The Hypersurface-Deformation Algebra
This appendix proves Proposition 30.43 of Constrained Hamiltonian Systems: the Dirac–Bergmann Formalism: the three brackets Equations (30.89), (30.90) and (30.91) of the smeared constraints of Theorem 30.42. With them the eight constraints of general relativity are shown to be first class in the sense of Definition 30.12, so that Equation (30.21) applies and the count of two propagating degrees of freedom per point of space is established; and the last of the three exhibits, on the right-hand side, the inverse spatial metric \(h^{ij}\) — a function on phase space where a Lie algebra would carry a constant.
Everything is computed with smeared constraints. That is not a matter of taste. The unsmeared densities have brackets proportional to \(\delta^{3}(\vect{x}-\vect{y})\) and to its first and second derivatives, and Equation (30.91) in unsmeared form is an identity between distributions in which the two sides must be compared after integrating twice by parts against test functions anyway. Smearing does that integration once and for all, at the start; every step below is then an ordinary integration by parts on smooth compactly supported data.
Two things are assumed rather than proved and are named here. The first is the constraint densities themselves, Equations (30.84) and (30.85), which Theorem 30.42 derives; nothing below depends on the overall sign discussed in Remark A52.11, because the brackets are computed from the densities as the chapter displays them. The second is that all fields fall off fast enough for the surface terms of the integrations by parts to vanish — on a region with a boundary they do not, and what they contribute is exactly the ADM charges of Remark 30.45.
Conventions: weights, smearing functions, and the bracket
The canonical pair is \(\left(h_{ij},\pi^{ij}\right)\) of Equation (30.82), with the field bracket Equation (30.47) read as
functional derivatives with respect to \(h_{ij}\) and \(\pi^{ij}\) being taken with both regarded as symmetric. The weights are: \(h_{ij}\) is a tensor of weight \(0\); \(\pi^{ij}\) is a tensor density of weight \(1\), as Equation (30.82) shows through its factor \(\sqrt{h}\); \(\Ham_{\perp}\) of Equation (30.84) is a scalar density of weight \(1\), its kinetic part carrying \(\pi\pi/\sqrt{h}\) and its curvature part \(\sqrt{h}\,{}^{(3)}\!R\); and \(\Ham_{i}\) of Equation (30.85) is a covector density of weight \(1\). The smearing functions \(f\), \(g\) are scalars and \(\xi^{i}\), \(\zeta^{i}\) vectors, all of weight \(0\), all smooth, all of compact support, and — this is used constantly and is easy to forget — all independent of the canonical variables, so that they pass through every functional derivative untouched. Finally
both of which are ordinary numbers because a weight-one density integrates without a metric factor. Rests on Notation 30.28, Equation (30.82) and Theorem 30.42.
Getting a weight wrong is the commonest way this computation fails, so the one fact used about them is stated once and checked: for a weight-one vector density \(V^{i}\),
for \(V\) of compact support. This is immediate from the covariant derivative of a density, \(D_{i}V^{i}=\pp_{i}V^{i} +\Gamma^{i}{}_{ik}V^{k}-\Gamma^{k}{}_{ki}V^{i}\), the last term being the weight term; the two cancel. Every integration by parts below is an application of Equation (A53.3). For a weight-zero vector the corresponding statement carries a \(\sqrt{h}\), \(\sqrt{h}D_{i}V^{i}=\pp_{i}\left(\sqrt{h}V^{i}\right)\), and that form is used where the integrand carries an explicit \(\sqrt{h}\).
The sign convention for the commutator of vector fields is
which is what makes the right-hand side of Equation (30.89) carry a plus sign rather than a minus.
The momentum constraint generates the spatial drag
For every smearing vector \(\xi\),
and consequently
the second Lie derivative being that of a weight-one contravariant density, \(\mathcal{L}_{\xi}\pi^{ij}=\xi^{k}\pp_{k}\pi^{ij} -\pi^{kj}\pp_{k}\xi^{i}-\pi^{ik}\pp_{k}\xi^{j} +\pi^{ij}\pp_{k}\xi^{k}\). Rests on Equation (30.85), Equation (A53.1) and Notation A53.1.
Derives Lemma A53.2. Equation (A53.5). By Equation (30.85), \(H_{\parallel}[\xi]=-2\int\dd^{3}x\;\xi^{i}D_{j}\pi^{j}{}_{i} =-2\int\dd^{3}x\;\xi_{k}D_{j}\pi^{jk}\), the index having been moved through \(D\) because \(D h=0\). The quantity \(\xi_{k}\pi^{jk}\) is a weight-one vector density, so Equation (A53.3) gives \(\int D_{j}\left(\xi_{k}\pi^{jk}\right)=0\), whence
the symmetrization being free because \(\pi^{jk}\) is symmetric. That \(D_{j}\xi_{k}+D_{k}\xi_{j}=\mathcal{L}_{\xi}h_{jk}\) is Equation (17.275).
First of Equation (A53.6). Take \(F=h_{ij}(\vect{x})\) in Equation (A53.1): only the first term survives, and \(\pb{h_{ij}(\vect{x})}{G} =\delta G/\delta\pi^{ij}(\vect{x})\). Applied to Equation (A53.5), in which \(\pi\) appears linearly and undifferentiated, this is \(\mathcal{L}_{\xi}h_{ij}\).
Second. Likewise \(\pb{\pi^{ij}(\vect{x})}{G} =-\delta G/\delta h_{ij}(\vect{x})\). Write out Equation (A53.5) in partial derivatives, \(\mathcal{L}_{\xi}h_{ab}=\xi^{k}\pp_{k}h_{ab}+h_{kb}\pp_{a}\xi^{k} +h_{ak}\pp_{b}\xi^{k}\), and vary. The first term contributes, after one integration by parts, \(-\pp_{k}\left(\xi^{k}\pi^{ij}\right)\); the second contributes \(\pi^{kj}\pp_{k}\xi^{i}\) and the third \(\pi^{ik}\pp_{k}\xi^{j}\). Hence
and the minus of the bracket cancels it. Note where the weight shows itself: the term \(\pi^{ij}\pp_{k}\xi^{k}\), which distinguishes the Lie derivative of a density from that of a tensor, comes from the integration by parts in the first term and from nowhere else. Had \(\pi^{ij}\) been treated as a weight-zero tensor, the density term would be spurious and Equation (30.89) would fail.
∎The first two brackets
Both follow from Lemma A53.2 and one observation, which is worth isolating because it does the work twice.
Let \(\Phi\) be a functional of the canonical variables and of one smearing field \(\sigma\) (a scalar \(f\) or a vector \(\xi\)),
with \(\mathcal{F}\) a scalar density of weight one built covariantly from its arguments. Then
Rests on Lemma A53.2 and Equation (A53.1).
Derives Lemma A53.3. By Equation (A53.1) and Equation (A53.6),
the change of \(\Phi\) when the canonical fields alone are dragged by \(\zeta\). But \(\Phi\) is invariant when everything is dragged: if all of \(\sigma\), \(h\), \(\pi\) are transported, \(\mathcal{F}\) changes by \(\mathcal{L}_{\zeta}\mathcal{F}=\pp_{k}\left(\zeta^{k}\mathcal{F}\right)\), because \(\mathcal{F}\) is a weight-one scalar density, and that integrates to zero. Hence dragging the fields alone gives minus the result of dragging \(\sigma\) alone, which is Equation (A53.10).
∎Equations (30.89) and (30.90) hold:
Rests on Lemma A53.3, Equation (30.84) and Equation (A53.4).
Derives Proposition A53.4. Both integrands satisfy the hypothesis of Lemma A53.3: \(\xi^{i}\Ham_{i}\) and \(f\Ham_{\perp}\) are weight-one scalar densities built covariantly from the smearing field and the canonical pair, by Notation A53.1.
For the first, apply Equation (A53.10) with \(\Phi=H_{\parallel}\), \(\sigma=\xi\): \(\pb{H_{\parallel}[\xi]}{H_{\parallel}[\zeta]} =-H_{\parallel}\!\left[\mathcal{L}_{\zeta}\xi\right] =-H_{\parallel}\!\left[\comm{\zeta}{\xi}\right] =H_{\parallel}\!\left[\comm{\xi}{\zeta}\right]\) by Equation (A53.4). Nothing about the detailed form of \(\Ham_{i}\) was used beyond Lemma A53.2; the identity is \(\comm{\mathcal{L}_{\xi}}{\mathcal{L}_{\zeta}} =\mathcal{L}_{\comm{\xi}{\zeta}}\) in disguise.
For the second, apply it with \(\Phi=H_{\perp}\), \(\sigma=f\) a scalar, so \(\mathcal{L}_{\xi}f=\xi^{i}\pp_{i}f\): \(\pb{H_{\perp}[f]}{H_{\parallel}[\xi]} =-H_{\perp}\!\left[\xi^{i}\pp_{i}f\right]\), and the antisymmetry of the bracket gives the stated form. The content is that \(\Ham_{\perp}\) is a scalar density of weight one — that dragging \(f\Ham_{\perp}\) produces \(\mathcal{L}_{\xi}\left(f\Ham_{\perp}\right)\), whose total-divergence part integrates away and whose remainder is \(\left(\xi^{i}\pp_{i}f\right)\Ham_{\perp}\).
∎The variation of the intrinsic curvature
The third bracket needs the piece a reader cannot supply from the chapter: how \(\int\sqrt{h}\,{}^{(3)}\!R\) responds to a variation of \(h_{ij}\).
For any smooth \(f\) of compact support, independent of \(h_{ij}\),
with \({}^{(3)}\!G^{ij}={}^{(3)}\!R^{ij} -\tfrac{1}{2}h^{ij}\,{}^{(3)}\!R\) the three-dimensional Einstein tensor and \(D^{2}=h^{ij}D_{i}D_{j}\). Rests on Definition 17.153, Equation (17.299) and Theorem 17.152.
Derives Lemma A53.5. Three pieces. First, \(\delta\sqrt{h} =\tfrac{1}{2}\sqrt{h}\,h^{ij}\delta h_{ij}\), from \(\delta\ln\det h=\tr\left(h^{-1}\delta h\right)\). Second, \(\delta h^{ij}=-h^{ik}h^{jl}\delta h_{kl}\), from varying \(h^{ik}h_{kj}=\delta^{i}_{j}\); so \(\delta\left({}^{(3)}\!R\right) ={}^{(3)}\!R_{ij}\delta h^{ij}+h^{ij}\delta\,{}^{(3)}\!R_{ij} =-{}^{(3)}\!R^{ij}\delta h_{ij}+h^{ij}\delta\,{}^{(3)}\!R_{ij}\).
Third, the term \(h^{ij}\delta\,{}^{(3)}\!R_{ij}\). The variation of a Christoffel symbol is a tensor, and from Equation (17.299) with vanishing torsion,
which one checks by evaluating both sides in normal coordinates at a point, where \(\Gamma=0\) and \(D=\pp\). Then Equation (17.306) gives \(\delta\,{}^{(3)}\!R_{ij} =D_{k}\delta\Gamma^{k}{}_{ij}-D_{i}\delta\Gamma^{k}{}_{kj}\), so \(h^{ij}\delta\,{}^{(3)}\!R_{ij}=D_{k}v^{k}\) with \(v^{k}=h^{ij}\delta\Gamma^{k}{}_{ij}-h^{kj}\delta\Gamma^{i}{}_{ij}\). Evaluate the two pieces from Equation (A53.14). In the second, the first and third terms of the bracket cancel on contraction, leaving \(\delta\Gamma^{i}{}_{ij} =\tfrac{1}{2}D_{j}\left(h^{il}\delta h_{il}\right)\). In the first, the two symmetric terms combine, leaving \(h^{ij}\delta\Gamma^{k}{}_{ij}=h^{kl}D^{j}\delta h_{lj} -\tfrac{1}{2}D^{k}\left(h^{ij}\delta h_{ij}\right)\). Hence
Collecting,
the first term assembling \(\tfrac{1}{2}h^{ij}\,{}^{(3)}\!R -{}^{(3)}\!R^{ij}\). Multiply by \(f\), integrate, and move the two derivatives of the second term onto \(f\) using \(\sqrt{h}\,D_{k}V^{k}=\pp_{k}\left(\sqrt{h}V^{k}\right)\) twice. The result is Equation (A53.13); note that \(D^{i}D^{j}f\) is automatically symmetric, \(f\) being a scalar.
∎The third bracket
Equation (30.91) holds:
Rests on Lemma A53.5, Equation (30.84) and Equation (30.85).
Derives Theorem A53.6. Step 1: the two functional derivatives. From Equation (30.84), in which \(\pi\) occurs only algebraically,
since \(\delta\left(\pi_{kl}\pi^{kl}\right)/\delta\pi^{ij}=2\pi_{ij}\) and \(\delta\left(\pi^{2}\right)/\delta\pi^{ij}=2\pi h_{ij}\). From Lemma A53.5 together with the variation of the kinetic part — which involves \(h_{ij}\) algebraically only, through \(\pi_{ij}\), \(\pi\) and \(\sqrt{h}\), and which therefore produces no derivative of \(f\) —
where \(A^{ij}\) is a local expression in \(h\) and \(\pi\) whose exact form is never needed: it collects the kinetic variation and the term \(+\sqrt{h}\,{}^{(3)}\!G^{ij}/2\kappa\), and the only property used is that it multiplies \(f\) undifferentiated.
Step 2: what cancels, and why. Insert Equations (A53.18) and (A53.19) into Equation (A53.1):
because the terms \(fgA^{ij}B_{ij}\) are symmetric under \(f\leftrightarrow g\) and cancel between the two halves of the bracket. This is the load-bearing observation of the whole computation and it deserves to be stated in words: everything ultralocal in the smearing functions drops out. The Einstein-tensor part of Equation (A53.13) goes with it, and so does the entire kinetic variation; what survives is only the group \(\Delta^{ij}\), which is precisely the part carrying two spatial derivatives. That is why the answer will carry a metric where a structure constant would sit: two derivatives must have their indices raised, and \(h^{ij}\) is what raises them.
Step 3: contract. With \(\sqrt{h}B_{ij}/2\kappa=2\left(\pi_{ij} -\tfrac{1}{2}\pi h_{ij}\right)\),
the three trace terms cancelling exactly, the last carrying \(h_{ij}h^{ij}=3\). Hence
Step 4: one integration by parts. \(\pi^{ij}\) is a weight-one density, so Equation (A53.3) applies directly to \(\pi^{ij}fD_{j}g\):
and likewise with \(f\) and \(g\) exchanged. In the difference, the term \(\pi^{ij}\left(D_{i}f\,D_{j}g-D_{i}g\,D_{j}f\right)\) is antisymmetric in \(i,j\) contracted with the symmetric \(\pi^{ij}\), and vanishes. What remains is
Step 5: recognize the momentum constraint. Lower the index on \(\pi\) and raise it on the bracket, which is legitimate because \(Dh=0\): \(\left(D_{j}\pi^{ji}\right)\left(fD_{i}g-gD_{i}f\right) =\left(D_{j}\pi^{j}{}_{k}\right)h^{ki} \left(f\pp_{i}g-g\pp_{i}f\right)\), the covariant derivatives of the scalars \(f\), \(g\) being ordinary ones. By Equation (30.85), \(-2D_{j}\pi^{j}{}_{k}=\Ham_{k}\), so
which is Equation (A53.17).
∎Every right-hand side in Equations (A53.12) and (A53.17) is a smeared constraint, and the primary constraints Equation (30.81) have vanishing bracket with everything, since no \(\Ham\) contains \(N\) or \(N^{i}\). So all eight constraints of Theorem 30.42 are first class in the sense of Definition 30.12, \(F=8\) and \(S=0\), and Equation (30.21) gives \(2\times10-2\times8=4\) per point of space. Rests on Theorem A53.6, Proposition A53.4 and Theorem 30.17.
Derives Corollary A53.7. By Proposition 30.13 it suffices that the brackets of the generators close on the constraint set, which is what the three displayed identities say — each right-hand side is \(H_{\perp}\) or \(H_{\parallel}\) of some smearing field, hence weakly zero. The smearing fields on the right of Equation (A53.17) depend on \(h_{ij}\), but that is irrelevant to the weak vanishing: a phase-space-dependent coefficient times a constraint still vanishes on \(\Sigma\). The count is Equation (30.21) with \(2n=20\) per point.
∎The geometric reading
The three brackets are the composition law for deformations of a spacelike surface embedded in a Lorentzian spacetime, and read that way they explain themselves.
\(H_{\parallel}[\xi]\) deforms the surface within itself: it slides the points along \(\xi\) and leaves the surface, as a subset of spacetime, where it was. Two such slides compose as diffeomorphisms of a three-manifold compose, which is Equation (A53.12) left, and their commutator is again a slide. That part of the algebra is a Lie algebra — the Lie algebra of vector fields on the leaf — and it is the only part that is.
\(H_{\perp}[f]\) pushes the surface off itself, by a proper time \(f/c\) per point along the normal. The mixed bracket says only that \(f\) is a scalar: sliding and then pushing differs from pushing and then sliding by pushing with the dragged function \(\xi^{i}\pp_{i}f\).
The third bracket is the one with content. Push by \(f\), then by \(g\); push by \(g\), then by \(f\). Each pair of normal pushes lands on the same surface — to first order the two orders of deformation reach the same place — but not on the same points of it: the two routes differ by a slide. The slide is easy to see. The normal at a point of the first intermediate surface is not the normal at the corresponding point of the second, because the two intermediate surfaces are tilted relative to one another by an amount proportional to the gradient of the pushing function; and tilting a normal is exactly what turns a normal deformation into a tangential one. The resulting tangent vector is \(f\,\nabla g-g\,\nabla f\) — the gradient of one function weighted by the other, antisymmetrized — and a gradient is a covector, so to be a deformation vector it must have its index raised. There is only one object available to raise it: the metric of the surface itself. Hence \(h^{ij}\) in Equation (A53.17).
And hence the sentence that closes Proposition 30.43. The coefficient on the right of the third bracket is a function on phase space, not a number, so Equation (30.16) for general relativity has structure functions and the first-class algebra is not a Lie algebra. This is not a technicality: it is the reason the constraint algebra of gravity cannot be treated as the algebra of a symmetry group with a fixed structure, why the BRST charge of Existence and Uniqueness of a Nilpotent BRST Charge needs terms beyond the two displayed in Equation (30.97), and why the comparison with Yang–Mills of Section 30.6.4, where the same brackets do close on constants, is the sharpest single contrast the chapter draws.
Nothing outside the treatise. The constraint densities Equations (30.84) and (30.85) are taken from Theorem 30.42, which proves them; the Levi-Civita apparatus is Theorems 17.150 and 17.152, and the component form of the Lie derivative is Proposition 17.127. Two assumptions are made and are not hidden. The smearing fields are independent of the canonical variables — for field-dependent smearings the brackets acquire extra terms and Equation (A53.12) is false as it stands. And the fields fall off fast enough that every integration by parts loses its surface term; on a region with a boundary they do not, the constraints are not even functionally differentiable without supplementary surface integrals, and those integrals are the ADM energy and momentum of Remark 30.45.
The Hypersurface-Deformation Algebra discharges the derivation owed at Proposition 30.43 of Section 30.6.5 in Constrained Hamiltonian Systems: the Dirac–Bergmann Formalism. It completes the last step of Theorem 30.42, which asserts that the eight constraints are first class and points forward to Equation (30.89) for the proof; with Corollary A53.7 the count of two degrees of freedom per point of space — the same count Proposition 30.32 gives for light, and the two polarizations detected in Experiment: Gravitational Waves — rests on nothing unproved. The Lagrangian from which those constraints were obtained is established in The Gauss–Codazzi Form of the Einstein–Hilbert Lagrangian.