Reduced Gravity as a Sigma Model

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A non-null Killing field converts part of the vacuum Einstein equation into a nonlinear sigma model on the space of its norm and twist. The target-space isometries are larger than the symmetries visible in the unreduced four-dimensional action, and acting with them can generate new exact metrics. The construction is local on a region where the chosen Killing field has fixed causal character and nonzero norm.

Required background. Killing Fields and Isometry Groups supplies the Killing equation, orbit geometry, and conserved generators.

Helpful background. Einstein’s Equation supplies the vacuum and electrovac systems; The Riemann Curvature Tensor fixes curvature conventions; Exact Solutions of Einstein’s Equations supplies benchmark seeds; Formulations of General Relativity supplies dimensional reduction and dualization; PDEs and Green Functions supplies elliptic and hyperbolic boundary-value distinctions.

Scope. The vacuum reduction and its target-space symmetry below assume four-dimensional $R_{ab}=0$ ; the electrovac extension assumes source-free Einstein–Maxwell theory, also with $\Lambda=0$ . Twist and Ernst Potentials supplies the earlier stationary-axisymmetric normalization used here.

Timelike stationary reduction

Let $\xi=\partial_t$ be timelike and define $f=-\xi^a\xi_a>0$ . On a simply connected stationary patch write

ds^2=-f(dt+A_i dx^i)^2+f^{-1}h_{ij}dx^idx^j.

Here $h_{ij}$ is a Riemannian metric on the three-dimensional orbit space. Fix its orientation so that the vacuum twist potential $\chi$ satisfies

d\chi=f^2 *_h dA.

Reversing the orbit-space orientation reverses $\chi$ and every formula odd in twist. Vacuum Einstein equations imply that the twist one-form is closed, so $\chi$ exists locally; nontrivial first cohomology can obstruct a global single-valued potential.

After discarding the Killing-coordinate volume and a boundary term, the reduced action is

S_3\propto\int d^3x\sqrt h \left[ R(h)-\frac{(Df)^2+(D\chi)^2}{2f^2} \right].

Had the four-dimensional action contained a nonzero cosmological constant, the same reduction would add the potential term $-2\Lambda/f$ inside the brackets. That term is not invariant under the full hyperbolic-plane isometry group displayed below. The $\Lambda=0$ boundary is therefore part of the generating theorem, not a convention that can be restored only in the final residual.

Thus $(f,\chi)$ define a harmonic map into the hyperbolic plane with target metric

d\ell^2=\frac{df^2+d\chi^2}{2f^2}.

With the Ernst potential $\mathcal E=f+i\chi$ , its equation is

(\operatorname{Re}\mathcal E)D^2\mathcal E =D_i\mathcal E D^i\mathcal E,

while the base equation is

R_{ij}(h)=\frac{1}{2f^2} \left(D_ifD_jf+D_i\chi D_j\chi\right).

Substituting these two equations back into the four-dimensional Ricci tensor is the independent check that the reduction has retained every vacuum equation.

Target-space symmetry

Set $\tau=\chi+if$ , so $\operatorname{Im}\tau>0$ . The target metric is invariant under fractional-linear transformations

\tau\longmapsto \frac{a\tau+b}{c\tau+d}, \qquad a,b,c,\ d\in\mathbb R, \qquad ad-bc=1,

modulo the common sign of the matrix. Writing the same action directly on the right-half-plane variable $\mathcal E=f+i\chi$ inserts corresponding factors of $i$ ; using a real-coefficient Möbius formula on $\mathcal E$ would be a convention error. Twist shifts and constant rescalings largely alter potential or Killing-coordinate normalization; the Ehlers generator mixes $f$ and $\chi$ nontrivially. The group can be described as $SL(2,\mathbb R)$ acting on the coset $SL(2,\mathbb R)/SO(2)$ .

This symmetry was not manifest in four dimensions because two special operations produced it: restriction to metrics admitting $\xi$ , and dualization of the three-dimensional one-form $A$ into the scalar $\chi$ . The transformation is consequently a symmetry of the reduced field equations, not a local off-shell symmetry acting on arbitrary four-metrics. Contopoulos et al. show the norm–twist construction and an explicit Ehlers map in the opposite-signature convention in Contopoulos et al. 2016, §§1 and 3, arXiv v3 pp. 1–2 and 7–11, Eqs. (4)–(12) and (44)–(75), PDF. Their stationary variable $\sigma$ is negative; reversing the metric signature gives our positive norm variable $f=-\sigma=-\xi^2$ .

Stationary Einstein–Maxwell theory adds electric and magnetic potentials. With one common normalization,

\Phi=v+iu, \qquad \mathcal E=f-|\Phi|^2+i\chi.

The scalar target enlarges to an $SU(2,1)$ coset; for timelike reduction its isotropy subgroup and target signature are commonly written $S(U(1,1)\times U(1))$ . Harrison transformations lie among its isometries and can mix gravitational and electromagnetic potentials. Factors of $2$ in $\Phi$ and $|\Phi|^2$ vary across sources, so a finite transformation is meaningful only together with this definition and the Einstein–Maxwell action normalization.

Why spacelike reduction is different

For a spacelike Killing field $\eta=\partial_z$ , use $f_s=\eta^2>0$ and write

ds^2=f_s(dz+A)^2+f_s^{-1}h_{ij}dx^i dx^j.

Now $h$ is Lorentzian. After dualizing $A$ to a real twist scalar $\chi_s$ , the vacuum action has the same positive target metric,

S_3\propto\int d^3x\sqrt{-h} \left[R(h)-\frac{(Df_s)^2+(D\chi_s)^2}{2f_s^2}\right].

The target is still $SL(2,\mathbb R)/SO(2)$ , not $SL(2,\mathbb R)/SO(1,1)$ . What changes is the signature of the base and hence the differential operator: the scalar fields form a wave map from a Lorentzian three-space into the Riemannian hyperbolic plane. This distinction is explicit in Andersson, Gudapati, and Szeftel 2017, §1 and Remark 1.2, arXiv v4 pp. 1–3, PDF.

The Hodge-star square does change with the base signature, so the dualization must be done consistently at the action or field-equation level. Substituting a duality relation directly into the Maxwell-like kinetic term without its first-order constraint term gives the wrong scalar sign. At a rotation axis $f_s=0$ , the reduced variables are singular even when the reconstructed four-metric is smooth; reduction applies on off-axis patches, and axis regularity is tested after lifting.

Exercise

Show that the inversion $\mathcal E' = 1/\mathcal E$ maps a solution of the vacuum Ernst equation to another solution wherever $\mathcal E\neq0$ .

Solution

Write $\mathcal E'=\mathcal E^{-1}$ . Then $D_i\mathcal E'=-\mathcal E^{-2}D_i\mathcal E$ and $D^2\mathcal E'=-\mathcal E^{-2}D^2\mathcal E+2\mathcal E^{-3}(D\mathcal E)^2$ . Substituting the seed equation and using $\operatorname{Re}(1/\mathcal E)=\operatorname{Re}\mathcal E/|\mathcal E|^2$ verifies the transformed equation after multiplying by the nonzero common denominator. The map can nevertheless fail where $\mathcal E=0$ and need not preserve asymptotic normalization.

References

Andersson, Lars, Nishanth Gudapati, and Jérémie Szeftel. “Global Regularity for the 2+1 Dimensional Equivariant Einstein-Wave Map System.” Annals of PDE 3 (2017): 13. DOI; Open PDF.
Contopoulos, I. G., F. P. Esposito, K. Kleidis, D. B. Papadopoulos, and L. Witten. “Generating Solutions to the Einstein Field Equations.” International Journal of Modern Physics D 25, no. 2 (2016): 1650022. DOI; Open PDF.