Dressing with Control:
Using integrability to generate
desired solutions to Einstein’s equations
Motivated by integrability of the sine-Gordon equation, we investigate a technique for constructing desired solutions to Einstein’s equations by combining a dressing technique with a control-theory approach. After reviewing classical integrability, we recall two well-known Killing field reductions of Einstein’s equations, unify them using a harmonic map formulation, and state two results on the integrability of the equations and solvability of the dressing system. The resulting algorithm is then combined with an asymptotic analysis to produce constraints on the degrees of freedom arising in the solution-generation mechanism. The approach is carried out explicitly for the Einstein vacuum equations. Applications of the technique to other geometric field theories are also discussed.
Can solutions to the Einstein Equations be “made to order,” i.e., is it possible to construct spacetimes with a prescribed set of asymptotic “observables” (such as total mass and angular momentum) and a prescribed causal and/or singular structure (e.g. number of components of event horizon, or number of ring singularities, etc.), given a specified set of initial parameters? We provide a first step in answering this question by investigating a solution-generation mechanism for integrable harmonic maps, known as the vesture, or dressing method.
The seeds of our discussion, however, are first sowed in sine-Gordon theory. The special class of nonlinear equations to which the sine-Gordon equation belongs is that of classically integrable partial differential equations (PDEs). Even though nonlinear, these PDEs have the property that new solutions can be formed from old ones, in what amounts to a nonlinear generalization of the superposition principle. We exploit this very feature in order to study the Einstein equations. As a consequence, the main thread of this communication can be summarized in the following diagram:
We shall first briefly introduce classically integrable PDEs. Then, building on the techniques developed for the sine-Gordon equation, we introduce Einstein’s gravitational equations and put them in a similar context as sine-Gordon, by briefly reviewing both the Zakharov-Belinski and Ernst formulations of the stationary axisymmetric Einstein vacuum equations. We will then realize this same system of PDEs as an axially symmetric harmonic map.
With background established, we place the gravitational field equations in the context of integrability by way of two theorems; informally, the results state that axially symmetric harmonic maps satisfy an integrable system of equations and that new axially symmetric harmonic maps with any number of prescribed singularities can be constructed from a given one. We implement the algorithm afforded by the proofs of these theorems explicitly in the case of the Einstein vacuum and Einstein-Maxwell equations. It is in the final section that we give a partial answer to the question posed at the beginning of this section, namely, by combining a novel asymptotic expansion with the calculated results of the vesture method, we show that, given any real numbers and , it is indeed possible to treat the arbitrary constants of the dressing procedure as control parameters and produce a 1-solitonic harmonic map that, when viewed as a stationary axisymmetric spacetime metric, has total ADM mass equal to and total ADM angular momentum equal to , and is otherwise free of unwanted pathologies (such as a non-zero NUT parameter), in other words, it is a member of the Kerr family of metrics. We conclude by proposing a strategy for generating solutions in more general cases and indicate possible future directions of study.
2. Integrability and Inverse Scattering
A nonlinear PDE (or system of PDEs) is said to be classically integrable if there exists an overdetermined linear system, called a Lax system, the compatibility condition of which is precisely the nonlinear PDE111The term integrable has been adopted in various contexts, and is ambiguous in characterizing the features of such an equation (e.g., existence of closed-form solutions by quadrature, possessing infinitely many conservation laws, exhibiting solitonic dynamics, etc.). We shall restrict our attention to those equations which are integrable in the Lax sense.. We describe this in further detail.
Start with a nonlinear evolution equation for . Associate to it a corresponding overdetermined system for an isospectral family of linear differential operators , with -dependent coefficients, whose eigenfunctions satisfy an additional linear evolution equation
where is another linear operator with coefficients depending on and its derivatives. Isospectrality implies a compatibility condition between and , namely
Under the appropriate choices222It is a long-standing open problem to characterize the PDEs which admit a Lax formulation. of , this equation agrees with the nonlinear PDE of interest, and the operators and are called a Lax Pair for the PDE .
One of the first equations to be successfully studied in this setting is the Korteweg-de Vries or KdV equation
first used in modeling shallow water wave dynamics. Defining the operators and , it is easy to verify that the compatibility condition (2) is satisfied if and only if satisfies (3). Notice that in this case the first equation in (1) is simply a classical eigenvalue problem for a linear Schrödinger operator having as potential.
As the above example demonstrates, the operator depends on the unknown , and may indeed be considerably more complicated than , so solving the Lax system (1) may not be a straightforward task, except possibly at . However, the appearance in (1) of time-evolution for the eigenfunction , provides a starting point for the Inverse Scattering Method (ISM), a nonlinear analogue of the Fourier Transform, to be applicable. The three steps of the ISM are depicted in Figure 1 (excerpted from ), in which evolution of the scattering data determined by the eigenvalue problem is used to recover a potential from the Cauchy data of the nonlinear PDE of interest.
In the first step, direct scattering refers to finding a matrix relating asymptotic eigenstates at by way of . Note that is a matrix since the asymptotic eigenspaces are multidimensional. This step requires identifying a Lax pair333It also hinges on space being non-compact and one dimensional.. For integrable equations, isospectrality of the operator implies that the time evolution of is linear; consequently, in the second step, the scattering matrix can be found (rightmost arrow). Upon integration of the Gelfand-Levitan-Marchenko equations associated with the system, the third step, inverse scattering, recovers a time-evolved solution to the original nonlinear PDE of interest. Standard techniques are described in [23, 1, 3] and references therein.
For what follows, we will be careful to distinguish between integrability of a PDE (or system of PDEs) and existence of a solution-generating mechanism for that PDE, noting that many of the examples of this discussion will possess both features.
2.1. Classical integrability, extended: the sine-Gordon equation
Expanding the scope of classical integrability to address second order evolution equations involves going from scalar linear operators to matrix operators, sometimes at the expense of introducing powers of , the spectral parameter444Although one is no longer considering an eigenvalue problem in the classical sense, is often still called a spectral parameter.. In this case, matrix equations comparable to (1) are given by
where and are matrix functions depending on . The compatibility or zero-curvature condition corresponding to (2) is
We take a moment to justify use of the phrase “zero-curvature” by describing the geometry afforded by (4) and (5). Observe that (4) can be used to define a connection on a two-dimensional vector bundle over by setting
and extending linearly to , for a vector field . In this manner, we may view the two equations in (6) as parallel transport of in the and directions, respectively. It is then straightforward to calculate the curvature of this connection for the basis vector fields to be
Thus, the connection has zero curvature if parallel transport of along any path connecting two points agrees, i.e., precisely if compatibility condition (5) is satisfied.
Matrix formulation of the ISM is used to address the integrability of the sine-Gordon equation (in null coordinates)
first used in the study of mechanical solitons and differential geometry of constant curvature surfaces555Using the transformation , one may easily recover the second recognizable form of the sine-Gordon equation, .. Defining as
the Lax system analogous to (4) is given by , and the zero curvature condition ensures that evolves according to (8). A comparable initial value problem to the one depicted in Figure 1 is explicitly solved for the sine-Gordon equation by the ISM in ; full details for the inverse scattering procedure may be found in [47, 48]. It is worth noting that various Lax formulations are possible, by choosing different in (4) (see, for instance, ).
2.2. From sine-Gordon to Chiral Fields
Progression from matrix Lax pairs for a single nonlinear PDE to matrix Lax pairs for nonlinear systems of PDEs occurs naturally when studying simple geometric field theories, as the next example demonstrates. What is surprising is that from the appropriate perspective, this matrix operator modification applies equally well to equations of gravitation and electrodynamics.
Let be an open subset of with coordinates and suppose is a smooth function of with values in a semisimple Lie group . Denote by and consider the following action
The Euler-Lagrange equations of this action are known as equations of the principal chiral field on the group –a free field in two-dimensional spacetime with values in [50, 47, 41]. The word chiral here refers to the fact that this Lagrangian is invariant separately under the left, and the right, action of the group on itself. With a non-commutative group, this action can thus be used to model physical phenomena that appear to break the chiral symmetry, such as the interaction of mesons in the chiral limit . In that case where is the number of quark flavors.
By introducing the compact notation , , and observing that , the Lagrangian density in (9) becomes . Consequently, the field equations are re-expressed as
where the second equation is the compatibility condition arising from the definitions of and .
To solve these equations for by means of inverse scattering, we cast (10) as the compatibility condition of a linear system of matrix equations of the form
for a vector function of and matrix functions of and a complex parameter . Equality of mixed partials in imposes the compatibility condition just as in (5), replacing and by and , respectively, yielding
There are several quantities to determine in (11), with varying degrees of freedom. The matrices are to be determined in terms of and , bearing in mind the compatibility conditions of both and ; furthermore, given a solutions of the linear system, the desired matrix should be recoverable by fixing a value of the parameter , under the definitions of and . Zakharov and Shabat carried out a vesture method (later called the dressing technique) to solve this overdetermined system of conditions by assuming have simple poles in :
Substituting these expressions into zero-curvature condition (12) and clearing denominators, one obtains a polynomial in of degree , which splits into equations corresponding to coefficients of the polynomial [47, 48].
In the special case , where each have one simple pole the compatibility condition (12) reduces to a degree 2 polynomial in from which three equations are obtained. For instance, assuming and under appropriate assumptions on , (11) can be reduced to a Lax formulation of sine-Gordon equivalent to the one given in Example 2. A complete analysis appears in [49, 21]. Note that just as in the previous examples, (11) naturally defines the covariant differentiation operations given by and .
Under the coordinate transformation , the Lagrangian density of the action (9) differs by a null Lagrangian from , where and . Noting that must lie in the Lie algebra of the Lie group , the transformed action may be viewed as an inner product of the matrix-valued one-form with itself, relative to the Minkowski metric on . As we shall see, this one form can be interpreted as the pull-back of the Maurer-Cartan form under the mapping . We shall also see below that the principal chiral field is simply a harmonic map into a Lie group. These two facts will combine in a particularly elegant manner where the integrability of such maps is studied.
3. Gravitational Equations
Motivated by the last example, we turn our attention to the Einstein vacuum equations with vanishing cosmological constant
This quasilinear system is satisfied by the metric tensor of a four-dimensional Lorentian manifold . Here, denotes the Ricci curvature tensor of . Complexity of these field equations is reduced considerably using symmetry reductions, corresponding to existence of Killing fields for the metric . In this context, existence of timelike Killing fields correspond to stationary metrics, and existence of spacelike Killing fields correspond to spherically symmetric or axisymmetric metrics (depending on whether the action generated is an or rotation, respectively). Under such symmetries, the simplest nontrivial asymptotically flat solutions666Asymptotically flat solutions represent vacuum outside an isolated body. are Schwarzschild spacetimes, which are static and spherically symmetric, and the Kerr spacetimes, which are stationary and axisymmetric [36, 27].
We shall describe two approaches for the analysis of a two-Killing field reduction of (14), namely the Zakharov-Belinski and the Ernst formulations, and establish a concrete connection between them by way of harmonic maps.
3.1. Integrable formulation of the Einstein vacuum equations
First, we briefly outline the techniques in [8, 9] used to establish integrability of the stationary, axisymmetric Einstein vacuum equations. Assuming the existence of two commuting Killing fields, the spacetime metric can be expressed in block-diagonal form
with and . Here the sign is to be used when both Killing fields are spacelike (and hence is Lorentzian), while the sign is for the case of one timelike and one spacelike Killing field (in which case the quotient metric is Riemannian). In the former case, let be null coordinates on the quotient manifold, and in the latter, set . The equations for are given by
for , , . Compare with the chiral field model (10). The task is to find a linear system for analogous to (11) for which the above PDE system appears as a compatibility condition; note that once is determined, can be found by quadrature (see, e.g. ).
Just as in (4) for the sine-Gordon equation and (11) for the chiral field, assume and have simple poles in . The key idea used in establishing integrability in this case is a generalization of to operators which include differentiation in the spectral parameter , namely
where , and is a rational function of . We shall interpret these generalized derivatives geometrically in Section 4. Note that here , thus in contrast to the chiral field model, the poles in (17) are not fixed, but moving.
With integrability established, the authors of [8, 9] implement a vesture or dressing method to generate new solutions to the Einstein vacuum equations. Formally, given a solution of the Lax system having initial data , dressing refers to the procedure of finding a matrix for which also solves the system. It is from this “dressed” generating matrix that new solutions (and hence ) are constructed. Determining the dressing matrix reduces to solving a finite system of algebraic equations. In the next section we shall describe a generalization of this technique in the context of dressing harmonic maps.
3.2. Ernst Formulation of the vacuum equations
Prior to the works of [8, 9], another approach to the two-Killing field reduction of EVE had been investigated in [19, 20], giving what is now known as the Ernst Equation for a complex potential function . We briefly review this now.
Let the two commuting Killing fields for the metric be denoted by and . Let us assume that is timelike and spacelike. Set
Thus, is the element of area of the cylindrical group orbits and is the twist form of . The line element of is then
where is a metric on the two-dimensional Riemannian quotient manifold conformal to .
Suppose now that the metric is a solution of Einstein vacuum equations. It then follows that (i) the function is harmonic , and (ii) is closed, i.e. . Thus (assuming the domain is simply connected) . The function is called the twist potential for the Killing field (see e.g. ). Choosing a conjugate harmonic function for , and using as isothermal coordinates on the quotient brings the line element to the form
This is known as the Lewis-Papapetrou form of the metric. There are now only three unknown metric coefficients left: , and . Next, by virtue of and being Killing fields (and the Einstein vacuum equations being satisfied), it is possible to show that the quantities and satisfy a coupled system of elliptic PDEs, which can be combined into a single equation by introducing a complex-valued potential , called the Ernst potential:
Once the potential is found, the remaining metric coefficients and can be found using quadratures .
Equation (22) is derivable from a variational principle, namely, it is the Euler-Lagrange equation of the following action
We will see below that this is also the action for a harmonic map into the hyperbolic plane.
3.3. Harmonic Maps and Gravitation
Recall that a harmonic map is a critical point of the Dirichlet Energy
This definition generalizes the notion of a geodesic to higher-dimensional domains: letting , the Euler-Lagrange equations for the harmonic map action reduce precisely to the equations of an energy-minimizing geodesic on . Harmonic maps also generalize harmonic functions to nonlinear targets: letting , the Euler-Lagrange equation for (24) corresponds to that of a harmonic function on the base manifold . With this notation in place, we can cast some of our prior examples from the previous sections in the context of harmonic maps777The sine-Gordon equation can also be put into this framework, see ..
Example 3, revisited. To express the chiral field appearing in Example 3 as a wave map, the hyperbolic analog of a harmonic map, let with a mapping from the -dimensional Minkowski space into the group , and null coordinates on the domain. The Maurer-Cartan form of is , a Lie-algebra-valued 1-form. Let denote its pull-back under . Thus where is a basis for the Lie algebra of the group , and are 1-forms on the domain: . On the other hand, if is semi-simple, the Killing-Cartan quadratic form is non-degenerate, and it thus endows with a pseudo-Riemannian888The metric is Riemannian if is a compact group. metric . With denoting the Minkowski metric, we thus have
which agrees with (9). More precisely, , so that , the chiral field Lagrangian, as before.
We now move on to the reduction of vacuum Einstein equations under the assumption of existence of two commuting Killing fields.
Let denote a solution of Einstein’s vacuum equations possessing two commuting Killing fields. We may then express the line element of in the Lewis-Papapetrou form (21). Let be coordinates on the real hyperbolic plane with its standard metric ,
Suppose is a harmonic map that is invariant under the action of the two given Killing fields, so that is well-defined on the quotient manifold, . The harmonic map Lagrangian density is therefore
which is the same as (23), the Lagrangian obtained for the Ernst equation. Indeed, since the action is independent of all other features of the spacetime metric, we can pretend that the domain is simply in cylindrical coordinates , and that we have an axisymmetric (i.e. -independent) harmonic map from into the hyperbolic plane.
Finally, observing that the real hyperbolic plane may be realized as a symmetric space , we may use the Cartan embedding of a symmetric space into its mother Lie group , to represent as a mapping into , namely
The fact that the target of the map is now embedded in a linear space999Note that by virtue of the Cartan embedding, a symmetric space can be embedded into a Lie group, for which one may choose a matrix representation, thus making addition and scalar multiplication of elements possible. is the key to the success of the vesture method, described in the next section.
The Einstein-Maxwell system, equations governing the interaction of the spacetime metric with an electromagnetic field , given by
also represent the equations of an axially symmetric harmonic map, assuming existence of two commuting Killing fields which also leave the field invariant. In this case, the mapping is given by , where is as in the previous example, and is the complex hyperbolic plane realized as with , and [20, 11, 29], and the corresponding Ernst formulation now involves a pair of complex potentials [19, 20, 12].
Appearances of harmonic maps in physics have been extensively documented, as has the integrability of harmonic maps into particular targets; see [30, 35, 45], for instance. The primary observation we now make is the following: the chiral field, Einstein vacuum and Einstein-Maxwell equations (under a two-Killing-field reduction) are integrable and admit a dressing mechanism simply because harmonic maps into symmetric spaces are integrable and admit a generalized dressing mechanism.
4. Integrability and vesture of axially symmetric harmonic maps
Motivated by previous work, we prove a pair of theorems in  which unify the integrability and solution-generating mechanism (vesture or dressing technique) of special cases considered in the literature for the Einstein vacuum, Einstein-Maxwell and Chiral Field models; the results further establish integrability and vesture for a broader class of geometric field theories. We shall outline the main points surrounding their proof, after which the second half of this communication will focus on control of the solution-generating mechanism.
4.1. Integrability of axially symmetric harmonic maps
(Integrability) Let be a real semisimple Lie group and a maximal compact subgroup. Then any axially symmetric harmonic map from into the Riemannian symmetric space satisfies an integrable system of equations.
The proof of this statement involves two main steps. Let where the metric is given in cylindrical coordinates . Suppose is an axially symmetric harmonic map and let be a parametrization of the symmetric space via the Cartan embedding , so that .
Step Rewrite the equations of a harmonic map as a Hodge System. Using the pull-back of the Maurer-Cartan form , the harmonic map equations are
where the domain is the right-half plane in with coordinates .
Step Exhibit a Lax Pair whose compatibility condition is the Hodge System. In our case, we generalize the linear operators appearing in (17) by defining
where , , and , , are three specific -valued functions on defined to be
The mapping , is a deck transformation on the universal cover of and for all . We prove in  that the compatibility condition of the is equivalent to a zero curvature condition , for the 1-form on the domain (cf. , p. 54).
4.2. Vesture of axially symmetric harmonic maps
(Vesture) If the Lie group is such that the two involutions and defining the symmetric space can be given by conjugation with the same element, then the vesture method can be used to construct new harmonic maps starting from any given harmonic map.
As stated earlier, vesture refers to the procedure of finding a dressing matrix such that if is a solution of (31) and
Step Construct a dressing matrix possessing the appropriate symmetries. An invertible mapping is a dressing matrix for , an axially symmetric harmonic map of into , if it satisfies
Here, are commuting involutions characterizing the symmetric space and is a complex group containing ; the second and third equations in (4.2) are referred to as the G-reality and involutive symmetry conditions, movitated by the study of wave maps into compact symmetric spaces in [40, 39]. The symmetry conditions in (4.2) ensure that the resulting axially symmetric harmonic map , found by setting in the generating matrix , takes its values in the appropriate target space, . We show that there always exists an element of the equivalence class of which satisfies such imposed symmetries .
Step Reduce the problem of finding the dressing matrix to an algebraic problem. Motivated by simplifications appearing in [8, 9, 18], we assume is a rational function of and unknown matrices of the form
The moving poles, , are specified in the next section. Under further assumptions on the involutions we prove solvability of the linear system in .
We summarize Theorems 1 and 2 in Figure 2. Steps and comprise the integrability component (top arrow) and steps and comprise the vesture component (rightmost and bottom arrows) of the results. Compare with Figure 1. In our case, the dressing procedure results in axisymmetric harmonic maps possessing ring singularities .
The linear system resulting from this reduction unfortunately still contains too many unknown (and seemingly free) parameters. Thus, the question posed at the beginning of this paper is restated as: Can we control the output of this mechanism to generate desired harmonic maps from a given initial seed map , and if so, how?
4.3. The Dressing Algorithm
In order to control the number of parameters appearing in the dressing mechanism, we will need to outline the solution-generating algorithm afforded by the proof of Theorem 2. We do so now.
Let be distinct non-real complex numbers in the upper half plane. For every , each corresponds to a pair of points on the Riemann surface defined in (33). Number these prescribed poles in such a way that they are related by the deck transformation on :
Suppose that the involutions characterizing the symmetric space target can be given by conjugation with respect to the same element . In conjunction with the G-reality and involutive symmetry conditions in (4.2), this assumption allows the algebraic reduction to be carried out efficiently101010In particular the pseudo-unitary groups satisfy this assumption. Moreover, groups possessing Dynkin diagrams with no symmetries may also qualify. Modification of this procedure to address other families of involutions will be taken up in a future work..
The overdetermined system in is further simplified using a rank-one assumption. Define , where and assume there exist non-zero vector functions for such that
and furthermore that the vectors satisfy
This rank-one ansatz is consistent with the prior conditions imposed on the system by symmetry, and the relation on the vectors imposes a relation on the vectors which does not overdetermine the problem.
Although the system is equivalent to a nonlinear system for the , it is shown in  that can be taken to be constant vectors, and that the rank one assumption reduces the system to a linear system for the unknown vector functions , given by
where we have set
Thus, a complete set of unknowns for the problem are the vectors , given an arbitrary set of constant vectors in . Equation (40) can be written more compactly in matrix form as
Here is the matrix whose columns are the vector functions , is a matrix function, and is the matrix whose columns are the vector functions defined above.
If the matrix can be shown to be invertible (at least in a neighborhood in ), then the above system has a unique solution in that neighborhood. From there, one can then calculate the matrices and the dressing matrix , and setting , the new solution is found to be
In general, is not invertible everywhere in the domain , and indeed the zero set of has a geometric significance for the dressed harmonic map . However, we can prove that under appropriate conditions on the arbitrary vectors , the matrix is, in general, invertible for large (i.e., in a neighborhood of infinity). Once a has been found, its domain of definition may be maximally extended by analytic continuation.
Observe that (40) and hence (43) satisfies scaling invariance: if we rescale the constant vectors by (nonzero) complex constants , , then the expressions for also rescale by a factor of . This suggests that by rescaling each by , one preserves the equality in (43). Thus, we may view not as vectors in , rather as elements of . This also means that the actual number of free real parameters in our dressing procedure is , and not as it initially appeared to be (from arbitrary constant vectors in ). We shall invoke the scale invariance of the system explicitly when it is convenient to do so in our subsequent calculations.
5. -solitonic solutions of Einstein’s Equations
The solution-generating algorithm outlined in the previous section is realized concretely for the case of the pseudo-unitary groups, in . Define involutions on by
Then , , and using the Cartan embedding, the symmetric space is the complex Grassmann manifold
In the special case where , one obtains , the target in the Ernst reduction of Einstein vacuum equations, appearing in Example 4, and for the case , one obtains , the target in the Ernst reduction of Einstein Maxwell equations, appearing in Example 5.
5.1. Kerr spacetimes from 1-soliton dressed Minkowski metrics
We now specialize to the case to demonstrate the dressing algorithm explicitly, showing that dressing Minkowski spacetime with one soliton results in a family of (naked) Kerr spacetimes.
(Dress ) Fix , and let be the two roots of so that
Solving for above, the new harmonic map in (43), expressed in terms of the complex constants and , is given by
(Change coordinates) We now change from Weyl coordinates to Boyer-Lindquist coordinates , defined as
for real parameters . Then the harmonic map is re-expressed as
At this point, we would like to systematically deduce whether particular choices of constants lead to recognizable solutions, such as Kerr spacetimes. We use this formulation of the harmonic map to develop our control theory approach.
5.2. Control Theory meets GR: an asymptotic calculation
In the solution-generating algorithm, it is necessary to invert the matrix to solve the linear system appearing in (40) and (48) for the unknown vectors . However, for each soliton added into the system, that is, for each , the matrices , and grow in considerably size (order ). The plan will be to replace with coordinates at infinity and write down an asymptotic expansion for in powers of , for which truncations of the matrix will be more readily inverted. This is made possible by that fact that since the roots can be expressed in terms of the elliptical coordinates
we have as (since as well) for all .
First observe that in the coordinates , the matrix can be expressed as , where is a diagonal matrix and is a nilpotent skew-hermitian matrix, . Consequently,
This, in turn, allows us to deduce constraint equations on the constants by explicitly inverting in (39) using truncations of this expansion. In particular
can be calculated to various desired orders of .
In our example, we shall truncate the expansion of at the term and, recalling the definitions of from (4), we use
Calculating explicitly to obtain the column vectors and combining them with the constant row vectors , we compute each component of the new (truncated) mapping . Note that each contains a prefactor of , resulting in the following components for