Stationary Dyonic Regular and Black Hole Solutions
We consider globally regular and black hole solutions in SU(2) Einstein-Yang-Mills-Higgs theory, coupled to a dilaton field. The basic solutions represent magnetic monopoles, monopole-antimonopole systems or black holes with monopole or dipole hair. When the globally regular solutions carry additionally electric charge, an angular momentum density results, except in the simplest spherically symmetric case. We evaluate the global charges of the solutions and their effective action, and analyze their dependence on the gravitational coupling strength. We show, that in the presence of a dilaton field, the black hole solutions satisfy a generalized Smarr type mass formula.
In Einstein-Maxwell (EM) theory the Kerr-Newman (KN) solutions represent stationary asymptotically flat black holes, characterized uniquely by their global charges: their mass , their angular momentum , their electric charge , and their magnetic charge nohair1 (); nohair2 (). Following Wheeler this uniqueness theorem of EM theory is often expressed as “EM black holes have no hair”.
In many unified theories, including string theory, dilatons appear. When a dilaton is coupled to EM theory, this has profound consequences for the black hole solutions. Not only do charged static Einstein-Maxwell-dilaton (EMD) black hole solutions exist for arbitrarily small horizon size emd (), but also the staticity theorem of EM theory wald-st () does not generalize to EMD theory for arbitrary dilaton coupling constant : at the Kaluza-Klein value stationary non-static black holes appear, whose horizon is non-rotating Rasheed (); KKN-c (), and beyond even counterrotating black holes arise, whose horizon angular velocity and global angular momentum have opposite sign KKN-c ().
The EM uniqueness theorem, on the other hand, does not readily generalize to theories with non-Abelian gauge fields coupled to gravity review (). The hairy black hole solutions of SU(2) Einstein-Yang-Mills (EYM) and Einstein-Yang-Mills-Higgs (EYMH) theory possess non-trivial magnetic fields outside their regular event horizon and are not uniquely characterized by their mass, their angular momentum, their electric and magnetic charge su2bh (); gmono (); kk (); hkk (); kkrot (); kknrot (); KKNmass (). Futhermore, black hole solutions arise, which are static and not spherically symmetric, showing that Israel’s theorem nohair1 (); nohair2 () does not generalize to non-Abelian theories, either kk (); hkk ().
The coupling to non-Abelian fields not only gives rise to new types of black hole solutions, but also allows for globally regular solutions, not present in EM theory either bm (); gmono (); review (). These are stationary solutions with a spatially localized energy density of the matter fields and a finite mass, and are referred to as solitons when they are stable, and sphalerons when they possess unstable modes. The known globally regular solutions of EYM theory represent sphalerons, whereas the globally regular magnetic monopoles of EYMH theory are solitons, whose topological charge is proportional to their magnetic charge tHooft (). Besides magnetic monopoles EYMH theory contains a plethora of further globally regular solutions, representing for instance monopole-antimonopole pairs, chains, and vortex ring solutions KKS ().
It is an interesting question whether such globally regular solutions can be endowed with rotation, like their black hole counterparts can. When EYM black holes start to rotate, the time component of their gauge potential is excited, as expected. Surprisingly, however, not only a magnetic moment is induced by the rotation but also an electric charge pert (); kkrot (), and this seems to preclude the existence of globally regular rotating EYM sphalerons bizon (); eugen ().
Globally regular EYMH solutions, on the other hand, can carry electric charge, and the presence of a time component of the gauge potential renders the solutions stationary. Together the electric and magnetic fields then give rise to an angular momentum density, except in the spherically symmetric case eugen (); eugen2 (); ulrike (). Still, globally regular EYMH solutions with a non-vanishing global magnetic charge cannot rotate: their angular momentum vanishes eugen (); eugen2 (); ulrike (). But globally regular EYMH solutions with no global magnetic charge do possess a finite angular momentum. In fact, it is proportional to their electric charge eugen (), giving rise to a quantization condition for the angular momentum,
where and are two integers, characterizing the EYMH solutions KKS ().
Here we derive a mass formula for the stationary globally regular EYMH solutions in the presence of a dilaton. Then we address the dependence of the global charges and of the effective action of these solutions on the gravitational coupling strength. For a given type of solution, typically two branches of solutions arise, which bifurcate at a maximal value of the coupling, . For static solutions, the mass exhibits a “spike” at gmono (); KKS (), since there the two branches must possess the same tangent w.r.t. peter (). When stationary and rotating solutions are considered, in contrast, the mass branches may exhibit a “loop”, when considered as a function of ulrike (). Here we show that for stationary and rotating solutions it is the effective action which may only exhibit a “spike” in the vicinity of the maximal value of the gravitational coupling constant. We illustrate this qualitative different behaviour of the mass and the effective action for several sets of numerically constructed stationary and rotating solutions.
where represents the temperature of the black holes and their entropy, denotes their horizon angular velocity, and and represent their horizon electric and magnetic potential, respectively.
In the presence of a dilaton an equivalent mass formula for EMD black holes is KKNmass ()
with dilaton charge and dilaton coupling constant . Interestingly, this second form of the mass formula also holds for the known non-Abelian black hole solutions of Einstein-Yang-Mills-dilaton (EYMD) theory KKNmass ().
Here we address stationary black holes of EYMH and Einstein-Yang-Mills-Higgs-dilaton (EYMHD) theory Forgacs2 (). For these black holes the zeroth law of black hole mechanics holds kknrot (), as well as a generalized first law heustrau (). We derive a mass formula for EYMHD black holes, based on the asymptotic expansion of the metric and the matter fields. The analytical mass formula represents a good criterion for the quality of numerically constructed EYMHD black hole solutions, also presented.
In section II we recall the SU(2) EYMHD action and the equations of motion. We discuss the stationary ansatz for the metric, the gauge potential, the Higgs field and the dilaton field, and we present the boundary conditions for globally regular and black hole solutions. In section III we address the physical properties of the solutions. We present the asymptotic expansion at infinity and the expansion at the horizon, needed to obtain the global charges and the horizon properties of the solutions. We evaluate the mass, the angular momentum and the effective action of the globally regular solutions in section IV, and discuss the dependence of these quantities on the coupling constant . We illustrate these results for a set of numerically constructed solutions. We then derive the mass formula for the stationary black hole solutions in section V, presenting also numerical results. In section VI we present our conclusions.
Ii Einstein-Yang-Mills-Higgs-dilaton solutions
After recalling the SU(2) EYMHD action and the general set of equations of motion, we discuss the ansatz for the stationary non-Abelian globally regular and black hole solutions. The ansatz for the metric represents the stationary axially symmetric Lewis-Papapetrou metric in isotropic coordinates. The ansatz for the gauge potential and the Higgs field includes two integers, and , related to the polar and azimuthal angles. For monopole-antimonopole chains the integer counts the total number of poles on the symmetry axis, while the integer gives the magnitude of the magnetic charge of each pole. As implied by the boundary conditions, the stationary axially symmetric solutions are asymptotically flat, and the black hole solutions possess a regular event horizon.
ii.1 SU(2) EYMHD Action
We consider the SU(2) Einstein-Yang-Mills-dilaton action
where is the scalar curvature, and the matter Lagrangian is given by
with dilaton field , gauge field strength tensor , gauge field , Higgs field in the adjoint representation , gauge covariant derivative , and Newton’s constant , dilaton coupling constant , Yang-Mills coupling constant , Higgs self-coupling constant , and Higgs vacuum expectation value .
The nonzero vacuum expectation value of the Higgs field breaks the non-Abelian SU(2) gauge symmetry to the Abelian U(1) symmetry. The particle spectrum of the theory then consists of a massless photon, two massive vector bosons of mass , and a massive Higgs field . In the limit the Higgs field also becomes massless. The dilaton is massless as well.
Including a boundary term Gibbons:1976ue (), variation of the action with respect to the metric and the matter fields leads, respectively, to the Einstein equations
with stress-energy tensor
and the matter field equations,
where , and
ii.2 Stationary Axially Symmetric Ansatz
The system of partial differential equations, Eq. (6), Eq. (9), Eq. (8), and Eq. (10) is highly non-linear and complicated. In order to generate solutions to these equations, one profits from the use of symmetries, simplifying the equations.
Here we consider solutions, which are both stationary and axially symmetric. We therefore impose on the spacetime the presence of two commuting Killing vector fields, (asymptotically timelike) and (asymptotically spacelike). Since the Killing vector fields commute, we may adopt a system of adapted coordinates, say , such that
In these coordinates the metric is independent of and . We also assume that the symmetry axis of the spacetime, the set of points where , is regular, and satisfies the elementary flatness condition
Apart from the symmetry requirement on the metric (, i.e., ), we impose that the matter fields are also symmetric under the spacetime transformations generated by and .
This implies for the dilaton field
and for the gauge potential ,
where and satisfy
Performing a gauge transformation to set , leaves , and independent of .
By virtue of the Frobenius condition and the circularity theorem, the metric can then be written in the Lewis-Papapetrou form, which in isotropic coordinates reads
where , , and are functions of and only.
The -axis represents the symmetry axis. The regularity condition along the -axis Eq. (12) requires
The event horizon of stationary black hole solutions resides at a surface of constant radial coordinate, , and is characterized by the condition kkrot (). The Killing vector field
is orthogonal to and null on the horizon wald (). The ergosphere, defined as the region in which is positive, is bounded by the event horizon and by the surface where
where and are integers. The symbols , and denote the dot products of the Cartesian vector of Pauli matrices, , with the spatial unit vectors
respectively. Like the dilaton field function , the gauge field functions and and the Higgs field functions depend only on the coordinates and .
ii.3 Dimensionless Quantities
Let us now introduce the dimensionless quantities, beginning with the dimensionless coupling constants , and
The dimensionless coordinate is given by
the dimensionless electric gauge field functions and are
and the dimensionless dilaton function is
Introducing these dimensionless quantities into the EOMs, the resulting equations depend only on the parameters , , and . Note, that in the limit the dilaton decouples and the equations of EYMH theory are obtained.
ii.4 Boundary Conditions
Boundary conditions at infinity
To obtain asymptotically flat solutions, we impose on the metric functions the boundary conditions at infinity
For the dilaton function we choose
since any finite value of the dilaton field at infinity can always be transformed to zero via , .
The asymptotic values of the Higgs field functions are
We further impose, that the two electric gauge field functions satisfy
where the asymptotic value is restricted to , and that the magnetic gauge field functions satisfy
Boundary conditions at the origin
To obtain globally regular solutions, we must impose appropriate boundary conditions at the origin. Regularity requires for the metric functions the boundary conditions
and for the dilaton function
the gauge field functions satisfy
while for even the gauge field functions and the Higgs functions satisfy
whereas for odd they satisfy .
Boundary conditions at the horizon
The event horizon of stationary black hole solutions resides at a surface of constant radial coordinate, , and is characterized by the condition kkrot ().
Regularity at the horizon then requires the following boundary conditions for the metric functions
for the dilaton function
while the Higgs and the magnetic gauge field functions satisfy
with the gauge condition taken into account kkrot (). The boundary conditions for the electric gauge field functions are obtained from the requirement that for non-Abelian solutions the electrostatic potential is constant at the horizon kknrot ()
Defining the dimensionless electrostatic potential ,
and the dimensionless horizon angular velocity ,
yields the boundary conditions
Boundary conditions along the symmetry axis
The boundary conditions along the -axis ( and ) are determined by the symmetries. For the positive -axis they are given by
The analogous conditions hold on the negative -axis. We note, that the globally regular solutions are symmetric w.r.t. the -plane. For the black hole solutions, this symmetry is broken via the boundary conditions of the time component of the gauge field.
In addition, regularity on the -axis requires condition Eq. (18) for the metric functions to be satisfied, and regularity of the energy density on the -axis requires
Iii Properties of Regular and Black Hole Solutions
We derive the properties of the stationary axially symmetric solutions from the expansions of their metric and matter field functions at infinity, at the origin and at the horizon. The expansion at infinity yields the global charges of the solutions, the expansion at the horizon yields the horizon properties of the black holes.
iii.1 Asymptotic Expansion
The asymptotic expansion depends on the integers and . Here we restrict to odd winding number , since the analysis for the even case seems to be ‘prohibitively complicated’. We then obtain for
For generic solutions, the expansions remain valid with . At first sight the power law decay of the Higgs field then appears surprising, since renders the Higgs massive and should thus lead to an exponential decay. However, this power law decay represents a gauge artifact and can be removed by the gauge transformation
Performing this gauge transformation leads to (note that the integer has been transformed away). We further obtain trivial gauge field functions and (up to order ).
iii.2 Global Charges
The expansion coefficients , , , and correspond to the global charges of the solutions. The dimensionless mass and angular momentum of the solutions are obtained from the asymptotic expansion of the metric
These correspond to the expressions obtained from the respective Komar integrals, as shown in sections IV and V for the globally regular and black hole solutions, respectively. Note that is the mass in units of , whereas corresponds to the mass in units of Planck mass. Likewise, the dimensionless electric charge and the dimensionless magnetic charge are given by
while the dimensionless dilaton charge is given by
iii.3 Expansion at the Horizon
Expanding the metric and matter field functions at the horizon in powers of
yields to lowest order
The expansion coefficients , , , , , , , , , are functions of the variable . Among these coefficients the following relations hold,
iii.4 Horizon Properties
With help of the above expansion we obtain the horizon properties of the SU(2) EYMHD black hole solutions. The first quantity of interest is the area of the black hole horizon. The dimensionless area is given by
and the dimensionless entropy by
The surface gravity of the black hole solutions is obtained from wald ()
As seen from Eq. (68), is indeed constant on the horizon, as required by the zeroth law of black hole mechanics. The dimensionless temperature of the black hole is proportional to the surface gravity,
iii.5 Electric and magnetic charge
A gauge-invariant definition of the electromagnetic field strength tensor is given by the ’t Hooft tensor tHooft ()
where is the normalized Higgs field, .
The ’t Hooft tensor yields the electric current
and the magnetic current
where represents the dual field strength tensor.
The electric charge is given by
where the integral is evaluated at spatial infinity.
To define the magnetic charge, we rewrite the ’t Hooft tensor as
with . Now it follows from Eq. (77) that the magnetic current and the topological current are related by
For globally regular solutions the integration of the magnetic charge density reduces to a surface integral at spacial infinity which yields
For black hole solutions we define the magnetic charge by its value on the horizon plus a volume integral,
where now denotes an asymptotically flat spacelike hypersurface bounded by the horizon , is the natural volume element on , and is normal to with .
In order to define the horizon magnetic charge we consider the normalized Higgs field at the horizon as a map between two two-dimensional spheres, which can be characterised by a topological number,
and obtain . For the evaluation of the magnetic charge we note that the volume integral reduces to a surface integral. Its contribution from the horizon cancels exactly the horizon magnetic charge, and the contribution from the asymptotic region yields . Note that for odd the horizon magnetic charge is either equal to the magnetic charge or to its negative value, depending on how often the Higgs field function changes sign on the symmetry axis. For even both the magnetic charge and the horizon magnetic charge are zero.
iii.6 Physical Interpretation of
The quantity is related to the asymptotic behaviour of the gauge potential , and therefore it is not defined in a gauge-invariant way. To find a physical interpretation of we apply a gauge transformation that leads to an asymptotically trivial gauge potential (for even ). Such a gauge transformation is given by
The transformed gauge potential and Higgs field are found to be
respectively, where now
We observe that in this gauge the fields are explicitly time dependent and rotate in internal space about the direction. The quantity is exactly the rotation frequency.
In the presence of magnetic charge, i.e. odd , the transformed gauge potential is singular on the negative axis. However, the physical interpretation of does not change.
Iv Stationary globally regular EYMHD solutions
iv.1 Global Charges
Mass, angular momentum and dilaton charge
We begin by recalling the general expressions wald () for the global mass
and the global angular momentum
Here denotes an asymptotically flat spacelike hypersurface, is normal to with , and is the natural volume element on wald ().
Now we express the Ricci tensor in terms of the Yang-Mills, Higgs and dilaton fields, using the Einstein equations, the definition of the stress energy tensor and the Lagrangian