Geometry of Spinning Ellis Wormholes
We give a detailed account of the properties of spinning Ellis wormholes, supported by a phantom field. The general set of solutions depends on three parameters, associated with the size of the throat, the rotation and the symmetry of the solutions. For symmetric wormholes the global charges possess the same values in both asymptotic regions, while this is no longer the case for non-symmetric wormholes. We present mass formulae for these wormholes, study their quadrupole moments, and discuss the geometry of their throat and their ergoregion. We demonstrate, that these wormholes possess limiting configurations corresponding to an extremal Kerr black hole. Moreover, we analyze the geodesics of these wormholes, and show that they possess bound orbits.
pacs:04.20.Jb, 04.40.-b, 04.40.Nr
General Relativity allows for Lorentzian wormholes Visser:1995cc (), which can be either non-traversable or traversable. Non-traversable wormholes were first discussed in 1935 by Einstein and Rosen Einstein:1935tc (). Here the famous ‘Einstein-Rosen bridge’ represents an interesting feature of the Schwarzschild spacetime, connecting two asymptotically flat universes by a throat. In the 1950’s Wheeler discussed the intriguing possibility to connect two distant regions within a single universe Wheeler:1957mu (); Wheeler:1962 (). Unfortunately, this kind of wormholes does not allow for their passage Kruskal:1959vx (); Fuller:1962zza (); Redmount:1985 (); Eardley:1974zz (); Wald:1980nk ().
Transversable wormholes in General Relativity were first obtained by Ellis Ellis:1973yv (); Ellis:1979bh () and Bronnikov Bronnikov:1973fh () (see also Kodama:1978dw ()). However, these need the presence of a new kind of matter field coupled to gravity. Such a phantom field carries the wrong sign in front of its kinetic term, and consequently its energy-momentum tensor violates the null energy condition. The conditions for the existence of traversable wormholes as well as the properties of such static wormholes were discussed later in detail by Morris and Thorne Morris:1988cz (), who addressed also the possibility of time-travel Morris:1988tu ().
While phantom fields have been employed frequently in cosmology in recent years, since they give rise to an accelerated expansion of the Universe Lobo:2005us (), it was also pointed out Hochberg:1990is (); Fukutaka:1989zb (); Ghoroku:1992tz (); Furey:2004rq (); Bronnikov:2009az (); Kanti:2011jz (); Kanti:2011yv (), that the presence of a phantom field can be evaded in the construction of wormholes, when gravity theories with higher curvature terms are considered.
Wormholes have also been considered from an astrophysical point of view, since they represent hypothetical objects that can be searched for observationally Abe:2010ap (); Toki:2011zu (); Takahashi:2013jqa (). Cramer et al. Cramer:1994qj () and Perlick Perlick:2003vg (), for instance, have investigated wormholes as gravitational lenses, Tsukamoto et al. have addressed the Einstein rings of wormholes Tsukamoto:2012xs (), Bambi Bambi:2013nla () and Nedkova et al. Nedkova:2013msa () have studied the shadow of wormholes, while Zhou et al. Zhou:2016koy () determined the iron line profile in the X-ray reflected spectrum of a thin accretion disk around wormholes. The astrophysical signatures of mixed neutron star–wormhole systems have been addressed by Dzhunushaliev et al. Dzhunushaliev:2011xx (); Dzhunushaliev:2012ke (); Dzhunushaliev:2013lna (); Dzhunushaliev:2014mza (); Aringazin:2014rva (); Dzhunushaliev:2016ylj ().
Here we consider rotating wormholes supported by a phantom field, which represent solutions of the coupled Einstein-phantom field equations. Slowly rotating perturbative solutions of these equations were obtained by Kashargin and Sushkov Kashargin:2007mm (); Kashargin:2008pk (). This was followed by a brief discussion of rapidly rotating non-perturbative wormholes given by two of us Kleihaus:2014dla (). In the present paper we give a detailed account of the properties of these rotating wormholes. Moreover, we discuss rotating non-symmetric wormholes for the first time. Let us note, that the rotating wormhole metric presented by Teo Teo:1998dp () was not obtained as a solution of a coupled system of Einstein-matter equations.
The paper is structured as follows. We begin with the theoretical setting in section II, which includes the presentation of the action and the Ansatz, the discussion of the equations of motion and the boundary conditions, the derivation of the global charges and mass formulae as well as the derivation of the quadrupole moment. We then give a discussion of the geometric properties of the wormholes, in particular, we define the position of the throat as a minimal surface, and we demonstrate the violation of the energy conditions.
Subsequently in section III we present our numerical results for symmetric wormholes, i.e., those wormholes whose global charges are the same on both sides of the wormhole. These wormholes then possess one less parameter. In particular, we consider families of wormhole configurations for fixed equatorial throat radius, and demonstrate that these solutions approach an extremal Kerr black hole, as the angular momentum tends towards its maximal value. We then exhibit embeddings of wormhole spacetimes and demonstrate, that the Gaussian curvature of the throat turns negative at the poles, when the rotation is sufficiently fast, analogously to the curvature of the horizon of Kerr black holes. Finally, we examine the effective potential and the geodesics of rotating wormholes, which in contrast to static Ellis wormholes allow for bound orbits.
Non-symmetric rotating wormholes are presented in section IV. They depend on three parameters, and have distinct properties when the asymmetry parameter is negative or positive. We analyze the domain of existence of these non-symmetric wormholes, and demonstrate that the families of non-symmetric wormholes also approach an extremal Kerr black hole. We then address the somewhat surprising fact, that the rotational velocity of the throat can exceed the velocity of light for positive . We show how the location of the throat changes as the asymmetry of the solutions is varied. Finally, we consider embeddings of non-symmetric wormholes and geodesics in these spacetimes. We end with our conclusions in section V, and present the metric functions for large positive asymmetry parameter in the Appendix.
Ii Theoretical Setting
ii.1 Action and Ansatz
We now turn to wormhole solutions in General Relativity, obtained with a phantom field . The corresponding action reads
leading to the Einstein equations
and the phantom field equation
To construct stationary rotating spacetimes we employ the line element
where , , and are functions only of and , and is an auxiliary function. We note that the coordinate takes positive and negative values, i.e. . The limits correspond to two distinct asymptotically flat regions. The phantom field also depends only on the coordinates and .
ii.2 Equations of Motion and Boundary Conditions
When we substitute the above Ansatz into the general set of equations of motion we obtain a system of non-linear partial differential equations (PDEs). As observed in Kleihaus:2014dla () the PDE for the function decouples, and has the simple form
A solution which satisfies the boundary conditions and is given by
Consequently, we obtain the phantom field equation
A first integral can be obtained under the assumption ,
where denotes the phantom scalar charge. This leads to the closed form solution for
where the integration constant was chosen to satisfy .
We now turn to the Einstein equations. With and the equations , , , and are independent of the scalar field. They yield three second order PDEs for the functions , and , and a constraint,
In order to obtain regular solutions we have to impose boundary conditions in the asymptotic regions and on the axis . For we require the metric to approach Minkowski spacetime, i.e.,
In the limit we allow for finite values of the functions and ,
We refer to the solutions as symmetric, when , and non-symmetric, when . Note, that in the limit the spacetime becomes Minkowskian after a suitable coordinate transformation is performed. Regularity on the symmetry axis requires
together with the analogous conditions at .
Static wormhole solutions correspond to . They are known in closed form,
Rotating wormhole solutions of the above equations have been studied in Kleihaus:2014dla () for the symmetric case only, while we here address the non-symmetric case, as well. Substitution of the solutions for , and in shows, that
where the scalar charge depends on the mass and the angular momentum of the spacetime. Explicitly we find
ii.3 Mass, Angular Momentum and Mass Formulae
The mass and angular momentum of the wormhole solutions are engraved in the asymptotic form of the metric tensor. For an asymptotically flat metric, the mass and the angular momentum can be read off from the components and ,
In the limit the wormhole metric becomes Minkowskian, as seen from the asymptotic behavior of the metric functions. In particular,
so we can read off the mass and the angular momentum directly.
The analogous asymptotic expansion at the other side of the wormhole reads
However, in order to identify the mass and the angular momentum at , we first have to perform a coordinate transformation to obtain an asymptotically flat spacetime in this limit. After applying
we obtain and in terms of the parameters and ,
Thus the angular momenta on both sides of the wormhole are equal.
For the static wormhole solutions (17), this relation reduces to , where the mass is related to by .
In the symmetric case, we regain a mass formula akin to the Smarr formula for black holes. Here we integrate both formulae from the throat () to infinity, and take into account that vanishes at the throat. Denoting , we find Kleihaus:2014dla ()
where follows from the symmetry and the choice of boundary conditions. Consequently, also the relation holds in this case. Thus on both sides of the wormhole the mass is the same. Note, that the mass relation (29) then agrees with the one for extremal Kerr black holes, when is identified with the horizon angular velocity.
To find a relation of to the mass we evaluate Eq. (19) in the limit . Here we must take into account the asymptotic behaviour of the function ,
which reduces to in the static case.
ii.4 Quadrupole Moment
In order to extract the quadrupole moment of the rotating wormhole solutions, we need to consider the expansion of the metric in the asymptotic regions. For simplicity, let us only consider here. The asymptotic expansion of the metric functions is then given by
where is the second Legendre polynomial and and are constants.
The quadrupole moment can now be derived employing the definitions of Geroch and Hansen Geroch:1970cd (); Hansen:1974zz (), and following later work Hoenselaers:1992bm (); Sotiriou:2004ud (); Pappas:2012ns () according to the steps outlined in Kleihaus:2016dui () we first transform the metric to quasi-isotropic coordinates. This yields
where is related to by , and . We compare this with
Using the expansion in the asymptotic region we find by comparison with Eqs. (A9)-(A11) in Kleihaus:2016dui ()
where we have slightly changed the notation of Kleihaus:2016dui (): , . Now , and can be expressed in terms of and ,
Thus we finally arrive at the quadrupole moment
(see e.g. Kleihaus:2016dui () for more details).
ii.5 Geometric Properties
Let us next address the geometric properties of the wormhole solutions. We first consider the equatorial (or circumferential) radius as a function of the radial coordinate
Because of the rotation, the throat of the wormholes should deform and its circumference should be largest in the equatorial plane. Therefore a study of should reveal the location of the throat in the equatorial plane.
Consequently, we define the throat of the wormhole as the surface of minimal area, which intersects the equatorial plane at the circle of minimal circumferential radius , i.e.,
for the line element Eq. (4).
We now parametrize the surface of the throat by . Then the line element on the surface reads
where we defined , and the functions , and are regarded as functions of and . The area of the surface is now given by the integral of the square root of the determinant of the metric tensor
The function is determined as solution of the Euler-Lagrange equation
This yields the ordinary differential equation
where we introduced . We solve this equation numerically for boundary conditions , required by regularity, and , where is the coordinate of the throat in the equatorial plane, determined by the minimum condition for the circumferential radius.
We note that in the symmetric case the throat is located at the constant coordinate , since and at for all . For the static wormholes and . In this case the minimum condition Eq. (44) yields , which implies in Eq. (49). Consequently the solution of Eq. (49) is in this case . Therefore the static non-symmetric wormholes also possess a throat with constant coordinate (as expected). For the rotating non-symmetric wormholes, however, it is not obvious, that the throat should be located at a constant radial coordinate, as well. Therefore the dependence of the coordinate on must be examined numerically. We further note, that we do not find rotating Ellis wormhole solutions with multiple throats, separated by bellies (or equators).
To gain further information on the geometry of the rotating symmetric wormholes, we consider in addition the polar radius
and the areal radius
Denoting the angular velocity of the throat by , the rotational velocity of the throat in the equatorial plane is given by
It was shown in Kleihaus:2014dla () that for the symmetric wormholes.
ii.6 Violation of the Null Energy Condition
Let us next demonstrate the violation of the Null Energy Condition (NEC) and consider the quantity
with null vector Kashargin:2008pk ()
Taking into account the Einstein equations and the phantom field equation we obtain
Since is non-positive the NEC is violated everywhere. In order to obtain a global scale invariant quantity as measure of the NEC violation we define for later reference for the symmetric wormholes
Iii Symmetric Wormholes
In the following we present the properties of symmetric rotating wormhole solutions in General Relativity, which are characterized by Kleihaus:2014dla (). We then continue with the non-symmetric case, where for fixed the solutions depend on the parameters and .
iii.1 Global Charges
Symmetric rotating Ellis wormholes depend on the throat parameter and the angular velocity of the throat . Let us first fix the throat parameter and vary , starting from the static Ellis wormhole. Then the mass and angular momentum increase from their Ellis value of zero with increasing angular velocity. At the same time, the rotational velocity of the throat in the equatorial plane increases as it should, since it is proportional to . The centrifugal force causes in addition an increase of the equatorial circumferential radius of the throat and thus of .
We exhibit in Fig. 1 the dependence of the mass and the angular momentum on the rotational velocity for these wormholes. As the rotational velocity tends to its limiting value of one, where the throat would be rotating with the velocity of light the mass and the angular momentum both diverge. Also shown in Fig. 1 is the dependence of the scalar charge (we have chosen to be non-negative) on the rotational velocity , and the relation between and .
The scalar charge is maximal for static wormholes and decreases monotonically to zero, as the family of wormhole solutions approaches its limiting configuration, where the rotational velocity reaches the speed of light. The angular velocity of the throat, however, does not increase monotonically with . Instead, it reaches a maximal value and then decreases again towards zero. The reason for the occurrence of two branches with respect to is our use of ‘isotropic’ coordinates. The analogous pattern arises for rotating black holes, when the isotropic horizon radius is held fixed, while the angular momentum increases monotonically (see e.g. Kleihaus:2000kg ()).
To get a more physical picture of the families of symmetric rotating wormholes and better understand the limiting behavior for , let us now fix the equatorial throat radius . Then the family of symmetric rotating wormholes evolves from the static Ellis wormhole with to a limiting configuration with . This is seen in Fig. 2, where the scaled mass , the scaled angular momentum and the scaled scalar charge are shown versus the rotational velocity of the throat in the equatorial plane. Again, the scalar charge decreases monotonically and tends to zero for . By exhibiting the r.h.s. of Eq. (29) as well, we demonstrate, that all solutions satisfy the Smarr-type relation (29) with high accuracy. Also shown in Fig. 2 are the corresponding quantities for the Kerr black holes for fixed equatorial horizon radius, , and .
Let us now demonstrate, that in the limit an extremal Kerr black hole is approached. We will consider only; the discussion for is analogous. First of all, the gobal charges mass and angular momentum precisely assume the respective Kerr values in the limit, while the scalar charge vanishes. (Note, that a circumferential radius corresponds to a Boyer-Lindquist horizon radius , and an extremal black hole has .) Thus asymptotically the extremal Kerr metric is approached. However, in the limit the metric tends not only asymptotically but everywhere to the extremal Kerr metric. This is demonstrated in Fig. 3. Here we fix the angular velocity of the throat and the horizon, . This requires a scaling of the radial coordinate and metric function , which leaves the product and invariant. Then the metric functions , and are shown versus the radial coordinate for and together with the limiting extremal Kerr solution.
As the dimensionless rotational velocity reaches its maximal value, , the hypersurface , which corresponds to the throat of the wormholes, must change its character. Since the limiting solution represents an extremal black hole, a degenerate horizon must form instead. Moreover, the phantom field vanishes identically in this limit, as it should for a Kerr black hole. Consequently, the Smarr relation (29) for extremal Kerr black holes is recovered, with denoting the horizon angular velocity.
iii.3 NEC, Quadrupole Moment and Moment of Inertia
In Fig. 4 we show the quantity , Eq. (55), which gives for the symmetric rotating wormhole solutions a scale independent measure for the violation of the NEC, as a function of the rotational velocity of the throat . We observe that the violation of the NEC is strongest in the static limit and disappears in the limit , where the extremal Kerr solution is approached.
The observation that with increasing rotation the violation of the NEC decreases and the family of rotating wormholes ends in an extremal black hole, appears to be a generic feature. It was also observed for rotating Ellis wormholes in 5 dimensions, which end in an extremal Myers-Perry black hole Dzhunushaliev:2013jja (). Moreover, the family of electrically charged static Ellis wormholes in 4 dimensions ends in an extremal Reissner-Nordström black hole with increasing electric charge Hauser:2013jea ().
Let us now turn to the quadrupole moment , given in Eq. (42), as obtained by following the procedure of Geroch and Hansen Geroch:1970cd (); Hansen:1974zz () by extracting it from the asymptotic expansion in appropriate coordinates (see e.g. Kleihaus:2014lba (); Kleihaus:2016dui ()). We exhibit the quadrupole moment in Fig. (4) for the family of symmetric rotating wormholes at fixed equatorial throat radius . Starting from zero in the static case, the scaled quadrupole moment increases monotonically to its maximum of , when the throat velocity approaches the speed of light. Also shown is the scaled quadrupole moment for the Kerr black holes, . We note that the scaled quadrupole moment is larger for the Kerr black holes for any value of the horizon resp. throat velocity (except for and , where both coincide).
One often considers the dimensionless quadrupole moment , which is constant for the Kerr solutions, . The dimensionless quadrupole moment of the wormhole solutions increases monotonically from zero in the static limit to the value of the Kerr solutions in the limit .
The moment of inertia of the wormholes is another quantity of interest, in particular, in comparison with other compact objects. We exhibit the moment of inertia for the same set of solutions in Fig. 4. The scaled moment of inertia is smaller than the corresponding value of the Kerr black holes, , except in the extremal limit, when they are equal.
The dimensionless moment of inertia is also exhibited in Fig. 4, but now versus the dimensionless quadrupole moment . For comparison, the Kerr solutions are also included, where with . Forming the vertical line at , they range from in the extremal rotating case to , when the limit is taken. In contrast, for rotating wormholes, the dimensionless moment of inertia diverges in the limit , since in the static limit the wormhole mass vanishes, .
iii.4 Geometry and Ergoregion
The shape of wormholes can be visualized with the help of embedding diagrams. Let us consider the wormhole metric at fixed in the equatorial plane, and embed it isometrically in Euclidean space
with , . Comparison yields
which leads to . We exhibit examples of such embedding diagrams in Fig. 5 for symmetric rotating wormholes at fixed equatorial throat radius with increasing rotational velocities .
We can also try to visualize the shape of the throat, i.e., the deformation of the hypersurface . For that end we try to embed this hypersurface in a Euclidean space. However, it is known for Kerr black holes, that beyond a critical value of the rotational velocity their horizon can no longer be embedded in a Euclidean space, and a pseudo-Euclidean space must be used for the embedding Smarr:1972kt (); Smarr:1973zz (). Since the family of rotating wormhole solutions approaches an extremal black hole, such a critical value of the rotational velocity is expected to arise for rotating wormholes as well.
To obtain the respective embedding, we equate the line element of the hypersurface , with the (pseudo-)Euclidean line element
with , . Now comparison yields
We show such embedding diagrams in Fig. 6 for the same set of wormholes with rotational velocities . The critical value corresponds to and marks the onset of a negative Gaussian curvature at the poles. Beyond parts of the hypersurface must be embedded in pseudo-Euclidean space, which is represented by the dashed curves in the figure.
We complete this demonstration of the throat geometry by considering the dependence of the throat radii on the rotational velocity . We show in Fig. 6 the ratio of the polar radius to the equatorial radius for fixed equatorial throat radius . As one would expect, this ratio decreases monotically with increasing rotation, and reaches the corresponding ratio of radii of the extremal Kerr horizon in the limit . The critical value , where the Gaussian curvature at the poles turns negative, is indicated by a cross in the figure. For comparison, also the respective curve for the Kerr solutions is shown. Interestingly, while the critical velocity is much smaller for the Kerr solution, the ratio has the same value for the wormhole and the Kerr critical velocities, as indicated in the figure by a thin horizontal line. The figure also exhibits the ratio of the areal radius to the equatorial radius , where the analogous findings hold.
Let us finally address the ergoregion of rotating wormholes. The ergoregion is defined by the condition ,
where the equality defines the ergosurface. Inspection of the rotating wormhole solutions reveals, that the location of the ergosurface in the equatorial plane is closely related to the angular velocity of the throat. Indeed, the relation holds with high accuracy. This is demonstrated in Fig 7.
For wormholes to possess an ergoregion in that part of the spacetime, where , which we have chosen to be asymptotically flat, they must rotate sufficiently fast. Note that the condition for the ergosphere at the throat in the equatorial plane can be written as
which yields for . Thus the throat must rotate with half the velocity of light. As the rotation velocity increases beyond this value, the ergoregion in this part of the spacetime increases. This is demonstrated in Fig. 7, where the ergosurfaces for increasing values of and thus decreasing values of are exhibited. However, there is always an ergoregion in the other part of the spacetime, where , and the function tends asymptotically to a finite value.
We now analyze the motion of particles and light in these rotating wormhole spacetimes. Interesting aspects emerging from this analysis are that massive particles can have stable bound states in rotating wormhole spacetimes. Moreover, there are unstable bound states for massless particles which indicates the presence of a photon region.
The motion of particles and light is governed by the Lagrangian
where the dot denotes the derivative with respect to an affine parameter, and for particles and zero for light. The symmetries of the spacetime lead to two cyclic coordinates and thus two constants of motion, the energy and the angular momentum .
Let us for simplicity consider only motion in the equatorial plane, . The Lagrangian then simplifies to
and the conserved charges become
Solving these equations for and yields
Insertion of these expressions into the Lagrangian leads to
When solving this equation for , it is instructive to introduce effective potentials as follows
where we abbreviated the right hand side by , and
implying . In fact the condition leads to a restriction for the allowed energies and angular momenta of the particles, or .
Thus the study of these effective potentials allows for a classification of the possible orbits in the equatorial plane. We exhibit in Fig. 8 examples of the effective potentials for massive and massless particles. In particular, we observe that there are bound orbits of massive particles in such rotating wormhole spacetimes. This is in contrast to the static Ellis wormhole, which does not possess bound orbits. For rotating wormholes there are even three types of bound orbits, those that always remain within a single universe, those that oscillate between the two universes, and those that remain at the throat ( only). Two examples of such single universe bound orbits for a massive particle are shown in Fig. 9 together with a two-world bound orbit.
For the case of massless particles the effective potentials shown in Fig. 8 are divided by the modulus of the angular momentum. We note that possesses a local maximum for retrograde motion for . This indicates that there is a photon region in this asymptotically flat universe. additionally possesses a local maximum for prograde motion for , which indicates the presence of a photon region in the other universe, as well. However, whereas the motion there looks like prograde motion in our coordinates, where the universe is asymptotically flat, it corresponds to retrograde motion, when an appropriate coordinate transformation is performed, making the universe asymptotically flat.
Of particular interest are the innermost stable circular orbits (ISCOs). To find the circular orbits we need to solve for , while stability requires at the same time . ISCOs are then characterized by the change of stability, i. e., . For these rotating wormholes there are both corotating and counterrotating ISCOs, just like for the Kerr black holes. Whereas the ISCO of the corotating orbits resides at the throat, the location of the ISCO of the counterrotating orbits depends on the rotation velocity. Here the ratio increases almost by a factor of two from the static limit to the limiting extremal Kerr black hole, as seen in Fig. 10. The figure also displays the ratio of the orbital period , where denotes the orbital period of the throat (or the horizon of the extremal black hole), which increases roughly by a factor of three from the static limit to the extremal Kerr black hole.
Iv Non-Symmetric Wormholes
We now turn to the discussion of the properties of non-symmetric rotating Ellis wormholes. In the non-symmetric case for fixed the wormhole solutions depend on two parameters, and . By varying these, the domain of existence can be mapped out. In the following we will first discuss the global charges of these non-symmetric rotating wormholes, and subsequently we will address their geometry.
iv.1 Global Properties
Let us start our discussion of the properties of non-symmetric rotating wormholes with their global properties mass , angular momentum and scalar charge , obtained from the asymptotic expansion at . To demonstrate these properties, we fix the throat parameter and consider families of solutions, where the remaining two parameters and are varied.
In Fig. 11a we exhibit the mass scaled by the equatorial throat radius versus the squared scalar charge for several fixed values of the parameter in the range , where the negative values of are indicated in the figure by dots. The curves are obtained by varying the value of for a fixed value of , starting from the static case .
The symmetric static solution has vanishing mass and scalar charge , while the mass of the rotating solutions increases monotonically towards the mass of the limiting extremal Kerr black hole as the scalar charge decreases monotonically to zero. As seen in the figure, for fixed and fixed equatorial throat radius , the mass always increases monotonically from the corresponding static value to the extremal Kerr value, while the scalar charge decreases monotonically to zero. In particular, for negative the mass is always positive. This is in contrast to positive , where the mass becomes negative in a part of the domain of existence. However, independent of , all families of non-symmetric wormhole solutions approach the extremal Kerr solution.
When we consider the angular momentum scaled by the squared equatorial throat radius, exhibited in Fig. 11b versus the