Lorentzian Wormholes in Lovelock Gravity
In this paper, we introduce the -dimensional Lorentzian wormhole solutions of third order Lovelock gravity. In contrast to Einstein gravity and as in the case of Gauss-Bonnet gravity, we find that the wormhole throat radius, , has a lower limit that depends on the Lovelock coefficients, the dimensionality of the spacetime and the shape function. We study the conditions of having normal matter near the throat, and find that the matter near the throat can be normal for the region , where depends on the Lovelock coefficients and the shape function. We also find that the third order Lovelock term with negative coupling constant enlarges the radius of the region of normal matter, and conclude that the higher order Lovelock terms with negative coupling constants enlarge the region of normal matter near the throat.
Wormholes are tunnels in the geometry of space and time that connect two separate and distinct regions of spacetimes. Although such objects were long known to be solutions of Einstein equation, a renaissance in the study of wormholes has taken place during 80’s motivated by the possibility of quick interstellar travel Mor1 (). Wormhole physics is a specific example of adopting the reverse philosophy of solving the gravitational field equation, by first constructing the spacetime metric, then deducing the stress-energy tensor components. Thus, it was found that these traversable wormholes possess a stress-energy tensor that violates the standard energy conditions (see, e.g., Mor2 (), Vis () or Lobo () for a more recent review). The literature is rather extensive in candidates for wormhole spacetimes in Einstein gravity, and one may mention several cases, ranging from wormhole solutions in the presence of the cosmological constant Mann (), wormhole geometries in higher dimensions high-dim (), to geometries in the context of linear and nonlinear electrodynamics Electro (). Also the stability of wormhole solutions has been analyzed by considering specific equations of state stability1 (), or by applying a linearized radial perturbation around a stable solution stability2 ().
One of the main areas in wormhole research is to try to avoid, as much as possible, the violation of the standard energy conditions. For static wormholes of Einstein gravity the null energy condition is violated, and thus, several attempts have been made to somehow overcome this problem. In order to do this, some authors resort to the alternative theories of gravity: the wormhole geometries of Brans-Dicke theory have been investigated in Brans (); of Kaluza-Klein theory in Kaluz (); and of a higher curvature gravity in HDgr (). In the latter, it was found that the weak energy condition may be respected in the throat vicinity of the wormholes of higher curvature gravity. A special branch of higher curvature gravity which respects the assumptions of Einstein –that the left-hand side of the field equations is the most general symmetric conserved tensor containing no more than two derivatives of the metric– is the Lovelock gravity Lov (). This theory represents a very interesting scenario to study how the physics of gravity are corrected at short distance due to the presence of higher order curvature terms in the action. Static solutions of second and third orders Lovelock gravity have been introduced in Boul () and Deh (), respectively. For wormholes with small throat radius, the curvature near the throat is very large, and therefore the investigation of the effects of higher curvature terms becomes important. The possibility of obtaining a wormhole solution from the instanton solutions of Lovelock gravity has been studied in Mar (). The wormhole solutions of dimensionally continued Lovelock gravity have been introduced in Li (), while these kind of solutions in second order Lovelock gravity and the possibility of obtaining solutions with normal and exotic matter limited to the vicinity of the throat have been explored in Kar (). Here, we want to add the third order term of Lovelock theory to the gravitational field equations, and investigate the effects of it on the possibility of having wormhole solutions with normal matter. We also want to explore the effects of higher order Lovelock terms on the region of normal matter near the throat.
The outline of this paper is as follows. We give a brief review of the field equations of third order Lovelock gravity and introduce the wormhole solutions of this theory in Sec. II. In Sec. III, we present the conditions of having normal matter near the throat and exotic matter everywhere. We finish our paper with some concluding remarks.
Ii Static Wormhole solutions
We, first, give a brief review of the field equations of third order Lovelock gravity, and then we consider the static wormhole solutions of the theory. The most fundamental assumption in standard general relativity is the requirement that the field equations be generally covariant and contain at most second order derivative of the metric. Based on this principle, the most general classical theory of gravitation in dimensions is the Lovelock gravity. The Lovelock equation up to third order terms without the cosmological constant term may be written as Hois ()
where ’s are Lovelock coefficients, is the energy-momentum tensor, is just the Einstein tensor, is the Gauss-Bonnet Lagrangian,
is the third order Lovelock Lagrangian, and and are
As in the paper of Morris and Thorne Mor1 (), we adopt the reverse philosophy in solving the third order Lovelock field equation, namely, we first consider an interesting and exotic spacetime metric, then finds the matter source responsible for the respective geometry. The generalized metric of Morris and Thorne in dimensions may be written as
where and are the redshift function and shape function, respectively. Although the metric coefficient becomes divergent at the throat of the wormhole , where , the proper radial distance
is required to be finite everywhere. The metric (5) represents a traversable wormhole provided the function is finite everywhere and the shape function satisfies the following two conditions:
where the prime denotes the derivative with respect to . The first condition is due to the fact that the proper radial distance should be real and finite for , and the second condition comes from the flaring-out condition Mor1 ().
The mathematical analysis and the physical interpretation will be simplified using a set of orthonormal basis vectors
Using the orthonormal basis (II), the components of energy-momentum tensor carry a simple physical interpretation, i.e.,
in which is the energy density, is the radial tension, and is the pressure measured in the tangential directions orthogonal to the radial direction. The radial tension , where is the radial pressure. Using a unit system with , and defining and for simplicity, the nonvanishing components of Eq. (1) reduce to
Iii Exoticity of the matter
To gain some insight into the matter threading the wormhole, one should consider the sign of , and . If the values of these functions are nonnegative, the weak energy condition (WEC) (, where is the timelike velocity of the observer) is satisfied, and therefore the matter is normal. In the case of negative , or , the WEC is violated and the matter is exotic. We consider a specific class of particularly simple solutions corresponding to the choice of const., which can be set equal to zero without loss of generality. In this case, and reduce to
iii.1 Positivity of and
Here, we investigate the conditions of positivity of and for different choices of shape function .
iii.1.1 Power law shape function:
First, we consider the positivity of and for the power law shape function with positive . The positivity of comes from the conditions (6) and (7). The functions and for the power law shape function are positive for provided , where is the largest positive real root of the following equations:
Of course if Eqs. (13) have no real root, then there is no lower limit for and and are positive everywhere.
iii.1.2 Logarithmic shape function:
Next, we investigate the positivity of and for logarithmic shape function, . In this case the conditions (6) and (7) include . The functions and are positive for provided , where is the largest real root of the following equations:
If , then and are positive for , but in the case that Eqs. (14) have no real positive root or their real roots are less than , then the lower limit for is just , and and are positive for
iii.1.3 Hyperbolic solution:
Finally, we consider the positivity of density and for the hyperbolic shape function, with , which satisfies the conditions (6) and (7). The functions and will be positive provided , where is the largest real root of the following equations:
Again for the case that Eqs. (15) have no real root, the functions and are positive everywhere.
iii.2 Positivity of
Now, we investigate the conditions of the positivity of . Since , as one may see from Eq. (6), the positivity of reduces to
One may note that when the Lovelock coefficients are positive, the condition (16) does not satisfy. For the cases that either of and or both of them are negative, the condition (16) is satisfied in the vicinity of the throat for power law, logarithmic and hyperbolic shape function provided that the throat radius is chosen in the range , where
For the choices of Lovelock coefficients where is not real, then the condition (16) does not hold. For the cases where is not real, then there is no lower limit for the throat radius that satisfies the condition (16). Even for the cases where exists and is chosen in the range , the condition (16) will be satisfied in the region , where and are the positive real roots of the following equation:
For negative , Eq. (18) has only one real root and the condition (16) is satisfied in the range . It is worth noting that the value of depends on the Lovelock coefficients and the shape function. The value of for the power law shape function is
which means that one cannot have a wormhole with normal matter everywhere. It is worth noting that , and therefore , as it should be.
iii.3 Normal and exotic matter
Now, we are ready to give some comments on the exoticity or
normality of the matter. First, we investigate the condition of having normal
matter near the throat. There exist two constraint on the value of for the power
law, logarithmic and hyperbolic shape functions, while for the logarithmic shape function
should also be larger than . The first constraint comes from the positivity
of and , which state that should be larger or equal to ,
where is the largest real root of
Eqs. (13), (14) and (15) for power law, logarithmic
and hyperbolic shape functions,
respectively. Of course, if there exists no real root
for these equations, then there is no lower limit for .
The second constraint, which come from the condition (16),
states that should be real. For positive Lovelock coefficients, there exists no real value
for , and therefore we consider
the cases where either of and or both of them are negative.
The condition (16) is satisfied near the throat for the following two cases:
1) and .
The root is real for the first case, while it is not real for the second case. In these two cases, one has normal matter in the vicinity of the throat provided , and is chosen in the range , where is the largest values of and .
The above discussions show that there are some constraint on the Lovelock coefficients and the parameters of shape function. In spite of these constraint, one can choose the parameters suitable to have normal matter near the throat. Even if the conditions of having normal matter near the throat are satisfied, there exist an upper limit for the radius of region of normal matter given in Eq. (18). Figures 1-4 are the diagrams of , and versus for various shape functions. In Figs. 2 and 4, the parameters have been chosen such that is real, while is not real in Figs. 1 and 3 and therefore has no lower limit. Note that for logarithmic shape function is larger than . All of these figures show that one is able to choose suitable values for the metric parameters in order to have normal matter near the throat. Also, it is worth to mention that the radius of normal matter increases as becomes more negative, as one may note from Eq. (18) or Fig. 5. That is, the third order Lovelock term with negative coupling constant enlarges the region of normal matter.
Second, we consider the conditions where the matter is exotic for with positive and . These functions are positive for provided , where is the largest real root of Eqs. (13), (14) and (15) for power law, logarithmic and hyperbolic solutions, respectively. Of course, there is no lower limit on , when these equations have no real root. On the other hand, the condition (16) does not hold for . Thus, if both of and are real, and one choose , where is the largest value of and , then the matter is exotic with positive and in the range . If none of and are real, then there is no lower limit for , and one can have wormhole with exotic matter everywhere.
Iv Closing Remarks
For wormholes with small throat radius, the curvature near the throat is very large, and therefore higher order curvature corrections are invited to the investigation of the wormholes. Thus, we presented the wormhole solutions of third order Lovelock gravity. Here, it is worth comparing the distinguishing features of wormholes of third order Lovelock gravity with those of Gauss-Bonnet and Einstein gravities. While the positivity of and does not impose any lower limit on in Einstein gravity, there may exist a lower limit on the throat radius in Lovelock gravity, which is the largest real root of Eqs. (13), (14) and (15) for power law, logarithmic and hyperbolic shape functions, respectively. Although the existence of normal matter near the throat is a common feature of the wormholes of Gauss-Bonnet and third order Lovelock gravity, but the radius of the region with normal matter near the throat of third order Lovelock wormholes with negative is larger than that of Gauss-Bonnet wormholes. That is, the third order Lovelock term with negative enlarges the radius of the region of normal matter. Thus, one may conclude that inviting higher order Lovelock term with negative coupling constants into the gravitational field equation, enlarges the region of normal matter near the throat. For th order Lovelock gravity with a suitable definition of in terms of Lovelock coefficients, the condition (16) may be generalized to
which can be satisfied only up to a radius . One may conclude from the above equation that as more Lovelock terms with negative Lovelock coefficients contribute to the field equation, the value of increases, but one cannot have wormhole in Lovelock gravity with normal matter everywhere for the metric (5) with . The case of arbitrary needs further investigation.
This work has been supported by Research Institute for Astrophysics and Astronomy of Maragha.
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