# Exact solutions for the Einstein-Gauss-Bonnet theory in five dimensions:

Black
holes, wormholes and spacetime horns

###### Abstract

An exhaustive classification of certain class of static solutions for the five-dimensional Einstein-Gauss-Bonnet theory in vacuum is presented. The class of metrics under consideration is such that the spacelike section is a warped product of the real line with a nontrivial base manifold. It is shown that for generic values of the coupling constants the base manifold must be necessarily of constant curvature, and the solution reduces to the topological extension of the Boulware-Deser metric. It is also shown that the base manifold admits a wider class of geometries for the special case when the Gauss-Bonnet coupling is properly tuned in terms of the cosmological and Newton constants. This freedom in the metric at the boundary, which determines the base manifold, allows the existence of three main branches of geometries in the bulk. For negative cosmological constant, if the boundary metric is such that the base manifold is arbitrary, but fixed, the solution describes black holes whose horizon geometry inherits the metric of the base manifold. If the base manifold possesses a negative constant Ricci scalar, two different kinds of wormholes in vacuum are obtained. For base manifolds with vanishing Ricci scalar, a different class of solutions appears resembling “spacetime horns”. There is also a special case for which, if the base manifold is of constant curvature, due to certain class of degeneration of the field equations, the metric admits an arbitrary redshift function. For wormholes and spacetime horns, there are regions for which the gravitational and centrifugal forces point towards the same direction. All these solutions have finite Euclidean action, which reduces to the free energy in the case of black holes, and vanishes in the other cases. The mass is also obtained from a surface integral.

Electronic addresses: gdotti-at-famaf.unc.edu.ar, juliooliva-at-cecs.cl, ratron-at-cecs.cl

###### Contents

## I Introduction

According to the basic principles of General Relativity, higher dimensional gravity is described by theories containing higher powers of the curvature Lovelock . In five dimensions, the most general theory leading to second order field equations for the metric is the so-called Einstein-Gauss-Bonnet theory, which contains quadratic powers of the curvature. The pure gravity action is given by

(1) |

where is related to the Newton constant, to the cosmological term, and is the Gauss-Bonnet coupling. For later convenience, it is useful to express the action (1) in terms of differential forms as

(2) |

where is the
curvature -form for the spin connection , is the vielbein and the wedge
product is understood ^{1}^{1}1The relationship between the constants
appearing in Eqs (1) and (2) is given by , ,
. . For a metric connection with vanishing
torsion, the field equations from (2) read

(3) |

The kind of spacetimes we are interested in have static metrics of the form

(4) |

where is the line element of a three-dimensional manifold that we call the “base manifold”. Note that is a timelike Killing vector field, orthogonal to 4-manifolds that are a warped product of with the base manifold .

If the Gauss-Bonnet coupling vanishes, General Relativity with a
cosmological constant is recovered. In this case the equations force the base
manifold to be of constant curvature (which can be normalized to
or zero) and ^{2}^{2}2The four dimensional case was discussed
previously in ehtop , ehtop1 , ehtop2 . Birmingham

(5) |

If , i.e., for , the Schwarzschild-anti-de Sitter
solution is recovered.

For spacetime dimensions higher than five, the
equations of General Relativity do not impose the condition that the base
manifold be of constant curvature. In fact, *any* Einstein base manifold
is allowed gh . For nonzero , however, the presence of the
Gauss-Bonnet term restricts the geometry of an Einstein base manifold by
imposing conditions on its Weyl tensor Dotti-Gleiser .

In this work we restrict ourselves to five dimensions *without assuming
any a priori condition on the base manifold* in the ansatz (4). We
show that in five dimensions, the presence of the Gauss-Bonnet term permits to
relax the allowed geometries for the base manifold , so that the
whole structure of the five-dimensional metric turns out to be sensitive to
the geometry of the base manifold. More precisely, it is shown that solutions
of the form (4) can be classified in the following way:

(i) Generic class: For generic coefficients, i.e., for arbitrary , , , the line element (4) solves the Einstein-Gauss-Bonnet field equations provided the base manifold is of constant curvature (that we normalize to ) and

(6) |

where is an integration constant Cai . In the spherically
symmetric case, (6) reduces to the well known
Boulware-Deser solution BD .

(ii) Special class: In the special case where the Gauss-Bonnet coupling is given by

(7) |

the theory possesses a unique maximally symmetric vacuum BH-Scan , and
the Lagrangian can be written as a Chern-Simons form Chamseddine . The
solution set splits into three main branches according to the geometry of the
base manifold :

(ii.a) Black holes:

These are solutions of the form (4) with

(8) |

( an integration constant). Their peculiarity is that with the above
choice of and , *any* (fixed) base manifold solves
the field equations. Note that for negative cosmological constant this solution describes a black hole Cai-Soh ,
ATZ , which in the case of spherical symmetry, reduces to the one found
in BD , BTZ .

(ii.b1) Wormholes and spacetime horns:

For base manifolds of constant nonvanishing Ricci scalar, , the metric (4) with

(9) | ||||

(10) |

( is an integration constant) is a solution of the field equations. In this
case, there are three subbranches determined by , or .
It is simple to show that, for negative cosmological constant and , the solution with corresponds to
the wormhole in vacuum found in DOTwormhole . The solution with
and corresponds to a brand new wormhole in vacuum (See Section
III).

If the base manifold has *vanishing* Ricci
scalar, i.e., , it must be

(11) | ||||

(12) |

with an integration constant. If and this solution
looks like a “spacetime horn”. If the base manifold is not
locally flat, there is a timelike naked singularity, but nevertheless the mass
of the solution vanishes and the Euclidean continuation has a finite action
(See Section IV).

(ii.b2) Degeneracy:

If is of constant curvature, , and given by Eq. (10), then the function is left undetermined by the field equations.

The organization of the paper is the following: in Section II we solve the field equations and arrive at the classification outlined above, Section III is devoted to describing the geometry of the solutions of the special class, including some curious issues regarding the nontrivial behavior of geodesics around wormholes and spacetime horns. The Euclidean continuation of these solutions and the proof of the finiteness of their Euclidean action is worked out in Section IV. The mass of these solutions is computed from surface integrals in Section V. Section VI is devoted to a discussion of our results, and some further comments.

## Ii Exact solutions and their classification

In this Section we solve the field equations and arrive at the classification
outlined in Section I. This is done in two steps. We first solve the
constraint equation , and find two different cases: (i) a
solution which is valid for any Einstein-Gauss-Bonnet theory, (ii) a solution
that applies only to those theories satisfying (7).

In a
second step we solve the remaining field equations and complete the
classification of the solution set.

The vielbein for the metric (4) is chosen as

(13) |

where stands for the vielbein on the base manifold, so that the indices run along . The constraint equation then acquires the form

(14) |

where is the Ricci scalar of the base manifold, and

(15) | ||||

(16) |

Since depends only on the base manifold coordinates, Eq.(14) implies that

(17) |

where is a constant. Hence, the constraint reduces to

(18) |

and implies that either

(i) the base manifold is of constant Ricci scalar , or

(ii) .

In case (i) the solution to (17) is

(19) |

( is an integration constant). Since this solution holds for generic
values of and we call case (i) the
*generic* branch.

Case (ii), on the other hand, implies (see equation (17)), and this system admits a solution only if the constants of the theory are tuned as in (7), the solution being

(20) |

Note that in case (ii) the constraint equation does not impose any condition
on the base manifold.

The radial equation , combined with the constraint in the form reduces to

(21) |

where

Finally, the three “angular” field equations are equivalent to the following three equations

(22) |

where

(23) |

and

(24) |

In what follows we solve the field equations (21) and (22), starting from the generic case (i), i.e., base manifolds with a constant Ricci scalar , and given by (19).

*Radial and angular equations, Generic case (i):* The
radial field equation allows to find the explicit form of
the function , whereas the components of the field
equations along the base manifold restricts its geometry to be of constant
curvature. This is seen as follows:

Since in case (i) the base
manifold has , where is a constant,
Eq.(21) reads

(25) |

its only solution being , where the constant can be absorbed into a time rescaling. Thus, in the generic case (i), the solution to the field equations for the ansatz (4) is given in (19)

The angular equations (22) imply

(26) |

for some constant , and then (22) is equivalent to

(27) |

Since for given by (19), the base manifold must necessarily be of constant curvature, i.e., the metric of satisfies , and, since , it must be . This takes care of the first of equations (27). The second one adds nothing new since

(28) |

is trivially satisfied because for ,

(29) |

and satisfies (17). This concludes the classification of
case (i).

*Radial and angular equations, Special case (ii):* From
the constraint equation , one knows that in this case, the
Gauss-Bonnet coefficient is fixed as in Eq. (7), and the metric
function is given by Eq. (20).

(ii.a) Having the first factor in (30) vanish, or by

(ii.b) Requiring the Ricci scalar of to be .

After a time re-scaling, the solution in case (ii.a), is , (given in Eq. (20)).

No restriction on is imposed in this case.

Case (ii.b), on the other hand, is solved by requiring , so that the scalar curvature of the base manifold is related to the constant of integration in (20). Note that, in this case, the metric function is left undetermined by the system .

The remaining fields equations, , can be written as

(31) |

For case (ii.a), the first factor of Eq. (31) vanishes,
and the geometry of base manifold is left unrestricted. We have a
solution of the full set of field equations of the special theories
(7) given by (4) with of Eq.
(20), and an arbitrary base manifold .

In
case (ii.b), Eq.(31) can be solved in two different ways:

(ii.b1) Choosing such that the first factor vanishes.

(ii.b2) Requiring the base manifold to be of constant curvature , i.e., .

Case (ii.b2) leaves the redshift function completely
undetermined.

Case (ii.b1) opens new interesting possibilities. The
vanishing of the first factor of Eq. (31) gives a
differential equation for the redshift function, whose general solution, after
a time rescaling, reads

(32) |

where is an integration constant. is not a constant curvature manifold, although it has constant Ricci scalar . Note that we do not loose generality if we set equal to .

For there are three distinct cases, namely , or
, with substantially different qualitative features. It is simple to
show that, for negative cosmological constant , the
solution with and corresponds to the wormhole in vacuum
found in DOTwormhole , whereas that with corresponds to a brand
new wormhole in vacuum (See Section III).

On the other hand, if
(base manifold with vanishing Ricci scalar), for negative
cosmological constant and nonnegative , the metric (4) describes
a spacetime that looks like a “spacetime horn”. We will see
in the next section that if the base manifold is not locally flat, there is a
timelike naked singularity. Yet, the mass of the solution vanishes and the
Euclidean continuation has a finite action (See Section IV).

This concludes our classification of solutions. Since case (i) has been extensively discussed in the literature, we devote the following sections to a discussion of the novel solutions (ii)a and (ii)b1/b2.

## Iii Geometrically well behaved solutions: Black holes, wormholes and spacetime horns

In this Section we study the solutions for the special case found above.

One can see that, when they describe black holes and wormholes, as goes to infinity the spacetime metric approaches that of a spacetime of constant curvature , with different kinds of base manifolds. This is also the case for spacetime horns, provided (See Sec. III. B). It is simple to verify by inspection that for , the solutions within the special case are geometrically ill-behaved in general. Hence, hereafter we restrict our considerations to the case , where is the anti-de Sitter (AdS) radius.

### iii.1 Case (ii.a): Black holes

According to the classification presented in the previous section, fixing an arbitrary base manifold , the metric

(33) |

solves the full set of Einstein Gauss Bonnet equations for the special theories (7). The integration constant is related to the mass, which is explicitly computed from a surface integral in Section V. For , the metric (33) describes a black hole whose horizon is located at . Requiring the Euclidean continuation to be smooth, the black hole temperature can be obtained from the Euclidean time period, which is given by

(34) |

For later purposes it is useful to express the Euclidean black hole solution in terms of the proper radial distance (in units of ), given by

with , so that the Euclidean metric reads

(35) |

The thermodynamics of these kind of black holes turns out to be very sensitive to the geometry of the base manifold, this is briefly discussed in Section IV.

### iii.2 Case (ii.b): Wormholes and spacetime horns

In this case the base manifold possesses a constant Ricci scalar , with normalized to or .

Let us first consider the case for which the base manifold has nonvanishing Ricci scalar, i.e., . By virtue of Eqs. (9), and (10) the spacetime metric (4) reads

(36) |

where is an integration constant and . The Ricci scalar of (36) is given by

(37) |

which generically diverges at and at any point satisfying and

(38) |

In the case the metric possesses a timelike naked singularity at , and if , an additional timelike naked singularity at . Due to this ill geometrical behavior, we no longer consider the spacetime (36) for the case .

*Wormholes: *The case is much more interesting. The
region must be excised since the metric (36) becomes
complex within this range, and the Schwarzschild-like coordinates in
(36) fail at . Introducing the proper radial distance
, given by

allows to extend the manifold beyond () to a geodesically complete manifold by letting . For the resulting metric for this geodesically complete manifold reads

(39) |

where , and the time coordinate has been rescaled. Note that since (36) is invariant under , the piece of (39) is isometric to (36) whereas the portion is isometric to the metric obtained by replacing in (36). In other words, (39) matches the region of the metric (36) with a given value of , with the region of the same metric but reversing the sign of . The singularity at in Eq. (38) is not present since , and that at is also absent since at all points.

For we obtain another wormhole in vacuum, by using again the proper distance defined above:

(40) |

In these coordinates it is manifest that the metrics (39) and (40) describe wormholes, both possessing a throat located at . No energy conditions are violated by these solutions, since in both cases, the whole spacetime is devoid of any kind of stress-energy tensor.

The spacetime described by Eq. (39) is the static wormhole solution found in DOTwormhole . This metric connects two asymptotically locally AdS regions, and gravity pulls towards a fixed hypersurface located at being parallel to the neck. This is revisited in the next subsection.

The metric (40) describes a brand new wormhole. Its Riemann tensor is given by

(41) |

where latin indices run along the base manifold. At the asymptotic regions the curvature components approach

(42) |

This makes clear that the wormhole (40) connects an asymptotically locally AdS spacetime (at ) with another nontrivial smooth spacetime at the other asymptotic region (). Note that although the metric looks singular at , the geometry is well behaved at this asymptotic region. This is seen by noting that the basic scalar invariants can be written in terms of contractions of the Riemann tensor with the index position as in (III.2), whose components have well defined limits (given in (III.2)), and . Thus, the invariants cannot diverge. As an example, the limits of some invariants are

(43) |

where is the Weyl tensor.

We have
also computed some differential invariants and found they are all well behaved
as .

Some features about the geodesics in these vacuum wormholes are discussed in the next subsection, their regularized Euclidean actions and their masses are evaluated in Sections V and IV, respectively.

*Spacetime horns: *Let us consider now the case when the
base manifold has vanishing Ricci scalar, i.e., .

In this case the metric (4) reduces to

(44) |

where is an integration constant. The Ricci scalar of this spacetime reads

(45) |

The timelike naked singularity at can be removed requiring ; however this condition is not strong enough to ensure that the spacetime is free of singularities. Indeed the Kretschmann scalar is given by

(46) |

where is the Kretchmann scalar of the Euclidean base manifold . Hence, for a generic base manifold with vanishing Ricci scalar, the metric possesses a timelike naked singularity at , unless the Kretchmann scalar of the base manifold vanishes. Since the base manifold is Euclidean, the vanishing of its Kretchmann scalar implies that it is locally flat. This drives us out of (ii.b1) to the degenerate case (ii.b2), for which the component of the metric is not fixed by the field equations, for this reason we will not consider the locally flat case.

If the base manifold is not locally flat, at the origin the Ricci scalar goes to a constant and the Kretschmann scalar diverges as . Therefore, the singularity at the origin is smoother than that of a conifold Candelas , whose Ricci scalar diverges as , and it is also smoother than that of the five-dimensional Schwarzschild metric with negative mass, that possesses a timelike naked singularity at the origin with a Kretschmann scalar diverging as . In spite of this divergency, the regularized Euclidean action and the mass are finite for this solution, as we show in Sections IV and V. In this sense this singularity is as tractable as that of a vortex.

In the case we are interested in, we introduce and a time rescaling; then the metric (44) expressed in terms of the proper radial distance is

(47) |

This spacetime possesses a single asymptotic region at where it approaches AdS spacetime, but with a base manifold different from . Note that as the warp factor of the base manifold goes to zero exponentially as , it actually looks like a “spacetime horn”.

For , the metric (44) can also be brought into the form of a spacetime horn,

(48) |

which also possesses a single asymptotic region at ,
which agrees with the asymptotic form of the new wormhole (40) as
.

The asymptotic form of the Riemann tensor is
not that of a constant curvature manifold, and can then be obtained from the
limit in (III.2).

### iii.3 Geodesics around wormholes and spacetime horns

The class of metrics that describe the wormholes and spacetime horns is of the form

(49) |

where the functions and can be obtained from Eqs. (39) and (40) for wormholes, and from Eqs.(47) and (48) for spacetime horns.

#### iii.3.1 Radial geodesics

Let us begin with a brief analysis of radial geodesics for the wormholes and spacetime horns. The radial geodesics are described by the following equations

(50) | ||||

(51) |

where dot stands for derivatives with respect to the proper time, the velocity is normalized as , and the integration constant corresponds to the energy. As one expects, Eq. (51) tells that gravity is pulling towards the fixed hypersurface defined by , where is a minimum of .

*Wormholes: * From (39) we have , then the equations for radial geodesics
(50) and (51) reduce to

(52) | ||||

(53) |

These equation immediately tell us that DOTwormhole : The coordinate of a radial geodesic behaves as a classical particle in a Pöschl-Teller potential; timelike geodesics are confined, they oscillate around the hypersurface . An observer sitting at lives in a timelike geodesic (here the proper time of this static observer); radial null geodesics connect both asymptotic regions (i.e., with ) in a finite span , which does not depend on (the static observer at says that this occurred in a proper time ). These observations give a meaning to : gravity is pulling towards the fixed hypersurface defined by , which is parallel to the neck at , and therefore is a modulus parameterizing the proper distance from this hypersurface to the neck.

The geodesic structure of the new wormhole (40) is quite different from the previous one. In this case, the radial geodesic Eqs. (50) and (51) read

(54) | ||||

(55) |

As expected, the behavior of the geodesics at is like in an AdS spacetime. Moreover, since gravity pulls towards the asymptotic region , radial timelike geodesics always have a turning point and they are doomed to approach to in the future. Note that the proper time that a timelike geodesic takes to reach the asymptotic region at , starting from is finite and given by

(56) |

It is easy to check that null radial geodesics can also reach the asymptotic region at in a finite affine parameter. This, together with the fact that spacetime is regular at this boundary, seems to suggest that it could be analytically continued through this surface. However, since the warp factor of the base manifold blows up at , this null hypersurface should be regarded as a spacetime boundary.

*Spacetime horns*: For the spacetime horn (47), the
() piece of the metric agrees with that of the wormhole (39). Hence, the structure of radial geodesics in both cases is the same, with
gravity pulling towards the surface. Timelike geodesics again
have a turning point, which, in this case, prevents the geodesics from hitting
the singularity at .

#### iii.3.2 Gravitational vs. centrifugal forces

In this Section we discuss an interesting effect that occurs for geodesics with nonzero angular momentum. One can see that for the generic class of spacetimes (49), which includes wormholes and spacetime horns, there is a region where the gravitational and centrifugal effective forces point in the same direction. These are expulsive regions that have a single turning point for any value of the conserved energy, and within which bounded geodesics cannot exist.

The class of metrics we consider are (49) with the further restriction that the base manifold have a Killing vector . Choosing adapted coordinates such that , the base manifold metric is and the spacetime geodesics with fixed are described by the following equations

(57) |

Here we have used the fact that, if is the geodesic tangent vector, then is conserved, and . If is a Killing vector then is a conserved angular momentum. Examples are not hard to construct, for spacetime horns, what we need is a base manifold with zero Ricci scalar and a Killing field. For wormholes, we need a nonflat 3-manifold with and a isometry, an example being , where is a freely acting discrete subgroup of , and the metric locally given by:

(58) |

The motion along the radial coordinate in proper time is like that of a classical particle in an effective potential given by the r.h.s. of Eq. (57). This effective potential, has a minimum at only if the following condition is fulfilled

(59) |

This expresses the fact that the gravitational effective force is canceled by the centrifugal force if the orbit sits at . The class of spacetimes under consideration have regions where the sign of is opposite to that of , i.e., the effective gravitational and centrifugal forces point in the same direction. Within these regions, there is at most a single turning point, and consequently bounded orbits cannot exist.

## Iv Regularized Euclidean action

Here it is shown that the geometrically well-behaved solutions discussed in the previous Section have finite Euclidean action, which reduces to the free energy in the case of black holes, and vanishes for the other solutions.

The action (2) in the case of special choice of coefficients can be written as

(61) |

and it has been shown that it can be regularized by adding a suitable boundary term in a background independent way, which depends only on the extrinsic curvature and the geometry at the boundary MOTZ . The total action then reads

(62) |

where the boundary term is given by

(63) |

and is the second fundamental form. The total action (62) attains an extremum for solutions of the field equations provided

(64) |

where . Therefore, the value of the regularized Euclidean action makes sense for solutions which are bona fide extrema, i.e., for solutions such that condition (64) is fulfilled.

The Euclidean continuation of the class of spacetimes described in Section III, including black holes, wormholes and spacetime horns, is described by metrics of the form

(65) |

where is the Euclidean time, and the functions and correspond to the ones appearing in Eq. (35) for the black holes; Eqs. (39) and (40) for the wormholes, and in Eqs. (47) and (48) for the spacetime horns.

Let us first check that these solutions are truly extrema of the total action (62).

### iv.1 Geometrically well-behaved solutions as extrema of the regularized action

For the class of solutions under consideration, the curvature two-form satisfies

(66) |

and the condition (64) reduces to

(67) |

where is the Euclidean time period, is the volume of the base manifold, and is the boundary of the spatial section. In Eq. (67) is defined by

(68) |

and the functions and in (67) are given by

(69) | ||||

(70) | ||||

Here we work in the minisuperspace approach, where the variation of the functions and correspond to the variation of the integration constants, and prime denotes derivative with respect to .

Now it is simple to evaluate the variation of the action (67) explicitly for each case.

*Black holes: *As explained in Section III, the Euclidean black
hole metric is given by

(71) |

with , and it has a single boundary which is of the form . In order to evaluate (67) it is useful to introduce the regulator , such that . It is easy to verify that the functions and defined in (69) and (70) respectively, satisfy

(72) |

and hence, the boundary term (67) identically vanishes. Note that it was not necessary to take the limit .