Gauge Orbits and Canonical Observables

Canonical general relativity represents one physical spacetime by many points of phase space: changing the embedding or coordinates of a slice moves along a gauge orbit. A canonical observable must therefore be constant along the directions generated by proper gauge transformations. This requirement is precise locally, but constructing a complete set of such observables is a global dynamical problem rather than a simple matter of removing coordinates.

Required background. Presymplectic Potentials and Currents supplies the degenerate directions that become gauge transformations, while The Gibbons-Hawking-York Boundary Term supplies the boundary conditions needed to distinguish gauge generators from physical boundary symmetries.

Helpful background. The Einstein Equation supplies the covariant solution space, Hypersurface Normals and Tangential Projectors supplies changes of slicing, and Formulations of General Relativity relates canonical gauge freedom to other formulations.

Gauge flows on the constraint surface

Let $\Gamma$ be the phase space of $(\gamma_{ij},\pi^{ij})$ and let $\bar\Gamma$ be the surface

$$\mathcal C_\perp=0, \qquad \mathcal C_i=0.$$

For lapse and shift smearings that vanish in the required way at the boundary,

$$G[N,\vec N]=H[N]+D[\vec N]$$

generates a tangent direction to $\bar\Gamma$. The integral curves of all such directions form a gauge orbit. If lapse and shift are retained as canonical variables, the full spacetime-diffeomorphism generator also contains their primary constraints $p_\alpha$ and $p_i$ with correlated coefficients. For a functional depending only on $(\gamma,\pi)$, the secondary-constraint conditions capture its invariance on the gravitational phase space.

The qualifier “proper” is essential. A smearing that approaches a nonzero time translation, rotation, or boost at an asymptotic boundary may carry a charge. Its generator moves between physically distinct boundary states and is not quotiented out as gauge.

Dirac observables and reduced phase space

A Dirac observable $O$ satisfies, weakly,

$$\{O,H[N]\}\approx0, \qquad \{O,D[\vec N]\}\approx0$$

for every proper gauge smearing. Weak equality means equality after imposing the constraints. Equivalently, $O$ descends to a function on the reduced phase space

$$\Gamma_{\mathrm{red}}=\bar\Gamma/\text{gauge orbits}.$$

A coordinate component such as $\gamma_{11}(x)$ is not an observable: a spatial diffeomorphism changes both its components and the coordinate label. Even a spacetime scalar evaluated at a coordinate point is generally not a Dirac observable, because the point itself has no invariant identification. Conversely, integrating a scalar density can remove spatial-coordinate dependence without ensuring invariance under normal deformations.

The constraint-generated gauge interpretation and its canonical origin are developed in Arnowitt, Deser, and Misner 1962/2004, §§3–4, PDF. The observable condition does not imply an absence of physical change: it removes unphysical coordinate time, while relational dependence can remain.

Gauge fixing and relational observables

Locally one may impose four gauge conditions $\chi_A=0$, one for each secondary constraint $C_A$. A valid local gauge requires the operator

$$\Delta_{AB}=\{\chi_A,C_B\}$$

to be invertible on the intersection of the gauge-fixing and constraint surfaces. Preservation of $\chi_A=0$ then fixes lapse and shift, and the Dirac bracket

$$\{F,G\}_{D} =\{F,G\} -\{F,\Phi_I\}(C^{-1})^{IJ}\{\Phi_J,G\}$$

implements the resulting second-class set $\Phi_I=(C_A,\chi_A)$ before the constraints are imposed strongly. Here $C_{IJ}=\{\Phi_I,\Phi_J\}$ is the full constraint matrix, distinct from the individual constraints $C_A$. In field theory these matrices are integral kernels: repeated constraint indices include integration over spatial points. An inverse therefore requires a domain and boundary conditions, not just a nonzero determinant of a finite matrix.

Relational constructions avoid choosing coordinate labels as physical locations. Select four phase-space functions $T^A$ as physical clocks and rods, and another function $f$. The complete observable $F_{f,T}(\tau)$ is the value of $f$ at the point on the gauge orbit where $T^A=\tau^A$. Although $T^A$ and $f$ are individually gauge dependent, their correlation is invariant. This construction and its weakly Abelianized constraint flow are detailed in Dittrich 2006, §2, pp. 3–6, PDF; the conceptual distinction between partial and complete observables is explained in Rovelli 2002, §§II–III, pp. 2–4, PDF.

For example, a parametrized free particle has the constraint $C=p_t+p_x^2/(2m)$. The coordinate $x$ is not invariant, since $\{x,C\}=p_x/m$, but its position when the clock reads $t=\tau$ is

$$X(\tau)=x+\frac{p_x}{m}(\tau-t).$$

Direct calculation gives $\{X(\tau),C\}=p_x/m-p_x/m=0$. Yet $\partial_\tau X=p_x/m$ describes ordinary motion. General relativity uses four clock-and-rod fields and infinitely many constraints, so its construction is harder; the distinction between gauge evolution and changing clock readings is already visible in this elementary model.

The parameter $\tau^A$ labels a physical clock reading. A family $F_{f,T}(\tau)$ can therefore encode change even though every member commutes weakly with the constraints. “Frozen” with respect to coordinate gauge flow does not mean devoid of relational dynamics.

Local validity and global obstructions

Gauge fixing is rarely global. The clock matrix $\Delta_{AB}$ can lose rank, a gauge orbit can cross the same gauge-fixing surface more than once, and solutions with Killing symmetries have stabilizers where the orbit dimension drops. A proposed clock may turn around or cease to distinguish points, and different topological sectors need not share one coordinate chart on reduced phase space. These failures are analogous to Gribov ambiguities.

Accordingly, a relational observable is normally valid only in a stated region of phase space. Establishing global existence, separation of gauge orbits, and smoothness requires additional analysis. At boundaries, one must also decide which transformations are proper gauge and which are physical symmetries before applying the Dirac criterion.

Exercise: is spatial volume a Dirac observable?

On a closed slice define $V=\int_\Sigma d^3x\sqrt\gamma$. Compute its brackets with $D[\vec N]$ and $H[N]$, and decide whether it is a Dirac observable.

Solution

Because $\sqrt\gamma$ is a weight-one density,

$$\{V,D[\vec N]\} =\int_\Sigma d^3x\,\mathcal L_{\vec N}\sqrt\gamma =\int_\Sigma d^3x\,\partial_i(N^i\sqrt\gamma)=0.$$

Thus $V$ is invariant under spatial diffeomorphisms on a closed slice. For a normal deformation,

$$\{\sqrt\gamma,H[N]\} =-N K\sqrt\gamma=-8\pi G N\pi,$$

so

$$\{V,H[N]\}=-\int_\Sigma d^3x\,N K\sqrt\gamma.$$

This is not weakly zero for generic constrained data. Spatial volume is therefore invariant under the vector constraint but is not, by itself, a Dirac observable. It can become part of a complete observable, for example the volume when a chosen physical clock has a specified value.

References

• Arnowitt, Richard, Stanley Deser, and Charles W. Misner. “The Dynamics of General Relativity.” In Gravitation: An Introduction to Current Research, edited by Louis Witten, 227–265. Wiley, 1962; reprinted as arXiv:gr-qc/0405109 (2004). arXiv.
• Dittrich, Bianca. “Partial and Complete Observables for Canonical General Relativity.” Classical and Quantum Gravity 23, no. 22 (2006): 6155–6184. DOI. Open PDF.
• Rovelli, Carlo. “Partial Observables.” Physical Review D 65, no. 12 (2002): 124013. DOI. Open PDF.