# Wormholes, the weak energy condition, and scalar-tensor gravity

###### Abstract

We obtain a large class of Lorentzian wormhole spacetimes in scalar-tensor gravity, for which the matter stress energy does satisfy the weak energy condition. Our constructions have zero Ricci scalar and an everywhere finite, non-zero scalar field profile. Interpreting the scalar-tensor gravity as an effective on-brane theory resulting from a two-brane Randall–Sundrum model of warped extra dimensions, it is possible to link wormhole existence with that of extra dimensions. We study the geometry, matter content, gravitational red-shift and circular orbits in such wormholes and argue that our examples are perhaps among those which may have some observational relevance in astrophysics in future. We also study traversability and find that our wormholes are indeed traversable for values of the metric parameters satisfying the weak energy condition.

###### pacs:

04.50.Kd, 04.50.-h, 11.25.-w## I Introduction

Curved spacetimes for which the Ricci scalar is identically zero have been known in General Relativity (GR) since the discovery of the Schwarzschild solution. While, in GR, the Schwarzschild is a vacuum spacetime for which both and are zero, the Reissner-Nordstrom geometry has but , thereby implying the presence of traceless matter. It has been further shown in R3 that a generalisation of Schwarzschild spacetime with everywhere finite and non-zero and can be obtained. More recently R4 , this wormhole spacetime has been found to be a solution in a scalar-tensor theory of gravity which is also the low energy, effective, on-brane gravity theory for the warped two-brane Randall-Sundrum model. The spacetime in R3 when viewed as a solution in GR, requires matter that violates the Weak Energy Condition (WEC). However, as shown in R4 , in the context of the scalar-tensor theory, the required matter does not violate the WEC and the scalar field (radion) is also finite and non-zero everywhere. In our work here, we further generalise the spacetime studied in R3 ; R4 . In particular, newer spacetimes with everywhere non-zero are constructed and shown to be solutions in scalar-tensor gravity. A sub-class of such spacetimes are Lorentzian wormholes with the required matter satisfying the WEC.

In our analysis here, we begin with a general static, spherisymmetric line element for which the constraint is written as a differential equation for . Obviously, this differential equation contains (or, where . Assuming as one should for a Lorentzian wormhole, we find that for some specific choices (eg. the Schwarzschild wormhole and the Ellis–Bronnikov wormhole EW1 ; EW2 ) the differential equation for can indeed be solved. We obtain a wide class of geometries in this manner. Thereafter, we show how these geometries fare in terms of the WEC violation issue. ) and its derivatives in its coefficients. It can therefore be solved once we provide our choice for

As mentioned above, in GR we require WEC violation for the matter that threads non-singular wormhole spacetimes. However, it has been shown recently that, within GR, a wormhole (which is topologically different from the Morris-Thorne class) can be supported by a negative cosmological constant (which is not quite exotic, WEC violating matter) GRW . Further, it is known that certain wormhole solutions which have been constructed in various modified theories of gravity such as Brans-Dicke theory (scalar-tensor gravity with constant coupling ) BDW , -gravity fRW , Gauss-Bonnet gravity GBW , third-order Lovelock gravity LGW , Eddington-inspired Born-Infeld gravity EiBIW , mimetic gravity MW , DGP gravity DGPW , do not require WEC violating matter.

In our work here, we consider scalar-tensor gravity with a non-constant coupling and in the presence of matter stress-energy. Through our specific solutions, we show that there may not be WEC violation for the matter required to support a Lorentzian wormhole. It must be noted though that the timelike convergence condition is still violated, as it must be, for wormholes. The essential point is that the relation between the convergence condition and the energy condition is not as it is in GR, because of which there is extra freedom to avoid any violation in a modified theory of gravity (here, scalar-tensor gravity) R4 .

Our paper is organised as follows. In Section II, we briefly recall scalar-tensor gravity both as an independent theory and in the context of braneworld gravity. We provide the details of our construction of spacetimes in Section III. Section IV and V deal with the status of the WEC. In Section VI, we study gravitational redshift in the wormhole spacetimes. The existence of stable and unstable circular orbits and the issue of traversability are addressed in Section VII. Finally, we conclude in Section VIII.

## Ii Scalar-tensor gravity

Scalar-tensor theories are well-studied from various angles and perspectives. The first such theory was, of course, the Jordan-Brans-Dicke theory jbd . A scalar field and an extra parameter characterises such theories. Though originally considered a constant, a modified version incorporates a -dependent ST . In general, the Einstein field equations for such theories are given as:

(1) | |||||

where the is the matter energy-momentum tensor and the metric is written in the Jordan frame where the action is not in the standard canonical (Einstein-Hilbert) form. Viewed as an independent competitor of GR, a scalar tensor theory could be specified by providing a form for . Note that must satisfy the equation,

(2) |

Various aspects of scalar-tensor theories have been extensively studied and we will not review them here. See STG and references therein for detailed reviews. Taking the trace of (1) and using (2), one obtains

Note that traceless matter implies in GR. But, in scalar-tensor gravity, traceless matter does not imply in general. But, for the specific form of given by

and hence, traceless matter implies . Here, is an integration constant. As we shall see, the case arises in the low energy, effective on-brane quasi-scalar-tensor gravity theory developed by Kanno and Soda KS in the context of the Randall-Sundrum two-brane model.

It is well-known that scalar-tensor theories arise as effective theories of gravity in diverse contexts. For example, the low-energy effective gravity theory which emerges out of superstring theory contains a scalar known as the dilaton and the theory is a Brans-Dicke theory bdstring . In the braneworld scenario, our four dimensional visible universe is considered as a lower dimensional hypersurface called a brane, which is embedded in a higher dimensional bulk. In the low energy limit, the field equation governing bulk gravity, leads to effective field equations for gravity on the brane. The presence of the extra dimensional bulk leaves its imprint by modifying the Einstein field equations on the brane. The most popular among the on-brane effective Einstein equations was obtained by Shiromizu-Maeda-Sasaki SMS in the context of a single-brane model and contains a non-local term (bulk-Weyl dependent ). The on-brane effective Einstein equations obtained by Kanno and Soda KS in the context of the two-brane Randall Sundrum model, however, does not contain any non-local contribution. Earlier, the non-local was used to obtain spacetime solutions R7 ; R8 ; R9 ; R10 (mostly wormhole solutions) in the single-brane effective theory of Shiromizu-Maeda-Sasaki. In the effective on-brane theory of Kanno and Soda, we have an effective energy momentum tensor (other than the matter on the branes) constructed from the scalar radion field (which measures the inter-brane distance) and its derivatives. The radion field plays a crucial role in obtaining both static, spherically symmetric spacetimes and cosmological solutions R4 ; R5 ; R6 . It has been shown that the presence of the radion field can correctly reproduce the observed virial mass of galaxy clusters and observed galaxy rotation curves. Thus, the radion field can act as a possible dark matter candidate DM .

Let us now briefly review the low energy effective on-brane quasi-scalar-tensor theory developed by Kanno and Soda KS . The action is given as,

where , and are the five dimensional gravitational coupling constant, the five dimensional curvature scalar and the induced metric on the branes, respectively. and are the brane-tensions. The positive tension brane A (Planck brane) and the negative tension brane B (visible brane) are respectively placed at fixed bulk locations ( and ). The bulk line element is,

where the bulk curvature radius is . Assuming the brane curvature radius as large compared to the bulk curvature radius , i.e., , Kanno and Soda used a low energy expansion scheme (the gradient expansion method) wherein the bulk metric and the extrinsic curvature are expanded in powers of . They obtained the effective field equations,

(3) | |||||

on the visible brane. Here, , , and are respectively the radion field, the matter on the Planck brane, the matter on the visible brane and the metric tensor on the visible brane. The proper distance between the branes is defined as

The radion field satisfies the following equation of motion on the visible brane:

(4) |

where both the traces and are taken with respect to the metric tensor and the coupling function is given by

(5) |

Using the radion equation of motion in the trace of the field equation (3), we obtain , where is the curvature scalar on the visible brane. Note that this is the case discussed earlier. Therefore, traceless matter implies in this theory. Using the transformation , we can express the radion equation of motion in the form

(6) |

In the subsequent sections, we obtain our new solutions. It is important to note that the solutions could either be viewed as solutions in a scalar-tensor theory which has no link with extra dimensions or as solutions in the context of braneworld gravity. In obtaining the solutions, we take the Planck brane to be devoid of matter, i.e., . Further, henceforth, we refer to this on-brane scalar-tensor gravity as Kanno-Soda theory of gravity because of the specific form of the coupling function given in (5).

## Iii R=0 spacetimes

We begin with the static, spherically symmetric line element given as

where and are unknown functions to be determined. We are interested in vanishing curvature scalar () solutions. The constraint yields the following second order differential equation for :

(7) |

The above equation can also be re-written as a first order differential equation for . Attempts have been made to find out spacetime solutions in the past R1 ; R2 ; R3 ; R4 ; R5 ; R6 ; R7 ; R8 ; R9 ; R10 . In most of these articles, the authors have (a) considered known forms of and obtained solutions by solving the first order differential equation for or (b) used some known solutions present in the literature and analysed them in the context of an effective theory of gravity. In our work, we take a different route. We specify the form of and solve the second order differential equation for . This way we hope to obtain a more general solution different from those obtained earlier. Another advantage of specifying the form of the shape function is that we can choose to have wormhole solutions with a desired spatial shape. One can take different forms of . In our work here, we take . Putting in the equation, we obtain

After some manipulations, the above equation can be re-written in the form

which can be integrated to obtain

(8) |

where and are integration constants. Therefore, the line element becomes

(9) | |||||

It is useful to note that, when or , the is of the general form, . However, when , are both non-zero, this generic form does not hold and we have . However, it is important to note that, in all cases, the red-shift function is finite and non-zero everywhere.

We now look at special cases of our general solution. Some of these special cases reproduce the spacetimes obtained earlier by other authors. For , and , the spacetime takes the form

(10) |

This spatially Schwarzschild spacetime has been obtained in R2 ; R3 ; R4 ; R7 . This spacetime represents either a wormhole or a naked singularity for . For and , the spatially Schwarzschild solution reduces to the Schwarzschild spacetime. Taking and in this solution, we obtain the spatial Schwarzschild wormhole obtained in R8 . For , , and , the general metric takes the form

(11) |

where . This metric has been obtained in R8 ; R9 . This metric reduces to the extreme Reissner-Nordstrm form for , i.e., . The above metric fails to produce spatial Ellis geometry because implies . However, we can obtain spatial Ellis geometry from our general solution by taking and . The line element becomes

(12) |

This new metric has not been obtained before. Note that, due to the presence of the factor in , the above metric is different from the spatial Ellis geometry obtained in TG for vanishing torsion scalar in the context of modified teleparallel gravity. For , the general metric reduces to spatial Reissner-Nordstrm metric. The spatial Reissner-Nordstrm metric further reduces to the Reissner-Nordstrm metric for and .

Our focus is on the general metric (9). Depending on the signs and values of , , and , the general spacetime may represent a black hole, a wormhole or a naked singularity. The nature of the solution depends on the presence of the zeroes of and . We will focus on solutions for which , and . The solution for such choices does represent a Lorentzian wormhole.

For later use, let us now write down the solution in isotropic coordinates. The transformation from the Schwarzschild radial coordinate to the isotropic coordinate is given as

(13) |

Using the transformation between and we obtain, in isotropic coordinates, the line element,

(14) |

where

(15) |

(16) |

and

(17) |

The constants and are given as

(18) |

and . It is easy to show that and are related to and as follows:

(19) |

It can be shown that, for a spacetime without a horizon is necessary. We will use the isotropic coordinate form of the line element while discussing the weak energy condition.

Defining one can solve the radion field equation as shown in R4 . Obviously one ends up introducing a new constant which we denote as . We have

(20) |

where we have omitted an overall additive constant. It is easy to see that is never zero as along as . Note that the domain of is from to , where corresponds to the wormhole throat.

We will now turn towards analysing the Weak Energy Condition inequalities and demonstrate that it can indeed be satisfied for the wormhole we have constructed.

## Iv Weak Energy Condition (WEC)

The Weak Energy Condition comprises the inequalities

for a diagonal energy momentum tensor with energy density , radial pressure and tangential pressures defined in the static observer’s frame. Physically, the WEC means that the matter energy density is always non-negative in any frame of reference.

For our case here, we have, for the L.H.S. of the WEC inequalities,

(21) |

(22) |

(23) |

where the last equation follows from the traceless requirement on the matter energy momentum tensor.

For arbitrary values of the parameters it is not possible to satisfy these inequalities. However, one can isolate the negativity by looking at the requirements that emerge near . In the limit , we have

(24) |

(25) |

(26) |

where . Note that, at the leading order, (i.e., terms which are ), and are opposite in sign. This violates the WEC in the limit . Therefore, we must set the coefficient of in to zero. This yields the following expression for .

(27) |

which reduces to the expression quoted in R4 for and . With the above condition satisfied, it can be shown that in , term vanishes as well and term is always positive for any positive and . Therefore, it is clear from the traceless condition that, at the , positivity of implies positivity of . Note that, at this point, and can be chosen so that and are positive. However, we choose those values of and for which the coefficient of the term in vanishes. This gives us the following requirement on .

(28) |

Equating the above two expressions for , one arrives at a relation between and which must be obeyed.

(29) |

Thus, once we are able to choose a and a which satisfies the above relation, we can use their values to find . The set of values for , and can therefore be used to write down the line element and the scalar field profile for a Lorentzian wormhole in scalar-tensor gravity which will satisfy all the WEC inequalities.

The central issue at this point is whether we can obtain an analytical handle on the relation between and . To get rid of the logarithm, we can assume . This will eventually lead us to a quartic equation in for which we can obtain the positive, real valued roots in terms of . Choosing one can then obtain , and then .

We will now illustrate the above statements with examples and also demonstrate our claim that the WEC is indeed satisfied for the wormholes we have constructed. Putting in (29), we obtain

(30) |

where,

(31) |

(32) |

Therefore, we fix and and hence , and by choosing a particular value of . It can be shown that for the whole range . Therefore, the definitions of and indicate that both are positive. Figure 1 shows the parametric plot for and . Figure 2 shows the dependence of on . Figures 3-5 show the L.H.S of the WEC inequalities for three sets of parameter values. It is to be noted that WEC inequalities are satisfied.

## V The limit and WEC

Let us now take the limit . In this limit and diverges. But, the product is finite. In terms of , we have

(33) |

Therefore, and have opposite signs. In this limit, the factor in the energy density and pressures must be replaced by . It can be shown that does not have zeroes in the range for , thereby representing a wormhole. But, for , it does have a zero and represents a naked singularity. Let us first consider the case . Like the case, here also, one must consider the expression (27). With this, it can be shown that, in the limit , the coefficient of the term in cannot be set to zero since it is negative for all , thereby violating the WEC. It also implies that the expression in (28) is not valid. However, we can make use of the freedom to add an additive constant to and make this coefficient positive. If we do so, then we find that in (27), i.e., is imaginary. On the other hand, it can be shown that if we force to be positive, then either or or both become negative in the limit . Therefore, the wormholes without the mass term ‘’ violate the WEC. Figure 6 shows the WEC violation for . We have carried out a similar analysis for the naked singularity case where we found that the WEC can be satisfied in the limit , but the radion field vanishes at a point between the singularity and spatial infinity. This gives an invalid solution. Therefore, WEC cannot be satisfied for the naked singularity.

## Vi Gravitational redshift

Let us now look at the gravitational redshift of a light signal propagating in the wormhole spacetimes discussed above. Note that by virtue of having a which is everywhere finite and non-zero, the gravitational redshift is always finite. By definition, wormhole spacetimes are horizon-free. Assume that a light signal is emitted at a frequency from the wormhole throat () and it travels to spatial infinity () where a static observer receives it at a frequency . Therefore, the fractional change in frequency due to the gravitational redshift is given by the standard formula.

(34) |

We choose values of and which satisfy Eqn. (29) and hence, the WEC. Figure 7 shows the fractional change in frequency as a function of . It is worth noting that the light signal may have a negative gravitational redshift, i.e., a gravitational blueshift () for certain values of and . For the three sets of and given in Figs. 3-5, are respectively , and . For and , vanishes. Therefore, the wormholes can exhibit both redshift and blueshift. It can be shown that, for those and satisfying the WEC, the maximum redshift is . This redshift does not exceed for such values of and . However, we can have more than redshift if we choose and arbitrarily, which obviously violates the WEC. Thus, in a certain sense, as mentioned above, the WEC does restrict the gravitational redshift to an amount below . It is possible to use this result (along with other signatures) to identify whether the cause of an observed gravitational frequency shift is indeed due to a wormhole.

## Vii Effective potentials, circular orbits and traversability

### vii.1 Effective potentials and circular orbits

The geodesic equations for a test particle moving in the equatorial () plane of the spacetime 9 reduce to the following first integrals:

(35) |

where the energy and the angular momentum are constants of motion. An overdot represents differentiation with respect to the parameter (affine for timelike and arbitrary for null geodesics). The effective potential is given by

(36) |

where the normalisation constant is for timelike geodesics and is for null geodesics. We now look for circular orbits which correspond to (i.e. ) and (i.e. extrema of the effective potential). Minima () and maxima () of the effective potential correspond to the stable and unstable circular orbits, respectively. For the line element in (9), the effective potential can be written as

(37) |

where

(38) |

and

(39) |

Using Eqs. (7) and (39), it can be shown that

(40) |

Therefore, minima () exist if are real and . Note that we must have . This is possible when . Figure 10 shows the plots for the potential for parameter values chosen arbitrarily. We have checked that the WEC is violated for these parameter values. Therefore, stable null circular orbits exist if the WEC is violated.

It should be noted that, for the parameter values used in Fig. 10, and become complex conjugates of each other because the square root terms in the expressions of and become imaginary. But, the metric function still remains real since and are real. It can also be shown that is real. To verify this, let where and are real. Therefore, . Writing and where , we obtain

(41) |

which is real.

### vii.2 Traversability

Traversability of a wormhole demands that the tidal force felt by a human traveller moving radially, must be within tolerable limits. In an orthonormal basis attached to the traveller frame, the tidal acceleration between two parts of his/her body, separated by the deviation vector is given by morris1

(42) |

where is the Riemann tensor. At the throat, the components of the tidal acceleration are given by

(43) |

(44) |

where is the the radial velocity of the traveler as measured by a static observer. denotes the radial velocity at the throat. For the wormhole solutions, we must have . Note that, the radial component of the tidal acceleration vanishes for . Therefore, it can be made arbitrarily small by choosing an appropriate value of . We now restrict the radial component below one Earth gravity, i.e., . Thus, for a traveller of typical size meters, we obtain and

(45) |

(46) |

where we have used the expressions and . Note that and are related through . For a given metric parameter , the constraint on (and hence on ) puts a constraint on the throat radius. The constraint on is given by

(47) |

For which corresponds to Fig. 4, we obtain , where km is the Earth radius. However, we can reduce the lower limit on if we choose to be close to .

If the wormhole is to be traversable, not only the tidal acceleration but also the magnitude of acceleration felt by the traveller as he or she travels through it, must be within tolerable limits. In the orthonormal basis of the traveler frame, the magnitude of the acceleration is given by morris1

(48) |

Let us now consider both and to be finite at the throat. Therefore, at the throat , we have

(49) |

For the wormhole solutions, we must have . Therefore, for a given and which restrict the tidal acceleration below one Earth gravity, the magnitude of the acceleration can be made arbitrarily small by taking to be very close to zero which, of course, satisfies the WEC (Fig. 4).

## Viii Conclusion

In this work, we have obtained a class of static, spherically symmetric spacetimes which generalise certain known line elements. We have shown that our spacetimes can arise in a scalar-tensor theory of gravity, an example of which is the low energy, effective on-brane gravity theory developed by Kanno and Soda KS in the context of the Randall-Sundrum two-brane model. Some special cases of the general solution reproduce the spacetimes obtained earlier by other authors in the context of GR and in the on-brane effective gravity theory due to Shiromizu-Maeda-Sasaki SMS for a single-brane scenario. A sub-class of the general spacetimes found by us represents Lorentzian wormholes. We have studied the WEC for the matter that supports such wormholes and have shown that they can satisfy WEC. This is in contrast to GR, where such wormholes must violate WEC. We have also shown that the mass term in the metric functions is necessary to satisfy the WEC for these wormholes. Note that most of the wormhole solutions obtained in BDW are in vacuum-Brans-Dicke theory. However, the solutions we have obtained include both matter and a non-constant, i.e., a -dependent coupling