Open questions on the even-odd parity technique for generating stationary axisymmetric vacuum solutions of Einstein's equation (Xanthopoulos and Chandrasekhar, 1979)

Xanthopoulos and Chandrasekhar (1979) introduced a discrete solution-generating technique for stationary axisymmetric vacuum spacetimes: starting from a one-parameter family whose quotient-space fields have a definite even-odd parity in the parameter, a linear operator A that sends beta^2 to -beta^2 produces a two-parameter family, and two applications of the resulting map reproduce an Ehlers transformation. The paper left open the structural theory of this technique: exactly which families admit the required even-odd form; whether Ehlers transformations along Killing fields outside the preferred pair preserve the even-odd condition (the author stated explicitly that he did not know); how to characterize the compositions that preserve asymptotic flatness; and whether an auxiliary ordinary differential equation arising in the paper's explicit-family constructions can be integrated in closed form to yield further two-parameter families.

Background

A stationary axisymmetric spacetime admits two commuting Killing fields (infinitesimal symmetry generators of the metric), one timelike and one spacelike with closed orbits, and in the vacuum case such spacetimes model the exteriors of isolated rotating masses. Because the two Killing fields commute, Einstein's vacuum equations reduce to a coupled nonlinear system on the two-dimensional orbit space (quotient manifold) $S$: the solution is encoded in a two-metric $h_{ab}$ on $S$, a conformal scale field $\tau$, the three inner products $\lambda_0,\lambda_1,\lambda_2$ built from the two Killing fields $\xi^a_0$ and $\xi^a_2$ (with $\lambda_0$ the squared norm of $\xi^a_0$, $\lambda_2$ that of $\xi^a_2$, and $\lambda_1$ their inner product), and the twist potentials $\omega_j$ - scalars whose gradients encode the rotation dragged by each Killing field, well defined because the twist one-forms are closed for vacuum solutions. Solving the reduced equations directly is hard, so from the 1960s onward an industry of solution-generating transformations developed. The Ehlers transformation acts fractionally-linearly (as a Moebius map) on the complex Ernst potential $E=\lambda+i\omega$ that combines a Killing norm $\lambda$ with its twist potential $\omega$; it maps vacuum solutions to vacuum solutions and, as shown by Geroch (1972), embeds into an infinite-dimensional transformation group of the reduced equations. Harrison (1968) and Kinnersley (1973) constructed the analogous generators for the Einstein-Maxwell (electrovac) equations, in which an electromagnetic field is included.

Xanthopoulos and Chandrasekhar (1979), building on Chandrasekhar (1978), isolated a discrete mechanism of a different kind. Suppose a one-parameter family of stationary axisymmetric vacuum solutions satisfies the even-odd condition: the quotient fields $\tau,\lambda_0,\lambda_2$ are even functions of the parameter $\beta$ and $\lambda_1$ is odd (equivalently, their power series in $\beta$ contain only even, respectively odd, powers, and converge), with the twist potentials $\omega_0,\omega_2$ odd and $\omega_1$ even in $\beta$, and the conformal structure of the quotient two-metric is parameter-independent. Define the linear operator $A$ on such series by $A(\sum_n\alpha_n\beta^{2n})=\sum_n(-1)^n\alpha_n\beta^{2n}$ and analogously on odd series, i.e. $A$ implements the substitution $\beta^2\mapsto-\beta^2$. Then for any Killing field $\eta$ of the family, with norm $\lambda_\eta$ and twist potential $\omega_\eta$, the primed fields

$\tau'=A(\tau),\quad \lambda_0'=\frac{A(\tau)}{A(\lambda_\eta)},\quad \lambda_1'=\frac{1}{\sqrt{2}}\frac{A(\tau)A(\omega_\eta)}{A(\lambda_\eta)},\quad \lambda_2'=\frac{1}{2}\frac{A(\tau)}{A(\lambda_\eta)}\bigl[(A(\omega_\eta))^2-(A(\lambda_\eta))^2\bigr],\quad h_{ab}'=\frac{A(\lambda_\eta)A(h_{ab})}{\sqrt{A(\tau)}}$

again solve the reduced vacuum equations (equations (7) of the paper). Two structural facts follow: applying the map twice, first with respect to one Killing field and then with respect to the linear combination $\xi^a_0+\beta\kappa\xi^a_2$ - which introduces a second, essential parameter $\kappa$ - reproduces an Ehlers transformation of parameter $\tfrac{1}{\sqrt{2}}\beta\kappa$, so the parity map is a 'square root' of the Ehlers map; and applying it twice with respect to the same Killing field returns an isometric solution. The map has no infinitesimal generator, and it does not in general preserve the Petrov type (the algebraic classification of the Weyl curvature tensor) or asymptotic flatness (the property that the metric approaches the Minkowski metric at large distance). The paper demonstrated the technique on an explicit family built from a pair of conjugate harmonic functions on $S$ and on the Schwarzschild family written in the gauge $r=2M\cosh^2u$ with suitably rescaled Killing fields, and stated that the technique applies to almost all known families of stationary axisymmetric solutions. Its reference list also cites the electrovac generating methods of Harrison (1968) and Kinnersley (1973) and the precursor constructions of Ernst (1977) and Demiański (1972).

Problem Statement

The problem is to complete the structural theory of the parity technique of Xanthopoulos and Chandrasekhar (1979) - to determine its exact scope, its closure properties under the transformations it is related to, and its behavior on physically significant classes. Concretely, in the quotient-space formulation defined in the background, for stationary axisymmetric vacuum solutions:

1. (Scope) Characterize precisely which one-parameter families of stationary axisymmetric vacuum solutions can be brought into the even-odd form required by the technique (parameter-independent conformal structure of the quotient two-manifold $S$, with $\tau,\lambda_0,\lambda_2$ even and $\lambda_1$ odd in the parameter, with convergent power-series expansions). In particular, determine whether the technique's applicability to almost all known families reflects a general property of such families or a bias of the known examples.

2. (Closure) Determine whether applying the Ehlers transformation to an even-odd family along a Killing field that is not one of the two preferred Killing fields preserves the even-odd condition; the source states 'I do not know whether the following is true: applying the Ehlers transformation...'.

3. (Global properties) Characterize the sequences or combinations of applications of the parity map and Ehlers transformations that preserve asymptotic flatness, given that a single application of either need not.

4. (Explicit families) In the paper's construction of explicit input families one starts from a pair $\tau,\sigma$ of conjugate harmonic functions on the quotient two-manifold $S$ (conjugate meaning $\mathrm{D}_a\sigma=\epsilon_a{}^m\mathrm{D}_m\tau$, where $\mathrm{D}_a$ is the covariant derivative of the quotient two-metric $h_{ab}$ and $\epsilon_{ab}$ its area form); one branch of the resulting ansatz is integrated in closed form in the paper, while the other branch is given by formulae of the form $\lambda_0=f(\tau)\,e^{\sigma}(f^{2}(\tau)+1)^{-1/2}$ and $\omega_0=e^{\sigma}(f^{2}(\tau)+1)^{-1/2}$, where the function $f$ of $\tau$ must satisfy the auxiliary ordinary differential equation (equation (19) of the 1979 paper)

$\frac{f^{2}f'^{2}}{(f^{2}+1)^{2}}-\frac{f^{2}}{f^{2}+1}\left[(\ln f)''+\frac{1}{\tau}(\ln f)'\right]=1,$

the prime denoting differentiation with respect to $\tau$. Determine whether this equation can be integrated in closed form; the paper's analysis indicates that its integration could yield an additional two-parameter family of stationary axisymmetric vacuum solutions, which the authors leave as an open task.

The common objective joining the four items is a complete theory of what the parity technique can and cannot generate; partial resolutions of individual items are meaningful but do not close the problem as a whole.

References

1. B. C. Xanthopoulos and S. Chandrasekhar, 'A technique for generating solutions of Einstein's equation', Proceedings of the Royal Society of London A 365, 381-394 (1979). DOI: 10.1098/rspa.1979.0024, https://doi.org/10.1098/rspa.1979.0024
2. S. Chandrasekhar, 'The Kerr metric and stationary axisymmetric gravitational fields', Proceedings of the Royal Society of London A 358, 405-420 (1978). DOI: 10.1098/rspa.1978.0019, https://doi.org/10.1098/rspa.1978.0019
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