Generalized Nariai Solutions for Yang-type Monopoles
Generalized Nariai Solutions for Yang-type
Monopoles
Pablo Diaz∗, Antonio Segui†
Departamento de Fisica Teorica, Universidad de Zaragoza, 50009-Zaragoza, Spain.
October 28, 2018
Abstract
A detailed study of the geometries that emerge by a gravitating generalized
Yang monopole in even dimensions is carried out. In particular, those which
present black hole and cosmological horizons. This two-horizon system is ther-
mally unstable. The process of thermalization will drive both horizons to coalesce.
This limit is what is profusely studied in this paper. It is shown that eventhough
coordinate distance shrinks to zero, physical distance does not. So, there is some
remaining space which geometry has been computed and identified as a generalized
Nariai solution. The thermal properties of this new spacetime are then calculated.
Topics, as the elliptical relation between radii of spheres in the geometry or a dis-
cussion about whether a mass-type term should be present in the line element or
not, are also included.
Keywords: Yang monopole, Nariai geometry, Horizon, Black hole.
∗pdiaz@unizar.es
†segui@unizar.es
http://arxiv.org/abs/0704.0366v2
Contents
1 Introduction 2
2 The gravitational coupling. Some geometrical features 3
3 The horizon coalecence geometry 4
3.1 Case m = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.2 Case m 6= 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4 Conclusions 11
A Proof of the finite nonzero physical distance 12
B Horizon coalescence as a flow on the line 13
1 Introduction
Monopoles have been subject of deep study and controversy all over the last century. This
is so because, although no experimental evidence of their existence has been found, many
theoretical issues make them almost unavoidable. They already appeared as solutions of
Maxwell equations as long as the null B-divergence condition was relaxed, that is, ∇·B 6=
0. It was Dirac [1] in the early thirties who first proposed the theoretical possibility of
creating an experiment to actually produce a “fake” monopole, in a way that its fakeness,
say, the Dirac string, was undetectable. As a consequence, the product of the electric
and the magnetic charges was quantized. Many years later, in 1959, the quantization
requirement was confirmed by the celebrated Aharonov-Bohm experiment [2].
Since 1954, owing to the papers by Yang and Mills [3] and by Utiyama [4], gauge
theories of a group of symmetry larger than U(1), in particular non abelian symmetry
groups SU(2) and SU(3) (which eventually would conform the Standard Model of par-
ticle physics) where gradually developped. In 1969, Lubkin [5] realized that monopoles
can be classified by the homotopy group of the gauge symmetry group of the theory,
so that the magnetic charge is replaced by the topological charge of the field configura-
tion. In the case of the Dirac monopole, the homotopy group π1 of U(1) is exactly Z.
However, it was not till 1975 that Yang [6] generalized the abelian monopole to the case
of an SU(2)-invariant gauge theory in six dimensions, see also [7]. Modern approaches
use the formalism of fiber bundles for a suitable description of monopoles. It generalizes
the traditional classification in terms of the homotopic group of the gauge theory. In
this way, magnetic monopoles are identified with the different instanton configurations
which come up basically as non trivial maps of the gauge group, usually SU(N), onto Sd,
where d is the spatial dimension. That is, magnetic monopoles are all those non trivial
principal bundles with group structure SU(N) that can be realized on the hypersurface
Sd. The classification coincides, as said before, with the different classes of homotopy
groups. The genalization of Yang monopoles to an arbitrary even dimension was carried
out in [8]. Using slightly different methods similar analysis have recently been done [9].
The reader can find good reviews on the subject in [10], [11] and the references therein.
As every existing object in nature, monopoles couple to gravity via their energy-
momentum tensor. The resulting geometry is obtained by solving the Yang-Mills-
Einstein equations, which get greatly simplified by imposing spherical symmetry (as
expected from a magnetic monopole field configuration). This geometry is fully specified
by choosing a point in the space of parameters {µ,m,Λ, k}, the meaning of which will
be explained in detail later on. For a given range of parameters, it is easy to prove
that the geometry presents both a cosmological and an event horizon. A full analogy
with the Schwarzschild-de Sitter solution reveals that, in these cases, the geometry is dy-
namically driven through the parameter space into a thermally stable point where both
horizons coalesce [12], the final line element being the analogue of Nariai’s spacetime in
four dimensions.
This paper is organized as follows: the next section sets a general framework and
fixes the notation used later. The main body of the article concerns the analysis of the
coalescence solutions. This is achieved in two subsections corresponding to the massless
and massive cases respectively. An explicit computation of the resulting geometry is
carried out in each case. A final section includes some conclusions and comments. Two
appendixes have been added to the article. They are topics which lie somehow out of the
main line of the paper, either for being technical aspects of a computation (Appendix
A) or for presenting a new idea the exposition of which would need a new section, as in
Appendix B. the absence of B, in turn, would not have prevented the reader from a full
understanding of the paper.
2 The gravitational coupling. Some geometrical fea-
tures
The gravitational effects of these monopoles have been recently studied [9]. It was done,
as usual, by minimally coupling the Yang-Mills energy-momentum tensor to gravity.
Variations of the Einstein-Hilbert action
− det g
(R− 2Λ)− 1
Tr|F |2
with respect to the metric tensor leads to
Gmn = 8πGTmn − gmnΛ, (2)
where
Tmn = γ
tr(F pm Fnp)−
gmntr(FpqF
is the energy momentum tensor of the YM strength field. The traces are taken in the
colour index and γ is the YM coupling constant. Finding general solutions for (2) is
a highly complicated problem. However, imposing spherical symmetry simplifies the
task enormously. According to this, the ansatz will be a spatially spherically symmetric
(2k + 2)-dimensional metric whose line element reads
ds2 = −∆dt2 +∆−1dr2 + r2dΩ22k. (4)
The last equation is consistent with (2) and (3) when [9]
∆(r) = 1− 2Gm
r2k−1
, (5)
where R =
k(2k+1)
is the de Sitter radius, µ2 is proportional to 1
and measures
the magnetic charge of the monopole, m comes up as a constant of integration with
dimensions of mass and G is the Newton constant in 2k + 2 spacetime dimension. At
first sight, (4) with (5) look like a Schwarzschild-de Sitter geometry in 2k + 1 spatial
dimensions with an extra term, the one involving µ, which seems to be independent of
the dimension of spacetime. It seems reasonable to think of this term as a contribution
of the magnetic monopole. This simple image, even if not exact1, is helpful and, unless
we face the vanishing limits, it may be kept in mind in the following.
The next step (and the next temptation) is to analyze how the causal structure
of this spacetime depends on given values of the parameters. The main body of this
work concerns a deep analysis of the solution in the case when parameters µ,Λ, m and
k allow the existence of two horizons. Then, inspired by the Schwarzschild-de Sitter
unstable solution, it is claimed that the system gets dynamically driven to a value of the
parameters where both horizons coalesce. Eventhough coordinate distance shrinks to
zero, physical distance does not. A generalized Nariai geometry “between” the horizons
is then explicitly obtained. The Nariai line element [13] is a nonsingular solution of the
Einstein’s vacuum equations with a positive cosmological constant, Rµν = Λgµν . It was
first found by Kasner [14] and its electrically charged generalization dates of 1959 [15].
However, the important fact that it emerges as an extremal limit of Schwarzschild-de
Sitter black holes was not noticed until 1983 [12].
Nariai spacetime in four dimensions is the direct product dS2 × S2, dS2 being no
more than the hyperbolic version of S2 as we change t → iτ . In 2k + 2 dimensions, the
solution gets generalized to dS2 × S2k. Again, it is the direct product of two constant
curvature spaces and admits a 3 + k(2k+ 1) group of isometries SO(2, 1)× SO(2k+ 1).
The space is homogeneous since the group acts transitively and is locally static, given
that a global dS-type spacetime cannot be described by merely one static coordinate
chart. In four dimensions, radii of curvature of the two product spaces are equal if the
black hole is neutral, and different in the charged case. If the black hole is electrically
charged, the respective radii a and b are different and related by the equation
a−2 + b−2 = 2Λ (6)
as shown in [15]. This relation will be generalized in the magnetic case, the object of
our study. A short but instructive recent work on the four dimensional geometry can be
found in [16].
3 The horizon coalecence geometry
Studying the horizons of a geometry like (4) is equivalent to searching the divergencies
of grr for finite values of the coordinates. This leads us to analyze the zeroes of function
∆(r), where the horizons will be located. For a certain range of values of {µ,Λ, k,m}
there will be two horizons. Finding this region in the parameter space will be the first
1The resulting geometry is, of course, not just the sum of terms of different geometries, but it casually
coincides. Differences are bound to exist on the limit of vanishing of a given contribution. For instance,
let us suppose that, given a set of parameters, say {m,µ,Λ, k }, we can switch off µ (by neglecting it
with respect to the others). The resulting geometry is topologically different to the one obtained by not
assuming any monopole at all at the beginning, that is, the limit does not coincide. However, in the
cases studied here, this is no more than an enough-to-be-aware-of subtlety.
task. After that, attention will be focused on the coalescence point of the horizons2. The
analysis consists of two steps, first, the parameterization of the coordinate separation
of the horizons (ǫ) and the calculation of the physical distance between them when
coalescence takes place (ǫ → 0). Then, following the strategy in [12], the computation
of the line element of the remaining geometry. This program is carried out on two cases:
m = 0 and m 6= 0, which are treated in the next subsections, respectively. The massless
case must be seen as a toy model of the massive one. This distinction is made not
merely for simplicity but also because, as will be explained, the mass parameter comes
out naturally for dynamical requirements.
3.1 Case m = 0
In the massless case, ∆(r) gets reduced to
∆(r) = 1−
. (7)
Solving ∆ = 0 is equivalent to finding the zeroes of a biquadratic equation as long as
r = 0 is not considered. We perform the change z ≡ r2 and solve a second order ordinary
equation. The horizons are found to be at
1− 4µ
z++ =
1− 4µ
. (9)
R > 2µ guarantees the existence of two positive solutions and, therefore, four solutions
for the quartic equation. Two of them, r+ = +
z+ and r++ = +
z++, correspond to the
radial coordinate of the inner (black hole) and outer (cosmological) horizon respectively.
If R = 2µ, both solutions coincide, which means that the horizons coalesce. As said
before, this does not mean that the geometry vanishes as a naive observation (given
a wrong choice of coordinates) would make one think. Physical distance between the
horizons, on the contrary, remains finite at the limit. In order to prove this, let us
compute it. For fixed time and angular coordinates, the physical distance is
D(µ,R) =
∫ r++
[−r4 +R2r2 − µ2R2]1/2
∫ z++
− µ2R2
z − R2
)2]1/2dz (10)
The requirement R > 2µ implies R
− µ2R2 > 0 so the above integral is exactly solved
as an cos−1-type. The result is
D(R) =
πR. (11)
Surprisingly, the physical distance does not depend on µ. It means that, given a cos-
mological constant Λ, one could “switch on” the monopole and go on till the horizons
coalesce but the distance would remain unchanged. However, because of quantization
2Coalescence as seen in Schwarzschild coordinates.
requirements, monopole charge µ cannot be tuned, but needs to have, instead, a fixed
value upto a sign. On the other hand, the cosmological constant, Λ, should be chosen
when writing the lagrangian. It means that changing its value does not drive us from
one model to another but implies an essential change in the theory [17]. Therefore, we
are not free to adjust any parameter arbitrarily as done with the mass of the black hole
in the Schwarzschild-de Sitter case. Then, eventhough physical reasons would lead the
horizons to coalesce, the absence of any free parameter in our model makes it impossible.
In the next section, m will come to our help as a free parameter for the model.
Despite the last remark, one could wonder about the kind of geometry that re-
mains when the horizons coalesce. This task, even if seems just a curious exercise now,
will be useful for the next section. Applying a technique similar to the one Gingspar
and Perry [12] used to study the geometry of Nariai’s solution, we proceed by, first,
parametrizing the separation of horizons as
R = 2µ(1 + ǫ2), (12)
in a way that coalescence corresponds to taking ǫ = 0. Then, we define a “wise” change
of coordinates
χ = cos−1
(r2 − r20)
τ = ǫ
, (13)
where A =
1− 4µ2
and r20 =
, and the angular coordinates remain unchanged. The
new coordinates (13) might seem randomly chosen at first sight. However, there are
some reasons that justify such a functional dependence. For instance, χ is nothing but
the physical distance between r+ and r. The timelike coordinate t is multiplied by i in
order to work in the Euclidean region3 and by ǫ because ∆/ǫ2 is expected to have a finite
limit when ǫ → 0. Now, we apply (12) and (13) and expand ∆(r(χ))dτ 2, ∆−1(r(χ))dχ2
and r2(χ) up to first order in ǫ. The line element (4) reads
ds2 = µ2dχ2 + µ2 sin2(χ)
1 + ǫ
2 cos(χ)
dτ 2 +
+ 2µ2
2 cos(χ)ǫ
dΩ22k. (14)
We take limit ǫ → 0 to obtain
ds2 = µ2
dχ2 + sin2(χ)dτ 2
+ 2µ2dΩ22k. (15)
As seen in (15), the 2k-sphere decouples from the rest. The resulting geometry is S2×S2k
for k ≥ 2. Notice the parallelism between this geometry and Nariai’s solution, which is
S2 × S2. The “classical” relation between radii (6) gets also generalized to
a−2 + b−2 = C0Λ, (16)
where C0 =
k(2k+1)
. The geometry (15) can be viewed as a “degenerate” black hole, in
which the two horizons have the same (maximum) size and are in thermal equilibrium.
This could be interpreted by an observer as a bath of radiation coming from both horizons
3τ will be periodic at both horizons, although different in each case. Equality will hold at the
coalescence point, when thermal stability is reached.
at a precise temperature [19]. The temperature can be calculated by means of surface
gravity κ, as computed in the new coordinates (13)
k(2k + 1)
. (17)
The entropy can also be computed as a quarter of the sum of the two horizons [18], so
k(2k + 1)
, (18)
where ω2k is the area of the 2k-dimension unit sphere.
3.2 Case m 6= 0
In the massive case we recover the full expression (5) for ∆. Since the singular point
r = 0 is not to be considered, we better analyze the function r2k−1∆(r)
∆̃ ≡ r2k−1∆ = −
r2k+1
+ r2k−1 − µ2r2k−3 − 2Gm. (19)
It is known that a polynomial equation with powers equal to or higher than five is not
generally solvable in a symbolical way. This happens for k ≥ 2. So, the purpose of doing
a study for the massive case analogous to that achieved in the first section is ruined.
Nevertheless, some information can be extracted from (19). We should first remember
the sign of the parameters: R2 > 0 (de Sitter), µ2 > 0 for k ≥ 2, and m will be free in
principle. Derivating (19) and equating to zero leads to a biquadratic equation of the
(2k + 1)r4 + (2k − 1)r2 − (2k − 3)µ2 = 0, (20)
which, as long as
Λµ2 ≤ k
(2k − 1)2
2k − 3
, (21)
has two positive (and two negative) roots, rmin and rmax ≡ rc. In terms of the cosmo-
logical constant
r2c ≡
k(2k − 1)
1− 4(2k − 3)Λµ
k(2k − 1)2
, (22)
rmin is obtained from (22) by swapping the sign of the square root. A quick look at (19)
shows that the smallest root is a minimum and the largest is a maximum of function ∆̃.
Now, let us plug rc into (19):
1. If m > 0, then (see fig.1)
a) ∆̃(rc) ≥ 0 implies that there are two event horizons, the black hole and the
cosmological horizon. The inequality gets saturated at the coalescence point.
b) ∆̃(rc) < 0 means that no horizon is found.
2. If m < 0, then (see fig.2)
a) ∆̃(rmin) < 0 together with ∆̃(rc) < 0 implies that there is just one Cauchy
horizon.
b) ∆̃(rmin) < 0 together with ∆̃(rc) > 0 assures the existence of a Cauchy horizon
and both black hole and cosmological horizon.
c) ∆̃(rmin) > 0 leaves us with the cosmological horizon only.
The case we will study is ∆̃(rmin) < 0 and ∆̃(rc) > 0 which, independently of the sign
of m, assures4 the existence of black hole and cosmological horizons. This corresponds
to values of m within range (see fig.3)
1.a 1.b
Figure 1: Case m > 0. The curve represents function ∆̃(r). Figure 1.a has two roots which
correspond to the black hole (r+) and cosmological horizon (r++) respectively. Figure 1.b
shows the absence of horizons.
2.a 2.b 2.c
Figure 2: Case m < 0. This time ∆̃(r) permits the existence of one (Cauchy) horizon as
in Figure 2.a, three horizons (Cauchy, black hole and cosmological) as in 2.b, or just the
cosmological horizon as shown in 2.c.
m− < m < m+, (23)
where
Gmc ≡ Gm+ =
1 + 2k
r2k−3c (r
c − 2µ2). (24)
The value of Gm− is obtained by replacing rc → rmin. In terms of Λ and µ we get
Gm± =
(2Λ)−k+1/2
1 + 2k
− k + 2k2 ±
k2(1− 2k)2 − 4Λµ2(2k − 3)k
]k−3/2
− k + 2k2 − 4Λµ2 ±
k2(1− 2k)2 − 4Λµ2(2k − 3)k
. (25)
4The value of m can be negative. That is because m should not be thought of as an entity with
physical meaning but as a geometrical parameter. Short calculation in (25) shows that m gets negative
values for Λµ2 ≥ k
(1 + 2k).
Hr,m+L
Hr,m-L
Figure 3: This figure shows the range of “masses” which are consistent with the existence of
both black hole and cosmological horizons. The curve ∆̃(r) “moves down” in the process of
coalescence.
The crucial point is that both horizons coalesce when rc is a root of (19) which
happens at m = mc(k,Λµ
2,Λ). Two relations have been imposed so far:
de∆(r;m)
|rc= 0,
that is, (20), which defines rc, and ∆(rc;mc) = 0 which leads to mc. In order for m to
be real, the bound which must be impossed on Λµ2 coincides with (21) which, in turn, is
nothing but the condition for the existence of two horizons. So, if a given a value for Λµ2
is low enough to produce two horizons, there always exists a real value of m which makes
them coalesce. Again, as in the Schwarzschild-de Sitter example, the system is unstable
and the equilibrium point is reached at m = mc. Unlike the massless case, plugging m
gives us enough room for maneuvre to drive the system to equilibrium.
At this point, we would like to remark that the procedure of horizon coalescence, as
studied in detail below, may be seen as a flow in a line which undergoes a Pitchfork
bifurcation at the coalescence point. Parameter m, moved by thermal instability, drives
the system to the critical situation. For concreteness see Appendix B.
Let us focus on the near coalescence point. This can be parameterized by
r = rc + δr = rc(1 + ǫ cosχ) (26)
m = mc − δm = mc(1 + bǫ2).
Parameterization of r also involves a change of coordinates r → χ and should be taken
as imposed at the moment although it will be justified later. The horizons will be
symmetrically located at: r+ = rc(1−ǫ) and r++ = rc(1+ǫ) which correspond to χ+ = π
and χ++ = 2π, respectively
5. The value of b as well as the absence of a linear term in ǫ of
the parameterization of m may be explained as follows. Near the coalescence point one
should Taylor expand ∆ around rc and have in mind that, for
≪ 1, ∆ is aproximately
parabolic, so that second order expansion is enough. By definition ∆(r+) = ∆(r++) = 0
5For a small enough ǫ, it is expected that the parabolic approach holds and, then, both horizons are
symmetrically located with respect to rc.
and ∆ reaches a maximum at rc. So,
0 = ∆(r++) = ∆(rc, m) + ∆
′(rc, m)(rcǫ) +
∆′′(rc, m)(rcǫ)
r2k−1c
∆′′(rc;mc)r
2, (27)
which means that
∆′′(rc;mc)r
. (28)
Calculating the physical distance near the coalescence point would, again, imply
solving the integration
D(ǫ) =
∫ r++
∆1/2(r)
, (29)
where r++ = r+ + 2rcǫ. Although the exact result is not computed, an explicit proof of
its finite nonzero value is given in Appendix A. The procedure of calculating the physical
distance also brings us some light on which is the change of coordinates that should be
made in order to understand the resulting geometry. It turns out to be
χ = cos−1
(r − rc)
t, (30)
where
k − 2k2 + 2Λr2c
is a dimensionless factor.
The coalescence of horizons takes place at ǫ = 0. In order to study the geometry at
the limit we proceed by calculating −∆dt2, ∆−1dr2 and r2 in the new coordinates (30)
and expand in ǫ around ǫ = 0. The new line element gets determined by taking the zero
order of the expansion. The relations for r and m in (26) are in accordance with (30),
where b takes the value of (28), by virtue of the parabolic approach. From (30), it is
straightforward to see that r2 takes a constant value r2c . Surprisingly, as in the massless
and Schwarzschild-de Sitter cases, the geometry splits in two disconnected parts which
lead to a product manifold S2 × S2k. The line element reads
ds2 = Br2c
dχ2 + sin2(χ)dτ 2
+ r2cdΩ
2k, (32)
where χ ∈ [π, 2π] and τ is periodic6.
As seen in (32), S2 has radius a2 = Br2c , and S2k has radius b2 = r2c . Now, the
generalized Bertotti relation (6) is
a−2 + b−2 =
2(1− k)
= CΛ, (33)
where C(k,Λµ2) is obtained by inserting (22) in (33). Note that C
k,Λµ2 = k(2k +
= C0, and then (33) turns into (16), that is, into the massless case. This is no
6τ is periodic on both horizon surfaces all over the process in order to avoid the conical singularity
at the horizons. At the coalescence point, however, both periods equal.
surprising since Λµ2 = k(2k + 1)/4 is the condition for coalescence in the massless case
(equivalent to R = 2µ), and, at the same time, it makes mc = 0. So, the massive
geometry is a consistent extension of the massless one. Now, fixing Λ does not determine
uniquely the geometry. Another dimensionless variable Λµ2 is required.
As in the last section, the geometry (32) can be viewed as a “degenerate” black hole,
in which the two horizons have the same (maximum) size and are in thermal equilibrium.
In the present case the temperature is given in terms of the surface gravity κ by
. (34)
In Planck units,the entropy associated with this solution may be calculated (given that
it is not extreme7) by means of the total area of the horizons as
c . (35)
4 Conclusions
The spherically symmetric solution of gravity due to a magnetic monopole in arbitrary
dimension has been studied, in particular, when the set of parameters {Λ, µ,m, k} allows
the existence of two horizons. In these cases, thermal instabilities drive a process of
horizon coalescence. Even though coordinate separation between the horizons shrinks
to zero, it has been proven in both the massless and the massive case that the physical
distance does not. The geometry of the remaining space between the horizons has been
calculated in both cases. They turned out to be Nariai-type solutions, that is, the product
of a 2-sphere and a 2k-sphere for a (2k + 2)-dimensional spacetime. In each solution,
the radii of the spheres are not independent. They are related by an elliptical equation
which should be understood as the generalization of the relation found by Bertotti. The
unique generalized equation involving these radii for both the massless and the massive
case has been given. After computing the line element in each case, the thermodynamical
properties (Hawking temperature and entropy) due to the existence of horizons have been
calculated.
The Yang monopole corresponds to the six dimensional case, where k = 2. The
geometry obtained after coalescence is S2 × S4 as can be explicitly read in (32). This
case is especially interesting since it may be described in String Theory (a realization
of the Yang monopole in Heterotic String Theory has recently been done [21] as well as
another complementary picture in Type-IIA String Theory [20]). In the same context,
it looks possible to find results (18) and (35) for the entropy by application of some
attractor mechanism [22, 23]. We believe that this would be an interesting topic to be
addressed in future research.
7A charge black hole is said to be extreme when it has the minimum mass. Then, as it cannot release
any energy without losing charge, it is supposed not to emit, and its associated Hawking temperature is
0. The black hole we are dealing with in this paper is extreme in the sense of carrying the “maximum
mass” allowed by the cosmological constant Λ. Obviously, the temperature will not be zero.
A Proof of the finite nonzero physical distance
Computing the physical distance is equivalent to performing the integration
∫ r++
∆1/2(r)
, (36)
where, for small ǫ, r++ = r+ + 2rcǫ. Divergencies might appear at the points where
∆ → 0. The case we have been considering all along section (3.2) concerns the existence
of two horizons which coalesce, that is, two single roots r+ and r++ of ∆ which join to
form a double one. Function ∆̃ can always be expressed as ∆̃ = (r − r+)(r++ − r)g(r),
where g(r) is a polynomial function of powers of degree 2k − 1 and no zeroes within the
range [r+, r++] are to be found by construction. Explicitly, equation (36) is
D(ǫ) =
∫ r++2rcǫ
(r − r+)1/2(r+ + 2rcǫ− r)1/2
rk−1/2
g1/2(r)
︸ ︷︷ ︸
. (37)
Now, h(r) is a continuous divergenceless strictly positive function in the compact [r+, r++],
which means that it will reach a positive maximum and minimum for certain r′s. Let us
call hmax and hmin the values of the function h in these points
8. Then
∫ r++2rcǫ
(r − r+)1/2(r+ + 2rcǫ− r)1/2
≤ D(ǫ) ≤
≤ hmax
∫ r++2rcǫ
(r − r+)1/2(r+ + 2rcǫ− r)1/2
. (38)
The integration can be performed:
∫ r++2rcǫ
(r − r+)1/2(r+ + 2rcǫ− r)1/2
= π. (39)
D(ǫ → 0) = πh(rc), (40)
where the value of rc is given in (22).
Integrations of form (39) are solved exactly by a cos−1 type function, and a nonzero
finite result is obtained. It is remarkable that the same can be said for any ∆ we would
choose, as long as no more than two single roots were to join to form a double one. The
key point is that (39), which could be problematic, is independent of ǫ and therefore the
distance is finite in the limit, when ǫ → 0. So, eventhough (39) was neither exactly the
physical distance in the massive case nor in Schwarzschild-de Sitter solution (however, it
was in the massless case as we have already seen in the first section), it is closely related
to it. This fact gives us a hint or, at least, justifies the change of coordinates we were
performing once and again to study the geometry at the limit ǫ → 0.
8These, in principle, depend on ǫ but coincide when ǫ → 0: hmin = hmax ≡ h(rc).
B Horizon coalescence as a flow on the line
The main phenomenon that concerns this paper, as said before, can be described in terms
of the dynamics of a vector field on the line. The coalescence point, in this picture, is no
more than a supercritical Pitchfork bifurcation. Let us remember some general features
of the dynamics of a one-dimensional flow. The equation of a general vector field on the
line can be expressed as:
ẋ = f(x, α) (41)
where f is any real function with real support, the dot means differentiation with respect
to t and α is a parameter of the model. Fixed points of (41) require ẋ = 0, which must
be obtained by finding the roots of f , that is
f(x∗, α) = 0. (42)
Equation (42) is solved by an n-collection of fixed points x∗i for a given value of α. Let
us suppose that f has three roots if α = α0. Fixed points come closer as α moves and
get “condensed” in a “fat” fixed point (bifurcation point) at α = αc. A paradigmatic
example of a Pitchfork bifurcation is shown by function
f(x) = x(α− x2). (43)
One question arises naturally now about the role the horizons play in this picture. Let
us claim that horizons are fixed points and the role of α is played by m. We will justify
this identification by constructing the vector flow.
Constructing a flow in a manifold (in our case it will be a line) is equivalent to giving
a family of curves r̄(t) which covers the manifold or part of it. Each of the curves gets
specified by the initial condition, say, r̄(t = 0). Now, let us consider geodesic motions.
Without loss of generality, the angular coordinates of our geometry will be frozen, θ and
φ are constants, and only radial curves r(t) are to be regarded. Static coordinate system
will serve us to describe the movement for any r ∈ (r+, r++). Let us invoke intuition at
this point. If r(0) is near the cosmological or the black hole horizon it is clear that a
test particle will move out of the region by approaching each horizon respectively. Then,
there is a point r = rg where the test particle will not “feel” any force and, consequently,
it will not move9. This is the first (unstable) fixed point.
Let us move the origin by defining r′ = r− rg, after this, primes will be dropped out
to simplify notation. The flow at each point will be determined by the physical velocity
ṙ(r) (as measured by an observer placed at r = 0) that a test particle would adquire at
r if it is dropped with ṙ = 0 at around r = 0 (as close as possible). It is not hard to see
that the velocity of the test particle, as seen by the static geodesic observer, is bound to
be zero at both horizons. So, horizons are fixed points. Now, our system can be treated
as a vector flow ṙ(r) which covers the region between the horizons. The vector flow has
three fixed points: {r+, r++, 0} where the first two are stable. As m runs towards mc,
the system shrinks into a Pitchfork bifurcation. Near the bifurcation point the flow can
be approximated by
ṙ = βr(r − r+)(r++ − r), (44)
9rg in our geometry, plays the role the asymptotic infinity does in Schwarzschild solution,that is, the
point where the time-like Killing vector should be normalized in order to define the horizon temperature.
Note that rg ≡ rc at the coalescence point, that is, when ǫ = 0.
where β is a positive constant which depends on µ, k and Λ. On the one hand, in the
coordinate system {χ, τ}, and using (30), we have
−→ ṙ =
. (45)
On the other hand, equation (44), expressed in the new coordinate system, reads ṙ =
ǫ3βr3c cosχ sin
2 χ, and so
= −iǫβr3cB cosχ sin2 χ. (46)
As expected, in the new coordinate system, every point converts into a fixed point as
horizons coalesce (ǫ → 0). Since the flux lines were identified with geodesics of test
particles, this can be understood as the abscence of forces at the end of the process.
Acknowledgment
We thank P. K. Townsend and Adil Belhaj for helpful discussions and Jean Nuyts for
critical reading of the manuscript. This work has been supported by MCYT ( Spain)
under grant FPA 2003-02948.
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http://arxiv.org/abs/hep-th/0609055
http://arxiv.org/abs/hep-th/0703255
http://arxiv.org/abs/hep-th/0607193
	Introduction
	The gravitational coupling. Some geometrical features
	The horizon coalecence geometry
	Case m=0
	Case m=0
	Conclusions
	Proof of the finite nonzero physical distance
	Horizon coalescence as a flow on the line
