Wormholes in generalized hybrid metric-Palatini gravity obeying the matter null energy condition everywhere
Wormhole solutions in a generalized hybrid metric-Palatini matter theory, given by a gravitational Lagrangian , where is the metric Ricci scalar, and is a Palatini scalar curvature defined in terms of an independent connection, and a matter Lagrangian, are found. The solutions are worked in the scalar-tensor representation of the theory, where the Palatini field is traded for two scalars, and , and the gravitational term is maintained. The main interest in the solutions found is that the matter field obeys the null energy condition (NEC) everywhere, including the throat and up to infinity, so that there is no need for exotic matter. The wormhole geometry with its flaring out at the throat is supported by the higher-order curvature terms, or equivalently, by the two fundamental scalar fields, which either way can be interpreted as a gravitational fluid. Thus, in this theory, in building a wormhole, it is possible to exchange the exoticity of matter by the exoticity of the gravitational sector. The specific wormhole displayed, built to obey the matter NEC from the throat to infinity, has three regions, namely, an interior region containing the throat, a thin shell of matter, and a vacuum Schwarzschild anti-de Sitter (AdS) exterior. For hybrid metric-Palatini matter theories this wormhole solution is the first where the NEC for the matter is verified for the entire spacetime keeping the solution under asymptotic control. The existence of this type of solutions is in line with the idea that traversable wormholes bore by additional fundamental gravitational fields, here disguised as scalar fields, can be found without exotic matter. Concomitantly, the somewhat concocted architecture needed to assemble a complete wormhole solution for the whole spacetime may imply that in this class of theories such solutions are scarce.
Within general relativity, wormholes were found as exact solutions connecting two different asymptotically flat regions of spacetime Morris:1988cz (); Visser:1995cc () as well as two different asympotically de Sitter (dS) or anti-de Sitter (AdS) regions lemoslobooliveira (). The fundamental ingredient in wormhole physics is the existence of a throat satisfying a flaring-out condition. In general relativity this geometric condition entails the violation of the null energy condition (NEC). This NEC states that , where is the matter stress-energy tensor and is any null vector. Matter that violates the NEC is denoted as exotic matter. Wormhole solutions have been also found in other theories, see, e.g., Camera:1995 (); Nandi:1998 (); Bronnikov:2002rn (); Camera:2003 (); Lobo:2007zb (); Garattini:2007ff (); Lobo:2008 (); Garattini:2009 (); Lobo:2010sb (); Garattini:2011fs () and Lobo:2017oab () for reviews. In these works the NEC for the matter is also violated. However due to its nature, it is important and useful to minimize its usage.
In fact, in the context of modified theories of gravity, it has been shown in principle that normal matter may thread the wormhole throat, and it is the higher-order curvature terms, which may be interpreted as a gravitational fluid, that support these nonstandard wormhole geometries. Indeed, in Lobo:2009ip () it was shown explicitly that in theories wormhole throats can be theoretically constructed without the presence of exotic matter, in Garcia:2010xb (); MontelongoGarcia:2010xd () nonminimal couplings were used to build such wormholes, and in Harko:2013yb () generic modified gravities were used also with that aim in mind, i.e., the wormhole throats are sustained by the fundamental fields presented in the modified gravity alone. This type of solutions were also found in Einstein-Gauss-Bonnet theory Bhawal:1992 (); Dotti:2007 (); Mehdizadeh:2015jra (), and are mentioned in brane world scenarios Lobo:2007 (), in Brans-Dicke theories Anchordoqui:1997 (), and in a hybrid metric-Palatini gravitational theory Capozziello:2012hr (). It is our aim to find wormhole solutions whose matter obeys the NEC not only at the throat but everywhere in a generalized hybrid metric-Palatini gravity, with action .
The action is well motivated. Indeed, a promising approach to modified gravity consists in having a hybrid metric-Palatini gravitational theory Harko:2011nh (), which consists in adding to the Einstein-Hilbert action , a new term , where is a curvature scalar defined in terms of an independent connection, and is some function of . In this approach, the metric and affine connection are regarded as independent degrees of freedom, and contrary to general relativity where the metric-affine, or Palatini, formalism coincides with the purely metric formalism, in an theory the two formalisms lead to different results Olmo:2011uz (). In the theory one retains through the positive results of general relativity, with further gravitational degrees of freedom being represented in the metric-affine component. One can further express the theory in a dynamically equivalent scalar-tensor representation which simplifies the analysis. In this representation, besides wormhole solutions Capozziello:2012hr (), solar system tests and cosmological solutions have been analyzed and found Capozziello:2012ny (); Capozziello:2012qt (); Capozziello:2013yha (); Capozziello:2013uya (); Capozziello:2015lza (), see also HarkoLobo () for a review. A natural generalization to is to consider an action, i.e., the gravitational action is taken to depend on a general function of both the metric and Palatini curvature scalars Tamanini:2013ltp (). One can also find the scalar-tensor representation of this generalized hybrid metric-Palatini gravity, which now has two scalar fields. Exact solutions were constructed representing an FLRW (Friedmann-Lemaître-Robsertson-Walker) universe in a generalized hybrid metric-Palatini theory Rosa:2017jld (). Among other relevant results, it was shown that it is possible to obtain exponentially expanding solutions for flat universes even when the cosmology is not purely vacuum. In addition, the junction conditions in this theory have been worked out rosalemos1 ().
In this work, our aim is to find static and spherically symmetric wormholes solutions in the generalized hybrid metric-Palatini matter theory in which the matter satisfies the NEC everywhere, from the throat to infinity, so there is no need for exotic matter. This fills a gap in the literature as most of the work that has been done in this area has been aimed at finding solutions where the NEC is satisfied solely at the wormhole throat paying no attention to the other regions. As far as we are aware our work is the first where the NEC is verified for the entire spacetime.
This paper is organized as follows. In Sec. II, we consider the action and write out the gravitational field equations, both in the curvature and in the equivalent scalar-tensor representation. In Sec. III, we present the equations of motion for static and spherically symmetric wormholes solutions. In Sec. IV, we impose specific choices for the metric redshift and shape functions and for the potential governing the scalar fields, and find solutions where the matter threading the wormhole satisfies the NEC in the whole spacetime. In Sec. V, we conclude.
Ii Generalized hybrid metric-Palatini gravity with matter and its scalar representation
ii.1 Action and field equations
Consider the general hybrid action given by
where , is the gravitational constant, the speed of light is put to one, is the spacetime volume, is the determinant of the spacetime metric , latin indices run from 0 to 3, is the metric Ricci scalar, is the Palatini Ricci scalar, where the Palatini Ricci tensor is defined in terms of an independent connection as, , denotes partial derivative, is a well behaved function of and , and is the matter Lagrangian density considered minimally coupled to the metric .
Variation of the action (1) with respect to the metric yields the following equation of motion
where is the covariant derivative and the d’Alembertian, both with respect to , and is the stress-energy tensor defined in the usual manner as
Varying the action (1) with respect to the independent connection provides the following relationship
where is the covariant derivative with respect to the connection . Now, recalling that is a scalar density of weight 1, we have that and so Eq. (4) simplifies to . This means that defined as
is a metric to the connection which then can be written as the following Levi-Civita connection
Note also from Eq. (5) that is conformally related to through the conformal factor .
ii.2 Scalar-tensor representation of the generalized hybrid metric-Palatini gravity with matter
In order to have a better grasp on the sytem of equations, it is useful to express the action (1) in a scalar-tensor representation. This can be achieved by considering an action with two auxiliary fields, and , respectively, in the following form
Using and we recover action (1). Therefore, we can define two scalar fields as and , where the negative sign is imposed here for convention. The equivalent action is of the form
where we defined the potential as
and here and should be seen as functions of and . Taking into account that defined in Eq. (5) is conformally related to and that it can now be written as , we have that the Ricci scalars and are related through
So, we can replace in the action (8), to obtain
Varying the action (II.2) with respect to the metric provides the following gravitational equation
Varying the action with respect to the field and to the field yields after rearrangements, see Rosa:2017jld (),
In the next section, we will use the equations of motion (12), (13) and (14) to find static and spherically symmetric wormhole solutions. We will do two things. We will impose the flaring out condition that has to be obeyed at a wormhole throat, and we impose that the NEC on the matter stress-energy tensor are obeyed.
Iii Wormhole ansatz and equations
iii.1 Wormhole ansatz and equations
We impose that the wormhole solutions are described by a static and spherically symmetric metric which in the usual spherical coordinates has components , and whose corresponding line element has then the form Morris:1988cz ()
where is the redshift function and is the shape function. The shape function should obey the boundary condition , where is the radius of the wormhole throat.
Since the geometry is static and spherically symmetric, we assume that the scalar fields are functions of the radial coordinate alone,
We further assume that matter is described by an anisotropic stress-energy tensor of the form
where is the energy density, is the radial pressure, and is the transverse pressure. We can now use the ansätze given in Eqs. (15)-(18) to derive from Eqs. (12)-(14) the system of equations appropriate to this problem.
The three independent components of Eq. (12) are given by
The scalar field equation for , Eq. (13), after rearrangements, can be written as
The scalar field equation for , Eq. (14), can be written as
To derive Eq. (22) we do the following. Equation (13) gives directly an equation for , i.e., . On the other hand, the trace of the stress-energy tensor is . From Eqs. (III.1), (III.1), and (21) we get . Subtracting Eq. (III.1) to the equation and replacing in the resulting equation the value of just obtained, the terms dependent on and cancel out, yielding Eq. (22).
We have now a system of equations to solve. There are five independent equations, given by the three field equations (III.1)-(21), and the two scalar field equations (22) and (III.1). On the other hand, there are eight variables, , , , , , , , and . Therefore, we can give three functions to the system to solve it.
iii.2 Strategy for wormhole solutions obeying the matter null energy condition
iii.2.1 Main aim: Wormhole matter fields do not violate the null energy condition anywhere
The main aim in our wormhole construction is that throughout the wormhole solution the matter must obey the NEC. This needs some explanation. In pure general relativity, it is known that at a wormhole troat, the NEC for the matter fields is violated in order to allow for the wormhole flare out. In the scalar representation of generalized hybrid metric-Palatini gravity we are studying, the gravitational field is complemented by the two other fundamental gravitational fields, namely, the scalar fields and , whereas the matter fields in this theory are still encoded in the stress-energy tensor . Thus, to allow for a mandatory flare out (see, e.g. Visser:1995cc ()), the matter can obey the NEC, as long as some appropriate combination of with the other fundamental fields does not obey the NEC. This is our main objective: to find wormhole solutions in a theory with extra fundamental gravitational fields, represented here by the two scalar fields, in which the matter obeys the NEC.
To build such a wormhole might be a difficult task as the NEC can be obeyed at a certain radius and then be violated at some other radius. Our strategy is then the following. The most critical radius is the wormhole throat radius. So we impose that the NEC for the matter fields is obeyed at the throat and its vicinity. We then look for a solution in the vicinity of the throat. When the solution starts to violate the NEC above a certain away from the throat, we cut the solution there. We then join it to a vacuum solution for larger radii and build a thin shell solution at a junction radius up to which the matter fields obey the NEC.
To be specific, we start by finding wormhole solutions for which the matter threading the throat is normal matter, i.e., the matter stress-energy tensor , for two null vectors and , must obey the NEC, i.e., . This NEC is to be obeyed at the throat and its vicinity. Noting that is of the form given in Eq. (18), taking into account Eqs. (III.1)-(21), and choosing in the frame of Eq. (15) a null vector of the form , one obtains from that
For a null vector of the form , we find that means
It might happen that the weak energy condition (WEC) for the matter alone is also be obeyed at the wormhole throat, although in our wormhole construction we do not impose it. To verify whether the WEC is verified or not, we choose a timelike vector of the form , and see if . For a perfect fluid becomes
iii.2.2 Flaring out condition at a wormhole throat
The shape function obeys two boundary conditions. First,
where is the radius of the wormhole throat. Second, the fundamental condition in wormhole physics is that the throat flares out which is translated by the following condition at the throat, . This imposes that at the throat we have Morris:1988cz (); Visser:1995cc ()
The flaring out condition, Eq. (28), is a geometric condition, indeed it is a condition on the Einstein tensor . In general relativity, since the field equation is , it is also a condition on the matter stress-energy tensor and it directly implies that violates the NEC. However, in this alternative theory, i.e., the scalar field representation of the generalized hybrid metric-Palatini theory, besides the metric gravitational field, there are two extra gravitational fundamental fields and . So, the flaring out condition on , in the hybrid generalization, translates into a condition on a combination of the matter stress-energy with an appropriate tensor built out of , , and geometric tensorial quantities. Let us see this in detail. Note that Eq. (12) can be written in the form where the tensor is given by We can define for clarity an effective stress-energy tensor as . Our aim, defined from the start, is that the NEC on the matter is obeyed everywhere, including the wormhole throat. But, since for wormhole construction one needs to flare out at the throat then the NEC on has to be violated there. This means that the contraction of the effective stress-energy tensor with two null vectors, and say, must be negative, i.e., , at the throat. From the definition of above, this condition can be converted into , where . So it gives . We assume that and write thus this condition as , at the throat, i.e., this condition has to be obeyed at the throat. Noting that is of the form given in Eq. (18), taking into account Eqs. (III.1)-(21), and choosing in the frame of Eq. (15) a null vector of the form , we find that at , the throat, is then . For a null vector of the form , we find that is then . The violation of the NEC implies the violation of the WEC at the wormhole throat Visser:1995cc (). We choose a timelike vector of the form , and we find that becomes . These inequalities at the wormhole throat, coming out of to enable flaring out, and the NEC and WEC on , Eqs. (24)-(26), are not incompatible at all.
iii.2.3 General remarks on the behavior of the fundamental fields
The fundamental gravitational fields are the metric fields, namely, the redshift function , the shape function , and the scalar fields and of which the potential is formed.
The redshift function must be finite everywhere so that the term in the metric (15) is always finite and does not change sign avoiding thus the presence of horizons. The shape function essentially should be well-behaved throughout and obey the flare out condition at the throat as we discussed above, see (28). The scalar fields and should be well-behaved throughout and the potential , dependent directly on them which in turn depend on , should be chosen appropriately.
Iv Wormhole solutions in the generalized hybrid theory with matter obeying the null energy condition everywhere
iv.1 Patch by patch building the wormhole solution
iv.1.1 Wormhole solutions in the neighborhood of the throat
We have the freedom to choose three functions out of the eight variables, , , , , , , , and , since there are only five independent equations (III.1)-(22). The three functions that we choose are , , and .
A quite general class of redshift and shape metric functions, see Eq. (15), that verify the traversability conditions for a wormhole is
where is a constant with no units, is the wormhole throat radius, and , , , and , are free exponents. We also assume that so that the redshift function given in Eq. (29) is finite and nonzero for every value of between and infinity, and is the value of at . We further assume that , so that the shape function given in Eq. (30) yields and , as it should for a wormhole solution, see Eqs. (27) and (28). We consider further a power-law potential of the form
where is a constant and some exponent. With these choices of Eqs. (29)-(31) and for various combinations of the parameters . and one can then find specific equations for and from Eqs. (22) and (III.1) and try analytical or numerical solutions which would then give wormholes.
Here we are interested in finding solutions that do not violate the matter NEC anywhere, Eqs. (24)-(25), in particular at the throat, and , and possibly at its neighborhood. As we will see, a solution that respects these conditions is obtained for , and . For these choices, Eqs. (29) and (30) yield
respectively. Thus, the line element Eq. (15) can be written for the interior wormhole region containing the throat as
Choosing Eq. (31) yields
respectively. Equation (37) can be solved analytically to obtain an expression for the scalar field , namely, , where and are integrating constants. Putting it into Eq. (36) we find the solution for . Thus, the solutions for and are
respectively. We can simplify further and still find nontrivial solutions. Let us choose , so that the solution for in Eq. (38) is
and the solution for in Eq. (39) is
So far we have the solution for the fundamental gravitational fields, displayed in Eqs. (32) and (33) or in the line element Eq. (34), for the scalar potential, Eqs. (35), and for the fundamental extra gravitational fields represented by the scalar fields, Eqs. (40) and (41). We can now find the matter fields and study in which cases they obey the matter NEC. Inserting these solutions into Eq. (III.1), (III.1), and (21), we obtain the energy density, the radial pressure, and the tangential pressure as
It is more useful to combine with and to obtain expressions directly connected to the NEC. These combinations give
At the throat, , one has . As we want to preserve the NEC at the throat we have to set negative,
In this case the NEC is valid in the throat’s neighborhood up to to the radius given by . The quantity is positive for every value of and zero at the throat . Thus the quantity sets the violation of the NEC. We then impose that the interior solution is valid within the following range of ,
So, the interior solution must stop at most at . Since , see Eq. (42), we have that in this range of the WEC, i.e., , is also obeyed.
The interior solution with its metric fields and , scalar fields and , and matter fields , , and are displayed for all in Fig. 1 for adjusted values of the free parameters. The quantity turns negative at which is precisely for . The interior solution should at most stop there.
iv.1.2 Solutions outside the throat’s neighborhood up to infinity: Schwarzschild spacetimes with a cosmological constant
To guarantee that the complete solution does obey the NEC for any value of , we need to match at some , say, less than the interior solution just found to an external vacuum spherically symmetric solution. To do so, one has to derive a vacuum and asymptotically flat exterior solution, and then use the junction conditions of this theory to perform the matching with some thin shell at the boundary between the interior and exterior. Having performed this, we still have to ensure that the matter of this thin shell obeys the matter NEC, in order that the complete spacetime obeys theis condition.
We now find an exterior vacuum solution so that we can use the junction conditions to match the interior wormhole solution to the exterior vacuum solution. To do so, we put the stress-energy tensor to zero, , as we want a vacuum solution. In addition, we chose the scalar fields to be constant in this exterior solution, i.e., and with and constants, , and where the subscript stands for exterior. For continuity we choose the potential to be . Note that from Eq. (9) it can be seen that choosing a particular form of the potential we are also setting a particular solution for the function . This means that both the interior and the exterior spacetimes must be solutions of the field equations with the same form of the potential , because otherwise they would not be solutions of the same form of the function . These choices imply that the field equation Eq. (12) can be written as
Considering that in the second term the multiplicative factors of the metric are constant, we see that the field equation is of the same form as the Einstein field equations in vacuum with a cosmological constant given by . Thus, the Schwarzschild solution with a cosmological constant of general relativity is a vacuum solution of the generalized hybrid theory we are studying. This class of solutions is also known as the Kottler solution, as well as Schwarzschild-dS solution if the constant cosmological term is positive and Schwarzschild–AdS solution if the constant cosmological term is negative. The metric fields and for the exterior region outside some radius are then
respectively. where is a constant of integration and represents the mass, and the factor is a useful constant. The line element of the generalized field equations given by Eq. (12) in the case where the scalar fields are constant is then
The sign of the term will determine whether the solution is Schwarzschild-dS or Schwarzschild-AdS. This sign will be determined by the matching surface and the imposition that the NEC holds everywhere. Wormholes in dS and AdS spacetimes in general relativity were treated in lemoslobooliveira ().
To complete, the potential, scalar fields, and matter fields have then for the outer solution the following expressions,
with and constants, and
iv.1.3 The shell at the junction: the surface density and pressure of the thin shell
To match the interior to the exterior solution we need the junction conditions for the generalized hybrid metric-Palatini gravity. The deduction of the junction equations has been performed in rosalemos1 (). There are seven junction conditions in this theory that must be respected in order to perform the matching between the interior and exterior solutions, two of them that imply the existence of a thin shell of matter at the junction radius . These conditions are
where is the induced metric at the junction hypersurface , with Greek indexes standing for and , the brackets denote the jump of any quantity across , is the unit normal vector to , is the trace of the extrinsic curvature of the surface (see, e.g., Visser:1995cc ()), is the trace of the stress-energy tensor of the thin shell, is the Kronecker delta, and the subscripts indicate the value computed at the hypersurface.
Now, the matter NEC given in Eqs. (24)-(25) should also apply to the shell, since we want this condition to be valid throughout the whole spacetime. When applied for the particular case of the thin shell the matter NEC is
This factor has been chosen so that the time coordinate for the interior, Eq. (34), is the same as the coordinate for the exterior, Eq. (IV.1.2). The angular part of the metrics Eq. (34) and Eq. (IV.1.2) are continuous. So the line element at the surface and outside it is
where we are using that the and are continous at the matching, see below.
The condition Eq. (58) means that
where we have used Eq. (40). Whenever it occurs we keep the notation .
The condition Eq. (59) means that
where we have used Eq. (41). Whenever it occurs we keep the notation .
The condition Eq. (60) is empty, since is constant everywhere.