## Abstract

Einstein’s equations with two commuting Killing vectors and the associated Lax pair are considered. The equations for the connection *A*(*ς*,*η*,*γ*)=*Ψ*_{,γ}*Ψ*^{−1}, where *γ* the variable spectral parameter are considered. A transition matrix *A*^{−1}(*ξ*,*η*,*γ*) for *A* is defined relating *A* at ingoing and outgoing light cones. It is shown that it satisfies equations familiar from integrable PDE theory. A transition matrix on *ς*= constant is defined in an analogous manner. These transition matrices allow us to obtain a hierarchy of integrals of motion with respect to time, purely in terms of the trace of a function of the connections *g*_{,ς}*g*^{−1} and *g*_{,η}*g*^{−1}. Furthermore, a hierarchy of integrals of motion in terms of the curvature variable *B*=*A*_{,γ}*A*^{−1}, involving the commutator [*A*(1),*A*(−1)], is obtained. We interpret the inhomogeneous wave equation that governs *σ*=*lnN*, *N* the lapse, as a Klein–Gordon equation, a dispersion relation relating energy and momentum density, based on the first connection observable and hence this first observable corresponds to mass. The corresponding quantum operators are ∂/∂*t*, ∂/∂*z* and this means that the full Poincare group is at our disposal.

## 1. Introduction

The general relativistic equations in the presence of two commuting Killing vectors have received a great amount of interest over the years. The first important contribution, relevant to the considerations here was the proof in [1] that there exists an infinite hierarchy of solutions which can be mapped to one another via transformations of what we now call the Geroch group. Later, it was shown [2–7] by different authors and in differing approaches that the field equations (2.2) are integrable (in the sense understood in the inverse scattering field) and a variety of solutions was obtained and analysed. These results are presented and reviewed in [8]. Relatively, more recently techniques of quantum inverse scattering were used [9] to quantize appropriate functions of the gravitational field appearing in (2.2) with the emphasis on the metric. The main motivation of this paper is to present an approach for future work in terms of connections which are appropriate variables for gravity.

The organization of this work is as follows. In §2, we introduce the equations and briefly present the solitonic technique. Further in §3, *A*(*γ*) is defined. Transition matrices for *A* relating it at points on ingoing and outgoing null coordinates is defined, and it is proved that it obeys equations similar to the ones of other integrable PDEs, a result absent from the literature thus far, for connection variables. This presents the possibility of obtaining integrals of motion, from the trace of the derivative of appropriate combinations of the two transition matrices, in terms of classical connections of General Relativity. These transition matrices first appeared in [10], where it is also noted that the possibility of observables in terms of connections can be attained. These may be appropriate for quantization as they are the fundamental variables of the classical theory and connections are fundamental also in Quantum Canonical General Relativity [11,12]. It is shown that Tr (*A*^{n}(1)*A*^{−n}(−1)) is an integral of motion. In §4, we consider the curvature *B*(*γ*)=*A*_{,γ}*A*^{−1} and obtain observables in a similar fashion as for the connection, but where now *B*(±1) involve the commutator [*A*(1),*A*(−1)]. We discuss these observables in §5, interpreting the first one of the connection observables as mass due to the fact that it appears, as the inhomogeneous term, in the wave equation governing the lapse *N* which is interpreted as a Klein–Gordon equation. This puts the associated Quantum Field Theory in our hands for quantization.

## 2. The Einstein equations with two commuting Killing vectors

The metric in the presence of two commuting Killing vectors, and assuming the existence of two-surfaces orthogonal to the group orbits, is given by [2,8,13]
*g*_{ab}(*t*,*z*) real and symmetric tensor, *a*,*b*=1,2.

Einstein’s equations corresponding to this metric are (in null coordinates *A*(±1) will become apparent in the following. Further from the above equations (taking the trace of (2.2)), it follows that *α* satisfies
*Ψ**(*γ**)=*Ψ*(*γ*). System (2.7) has compatibility conditions equation (2.2) and the zero-curvature condition [15] associated with this integrable system [8], p. 15, eqn 1.48. The differentials are given by
*α*=*a*(*ς*)−*b*(*η*) is a solution of (2.6) and *γ* is a ‘variable’ spectral parameter satisfying
*w* is a complex constant and *β*=*a*(*ς*)+*b*(*η*) is a second solution of (2.6). The choice *α*=*t*, *β*=*z* corresponds to the cosmological case, while *α*=*ρ*, *β*=*t* corresponds to cylindrically symmetric gravitational waves.

Solutions of equations (2.7) can be reproduced from a known background solution according to the BZ (Belinski–Verdaguer) dressing procedure [2,8,14]: One starts with *Ψ*_{0} a solution of (2.7) (corresponding to a background metric *g*_{0}) and form
*γ*_{k}, which have to be solutions of (2.10), that is, are given by (2.11) for *w*=*w*_{k}, correspond to solitons in the sense understood in the inverse scattering literature. Reality is ensured via *χ**(*γ**)=*χ*(*γ*).

It is important that the variables *R*_{k} have no *γ* dependence and are only functions of *ς* and *η*. The poles *γ*_{k} can be interpreted as the null trajectories of perturbations propagating on the background solution and can be thought of as gravitational solitons [8] although they are not fully analogous to solitons as they are known in other integrable PDEs [16–18]. The poles come in pairs either real

It is ensured that *g* is symmetric via

## 3. The connection *A*, the transition matrix and observables

Following [9], we consider the Lie algebra-valued connection or logarithmic derivative of *Ψ* *A*(*γ*) given by
*γ*=±1, we see how the definitions (2.5) arise. Now we consider the PDEs for *A*(*γ*). Differentiating the r.h.s. of (3.1) w.r.t. *η*, *ς* and using the Lax pair (2.7), we obtain
*ς* and *η* a fact first observed in [19]. These equations have appeared in ch. 6 of [20]. *A*_{±} satisfy the zero curvature condition
*A*(*γ*) satisfies

Also it can be shown from (2.5), *A*(±1) satisfies
*α*=*ς*−*η*. This relation appears in [9] as a reality condition for *A*(±1) since for stationary axisymmetric systems *α*=*iρ*. Now observing (3.2), (3.3) and (2.9), we see that the transformation

It may be further noted that the transformation
*α*=*t*, to *A*(*γ*) (from (3.1) and (3.9)), that is it satisfies
*η* playing the role of time. It should be mentioned that such a matrix was lacking for the equations of gravity in the presence of two commuting Killing vectors in the connection *A* formulation. Of course, the roles of *η* and *ς* can be reversed with the definition *χ* and hence *A*(*γ*) correspond essentially to the null trajectories of the solitons and are the light cones *w*_{k}−*β*=±*α* [8].

It is clear that for *η*= w_{2}, *γ*=−1 from (2.11) with w= w_{2} and *γ*=1 for *ς*= w_{2}. This holds in general for appropriate expression of *γ* for *η*, *ς* constant.

In figure 1, we see the path, on which the transition matrix carries *A*. We define *γ* in *A* along the path from *γ*=*e*^{iϕ} *ϕ*∈[−*π*,0] which corresponds, from (2.10) to the branch cut *w*∈[*β*−*α*,*β*+*α*]. (It should be stressed that at no point do we attain the singularity *α*=0 which corresponds to *η*=*ς* and from (2.11) to either *γ*=0 or *w*=*w*_{2}, *ξ*=*ϑ* and *η*=*ς* in the boundaries of the above path integrals and the basic feature of path integrals that

Also, in a similar way
*γ*(*ς*,*η*,*w*)=−*γ*(*η*,*ς*,*w*) for the case *α*=*t*. Since

In the case *α*=*ρ*,*β*=*t* which corresponds to cylindrically symmetric gravitational waves, i.e. *α* spacelike, we have
*τ*^{′} defined by
*A*(*γ*_{−}) satisfies the same differential equation as *A*(*γ*_{+}) (because (2.11) are the two solutions of (2.10)). Hence *τ*^{′} is an involution of the linear system (2.7)
*α*=*ρ* (i.e. *α* spacelike) as
*α* spacelike in a similar way to the timelike case above. (In this and the next chapter, all commutators are matrix commutators.)

The metric (2.1) corresponds to a wide variety of solutions including cosmological, cylindrically symmetric gravitational waves and stationary axisymmetric space–times (with appropriate transcription of the coordinates). The solitonic ansatz, among other methods, may be employed to obtain solutions (from a diagonal background usually but not exclusively) from a seed solution including Schwarzschild and Kerr among many others. Although the considerations here involve mainly space–times with one of the two significant coordinates timelike, the observables may be relevant for the axistationary case, e.g. inside a Black Hole horizon where one of the two significant coordinates becomes timelike.

A generic diagonal seed space–time for the solitonic technique [8] is
*α*=*ς*−*η*, *β*=*ς*+*η*), (∂/∂*ς*−∂/∂*η*) *d*-dimensional general relativity in the presence of *d*−2 Killing vectors. The considerations here are valid in that case also and the observables are available to use in string-theoretic considerations.

## 4. The curvature *B* and observables

We now consider
*B* with respect to *ς*, *η* turning the partial derivatives on the r.h.s. of (4.1) and using (3.2), (3.3), one obtains

Further, we want to calculate the limit

We see from (3.11) and (4.1) (in the case *α*=*t*, i.e. *α* timelike)
*τ* is again an involution of equations (4.2) and (4.3) so like in §3, we can define transition matrices for *B*. The process of obtaining observables this way must involve the derivatives of *A*_{±},*B*_{±} and commutators thereof, always modulo the integrable systems zero curvature condition, the field equations (2.2) and Bianchi identity.

So we have (for *α*=*t* timelike)

We may consider obtaining observables directly from

So we have obtained in (4.25) and a relation giving observables in terms of the curvature variables *B*_{±}(∓1) *B*(*γ*) in what is a form of generalized zero curvature condition. It is also clear that there exists a countable infinite hierarchy of hierarchies of constants of motion built from *A*(*γ*),*B*(*γ*),….

## 5. Field equations and interpretation of the connection observable

First consider (3.27). Using the fact that the inverse of a 2×2 matrix *M* can be written as
*α*=0, which for *α* timelike is the cosmological singularity and in the cylindrically symmetric case is the axis of symmetry. Further, we may traverse, in figure 1, the path in the opposite direction *A*(−1)*A*^{−1}(1))). This implies that
*E*(*ς*,*η*)=−*E*(*ς*,*η*) hence *E*(*ς*,*η*)=0 and so *A*(1)*A*(−1)) is a constant of motion. Of course upon quantization if *A*,*B*∈ su(2) or so(2), we have (Tr (*AB*^{−1})=Tr (*AB*)).

The Einstein–Hilbert action is [25]
*K*_{μν} is the extrinsic curvature, *s* is the signature of the metric and *n*^{μ}=(1/*N*)(∂/∂*t*) is the normal of the *D*-surface for *D*=3, *n*^{μ}=(1/*N*)(∂/∂*z*) for *D*=2 (we view the two hypersurface orthogonal Killing vectors metric (2.1) as 1−1−2 metric and apply (5.7) twice), we obtain
*g*_{ab}(*z*,*t*) give
*z*,*t*). The trace of (5.9) gives
*α*=*a*(*ς*)−*b*(*η*) is a timelike solution of (2.6), then *β*=*a*(*ς*)+*b*(*η*) is an a second independent spacelike solution of (2.6), (5.10). The Euler–Lagrange equations give the equation for *σ*
*α*_{,z}=*a*^{′}−*b*^{′}=*β*_{,t}, *β* spacelike. Since *N* is the lapse that is it measures proper time and also proper distance in the *z*-direction, we interpret (5.11) as a dispersion relation that is an equation of the form *E*^{2}−*p*^{2}=*m*^{2}. That means that our connection observable is interpreted as a constant that corresponds to mass. This makes equation (5.11) a typical Klein–Gordon equation and makes available all the tools of the corresponding field theory for quantization. With the definite choice of variables *α*=*t*, *β*=*z*, equation (5.11) is solved by Belinski & Verdaguer [8] (transcribed to the variables used here)
*C*(*z*,*t*). To stress the interpretation of (5.11), with variables and conjugate momenta to be (*α*,*p*_{α}), (*β*,*p*_{β}), (*g*,*p*_{g}), where
*C*_{,t}=−1/*α*. So the energy of the system is *p*_{t} and is positive definite. Further, it is clear that upon quantization *σ*_{,t}, *σ*_{,z} are *i*(∂/∂*t*), *i*(∂/∂*z*) which clearly commute by construction here. The other translation generators are trivial. Rotation generators are

## 6. Conclusion

We have obtained transition matrices (3.13) in terms of connection variables satisfying equations similar to the ones satisfied by the transition matrices for other integrable PDEs. Using these, hierarchies of observables in terms of connection and curvature variables have been obtained. The first in the hierarchy of connection observables has been shown to correspond to the mass of a Klein–Gordon equation that governs the lapse variable *N*.

## Competing interests

I declare I have no competing interests.

## Funding

I received no funding for this study.

- Received May 28, 2015.
- Accepted October 12, 2015.

- © 2015 The Author(s)