of coordinates, we will also find objects with other transformation properties throughout this review. In particular, we will encounter vector densities, which pick up some power of the Jacobian under a change of coordinates. If is a vector density of weight , it transforms as222This is true for true tensorial densities. For pseudo-tensorial densities the transformation also picks up a sign for parity odd transformations.
This modified transformation property makes necessary to add a piece to the definition of the covariant derivative to maintain its tensorial character, that reads
Again, this formula can be generalized for an arbitrary tensorial density by adding a term in (1.3).
After introducing the connection, we can start computing geometrical objects from the commutator of covariant derivatives acting on different tensorial fields. The first commutator we can compute is that of two covariant derivatives acting on a scalar field, which reads
the torsion tensor. Let us notice that it has tensorial transformation properties because it can be seen as the difference of two connections. The next geometrical important object is obtained by computing the commutator of two covariant derivatives acting on a vector field, which can be written as
where we have introduced the curvature Riemann tensor, defined as
Out of this general Riemann tensor, we can build two independent traces, namely the Ricci tensor defined as usual and the homothetic tensor given by . While the Ricci tensor does not have any symmetry (even for a torsion-free connection), the homothetic tensor is antisymmetric. A quantity that we will need to compute field equations is the variation of the Ricci tensor under an infinitesimal displacement of the connection , which reads
where the bars denote quantities corresponding to the background connection . This relation reduces to the usual Ricci identity for torsion-free connections.
After setting-up the notation and convention for the objects directly related to the connection, we will turn to the conventions for the metric tensor . This object is assumed to be non-degenerate and its inverse is denoted with upper indices so that and so on. Furthermore, this object is used to raise and lower indices of arbitrary tensors (i.e. it establishes an isomorphism between the tangent and the co-tangent spaces). We will use the mostly plus signature for the metric so that the Minkowski metric is . The covariant derivative of the metric defines the non-metricity tensor as
Notice that the non-metricity is symmetric in the last two indices. This expression can be solved in the usual way to write the connection as
where the first term is the standard Levi-Civita piece, the second term depends on the non-metricity and the last term (usually called contorsion) is determined by the torsion. If the non-metricity vanishes and the connection is symmetric (i.e. vanishing torsion), the connection reduces to the Levi-Civita connection given by the Christoffel symbols. With a metric at hand, there is yet a third rank-2 tensor we can construct from the Riemann tensor of the full connection, known as co-Ricci tensor and defined as . Of course, for the Levi-Civita connection all three objects coincide up to a sign so the only independent trace of the Riemann is the Ricci tensor . Throughout this review we will denote with calligraphic letters the objects corresponding to an arbitrary connection, while the curvature objects associated to the Levi-Civita connection will be denoted with normal characters
The determinant of the metric is a tensorial density of weight so that is a tensorial density of weight whose covariant derivative is given by
We can thus use to tensorialize tensorial densities. For instance, if is a tensorial density of weight , then has weight zero. Another important use of this object is to construct invariant volume elements. Since generates a Jacobian under a change of coordinates, we can compensate for that by adding a factor of so that will be invariant. Let us notice that this is a choice and actually we could use with being whatever scalar density of weight . For instance, with being an arbitrary rank 2 tensor will do the job.
The totally antisymmetric tensor is defined as
with the totally antisymmetric Levi-Civita symbol with . The contravariant version of it is
The Levi-Civita tensor allows to introduce the Hodge dual that establishes an isomorphism between333Let us remember that a form is nothing but a completely antisymmetric tensor. -forms and -forms. If is a -form, its dual is defined as
As a specific example that we will use throughout the review, the dual of a 2-form in four dimensions is given by
For an antisymmetric rank 2 tensor we can introduce the so-called electric and magnetic components relative to an observer with 4-velocity as
For an observer with these definitions reduce to the usual expressions and .
An alternative language to describe the geometrical framework of gravity theories is provided by the formalism of frames. We start by introducing a set of vectors defined on the tangent space with a Lorentz index so that they satisfy the following orthonormality condition
with respect to the Minkowski metric . These objects receive several aliases in the literature: tetrads, vierbein or frames. The corresponding dual objects belonging to the cotangent space are defined in the usual way . This relation in turns also implies . They are sometimes interpreted as the square root of the metric because can be expressed as
The vierbein can be used to transform tangent space indices into spacetime indices for arbitrary tensors. All the geometrical objects introduced above thus have their corresponding object in the tetrads formulation. If we introduce the so-called spin connection given by the set of 1-forms , the associated curvature 2-form is given by
where is the exterior derivative and stands for the exterior product. The existence of the tetrad allows to define the torsion 2-form as
Applying the exterior derivative on this expression we obtain a consistency condition
that relates all the relevant objects, namely, the tetrads, the spin connection, the torsion and the curvature. Taking a second exterior derivative of this expression will yield the usual Bianchi identities, which we do not need to display here. Instead, let us focus on two special connections that will be of relevance for this review. The first one is defined by the condition of being torsion-free, so it is defined by and it is the relevant one for the usual formulation of General Relativity. The second connection is curvature-free so we have and defines the so-called Weitzenböck space. This is the natural place for the Teleparallel formulation of GR.
A perfect fluid can be defined as one in which the energy-momentum tensor is locally seen as isotropic and it is fully determined by its density and its pressure . According to this definition, the energy-momentum tensor of a perfect fluid as seen by an observer with 4-velocity () is given by.
where it is immediate to see that and . For a comoving observer with we have that and .
For a general energy-momentum tensor, there is a set of conditions known as energy conditions that play an important role in theories of gravity in relation with singularity theorems, instabilities, superluminal propagation or entropy bounds. In the following we list them for future reference:
Weak Energy Condition (WEC). This condition states that for every time-like vector (). For a perfect fluid, it implies the positivity of the energy density as measured by any observer and .
Dominant Energy Condition (DEC). This condition is satisfied if for every causal vector () and is a future-oriented causal vector. For a perfect fluid, this condition translates into .
Strong Energy Condition (SEC). The SEC is satisfied if for every time-like vector (). A perfect fluid satisfies this conditions if and .
Null Energy Condition (NEC). The NEC is satisfied if for any null vector () the condition holds. For a perfect fluid this implies . This condition is satisfied for all known types of matter and it is saturated by a cosmological constant.
Given a rank-2 tensor, we will often use a hat to denote the corresponding matrix. Thus, the metric tensor will also appear as and its inverse will be denoted by and similarly for other objects. The determinant of a matrix will be explicitly spelled out as or will be alternatively denoted as where no confusion with absolute value should occur. In the special case of a metric , we will alternatively use the broadly used notation for its determinant. Analogously, for the trace of a matrix we will use either the explicit notation or the more compact notation where, again, the context should clarify when the square brackets stand for the trace or simply play the role of actual brackets.
A recurrent matrix formula that we will use throughout this review is the expansion valid for an arbitrary matrix given by
where is the identity and the elementary symmetric polynomials which, for the case of interest here of , read:
It is useful to notice that the last elementary symmetric polynomial coincides with the determinant of . Moreover, if is antisymmetric its trace is identically zero and, thus, and vanish.
Units and constants
Unless otherwise stated, we will use units with . We will mostly use the reduced Planck mass, related to Newton’s constant as . We will also make use of the Einstein’s constant .
2 Born-Infeld theories
The class of theories that generally go under the name of Born-Infeld all share the same basic feature of being defined in terms of some square root structure aimed at regularising the presence of divergences. The inception of these theories originated from the pioneering works by Born and Infeld in the 1930’s Born:1933qff (); Born410 (); 1933Natur.132.1004B (); Born:1934gh () where they assumed a principle of finiteness, according to which physical quantities are always bounded and can never become infinite. The self-energy of the electron, or a general point-like charged particle, is infinite in the classical Maxwell’s theory so they searched for a non-linear modification capable of regularising this divergence as to comply with the principle of finiteness, i.e., a non-linear theory where point-like charges had finite self-energy444We should perhaps remark here that, at the time when Born and Infeld developed their theory for electromagnetism, the full machinery of quantum electrodynamics and the renormalization techniques were not available. Today we know that quantum electrodynamics is a renormalizable quantum field theory where physical quantities are finite and, in particular, the charge of a particle acquires radiative corrections at high energies owed to virtual processes.. Motivated by the existence of an upper bound for the velocities of particles in relativistic mechanics, in the summer of 1933 Born proposed to introduce the same square root structure for electromagnetism in order to have an upper bound for the electric fields Born:1933qff (); Born410 (). A few months later Infeld joined Born and together worked on a better version of this construction because they wanted a theoretically better motivated argument for such a theory and, then, they argued that the square root structure should come in from symmetry arguments. In analogy with mechanics where going from Newtonian to relativistic mechanics means upgrading Galilean transformations to the fully relativistic Lorentz group, Born and Infeld assumed that the Lorentz symmetry of Maxwell’s theory should be enlarged in the new theory. They considered the new symmetry to be the full group of coordinate transformations which, after imposing the recovery of Maxwell’s theory in the appropriate limit, led to the non-linear theory now known as Born-Infeld electromagnetism, expressed as the square root of a certain determinant 1933Natur.132.1004B (); Born:1934gh (). It is no surprise that the use of symmetries as a guiding principle gave rise to a remarkable theory of non-linear electromagnetism which, not only classically regularises the self-energy of point like charges, but it also shares some interesting features with Maxwell’s theory and found a natural arena in the realm of other theoretically appealing theories, like e.g. string theory polchinski1998stringI (); polchinski1998stringII (); zwiebach2009first ().
Given the success of Born-Infeld theory to classically regularise divergences in electromagnetism, it is perhaps surprising that the same ideas were applied to resolve the singularities of General Relativity (GR) only in the late 1990’s555A possible reason for this was the relative lack of interest in these topics until the seminal works by Hawking and Penrose Penrose:1964wq (); Penrose:1969pc (); Hawking:1966vg () concerning the singularity theorems in GR. On the other hand, the extraordinary success of quantum field theory perhaps motivated to invoke quantum gravity effects as the most likely mechanism that should regularise gravity in the high curvatures regime.. The first attempt in this direction came about in a work by Deser and Gibbons Deser:1998rj (), where they finally took over the idea and tried to apply it to the case of gravity. However, as usual with gravity, things can very quickly go wrong when one tries to modify the Einstein-Hilbert action. The most straightforward application of the Born-Infeld philosophy by introducing a square root structure of a determinant involving the Ricci tensor gives rise to the presence of ghosts owing to the Ostrogradski instability associated to higher order equations of motion Woodard:2006nt (); Woodard:2015zca (). In order to resolve the ghost problem, they proposed to add an additional term to remove the ghost order by order so that, when expanding the full action in the curvature, only the corresponding Lovelock term remains. This avoids the problem of the ghost, but the large freedom remaining in the choice of the additional piece and the lack of any guiding principle, makes the construction less appealing than the case of Born-Infeld electromagnetism. An obvious way to get around the ghost problem is to only use the Ricci scalar and apply the Born-Infeld construction to this quantity. This would lie within the class of theories that contain one extra degree of freedom with respect to GR and, thus, it would deviate from the original Born-Infeld spirit where the theory is modified in some high energy regime by changing the structure of the theory in that regime instead of adding additional modes.
Some years later, Vollick re-considered Born-Infeld type of actions for gravity from a different perspective Vollick:2003qp (). Similarly to Deser and Gibbons, Vollick also resorted to a straightforward translation of the Born-Infed action to the case of gravity. However, instead of adopting the metric formalism, he considered the action within a metric-affine approach so that the connection is left arbitrary and promoted to an independent field. Within that formalism, the problem of the ghosts encountered in the metric formalism are avoided and, thus, no additional terms to remove undesired interactions are needed. This approach can actually be seen as a combination of the Born-Infeld ideas together with the original purely affine theory of gravity proposed by Eddington. Later on, Bañados and Ferreira took on Vollick’s approach with a slight modification of the original action, that now goes under the name of Eddington-inspired Born-Infeld gravity (EiBI), and showed the existence of non-singular cosmological and black hole solutions. This particular realisation of Born-Infeld gravity theories has since then received a considerable attention and has been extensively explored in different contexts with promising results.
The proposal by Vollick and its relative by Bañados and Ferreira finally succeeded to implement the ideas of Born-Infeld electrodynamics to the case of gravity. However, it is fair to say that this initial proposal merely consisted in obtaining a gravitational action à la Born-Infeld, but it lacked any underlying guiding principle, based on symmetries like in Born-Infeld electrodynamics or any other equally valid motivation. In fact, it is very simple to come up with more general actions that could also be catalogued as Born-Infeld theories and could be considered on the same footing as EiBI. It does not come as a surprise then that very soon, modifications, extensions or alternative implementations of the Born-Infeld ideas to gravity appeared in the literature.
In this section we will review in detail the developments discussed above that led to the formulation of Born-Infeld gravity theories. We will start by reviewing Born-Infeld electrodynamics as a good starting point to motivate the search for analogous theories within gravitational contexts. We will show how the first attempts formulated in the metric formalism did not succeed due to the presence of ghosts. After that, we will turn to the formulation of Born-Infeld actions for gravity within a metric-affine approach and explain how the ghost issue is avoided. The general properties of these theories will be discussed in detail and, in particular, we will explain the existence of two frames. We will end this section by performing a classification of the different Born-Infeld inspired theories of gravities considered in the literature so far and briefly discuss them.
2.1 Born-Infeld electromagnetism in a nutshell
The underlying idea used by Born and Infeld to develop a modification of the Maxwell action as a potential mechanism to regularise some divergences associated to point-like charges was motivated by the appearance of an upper bound for the speed of particles when upgrading Newtonian mechanisms to relativistic mechanics. In that case, the Newtonian Lagrangian for a massive particle of mass is simply , where is its velocity and can take any value. When including the principles of relativistic mechanics, the Lagrangian for the massive particle becomes , where the speed of light makes its appearance as an upper bound for the velocities due to the square root. Taking inspiration from this, Born came up with the idea of modifying Maxwell’s Lagrangian in such a way that the divergences of the Coulomb potential are automatically regularised due to the existence of a natural upper bound in the theory. In Born:1933qff (); Born410 (), he followed the most straightforward application of this idea and proposed the following replacement of Maxwell’s Lagrangian:
with representing the desired upper limit of possible electric fields. Although this simple replacement could do the job of regularising the infinities associated to point-like charges, it is not completely satisfactory from a theoretical point of view since there is no guiding principle for it other than the principle of finiteness. That is the reason that motivated Born, this time in collaboration with Infeld, to look for a more theoretically appealing modification of Maxwell electromagnetism. They noted that, when going from classical mechanics to relativistic mechanics, the symmetry group is enlarged from the Galileo to the Lorentz group and it is precisely this group structure that nicely introduces the desired square root. Born and Infeld embraced this line of reasoning and looked for a non-linear theory of electromagnetism enlarging the group of special relativity as the relevant one. The idea is then that, very much like Newtonian mechanics is the limit of special relativity for small velocities and the Lorentz group reduces to the Galilean transformations, Maxwell electromagnetism should be the limit of some theory with a larger group of symmetries which, in some suitable limit, should reduce to the usual relativistic Lorentz transformations. Motivated by recent developments in gravity where the relevant group was shown by Einstein to be general coordinate transformations, they opted by enlarging the symmetry group of electromagnetism from the Lorentz group to the full group of general coordinate transformations666Incidentally, they were aware and noticed similarities with earlier attempts by Einstein, Weyl and Eddington, among others, in the same direction as a way to unify gravity and electromagnetism in a geometrical theory. However, Born and Infeld motivation was completely different and, as themselves claimed, their approach had nothing to do with those theories, except for some formal analogies, specially with Eddington’s developments in eddington1924mathematical (). Remarkably, Eddington’s theory eventually served as guidance to develop gravity theories à la Born-Infeld, as we will see in the section 2.4.. Then, to have general covariance, the action should be constructed as , with some rank-2 covariant tensor whose symmetric part can be identified with the metric tensor and its antisymmetric part is identified with the electromagnetic field strength . After imposing that Maxwell’s theory should be recovered for small electromagnetic fields and neglecting some boundary terms, they arrived at the celebrated Born-Infeld action
This action has the properties they were after, namely, it introduces the square root structure by means of enlarging the symmetry group of Maxwell’s theory. The constant is the only free parameter of the theory and it precisely gives the maximum allowed value for electric fields. Born and Infeld assumed the value of to be such that the corrections arise at the electron radius, although that value is now experimentally ruled out (see PhysRevD.93.093020 () for a recent review on experimental bounds for non-linear electromagnetism). In order to see the appearance of a maximum value for the electric field, let us notice that the action can be written in several useful ways by expanding the determinant in (2.2) to obtain
with the dual of the field strength, and the corresponding electric and magnetic parts and we have used the matrix identity
Notice that this implies a symmetry owed to the property of the determinant for an arbitrary matrix . From the above expression it is straightforward to see that Maxwell’s electromagnetism is recovered for electromagnetic fields much smaller than and that, for configurations without magnetic field, we also recover the first Lagrangian (2.1) considered by Born. Furthermore, written in this way, we can easily understand why the self-energy of point-like charged particles is regularised. Since a particle at rest (or in its own rest frame) only generates electric field, the Lagrangian reduces to
and we clearly see that the electric field is bounded by . Given the gauge character of the theory, we still have the constraint equation generating the gauge symmetry (or the equivalent of Gauss’ law) given by
and is the density of electric charge. As usual, for a point-like particle of charge we can integrate the equation over a sphere to obtain
where is the total charge enclosed by the sphere . By inverting the relation (2.6) between and we can obtain the solution for the electric field generated by the particle
As promised, for we have which is the usual result in Maxwell’s electromagnetism, while in the opposite regime with the electric field saturates to the value . This saturation is in turn the responsible for the regularization of the self-energy of the particle, that is given by
where we have used the expression for the Hamiltonian density777For the more careful reader, let us clarify that the Hamiltonian density including the interaction between the electric potential and the charge is . However, we can use the definition of the electric field to express the Hamiltonian density, up to total derivatives giving rise to boundary terms, as . The term depending on will then be responsible for the gauge constraint giving Gauss’ law that vanishes on-shell, so that it will not modify the self-energy of the particle. and the corresponding solution (2.7). The integral diverges in the case of Maxwell electromagnetism due to the unbounded contributions from the small scales where one has . In the Born-Infeld case however, the small scales region is modified and we have which makes the integral convergent (see Fig. 1). The integral can be exactly computed in terms of the gamma function and the total finite result is
Let us return to the solution for the electric field given in (2.11) and express it directly in terms of the generating charge as888For the amusement of the reader familiarised with screening mechanisms in modified gravity, let us notice that this solution realises a screening mechanism for the electromagnetic interaction resembling the so called K-mouflage or Kinetic screening of scalar fields.
This expression allows for an alternative interpretation of Born-Infeld electromagnetism. Instead of having modified Maxwell equations in the sector of the electromagnetic field, we can equivalently interpret Born-Infeld electromagnetism as a modification in the source term, i.e., the way in which charges generate electric fields is modified on small scales. In other words, we can interpret it as an effective scale-dependent charge, showing a certain formal resemblance with the renormalisation of the charge when radiative corrections are included in standard QED, but here from a purely classical standpoint without any quantum effect. This re-interpretation of Born-Infeld electromagnetism will be useful for the case of gravity where the Born-Infeld inspired modified gravity theories will also admit an interpretation as a modification of the way in which matter gravitates at high energies.
We will conclude by stressing that the resulting theory turned out to have a series of remarkable features that make the Born-Infeld action be very special among all possible non-linear extensions of electrodynamics. Such properties are related to its special structure, giving additional motivation and support to the idea of implementing the principle of finiteness by enlarging the symmetry group of Maxwell theory. This is nothing but another example of the power of using symmetries as guiding principles to formulate physical theories. In order to avoid further delays in entering into the main topic of this review, namely Born-Infeld inspired theories of gravity, we will abstain our desire of going through all the fascinating features of Born-Infeld electromagnetism and we will content ourselves with briefly enumerating some of its more remarkable properties. For more detailed information we refer to Plebanski:106680 (); Gibbons:2001gy (); Ketov:2001dq () or standard textbooks on string theory where the Born-Infeld Lagrangian naturally appears, as e.g. polchinski1998stringI (); polchinski1998stringII (); zwiebach2009first ():
The Born-Infeld action arises in string theory from -duality invariance when describing an open string in an electromagnetic field, i.e., the Born-Infeld action is the appropriate one to couple strings to electromagnetic (or more general gauge) fields.
Born-Infeld electromagnetism shares with its Maxwellian relative (and other non-linear theories of electromagnetism) the so-called electric-magnetic self-duality BialynickiBirula:1984tx (); Gibbons:1995cv (). This is a highly non-trivial invariance of the theory corresponding to a dual transformation of the electric and magnetic fields. See Aschieri:2008ns () for a review on many interesting aspects of duality rotations and theories with duality symmetry.
Despite the highly non-linear character of the Born-Infeld action, the corresponding equations of motion give rise to causal propagation and avoid the presence of shock waves and birrefringence phenomena.
As we can see, the Born-Infeld theory for electromagnetism not only conforms to the task it was devised for, namely the regularisation of divergences associated to point-like charges, but it is kind enough as to also provide a number of additional gifts that were not required a priori. In the remaining of this section we will overview the attempts to apply similar ideas to the case of gravity. In general, we could say that, by the time of writing, there is not a gravitational analogue of Born-Infeld electromagnetism exhibiting all the successes and remarkable properties discussed above, but the search for it has nevertheless yielded very interesting gravitational theories à la Born-Infeld, both from a theoretical and a phenomenological points of view. We will start our tour however by reviewing the first attempts in this direction that led to pathological theories.
2.2 The Deser-Gibbons proposal: The ghost problem of the metric formalism
The original idea by Born and Infeld to regularise divergences in electromagnetism was taken over by Deser and Gibbons Deser:1998rj () as a potential mechanism to regularise the singularities that commonly appear in General Relativity, like e.g. the divergences at the center of black holes or the original Big Bang singularity. Following the same scheme, they considered an action for the gravitational interaction including the same determinantal and square root structures that appear in Born-Infeld electromagnetism. A straightforward translation of the Born-Infeld action for electromagnetism to the case of gravity would be the naive replacement of field strength by the Ricci tensor so that the first naive tentative action for a gravitational version of Born-Infeld electromagnetism would be
where and some parameters, the spacetime metric and the Ricci tensor of the corresponding Levi-Civita connection. However, this naive procedure leads to a theory plagued by ghost-like instabilities. The reason is clear from the well-known fact that an arbitrary action containing a non-linear function of the Ricci tensor will give rise to higher order gravitational field equations and, thus, it will be prone to the Ostrogradski instability Woodard:2006nt (). In order to avoid the presence of ghosts in the theory, Deser and Gibbons considered instead the action
where the fudge tensor must be tuned in order to get rid of the ghost. The form can be obtained perturbatively to remove the ghost at a given order and its effects are then pushed to higher orders. We can use the identity
valid for an arbitrary matrix , to expand the action in powers of the curvature as
where is the Ricci scalar and . In this expression we can see that, omitting for a moment, we have a cosmological constant at zeroth order, while at first order we recover the usual Einstein-Hilbert term. At higher orders however the appearance of the quadratic terms will lead to higher order equations of motion, thus rendering the theory unstable due to the presence of ghosts. Since we know that, at quadratic order, only the Gauss-Bonnet prevents the appearance of such ghosts, we must use the leading order contribution from in order to remove the undesired terms. We can then assume an expansion in curvatures starting at quadratic order999We could also add lower order terms for the fudge tensor as, e.g. , but that will not introduce the discussion other than adding some more terms in the equations. for the fudge tensor of the form and choose to satisfy
with some constant. The above choice thus only leaves the Gauss-Bonnet contribution at second order. By iterating this procedure one could remove the ghosts at any desired order. However, we already see at quadratic order that only the trace of is determined and, therefore, a large variety of fudge tensors can do the job (see Gullu:2015cha (); Gullu:2014gza () for explicit constructions). In fact, except for some singular actions, one can presumably write almost any gravitational action in the form of (2.13) by means of an appropriate choice of . We can exemplify this by taking the Born-Infeld gravity theory developed by Nieto in Nieto:2004qj (). Motivated by the MacDowell-Mansouri formalism, Nieto considered a spacetime manifold endowed with a connection giving rise to a total curvature that can be split as , where is the usual curvature of the Levi-Civita connection, is the vielbein field and a constant parameter. For this connection, he then considers a Lagrangian in dimensions given by
If we use the previous splitting, we can write the Lagrangian as
where and we have used the matrix identity , with denoting the -th elementary symmetric polynomial of the matrix (see (1.25)). In the present case, the matrix is the Ricci tensor and its elementary symmetric polynomials are precisely the Lovelock invariants, that we denote by , so that the considered action is nothing but a combination of all the Lovelock terms and, thus, the theory is ghost-free. One can then rewrite this Lagrangian in the Deser-Gibbons form by simply defining a matrix given by so that the Lagrangian can be alternatively written as
where we have used the commutativity of the determinant and the square root (whenever it exists). This is the form found by Nieto and which he then related to Born-Infeld gravity. However, as we have seen, it is nothing but Lovelock gravity written in an obscure way. Furthermore, no additional work is necessary to know that the theory does not contain any ghosts. This example perfectly illustrates the necessity of a better defined strategy to construct theories of gravity à la Born-Infeld in order not to be deluded with well-known healthy theories in mysterious disguises.
2.3 Other proposals in the metric formalism
In the procedure presented in the precedent section, we have been careful to impose that only the Lovelock invariants should remain at a given order in the expansion. This is a crucial requirement for the consistency of the theory, as the presence of ghosts invalidates any background classical solution. The approach followed by Deser and Gibbons can be seen as a way to make sense of the theory by pushing the scale at which the ghost becomes relevant at higher scales, but the lack of any other guiding principle obstructs the construction of an appealing and well-defined full theory.
One might however take a less demanding approach and impose instead a weaker condition without compromising the stability of the theory due to the presence of ghosts, but at the expense of partially abandoning the original Born-Infeld spirit. For instance, instead of using the fudge tensor to only leave Lovelock invariants at each order, one could allow for some arbitrary functions of them. Thus, at quadratic order we could have allowed for terms involving some linear combination of the squares of the Ricci scalar and the Gauss-Bonnet term. This would find motivation in the fact that arbitrary functions of these two scalar quantities are known to be particular cases where the Ostrogradski instability is bypassed. In the end, this would be nothing but a complicated way of rewriting the class of theories described by an arbitrary function , with and the Ricci scalar and the Gauss-Bonnet term respectively. Although perfectly legitimate, these theories introduce additional scalar degrees of freedom and, thus, they can hardly be considered as genuine Born-Infeld modifications of gravity. Of course, this does not mean that those alternatives are uninteresting, but rather they should be regarded as belonging to another class of theories.
In case one is interested in obtaining gravitational theories with an upper bound for the curvature, then one can simply write a specific model of an theory where the function presents a branch cut at some high but finite curvature . The square root function typical from Born-Infeld would achieve this, but other functions involving e.g. logarithms could serve as well. Feigenbaum et al.Feigenbaum:1997pf () explored this route in two dimensions where the curvature is fully determined by the Ricci scalar and they studied some black holes solutions. In a subsequent work Feigenbaum:1998wy (), Feigenbaum extended the analysis to four dimensions where he considered an action of the following type:
with and some constants. Again he studied black hole solutions that we will briefly review in section 4.1. However, the problem of ghosts arising from the explicit dependence on the full Riemann and the Ricci tensors is not discussed. In fact, from the own equations of motion given in Feigenbaum:1998wy (), one can see that they are fourth order and, thus, it would be expected to have ghosts. This pathology renders the black hole solutions of limited physical interest, as the perturbations around them are likely to be unstable. The same problem applies to the theories considered by Comelli and Dolgov in Comelli:2004qr () constructed in terms of the Lagrangian
with and some given functions of the Ricci scalar. This Lagrangian combines the Deser and Gibbons proposal with -type of theories, but without taming the presence of ghosts so that the obtained cosmological solutions are again of limited realistic applicability.
A more interesting proposal that is also closer to the Born-Infeld spirit was given by Wohlfarth in Wohlfarth:2003ss (). The theory is based on a symmetric object defined as
where the indices , should be regarded as ordered pairs of indices. He then introduces the new metric and Kronecker delta
that are then used in the usual way to manipulate capital indices. Moreover, one has the identity valid in dimensions. The proposed Lagrangian within this formalism is
with some constant and a parameter with the only restriction to be a fractional number in order to allow for a regularization of curvature divergences. This represents an extension of Deser and Gibbons construction since more general curvature invariants appear in the Lagrangian. However, it shares the same problematic of containing ghosts (typically appearing at the scale determined by ) which is then resolved in a similar fashion, i.e., the Lagrangian is corrected as
where and are general expressions containing linear and quadratic curvature terms, respectively. The relative parameters among all the terms must be tuned to remove the ghosts at quadratic order, although one would expect to find again the ghost at higher orders. Thus, similarly to the Deser and Gibbons construction, additional requirements are necessary to find a satisfactory Born-Infeld theory of gravity within this formalism.
Another approach that has been taken in the literature consists in choosing the fudge tensor such that some specific gravity theories are recovered in the low curvatures regime. In Gullu:2010pc (), the authors followed this path to construct a Born-Infeld extension of the so-called New Massive Gravity theory Bergshoeff:2009hq (), whose action is given by
and describes a massive graviton in 3 dimensions101010Since the graviton propagator trivialises in 3 dimensions, the problem of the potential ghosts discussed above are less virulent.. One can then see that this action is recovered at quadratic order from (2.13) in 3 dimensions by choosing proportional to and appropriately tuning the parameters (with the possible addition of a cosmological constant). Interestingly, the resulting action that they consider recovers at cubic order the extension of New Massive Gravity found in Sinha:2010ai () by imposing the existence of a theorem. The same authors pursued a similar approach in Gullu:2010wb () to construct theories that recover Horava’s gravity Horava:2009uw (); Horava:2009if () in 3 dimensions at quadratic order.
2.4 Eddington-Born-Infeld gravity
In the previous sections we have seen that a straightforward implementation of the Born-Infeld idea to the case of gravity is not an obvious task. It is not difficult to convince oneself that the main difficulty is the avoidance of ghosts and this is hardwired in the use of the metric formalism in the action. One can however seek for Born-Infeld inspired modifications of gravity within the realm of affine theories of gravity where the connection is regarded as an independent object. Within this framework, it is very natural to remember the purely affine theory of gravity introduced by Eddington and described by the following action111111Deser and Gibbons already made reference to this approach in Deser:1998rj (), but they did not consider it any further in favour of a metric formalism. eddington1924mathematical ():
where is the symmetric part of the Ricci tensor of an arbitrary connection . In vacuum, this theory is equivalent to GR121212The recovery of the GR equations in vacuum is not specific of Eddington’s theory and, in fact, it is a general result for any theory of gravity. The generality of this result actually boils down to the covariance of the field equations which imposes that, in vacuum, the Ricci tensor must be proportional to the metric. In theories of gravity with additional degrees of freedom, the extra fields should be regarded as matter fields and the recovery of GR in vacuum also applies. Another complementary way of understanding this general result is provided by the fact that GR is the only Lorentz invariant and unitary theory for a self-interacting massless spin-2 field in 4 dimensions, usually called graviton. Thus, if by gravity we understand a theory for such a particle, we will inevitably find GR in vacuum. Differences can however show up when including matter fields, as we will discuss later.. This is easy to understand, since this theory can be seen as GR after integrating out the metric tensor. If we start with GR in the presence of a cosmological constant and in the Palatini formalism, we have
that gives the Einstein equations
We can now take the trace to obtain , which allows to rewrite the equations as . This relation can be used in the action to remove the dependence on the metric tensor and we then recover the Eddington action. This procedure of integrating out the metric tensor is also valid when including matter fields as long as they couple minimally, i.e., the metric tensor will only enter algebraically. In that case, the resulting action will be more involved, but it allows to write a fully affine theory of gravity, as it was Eddington’s original idea.
An important consequence of using the connection as a fundamental geometrical object in Eddington’s theory is the avoidance of introducing ghosts associated to higher order equations of motion for the metric tensor. This is not a specific feature of Eddington’s theory, but it is a general result for theories formulated à la Palatini. In view of these results, Eddington’s action seems to be a better suited starting point to implement the Born-Infeld construction for theories of gravity. This approach was taken by Vollick Vollick:2003qp (), who considered the action131313Here we use the dimension 1 parameter as the Born-Infeld scale instead of the constant used in Vollick:2003qp (). The relation between both is .
where is a mass scale determining when high curvature corrections are important. The second term is introduced to remove a cosmological constant, thus allowing for Minkowski solutions in vacuum. The above action for a theory of gravity combines the ideas of Eddington’s theory with the Born-Infeld construction, resulting in a theory of gravity formulated in a metric-affine approach and incorporating the square root and determinantal structures characteristic of Born-Infeld electrodynamics.
Before entering into further developments, let us check that GR is indeed recovered when the curvature is much smaller than the scale . When taking that limit, the leading order correction is
thus reproducing the Einstein-Hilbert action in the first order formalism, which is known to coincide with GR on-shell and provided the matter fields couple minimally141414The equivalence between the metric and the Palatini approaches has also been considered for more general actions in, e.g. Exirifard:2007da (); Borunda:2008kf (); Dadhich:2010dg (). A particularly interesting result is that the equivalence of both formulations extends to the whole series of Lovelock invariants, among which the Einstein-Hilbert action represents nothing but the lowest order term. (see for instance Hehl1978 (); ortin2007gravity ()). Let us pause here for a moment and seize the opportunity to clarify some subtleties concerning this point which are well-known in the community but are still source of a little confusion in some works (see for instance the discussion at this respect in section 2.3.1 of Clifton:2011jh ()). When considering the Einstein-Hilbert action in the Palatini formalism in the presence of minimally coupled fields, the field equations of the connection can be recast as a metric compatibility condition for the metric tensor151515See for instance ortin2007gravity () for details. We will also show more details on how this is achieved in section 2.7.1 within the context of more general theories. and, thus, a solution of the equations is the Levi-Civita connection of the spacetime metric. An important point to note however is that such a solution represents a solution, but the most general solution for the connection field equations involves an arbitrary 1-form, which can be taken to be the trace of the non-metricity or the trace of the torsion tensor. This is of course nothing but a reflection of the fact that the metric compatibility condition obtained from the connection field equations does not fully determine the connection and the Levi-Civita connection is only obtained after imposing a symmetric condition. It is sometimes stated that such a condition must be supplemented for the Einstein-Hilbert action to give GR in the Palatini formalism. However, one must also notice that the Einstein-Hilbert action has a projective invariance161616In section 2.5 we will show that this symmetry is shared by all theories defined in terms of the symmetric part of the Ricci tensor and we will compute the associated conserved current. which also involves an arbitrary 1-form , and this is precisely the undetermined mode obtained when solving the connection equation. The gauge character of the projective invariance is discussed in great detail in Julia:1998ys (); Dadhich:2010xa ().
In the case of the action (2.31), the projective invariance is only obtained as a low curvature accidental symmetry, but it is generally broken by higher order interactions, unless the initial theory is defined only in terms of the symmetric part of the Ricci tensor, in which case the projective invariance is a symmetry of the full theory. Considering only the symmetric part of the Ricci tensor is a widely adopted (and very convenient) option in the literature and, in addition, it would be closer to Eddington’s original theory. This is the option adopted by Bañados and Ferreira in Banados:2010ix ()171717Here we prefer to restore all the dimensionful constants as opposed to Banados:2010ix (), where the authors set . Furthermore, we correct a typo in form of a factor of 2 appearing there, which has propagated in the literature., where they considered the action
that has now become the standard version of the so-called Eddington-inspired-Born-Infeld gravity (EiBI). In this version, it is customary to let a cosmological constant term be encoded in the parameter as . An important notational convention that might lead to some misinterpretations but is very common in the community is to use to denote the symmetric part of the Ricci tensor without the explicit symmetrisation. To avoid any confusion, we will always make explicit the corresponding symmetrisation.
2.5 Field equations
In the literature there is a number of subtle points in the derivation of the field equations that are sometimes overlooked or omitted, so we will provide a detailed derivation here. The main differences that one can encounter eventually boil down to whether only the symmetric part or the full Ricci tensor is considered and if the connection is assumed to be symmetric a priori or not. The former condition is related to the presence of a projective invariance, while the latter has to do with the presence of torsion. In many practical applications, these differences do not make a huge impact in the results, but one should nevertheless be careful to obtain the correct field equations. Let us then consider the action
where no assumptions are made a priori on the connection and the full Ricci tensor with both its symmetric and its antisymmetric parts. Let us stress again that Vollick Vollick:2003qp () used the full Ricci tensor but constrained the connection to be symmetric, while Bañados and Ferreira left the connection fully undetermined but considered only the symmetric part of the Ricci. We have also added general matter fields that can, in principle, couple to both the metric and the connection. Then, we will detail where the differences arise when making the different assumptions. For later convenience and to comply with standard notation in the literature, let us introduce the notation
Then, the variation of the action (2.34) can be expressed as181818In the literature of Born-Infeld theories it is customary to denote the inverse of the matrix simply as , in accordance with the usual convention of denoting the inverse of a metric with upper indices. Since we will have two metrics, we prefer to explicitly keep the inverse for the moment in order to avoid any confusion to the unfamiliar reader in these first steps into the formalism of Born-Infeld theories, since could very well be confused with . We will eventually drop the explicit mention for the inverse of to alleviate the notation and whenever there is no risk of confusion.
where and we have used the formula
valid for an arbitrary matrix . The field equations for the metric tensor are then immediately seen to be
with the energy-momentum tensor of the matter fields defined as
Notice that this energy-momentum tensor is defined at constant connection. For minimally-coupled bosonic fields this is not relevant and the energy-momentum tensor will have the standard form. However, when considering fermionic and non-minimally coupled bosonic fields, the expression for the energy-momentum tensor will be in general different from the one that would be obtained in a purely metric formalism. It is important to note the symmetrisation of the object in the field equations as a consequence of the symmetry of the metric tensor. Had we considered only the symmetric part of the Ricci tensor in the starting action, this symmetrisation would be innocuous. Furthermore, as said before, in most practical applications in cosmological contexts or spherically symmetric solutions, the matrix is symmetric and then one could omit the symmetrisation, but in the general case it is important to properly include it. We will come back to this point in section 2.7.1 for more general Lagrangians.
The derivation of the connection field equations requires a bit more of work. In order to obtain them, we need the variation of the Ricci tensor: