Wormhole geometries in modified gravity

where Φ(r) and b(r) are arbitrary functions of the radial coordinate, r, denoted as the redshift function, and the shape function, respectively [1]. The radial coordinate r decreases from infinity to a minimum value r 0 , representing the location of the wormhole throat, where b(r 0 ) = r 0 , and then increases from r 0 back to infinity. For the wormhole to be traversable, one must demand that there are no horizons present, which are identified as the surfaces with e 2Φ → 0, so that Φ(r) must be finite everywhere. Note that in any region where the t coordinate is timelike requires that r > b(r). Now, a fundamental property of wormholes is the flaring out of the throat, which is translated by the condition (b -b ′ r)/b 2 > 0 [1]. At the throat b(r 0 ) = r = r 0 , the condition b ′ (r 0 ) < 1 is imposed in order to have wormhole solutions. It is precisely these restrictions that impose the violation of the null energy condition (NEC) in classical general relativity. At this stage, it is important to emphasize that the NEC violation involves a stress-energy tensor such that T µν k µ k ν < 0, where k µ is any null vector [1].

However, recently it has been shown in the context of modified theories of gravity that the matter threading the wormhole may satisfy the energy conditions, and it is precisely the effective stress-energy tensor involving higher order derivatives that is responsible for the violation of the NEC. This approach has been explored in f (R) gravity [2], curvature-matter couplings [3,4], conformal Weyl gravity [5] and in braneworlds [6], amongst other contexts. In this paper, we review some of these scenarios.

In first place, we consider f (R) modified theories of gravity, which has recently been extensively explored as a possible cause of the late-time cosmic acceleration (see [7] for a review). The action is given by

where κ 2 = 8πG, and for notational simplicity we consider κ 2 = 1. S M (g µν , ψ) is the matter action, defined as

, where L m is the matter Lagrangian density, in which matter is minimally coupled to the metric g µν and ψ collectively denotes the matter fields.

Varying the action with respect to the metric, one deduces the gravitational field equation written in the following form

where the effective stress-energy tensor, T eff µν , is given by

where F = df /dR. The flaring out condition for wormhole geometries imposes T eff µν k µ k ν < 0, which yields the following generic condition in f (R) gravity

This condition has been extensively explored in [2], and specific wormhole solutions have been found. In principle, one may impose the condition T (m) µν k µ k ν ≥ 0 for the normal matter threading the wormhole. This is extremely useful, as applying local Lorentz transformations it is possible to show that the above condition implies that the energy density is positive in all local frames of reference. Note that in general relativity, i.e., f (R) = R, we regain the condition for the matter stress-energy tensor violation of the NEC, i.e., T (m) µν k µ k ν < 0. Thus, in the context of f (R) modified theories of gravity it is the higher order curvature terms, interpreted as a gravitational fluid, that sustain these non-standard wormhole geometries, fundamentally different from their counterparts in general relativity.

An interesting generalization of f (R) gravity is a model described by a curvature-matter coupling considered in [8,9]. The action is given by

where f i (R) (with i = 1, 2) are arbitrary functions of the Ricci scalar R and L m is the Lagrangian density corresponding to matter. The coupling constant λ characterizes the strength of the interaction between f 2 (R) and the matter Lagrangian. The effective gravitational field equation is given by Eq. ( 3), with the effective stress-energy tensor provided by

where for simplicity we have considered

µν k µ k ν ≥ 0 for the normal matter threading the wormhole, the NEC violation condition T eff µν k µ k ν < 0, and considering that 1 + λf 2 (R) > 0, imposes the restriction

In an analogous manner to f (R) modified theories of gravity, it is the higher order curvature terms in the curvature-matter coupling that sustain these non-standard wormhole geometries. These conditions have been extensively analysed and specific solutions have been found in [3,4].

Braneworld cosmology is an interesting scenario based on the idea that our Universe is a 3-brane embedded in a five-dimensional bulk. Considering that the bulk is vacuous, the induced field equations on the brane may formally be rewritten by Eq. ( 3), where the effective stress-energy tensor is given by

with κ 2 = λk 2 5 /6; κ 2 and k 2 5 are the gravitational coupling constants, on the brane and in the bulk, respectively; λ is the tension on the brane and T µν is the stress energy tensor confined on the brane. The first correction term re