The Hypersurface-Deformation Algebra

The canonical constraints do more than vanish on admissible initial data. Their Poisson brackets reproduce the noncommuting ways in which a spacelike hypersurface can be deformed normally and tangentially in spacetime. The result is the hypersurface-deformation algebra, whose metric-dependent structure functions distinguish it from a Lie algebra with fixed structure constants.

Required background. Presymplectic Potentials and Currents supplies the Poisson structure derived from the action, while The Gibbons-Hawking-York Boundary Term explains why surface terms must be controlled before a generator is differentiable.

Helpful background. The Einstein Equation supplies the covariant dynamics, Hypersurface Normals and Tangential Projectors supplies the split into normal and tangential directions, and Formulations of General Relativity places the canonical brackets in their broader geometric setting.

Smeared normal and tangential generators

On a closed slice, or for smearings of compact support, define

$$H[N]=\int_\Sigma d^3x\,N\mathcal C_\perp, \qquad D[\vec N]=\int_\Sigma d^3x\,N^i\mathcal C_i.$$

Here $N$ is a scalar lapse smearing and $N^i$ is a vector shift smearing. For pure gravity,

$$\mathcal C_i=-2D_j\pi^j{}_i.$$

An integration by parts gives

$$D[\vec N] =\int_\Sigma d^3x\,\pi^{ij}\mathcal L_{\vec N}\gamma_{ij}.$$

The fundamental bracket therefore implies

$$\{\gamma_{ij},D[\vec N]\}=\mathcal L_{\vec N}\gamma_{ij}, \qquad \{\pi^{ij},D[\vec N]\}=\mathcal L_{\vec N}\pi^{ij},$$

where the second Lie derivative is that of a weight-one tensor density. Thus $D[\vec N]$ generates a change of spatial coordinates. Since $\mathcal C_\perp$ is a scalar density of weight one, its transformation law immediately determines the tangential–normal bracket.

Computing the three brackets

For field-independent smearings, the two brackets involving a tangential generator are

$$\{D[\vec N],D[\vec M]\} =D[[\vec N,\vec M]],$$ $$\{D[\vec N],H[M]\} =H[\mathcal L_{\vec N}M],$$

with $[\vec N,\vec M]^i=N^j\partial_jM^i-M^j\partial_jN^i$. These follow by substituting the Lie derivatives above and integrating a spatial divergence.

The normal–normal bracket contains the distinctive information. For the vacuum scalar constraint,

$$\mathcal C_\perp =\frac{16\pi G}{\sqrt\gamma} \left(\pi_{ij}\pi^{ij}-\frac12\pi^2\right) -\frac{\sqrt\gamma}{16\pi G} \left({}^{(3)}R-2\Lambda\right),$$

the momentum derivative is

$$\frac{\delta H[N]}{\delta\pi^{ij}} =\frac{32\pi G N}{\sqrt\gamma} \left(\pi_{ij}-\frac12\pi\gamma_{ij}\right).$$

In the metric derivative, only the differentiated-lapse part of the curvature variation survives antisymmetrization in $N$ and $M$:

$$\left.\frac{\delta H[N]}{\delta\gamma_{ij}}\right|_{D N} =\frac{\sqrt\gamma}{16\pi G} \left(\gamma^{ij}D^2N-D^iD^jN\right).$$

All ultralocal terms are symmetric under exchanging the two smearings. Insert these two functional derivatives into the Poisson bracket. Their contraction first simplifies to

$$\{H[N],H[M]\} =2\int_\Sigma d^3x\,\pi^{ij} \left(ND_iD_jM-MD_iD_jN\right).$$

Let $v_i=ND_iM-MD_iN$. In $\pi^{ij}D_i v_j$, the two products of first derivatives cancel because $\pi^{ij}$ is symmetric. Integrating the remaining divergence by parts gives $-2\int_\Sigma d^3x\,v_iD_j\pi^{ij}$. This is precisely the momentum constraint smeared with the raised vector:

$$\{H[N],H[M]\} =D[\vec v], \qquad v^i=\gamma^{ij} \left(N\partial_jM-M\partial_jN\right).$$

Minimally coupled generally covariant matter contributes additional terms to the individual constraints, but the total constraints obey the same algebra. The geometric derivation from path-independent hypersurface evolution is given in Teitelboim 1973, §§2–4, pp. 546–553, PDF; the canonical gravitational generators themselves are constructed in Arnowitt, Deser, and Misner 1962/2004, §§3–4, PDF.

Why the coefficients depend on the metric

Two tangential deformations remain tangential, and commuting a tangential deformation past a normal one changes the lapse as a scalar. Two position- dependent normal deformations, however, end on slices whose points are relabeled relative to one another. Their mismatch is tangential. Raising the gradient of a lapse requires the inverse spatial metric, which is why $\gamma^{ij}$ appears in $v^i$.

Because $\gamma^{ij}$ is itself a canonical variable, these are structure functions rather than constants. The algebra is therefore more precisely a Lie algebroid of hypersurface deformations. It nevertheless closes on the constraints: no new bulk constraint is generated. This closure is the canonical expression of spacetime embeddability and ensures that different sequences of infinitesimal deformations describe compatible spacetime evolution.

The displayed equations assume that integrations by parts have no remainder. On a finite or asymptotic boundary, one must improve $H[N]$ and $D[\vec N]$ by surface terms and restrict the allowed smearings. Brackets of the improved generators can then acquire boundary charges or, in some settings, central terms. Those are properties of the specified boundary conditions, not failures of the bulk constraint algebra.

Common pitfalls

Calling this a Lie algebra with structure constants. The normal–normal bracket contains $\gamma^{ij}$, so its coefficient varies over phase space.

Dropping boundary terms without specifying support. The compact-support or closed-slice assumption is part of the derivation. Nonvanishing asymptotic smearings may generate physical symmetries rather than gauge redundancies.

Exercise: compose two normal deformations

On a locally Cartesian patch with $\gamma_{ij}=\delta_{ij}$, take lapse smearings $N=f(x)$ and $M=g(x)$ that depend on one coordinate. Determine the shift in $\{H[N],H[M]\}$, and evaluate it for $f=x$ and $g=x^2$ within the patch.

Solution

Only the $x$ component is nonzero:

$$v^x=f\,g'-g\,f', \qquad v^y=v^z=0.$$

For $f=x$ and $g=x^2$,

$$v^x=x(2x)-x^2(1)=x^2.$$

Hence

$$\{H[x],H[x^2]\}=D[x^2\partial_x]$$

on that patch. If globally compact support is required, multiply both profiles by a common bump function that equals one on the region being examined. The calculation shows concretely that two nonuniform normal moves generally differ by a tangential relabeling; constant lapse profiles instead give $v^i=0$.

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.
• Teitelboim, Claudio. “How Commutators of Constraints Reflect the Spacetime Structure.” Annals of Physics 79, no. 2 (1973): 542–557. DOI.