Palatini wormholes and energy conditions from the prism of general relativity
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Abstract
Wormholes are hypothetical shortcuts in spacetime that in general relativity unavoidably violate all of the pointwise energy conditions. In this paper, we consider several wormhole spacetimes that, as opposed to the standard designer procedure frequently employed in the literature, arise directly from gravitational actions including additional terms resulting from contractions of the Ricci tensor with the metric, and which are formulated assuming independence between metric and connection (Palatini approach). We reinterpret such wormhole solutions under the prism of General Relativity and study the matter sources that thread them. We discuss the size of violation of the energy conditions in different cases and how this is related to the same spacetimes when viewed from the modified gravity side.
1 Introduction
There exist numerous exact solutions of the Einstein field equations in which the physical relevance is not always clear and which, consequently, are often regarded as exotic [1]. A wellknown example is the static and spherically symmetric Schwarzschild solution, where the importance of this geometry was not fully understood until the observational discovery of very compact objects such as neutron stars. In the 1960s, the Schwarzschild solution began to be considered seriously not only as an approximate description of the exterior geometry of (nonrotating) stars but also as the final outcome of gravitational collapse [2]. Nowadays, black holes are an intensely studied topic of research, as e.g. the sources of gravitational waves [3, 4], and they are no longer regarded as exotic solutions.
Wormholes are also regarded as an exotic family of solutions of Einstein’s equations [5, 6, 7] and have been extensively analyzed for a number of reasons. From a quantum gravitational perspective, they are seen as a natural consequence if the topology of spacetime fluctuates in time [8]. Recently, a new connection between specific types of (nontraversable) wormholes [9] and quantum systems was introduced with relevant implications in high energy physics [10, 11, 12, 13, 14, 15]. From an astrophysical and cosmological viewpoint, the latetime cosmic speedup [16, 17, 18] suggests that the source driving the expansion [19, 20, 21, 22, 23] could be compatible with the kind of matter necessary to generate traversable wormholes. In fact, several cosmological models with exotic sources supporting wormholes have been proposed, including phantom energy [24, 25, 26, 27, 28], the Chaplygin gas [29, 30, 31, 32], or its generalizations [33, 34, 35, 36]. This has led many researchers to explore the astrophysical consequences and characterization of macroscopic wormholes [37, 38, 39, 40, 41].
In the context of general relativity (GR) wormhole spacetimes are specifically designed, where Einstein’s equations are solved in a reversed manner. This means that a suitable wellbehaved geometry is defined a priori and the matter sources that generate it are derived a posteriori. The outstanding feature was the requirement of exotic matter, defined as matter that violates the null energy condition, i.e., \(T_{\mu \nu } k^{\mu }k^{\nu }<0\), where \(T_{\mu \nu }\) is the stressenergy tensor and \(k^{\mu }\) is any null vector [5, 6, 42, 43]. Although exotic matter is considered classically nonviable, there is experimental evidence of such energy conditions violations in several systems where quantum effects such as the Casimir effect, Hawking evaporation or vacuum polarization take place [6, 45, 46]. Therefore, since the existence of exotic matter is classically a problematic issue, it is particularly important to find solutions which minimize the violation of the energy conditions. In fact, the amount of violation can be made infinitesimally small by choosing the geometry of the wormhole in a very specific and appropriate way [47]. The thinshell formalism is another approach to minimization of the exotic matter, as the latter is now localized on the thin shell [48, 49, 50, 51, 52, 53, 54, 55, 56].
However, in the context of modified gravity, it was shown explicitly that the matter threading the wormhole throat may, in principle, satisfy all of the energy conditions, and it is the higherorder curvature terms, interpreted as a gravitational fluid, that support the wormhole geometry [57]. In fact, a plethora of wormhole solutions beyond GR have been found in a variety of theories, such as in conformal Weyl gravity [58], Kaluza–Klein [59, 60, 61], Brans–Dicke [62, 63, 64, 65], Gauss–Bonnet [66, 67, 68, 69], Lovelock [70, 71] braneworlds [62, 72, 73, 74], Hor̆ava–Lifshitz [75, 76], Eddingtoninspired Born–Infeld [77, 78, 79], metric [80, 81, 82, 83] and Palatini [84, 85] approaches of f(R) gravity and extensions [86, 87, 88], the modified teleparalellism framework [89, 90] or Einstein–Cartan theory [91].
The purpose of this work is to consider specific wormhole solutions that can be obtained in gravitational theories formulated in the Palatini approach, and interpret them from the prism of standard GR. This means that considering specific wormhole solutions, and taking into account the modified Einstein equation, it is then possible to interpret its associated effective stressenergy tensor from a standard perspective and determine if it violates the generalized energy conditions [57]. In this sense, the relevance and novelty of this work lies in the fact that the wormhole solutions considered here do not arise as a result of the reversed philosophy procedure, where the spacetime geometry is given first and then the Einstein equations are driven back to find the matter source threading such a geometry, but instead flows from the resolution of the modified Einstein’s equations associated to a welldefined gravitational actions without ghosts (see Sects. 4 and 5). More specifically, a thorough analysis of the specific solutions dictates that these wormhole solutions are inherent in the Palatini theories considered in this work. Thus this approach allows one to study wormhole geometries in less artificial scenarios and to investigate such geometries from the GR point of view in relation to violation of the energy conditions.
The paper is organized as follows. We begin by introducing the basics of Palatini theories in Sect. 2. In Sect. 3, we briefly review wormholes physics in GR. In Sect. 4, we present specific Palatini wormholes, namely, wormholes obtained in Born–Infeld gravity formulated in the Palatini approach, and we discuss them under the prism of GR, identifying the matter source threading the wormhole and discussing the violation of the energy conditions. In Sect. 5, this analysis is extended to wormholes obtained in the context of f(R) gravities, and we conclude in Sect. 6 with a summary and discussion.
2 Palatini theories of gravity
2.1 f(R) theories
2.2 Beyond the f(R) case
In these theories, the nonmetricity tensor takes the form \(Q_{\alpha \mu \nu }=g_{\rho \nu }(\nabla _\alpha {{\Omega ^{1}}_{\mu }}^\lambda ) {\Omega _\lambda }^\rho \) (symmetrized over \(\mu \) and \(\nu \)), where \({\Omega _{\mu }}^\alpha \) is a function of \(T_{\mu \nu }\). In the f(R) case, one has \({\Omega _{\mu }}^\alpha =f_R \delta _\mu ^\alpha \) (conformal transformation), which leads to a simple nonmetricity tensor. In general, however, the relation between the metrics is not conformal and a clean representation of the field equations in terms of \(g_{\mu \nu }\) is not immediate. For this reason, Eq. (9) is typically preferred. Nonetheless, for the spherically symmetric scenarios that will be considered here, the Einstein tensor \(G_{\mu \nu }(g)\) can always be computed straightforwardly, which facilitates the interpretation of the solutions in terms of an effective \(\tau _{\mu \nu }\) in much the same way as in f(R) theories.
As the field equations can be written in the Einsteinlike form (7), a given spacetime (wormhole or not) can be interpreted as a solution of an (in principle) infinitely degenerate family of theories of gravity. Here, we will show that these wormhole solutions are not designed, but that they follow directly from wellmotivated actions. This could shed light on obtaining analytic wormhole solutions and on the kind of energy sources that can generate them in the context of GR. The reinterpretation of these wormhole solutions from the prism of GR might be useful to understand their properties from a more standard perspective. To proceed, we review wormholes physics in the next section.
3 General theory of wormholes
In 1955, Wheeler introduced the term geon to denote a hypothetical gravitationalelectromagnetic object with a nontrivial topological structure [8], and later with Misner [97] coined such a construction a wormhole. These concepts extended the very first calculations tracing back to the Einstein and Rosen bridge [9]. But it was the seminal work of Morris and Thorne [5] in 1988 which gave rise to the great interest on traversable Lorentzian wormholes which continues today. In this section, such concepts will be introduced, and the transition from wormholes in GR to those in metricaffine geometries will become clear.
3.1 Wormhole geometry
The functions \(\Phi (r)\) and b(r) fulfill further properties in order to describe a wormhole spacetime. The asymptotically flat limit imposes \(b(r)/r \rightarrow 0\), as \(r \rightarrow \infty \). Moreover, for the wormhole to be traversable there should be no event horizons so that \(\Phi (r)\) must be finite everywhere. Note that it is straightforward to define the mass of the wormhole as the finite limit for the function b(r). By comparison with the Schwarzschild geometry, an observer from spatial infinity sees that \(b=2G_NM\).
A useful way to construct wormhole spacetimes resides in the cutandpaste procedure, which is a mathematical construction where two spacetimes are matched at a given junction interface. If this hypersurface contains nonzero surface stresses it is called a thin shell, otherwise it is just a boundary surface. Following the standard junctioncondition formalism [98, 99, 100, 102] one cuts and pastes two manifolds to form a geodesically complete one with the throat located at the joining shell, where the exotic matter is located [48, 49, 50, 51, 52, 53, 54, 55, 56]. Beyond GR, the junction formalism is required to be generalized for the specific theory of gravity under consideration, for example, such as in metric f(R) gravity [103, 104].
At this point we would like to point out a special feature of the thinshell structure. It is well known that thinshell wormholes are geometric constructions that turn out to be geodesically complete although the Riemann tensor is divergent at the thin shell where the throat is located [105]. Thus, the spacetime curvature becomes divergent at the nonnull hypersurface layer, however, this divergence is physically interpreted as a surface layer with a stressenergy tensor on it. Therefore, the existence of curvature divergences at the wormhole throat should not be surprising at all, and this does not necessarily entail the presence of spacetime singularities. Note, in this sense, that we are using the term singularity in a way that transcends the notion of divergence. A spacetime is said to be singular when it is geodesically incomplete [106, 107], regardless of the existence or not of curvature divergences. Thus, these terms will not be interchangeable in our discussion. In fact, thinshell wormholes are examples of spacetimes which are geodesically complete (hence nonsingular) but which, by construction, contain curvature divergences.
3.2 Energy conditions

The weak energy condition (WEC) states that the energy density measured by an arbitrary observer must be nonnegative, \(\rho \ge 0\) and \(\rho + p_{i} > 0\).

The strong energy condition (SEC) asserts that gravity is attractive, \(\rho + \sum p_{i}\ge 0\) and \(\rho + p_{i} \ge 0\). Note that this condition is violated in many current models of accelerated cosmic expansion as well as in inflationary models.

The dominant energy condition (DEC) expresses that the energy density measured by any observer is positive but also that its flux propagates in a causal way (i.e., it is null or timelike) so that \(\rho \ge 0\) and \(\rho \ge p_{i}\).

The null energy condition (NEC) implies that \(\rho + p_{i} \ge 0\), \(i=1,2,3\).
Note that the energy conditions in GR can be traced back to the Raychaudhuri equation, where it is straightforward to determine that the attractive nature of gravity is represented by the condition that the Ricci tensor fulfils \(R_{\mu \nu } k^\mu k^\nu \ge 0\) for any null vector \(k^{\mu }\). This condition ensures the focusing of the geodesic congruence, which in turn can be written, via Einstein’s field equations, as a condition over the stressenergy tensor \(T_{\mu \nu } k^\mu k^\nu \ge 0\) [108]. On the contrary, the wormhole structure requires that the null geodesic congruence must be defocused at the throat in order for geodesic completeness to hold. This is precisely the physical meaning encoded in the flaringout condition.
In extended theories of gravity, it is possible to express the gravitational field equations as in Eq. (7) where the effective stressenergy tensor \(\tau _{\mu \nu }\) includes all new theorydependent terms as well as the corresponding stressenergy tensor of the matter [109, 110]. Thus, if one has repulsive gravity, \(R_{\mu \nu } k^\mu k^\nu < 0\), this implies \(\tau _{\mu \nu } k^\mu k^\nu < 0\), but the matter stressenergy tensor can, in principle, be imposed to obey the energy conditions, or more specifically, in this case the NEC, i.e., \(T_{\mu \nu } k^\mu k^\nu \ge 0\). Thus, in the context of modified gravity it has been shown that wormhole geometries are in fact supported by the effective stressenergy tensor, which actually plays the role of exotic matter and violates the energy conditions, while the physical matter satisfies them [57].
4 Wormholes in Born–Infeld and quadratic Palatini gravity
We will now study a family of wormholes which arise as an exact solution of an extension of GR containing quadratic curvature scalar and Riccisquared terms [111, 112, 113] and also of Born–Infeld gravity [114], a theory that has attracted a good deal of attention in astrophysics and cosmology in the last few years [115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128] (see [129] for a recent review). Quadratic extensions of GR are supported by the quantization of fields in curved spacetimes [130, 131] by unifying approaches to quantum gravity [132, 133] from effective Lagrangians methods [134]. They are also employed as phenomenological tools to address issues like singularities, early cosmology, astrophysics, and alternatives to the dark matter/energy paradigm [see e.g. [135, 136, 137, 138] for some reviews].
4.1 Wormhole geometry

\(\delta _1>\delta _c\): this is essentially similar to the standard Reissner–Nordström solution of GR, in that there may be two, one (degenerate) or no horizons on each side of the wormhole.

\(\delta _1<\delta _c\): a single (nondegenerate) horizon always exists on each side of the wormhole, resembling the Schwarzschild black hole.

\(\delta _1=\delta _c\): this configuration exhibits a single horizon (on each side) which disappears if \(N_q<N_c^\epsilon \). For these configurations the metric at \(r=r_c\) is finite (i.e. Minkowskianlike).
4.2 Energy conditions
Let us now assume that the line element (15) is a solution of Einstein’s equations coupled to a certain fluid: \(G_{\mu \nu }(g)=\kappa ^2\tau _{\mu \nu }\), recalling that \(\tau _{\mu \nu }\) is the effective stressenergy tensor. As emphasized above, in such a reinterpretation of the theory, the existence should not come as a surprise of matterenergy sources in which \(T_{\mu \nu }\) satisfies the energy conditions, while \(\tau _{\mu \nu }\) does not [109, 110]. Let us apply such a procedure to the spacetime metric (15).
Among the spectrum of solutions of our theory, let us just focus on traversable wormholes, which correspond to i) solutions with \(\delta _1=\delta _c\) and \(N_q<N_c^\epsilon \), and ii) solutions with \(\delta _1>\delta _c\) and no horizons (naked configurations).
In Fig. 4 we illustrate how the energy density near the wormhole throat changes sign as the number of charges decreases from its critical value \(N_c^\epsilon \). It should be noted that this figure represents the (dimensionless) energy density \({\tilde{\rho }}\), which hides a certain dependence on the number of charges \(N_q\). The effect of this dependence is relevant, as \(\rho _{\mathrm{eff}}\sim {\tilde{\rho }}/N_q\), thus implying that the energy density becomes very negative as \(N_q\rightarrow 0\). Note also that the xaxis should also be rescaled to account for the dependence on \(N_q\), which leads to \(x\rightarrow x N_q^{1/2}\). Thus, the region where the energy density becomes very negative is also restricted to a tiny region near the throat. In the limit \(N_q\rightarrow 0\) then one finds that this region is of zero size, which is consistent with the fact that without electric charge there is no wormhole and the Schwarzschild solution is recovered.
From our analysis it is clear that the emergence of the wormhole structure from the prism of GR is related to the deviations that the stressenergy tensor suffers with respect to that of a Maxwell field as the innermost region \(r=r_c\) is approached. This is so because the symmetry satisfied by the stressenergy tensor characterizing electromagnetic fields, namely, \({T_t}^t={T_r}^r\) and \({T_\theta }^{\theta }={T_\varphi }^{\varphi }\) cannot support wormhole solutions in the context of GR, even for nonlinear electromagnetism [141]. Thus, one is forced to consider exotic sources of matter to generate the wormhole. Alternatively, one can reconsider this scenario and interpret these wormholes as emerging out of modified gravity effects, in which case electromagnetic fields (either Maxwell or nonlinear [143]) with \({T_t}^t={T_r}^r\) and \({T_\theta }^{\theta }={T_\varphi }^{\varphi }\) but \({\tau _t}^t\ne {\tau _r}^r\) and \({\tau _\theta }^{\theta }\ne {\tau _\varphi }^{\varphi }\) can naturally sustain them without violation of the energy conditions.
Note that in all cases the typical size of the region where these violations of the energy conditions occur is determined by the scale \(r_c=\sqrt{r_q l_{\epsilon }}=l_\epsilon \sqrt{2N_q/N_c^\epsilon }\) (observe that the xscale and the vertical scale in Figs. 2–3 are both measured in units of \(r_c\)). This implies that violations of the energy conditions are restricted to a region of order \(\sqrt{l_Pl_\epsilon }\) which may grow with the charge as \(N_q^{1/2}\) (recall that \(N_c^\epsilon \approx 16.55 l_\epsilon /l_P\)). Thus, unless \(l_\epsilon \) be much greater than \(l_P\) and/or one considers huge amounts of charge, the size of this region will typically be very small.
Let us stress that though curvature divergences may arise at the wormhole throat \(r_c\), these wormholes are geodesically complete for all spectra of mass and charge and physical observers can safely traverse them [144]. This should not come as a surprise since the same feature is quite common in the thinshell approach discussed above.
5 f(R) wormholes
5.1 Wormhole geometry
5.2 Energy conditions
Let us stress that, like in the Born–Infeld gravity case, for the existence of these wormhole solutions it is not enough to have a nonlinear electrodynamics source. This follows from the fact that in GR no wormholes supported by this kind of matter source are allowed [141]. Indeed, as follows from Eqs. (49)–(51) the effective stressenergy tensor that sources our wormholes does not have the symmetries of nonlinear theories of electrodynamics, namely, \(\tau _{t}^t \ne \tau _x^x\) and \(\tau _{\theta }^{\theta } \ne \tau _{\varphi }^{\varphi }\), due to the presence of \(\lambda \)corrections. When \(\lambda \) vanishes, the wormhole throat closes and one recovers the GR results. Note that the preceding discussion could be further extended to include other theories of (nonlinear) electrodynamics, where wormholes have been recently found [84].
6 Conclusion and discussion
In this work, we have presented two classes of traversable wormhole spacetimes which are supported by a single matter source given in the action from which the geometry is derived and which turns out to be well defined everywhere. These spacetimes are exact solutions of certain extensions of GR including higherorder powers and contractions of the Ricci tensor, and they are formulated in the Palatini approach. Here we have shown that they can also be interpreted as exact solutions of GR with a modified or effective stressenergy tensor. We stress that these solutions are not constructed as a result of the standard reversephilosophy in general relativistic wormhole physics, but instead directly arise from welldefined actions. Indeed, these features have been obtained not as a consequence of any designer or engineering process but as the result of a direct derivation from wellmotivated gravitational actions. The keystone seems to be the Palatini framework in which the original theories were formulated.
From the GR viewpoint, we have found that the generalized energy conditions for wormholes coming from Born–Infeld and f(R) gravity are violated at/near the wormhole throat, which is in perfect agreement with the current knowledge of wormholes physics. Although the kind of matter source, needed to sustain a wormhole from this point of view, is quite peculiar, it is confined to a restricted region around the wormhole throat and, more important, it does not seem to be in contradiction with any fundamental physics principle. However, from the modified gravity side there is a straightforward way to reformulate the standard view by describing a wormhole spacetime actually threaded by ordinary matter but supported by the novel gravitational effects attached to the particular extended gravity and whose size is determined by a fundamental length scale, which in turn defines the size of the violations of the energy conditions from the GR viewpoint. Certainly, the properties or the behavior of matter could be dependent on the theoretical framework adopted. Therefore, it is an overriding issue to keep exploring alternative theories of gravity. The Palatini approach is indeed an encouraging new perspective by making it possible to deal with the problem of singularities by replacing them with microscopic wormholes sustained by ordinary matter.
In summary, the wormhole spacetimes presented in this work arise in a more natural (less artificial) way in the Palatini formalism where metric and connection are assumed as independent dynamical variables. This novel approach may bring about new avenues to enlarge our knowledge of wormhole physics, offer new insights for constructing wormhole solutions in the context of GR, and find reasonable scenarios where they might take place.
Footnotes
Notes
Acknowledgements
C. B. is financially supported by the National Scientific and Technical Research Council (CONICET). F. S. N. L. acknowledges financial support of the Fundação para a Ciência e Tecnologia (FCT) through an Investigador Research contract, with reference IF/00859/2012, funded by FCT/MCTES (Portugal). G. J. O. is supported by a Ramon y Cajal contract, the grant FIS201457387C31P (MINECO/FEDER, EU), and the project SEJI/2017/042 (Generalitat Valenciana). This work has also been supported by the iCOOPB20105 grant of the Spanish Research Council (CSIC), the Consolider Program CPANPHY1205388, the Severo Ochoa grant SEV20140398, and the CNPq (Brazilian) Project No.301137/20145. D. R.G. is funded by the FCT postdoctoral fellowship No. SFRH/BPD/102958/2014 and the FCT research grant UID/FIS/04434/2013. C. B. and D. R.G. thank the Department of Physics of the University of Valencia for hospitality during the initial stage of this work. This article is based upon work from COST Action CA15117, supported by COST (European Cooperation in Science and Technology).
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