The Gauss–Codazzi Form of the Einstein–Hilbert Lagrangian
This appendix proves Proposition 30.41 of Constrained Hamiltonian Systems: the Dirac–Bergmann Formalism: that the \(3+1\) decomposition Equation (30.78) gives \(\sqrt{\abs{g}}=N\sqrt{h}\), that the four-dimensional curvature scalar splits into the intrinsic curvature of the leaf, a quadratic in the extrinsic curvature and two divergences, and — the conclusion the chapter actually consumes — that no time derivative of the lapse or of the shift survives anywhere in the Einstein–Hilbert Lagrangian.
The mathematics is the Gauss–Codazzi decomposition of the Riemann tensor of an ambient manifold along a hypersurface. The difficulty is not the decomposition, which is four projections of one identity; it is the signature bookkeeping, and this treatise sits in exactly the place where that bookkeeping is worst. The metric of Minkowski Space and Its Symmetries is mostly minus, so the unit normal to a spacelike leaf satisfies \(n\cdot n=+1\); the curvature conventions of Theorem 17.152 and Definition 17.153 are the ones for which the round two-sphere has \(R>0\); and Definition 30.40 takes \(h_{ij}\) to be the positive-definite Riemannian metric of the leaf, so that the metric \(g_{ij}\) actually induced on the leaf by Equation (30.78) is \(-h_{ij}\). Three sign conventions, and every formula in the literature is written with a different combination of them. Nothing below is inherited from a mostly-plus source. The discipline used instead is this. Write
carry \(\epsilon\) as a symbol through every step, do the whole computation with the induced metric \(\gamma_{\mu\nu}=g_{\mu\nu}-\epsilon n_{\mu}n_{\nu}\) — whatever its sign — and only at the very end substitute \(\epsilon=+1\) and \(\gamma_{ij}=-h_{ij}\). A reader checking against a mostly-plus text sets \(\epsilon=-1\) and \(\gamma_{ij}=+h_{ij}\) and compares line by line; Remark A52.11 records what happens when the substitution is made, which is the one place where the answer differs from the printed statement of Proposition 30.41 and the reason the chapter's prose insists that the sign be settled here rather than assumed.
Throughout, the ambient connection \(\nabla\) is the Levi-Civita connection of \(g_{\mu\nu}\) (Theorem 17.150 with vanishing torsion), \(R^{\lambda}{}_{\rho\mu\nu}\) is Equation (17.306), and \(R_{\mu\nu}\), \(R\) are the contractions Equation (17.307). Indices \(\mu,\nu,\ldots\) run over \(0,1,2,3\) and \(i,j,k,\ldots\) over \(1,2,3\); \(x^{0}=ct\), as in Notation 30.28. Since \(h_{ij}\) and \(N\), \(N^{i}\) are dimensionless and the \(x^{i}\) carry metres, every term of Equation (30.80) has the SI dimension \(/\mathrm{m}^{2}\), and the identity is dimensionally homogeneous as it stands; no factor of \(c\) or \(G\) enters until the Lagrangian is multiplied by \(1/2\kappa\).
The kit: normal, inverse metric, projector, determinant
Write Equation (30.78) in terms of the coframe
Then, with \(N>0\):
-
the dual frame is \(e_{0}=\pp_{0}-N^{i}\pp_{i}\), \(e_{i}=\pp_{i}\), and the inverse metric is
\begin{equation}\tag{A52.3} g^{00}=\frac{1}{N^{2}}\ec\qquad g^{0i}=-\frac{N^{i}}{N^{2}}\ec\qquad g^{ij}=\frac{N^{i}N^{j}}{N^{2}}-h^{ij}\ec \end{equation}with \(h^{ij}\) the inverse of \(h_{ij}\);
-
the future-directed unit normal to the leaves \(x^{0}=\text{const}\) is
\begin{equation}\tag{A52.4} n_{\mu}=N\delta^{0}_{\mu}\ec\qquad n^{\mu}=\left(\frac{1}{N},\,-\frac{N^{i}}{N}\right) =\frac{1}{N}e_{0}\ec\qquad \epsilon=n^{\mu}n_{\mu}=+1\ec \end{equation}so the normal is timelike and the leaves are spacelike;
-
\(\det g=-N^{2}h\) with \(h=\det h_{ij}>0\), hence
\begin{equation}\tag{A52.5} \sqrt{\abs{g}}=N\sqrt{h}\ep \end{equation}
Rests on Definition 30.40, Equation (30.78) and Definition 17.117.
Derives Lemma A52.1. (i). That Equation (A52.2) reproduces Equation (30.78) is the definition of \(\theta^{i}\). For the dual frame, \(\theta^{0}(e_{0})=1\), \(\theta^{i}(e_{0})=-N^{i}+N^{i}=0\), \(\theta^{0}(e_{i})=0\) and \(\theta^{j}(e_{i})=\delta^{j}_{i}\), so \(\set{e_{0},e_{i}}\) is indeed dual to \(\set{\theta^{0},\theta^{i}}\). In that basis the metric is the block array \(\diag\left(N^{2},-h_{ij}\right)\), whose inverse is the block array \(\diag\left(N^{-2},-h^{ij}\right)\); hence
and expanding \(e_{0}=\pp_{0}-N^{k}\pp_{k}\) in coordinates gives Equation (A52.3).
(ii). A covector annihilating every vector tangent to a leaf — that is, every \(\pp_{i}\) — is proportional to \(\delta^{0}_{\mu}\). Normalizing, \(g^{\mu\nu}\left(N\delta^{0}_{\mu}\right) \left(N\delta^{0}_{\nu}\right)=N^{2}g^{00}=1\) by Equation (A52.3), so Equation (A52.4) is a unit covector and \(\epsilon=+1\). Raising with Equation (A52.3), \(n^{\mu}=Ng^{\mu0}\) gives the components stated, and \(n^{0}=1/N>0\) makes it future directed.
(iii). This is the step Proposition 30.41 asserts and does not prove. The change of coframe Equation (A52.2) has the matrix \(\theta^{a}=\Lambda^{a}{}_{\mu}\dd x^{\mu}\) with \(\Lambda^{0}{}_{0}=1\), \(\Lambda^{0}{}_{i}=0\), \(\Lambda^{i}{}_{0}=N^{i}\), \(\Lambda^{i}{}_{j}=\delta^{i}_{j}\): it is lower triangular with unit diagonal, so \(\det\Lambda=1\). Since \(g_{\mu\nu}=\Lambda^{a}{}_{\mu}\hat{g}_{ab}\Lambda^{b}{}_{\nu}\) with \(\hat{g}=\diag\left(N^{2},-h_{ij}\right)\),
the factor \(\left(-1\right)^{3}\) being the three minus signs the signature puts on the spatial block. Completing the square in \(\dd x^{i}+N^{i}\dd x^{0}\) is thus exactly the statement that the shift can be removed from the determinant by a unimodular change of coframe.
∎The projector onto a leaf and the induced metric are
A tensor is tangential if every index is annihilated by contraction with \(n\), equivalently if projecting every index returns it. For a tangential tensor \(T\) the intrinsic covariant derivative is
Rests on Equation (A52.4) and Definition 17.145.
Three facts about Equation (A52.8) are used without further comment. First, \(\gamma^{\mu}{}_{\nu}n^{\nu}=n^{\mu}-\epsilon^{2}n^{\mu}=0\) and \(\gamma^{\mu}{}_{\nu}v^{\nu}=v^{\mu}\) for \(v\) tangent to a leaf, since such a \(v\) has \(n_{\nu}v^{\nu}=0\); so \(\gamma\) is the projection along \(n\), and \(\gamma^{\mu}{}_{\alpha}\gamma^{\alpha}{}_{\nu} =\gamma^{\mu}{}_{\nu}\). Second, \(D\) is the Levi-Civita connection of \(\gamma\) on the leaf: it is torsion free, and \(D_{\mu}\gamma_{\nu\rho}=\gamma\gamma\gamma\nabla \left(g-\epsilon nn\right)=0\), the first term because \(\nabla g=0\) and the second because each projector annihilates the \(n\) it meets. Third — and this is the sign that will have to be paid for at the end — in the coordinates of Lemma A52.1 one has \(n_{i}=0\), so
where \(\gamma^{ij}\) means the inverse of \(\gamma_{ij}\) as a three-by-three matrix, which is also \(\gamma^{\mu\nu}\) restricted to spatial indices. The induced metric of Equation (30.78) is negative definite; the positive-definite \(h_{ij}\) of Definition 30.40 is its negative.
The extrinsic curvature, and the dictionary
Put
Rests on Equation (A52.8) and Definition 17.145.
\(\mathcal{K}_{\mu\nu}\) and \(a_{\mu}\) are tangential, \(\mathcal{K}\) is symmetric, and
Rests on Definition A52.3, Lemma A52.1 and Equation (17.295).
Derives Lemma A52.4. Tangentiality of \(\mathcal{K}\) is built into Equation (A52.11). For \(a\), differentiate \(n^{\nu}n_{\nu}=\epsilon\): \(n^{\nu}\nabla_{\mu}n_{\nu}=0\), and contracting with \(n^{\mu}\) gives \(n^{\nu}a_{\nu}=0\).
Equation (A52.12). Insert \(\delta^{\alpha}_{\mu}=\gamma^{\alpha}{}_{\mu} +\epsilon n^{\alpha}n_{\mu}\) in both slots of \(\nabla_{\mu}n_{\nu}\). The \(\gamma\gamma\) term is \(\mathcal{K}_{\mu\nu}\). Any term carrying \(n^{\beta}\) in the second slot vanishes, because \(n^{\beta}\nabla_{\alpha}n_{\beta}=0\). The remaining term is \(\epsilon n^{\alpha}n_{\mu}\gamma^{\beta}{}_{\nu}\nabla_{\alpha}n_{\beta} =\epsilon n_{\mu}\gamma^{\beta}{}_{\nu}a_{\beta} =\epsilon n_{\mu}a_{\nu}\), since \(a\) is tangential.
Symmetry of \(\mathcal{K}\) follows: the leaves are level sets of the coordinate \(x^{0}\), so \(n_{\mu}/N=\pp_{\mu}x^{0}\) is a gradient and \(\nabla_{[\mu}\left(n_{\nu]}/N\right)=0\), whence
Projecting both slots kills the right-hand side, so \(\mathcal{K}_{\mu\nu}=\mathcal{K}_{\nu\mu}\).
Equations (A52.13) and (A52.14). Trace Equation (A52.12) with \(g^{\mu\nu}\): the second term gives \(\epsilon n^{\nu}a_{\nu}=0\) and the first gives \(g^{\mu\nu}\mathcal{K}_{\mu\nu}=\gamma^{\mu\nu}\mathcal{K}_{\mu\nu} =\mathcal{K}\), because \(\mathcal{K}\) is tangential and \(g\) and \(\gamma\) agree on tangential tensors. Squaring Equation (A52.12),
each cross term carrying an \(n\) contracted with a tangential index and the last term carrying \(a^{\nu}n_{\nu}=0\).
Equation (A52.15). Compare Equation (A52.16) with the antisymmetric part of Equation (A52.12), which is \(\epsilon\left(n_{\mu}a_{\nu}-n_{\nu}a_{\mu}\right)\), and contract with \(n^{\mu}\). The left-hand side gives \(\epsilon\left(\epsilon a_{\nu}-0\right)=a_{\nu}\) and the right-hand side gives \(n_{\nu}\left(n^{\mu}\pp_{\mu}\ln N\right) -\epsilon\pp_{\nu}\ln N\). Projecting with \(\gamma\), which fixes the tangential \(a_{\nu}\) and kills \(n_{\nu}\), leaves Equation (A52.15).
∎In the coordinates of Lemma A52.1, with \(K_{ij}\) the extrinsic curvature Equation (30.79) of Definition 30.40, \(K=h^{ij}K_{ij}\), \(K^{ij}=h^{ik}h^{jl}K_{kl}\) and \({}^{(3)}\!R\) the Ricci scalar of \(h_{ij}\):
\(R[\gamma]\) denoting the curvature scalar Equation (17.307) of the induced metric \(\gamma_{ij}\) of Equation (A52.10). The Levi-Civita connections of \(\gamma_{ij}\) and of \(h_{ij}\) coincide, so \(D_{i}\) of Equation (A52.9) is the \(D_{i}\) of Definition 30.40. Rests on Equation (30.79), Equation (A52.10) and Definition 17.153.
Derives Lemma A52.5. Connections and Riemann tensors. \(\gamma_{ij}=-h_{ij}\) differs from \(h_{ij}\) by a constant factor, and the Christoffel symbols Equation (17.299) are homogeneous of degree zero in the metric — one inverse metric against one derivative of the metric — so they are the same for \(\gamma\) and for \(h\). By Equation (17.306) the Riemann tensors \(R^{i}{}_{jkl}\) and hence the Ricci tensors \(R_{jl}\) agree. The scalars do not: \(R[\gamma]=\gamma^{jl}R_{jl}=-h^{jl}R_{jl}=-{}^{(3)}\!R\), which is the fourth relation.
Extrinsic curvature. Since \(n_{i}=0\), \(\gamma^{\alpha}{}_{i} =\delta^{\alpha}_{i}\), so \(\mathcal{K}_{ij}=\nabla_{i}n_{j} =\pp_{i}\left(N\delta^{0}_{j}\right) -\Gamma^{\lambda}{}_{ij}n_{\lambda}=-N\Gamma^{0}{}_{ij}\). Compute \(\Gamma^{0}{}_{ij}\) from Equation (17.299) with \(g_{0i}=-N_{i}\), \(g_{ij}=-h_{ij}\) and Equation (A52.3):
every term having picked up one overall minus from \(g_{ij}=-h_{ij}\), \(g_{0i}=-N_{i}\) and, in the second group, a further minus from \(g^{0k}=-N^{k}/N^{2}\). Now the Christoffel symbols of \(h_{ij}\) give \(D_{i}N_{j}+D_{j}N_{i}=\pp_{i}N_{j}+\pp_{j}N_{i} -2\Gamma^{(3)k}{}_{ij}N_{k}\) and \(2\Gamma^{(3)k}{}_{ij}N_{k} =N^{l}\left(\pp_{i}h_{lj}+\pp_{j}h_{li}-\pp_{l}h_{ij}\right)\), so the bracket of Equation (A52.19) is \(\pp_{0}h_{ij}-D_{i}N_{j}-D_{j}N_{i}\) and
by Equation (30.79). Hence \(\mathcal{K}_{ij}=-K_{ij}\), the first relation. This is not a choice: it is forced, because \(\mathcal{K}\) is one half the Lie derivative of \(\gamma\) along \(n\) while \(K\) is one half the Lie derivative of \(h\), and \(\gamma=-h\).
Trace and square. \(\mathcal{K}=\gamma^{ij}\mathcal{K}_{ij} =\left(-h^{ij}\right)\left(-K_{ij}\right)=K\): the two minus signs cancel, so the traces agree even though the tensors differ in sign. Likewise \(\mathcal{K}_{\mu\nu}\mathcal{K}^{\mu\nu} =\gamma^{ik}\gamma^{jl}\mathcal{K}_{ij}\mathcal{K}_{kl} =h^{ik}h^{jl}K_{ij}K_{kl}=K_{ij}K^{ij}\), four minus signs cancelling in pairs. Only the curvature scalar, which carries one inverse metric rather than an even number, survives the substitution with its sign changed — and it is exactly that asymmetry which Remark A52.11 is about.
∎The Gauss equation
For a hypersurface with unit normal \(n\), \(\epsilon=n\cdot n\),
\(R[\gamma]\) being the Riemann tensor of the induced metric. Rests on Equation (17.305), Definition A52.3 and Equation (A52.9).
Derives Lemma A52.6. Let \(\omega_{\mu}\) be a tangential covector field. By Equation (A52.9), \(\left(D\omega\right)_{\alpha\beta} =\gamma^{\sigma}{}_{\alpha}\gamma^{\tau}{}_{\beta} \nabla_{\sigma}\omega_{\tau}\), and applying \(D\) once more,
Expand the derivative by the Leibniz rule. Because \(\nabla_{\lambda}\gamma^{\sigma}{}_{\alpha} =-\epsilon\left(n^{\sigma}\nabla_{\lambda}n_{\alpha} +n_{\alpha}\nabla_{\lambda}n^{\sigma}\right)\), and because every \(n_{\alpha}\) or \(n_{\beta}\) meets a projector and dies, the two terms in which \(\nabla\) falls on a projector are
The first is symmetric in \(\rho\leftrightarrow\mu\) and therefore drops out of the commutator \(D_{\rho}D_{\mu}-D_{\mu}D_{\rho}\). In the second, differentiate \(\omega_{\tau}n^{\tau}=0\) to get \(n^{\tau}\nabla_{\sigma}\omega_{\tau} =-\omega_{\tau}\nabla_{\sigma}n^{\tau}\), and then \(\gamma^{\sigma}{}_{\mu}\omega_{\tau}\nabla_{\sigma}n^{\tau} =\omega_{\tau}\mathcal{K}_{\mu}{}^{\tau}\) because \(\omega\) is tangential; so the second term equals \(+\epsilon\,\mathcal{K}_{\rho\nu}\mathcal{K}_{\mu}{}^{\tau} \omega_{\tau}\).
The remaining term of Equation (A52.22) is \(\gamma^{\lambda}{}_{\rho}\gamma^{\sigma}{}_{\mu}\gamma^{\tau}{}_{\nu} \nabla_{\lambda}\nabla_{\sigma}\omega_{\tau}\). Antisymmetrizing in \(\rho\leftrightarrow\mu\) and using the Ricci identity Equation (17.305) in its covector form, \(\comm{\nabla_{\lambda}}{\nabla_{\sigma}}\omega_{\tau} =-R^{\kappa}{}_{\tau\lambda\sigma}\omega_{\kappa}\) (which follows from Equation (17.305) by applying the commutator to the scalar \(\omega_{\lambda}V^{\lambda}\)), and the same identity on the leaf for \(D\),
Both sides are tangential in the free index \(\kappa\) and \(\omega\) was an arbitrary tangential covector, so the coefficients agree after projecting \(\kappa\), which is Equation (A52.21).
∎The sign of the quadratic term in Equation (A52.21) can be checked on the one case every reader already knows, and in the signature that matters here. Take the sphere of radius \(r\) in Euclidean \(\R^{3}\): the ambient is flat, \(\epsilon=+1\) because the normal is a unit vector of a positive-definite metric, and \(\mathcal{K}_{ij}=\nabla_{i}n_{j}=\gamma_{ij}/r\) for the outward normal. Then Equation (A52.21), with indices lowered, gives \(R[\gamma]_{\kappa\nu\rho\mu} =r^{-2}\left(\gamma_{\mu\nu}\gamma_{\rho\kappa} -\gamma_{\rho\nu}\gamma_{\mu\kappa}\right)\), whose double contraction is \(R[\gamma]=2/r^{2}\). That is \(2\) times the Gaussian curvature, which is what Definition 17.153 says a two-dimensional curvature scalar must be, and it is Gauss's theorema egregium: the sphere's intrinsic curvature is fixed by its second fundamental form alone. A sign error in Equation (A52.21) would make the sphere intrinsically hyperbolic.
The Codazzi equation
With the same notation,
and contracting \(\nu\) with \(\rho\),
No factor of \(\epsilon\) appears in either. Rests on Lemma A52.4, Equation (17.305) and Definition A52.3.
Derives Lemma A52.8. By Equation (A52.12), \(\mathcal{K}_{\beta\lambda}=\nabla_{\beta}n_{\lambda} -\epsilon n_{\beta}a_{\lambda}\), so
the second term because \(\gamma^{\beta}{}_{\nu}\nabla_{\alpha}\left(n_{\beta}a_{\lambda}\right) =a_{\lambda}\gamma^{\beta}{}_{\nu}\nabla_{\alpha}n_{\beta}\), the other piece dying on \(\gamma^{\beta}{}_{\nu}n_{\beta}=0\). Antisymmetrizing in \(\mu\leftrightarrow\nu\), the term \(a_{\rho}\mathcal{K}_{\mu\nu}\) is symmetric and drops, and the Ricci identity in covector form leaves Equation (A52.25).
For the contraction, apply \(\gamma^{\nu\rho}\). On the left it gives \(D_{\mu}\mathcal{K}-D_{\nu}\mathcal{K}^{\nu}{}_{\mu}\), which is minus the left-hand side of Equation (A52.26). On the right, \(\gamma^{\beta\lambda}=g^{\beta\lambda}-\epsilon n^{\beta}n^{\lambda}\); the \(g^{\beta\lambda}\) piece contracts the second and fourth slots of \(R_{\kappa\lambda\alpha\beta}\), which by the antisymmetry within each index pair (Proposition 17.154) is the same as contracting the first and third, that is again the Ricci tensor, giving \(n^{\kappa}R_{\kappa\alpha}\), while the \(n^{\beta}n^{\lambda}\) piece vanishes because \(R_{\kappa\lambda\alpha\beta}n^{\kappa}n^{\lambda}=0\) by antisymmetry in the first pair. Rearranging gives Equation (A52.26).
∎Equation (A52.26) is worth reading before it is used. Under the dictionary Equation (A52.18) the mixed-index extrinsic curvature carries two inverse metrics' worth of sign and is unchanged, \(\mathcal{K}^{j}{}_{i}=\gamma^{jk}\mathcal{K}_{ki} =\left(-h^{jk}\right)\left(-K_{ki}\right)=K^{j}{}_{i}\), so the left-hand side is \(D_{j}K^{j}{}_{i}-D_{i}K\). By Equation (30.82) that is \(2\kappa h^{-1/2}D_{j}\pi^{j}{}_{i} =-\kappa h^{-1/2}\Ham_{i}\) with \(\Ham_{i}\) the momentum constraint Equation (30.85); and the right-hand side is a projection of the Ricci tensor onto one normal and one tangential index. So the momentum constraint of Theorem 30.42 is the contracted Codazzi equation, which is the precise content of the first paragraph of Remark 30.44: the constraints are the \(G^{0}{}_{\mu}\) components of the field equations, and they are conditions on data laid down on one leaf because Codazzi's equation contains no derivative off the leaf.
The twice-normal contraction
The double contraction of Equation (A52.21) will leave behind the term \(R_{\mu\nu}n^{\mu}n^{\nu}\), which is neither intrinsic nor a projection of anything on the leaf. Converting it is the step that produces the total derivatives, and it is the one usually passed over. It is not passed over here.
For any unit normal field,
Rests on Equation (17.305), Lemma A52.4 and Definition 17.153.
Derives Lemma A52.9. Apply the Ricci identity Equation (17.305) to \(n^{\lambda}\), set \(\lambda=\mu\) and contract: since \(R^{\mu}{}_{\rho\mu\nu}=R_{\rho\nu}\) by Equation (17.307),
Contract with \(n^{\nu}\) and rewrite each side as a divergence minus the term the Leibniz rule left over:
Subtracting, and inserting Equations (A52.13) and (A52.14) for the two quadratic terms and \(a^{\mu}=n^{\nu}\nabla_{\nu}n^{\mu}\) for the first divergence, gives Equation (A52.28).
∎Equation (A52.28) is the trace of what is called the Ricci, or Mainardi, equation — the twice-normal projection of the Riemann tensor. Only the trace is needed below, and only the trace is proved: the untraced equation would be an identity for \(\gamma^{\alpha}{}_{\mu}\gamma^{\beta}{}_{\nu}n^{\lambda}n^{\sigma} R_{\alpha\lambda\beta\sigma}\) in terms of the Lie derivative of \(\mathcal{K}\) along \(n\), and nothing in this appendix or in Constrained Hamiltonian Systems: the Dirac–Bergmann Formalism consumes it.
Assembling the Lagrangian
For a foliation by hypersurfaces with unit normal \(n\) and \(\epsilon=n\cdot n\),
Specialized to Equation (30.78) by Lemmas A52.1 and A52.5, that is \(\epsilon=+1\), \(\gamma_{ij}=-h_{ij}\),
with \(\sqrt{\abs{g}}=N\sqrt{h}\). In particular \({}^{(3)}\!R\) contains \(h_{ij}\) and its spatial derivatives only, and \(K_{ij}\) contains \(\pp_{0}h_{ij}\) linearly and no other time derivative, so neither \(\pp_{0}N\) nor \(\pp_{0}N^{i}\) occurs anywhere on the right. Rests on Lemmas A52.5, A52.6 and A52.9.
Derives Theorem A52.10. Step 1: contract the Gauss equation once. Put \(\rho=\kappa\) in Equation (A52.21) and sum. On the right, \(\gamma^{\kappa}{}_{\alpha}\gamma^{\lambda}{}_{\kappa} =\gamma^{\lambda}{}_{\alpha}\), so, lowering the free index,
Writing \(\gamma^{\lambda\alpha}=g^{\lambda\alpha} -\epsilon n^{\lambda}n^{\alpha}\) and using \(g^{\lambda\alpha}R_{\alpha\tau\lambda\sigma}=R_{\tau\sigma}\) from Equation (17.307),
Step 2: contract again. Apply \(\gamma^{\nu\mu}\). The first bracket gives \(\gamma^{\tau\sigma}R_{\tau\sigma}=R-\epsilon R_{\tau\sigma} n^{\tau}n^{\sigma}\). The second gives \(\gamma^{\tau\sigma}n^{\alpha}n^{\lambda} R_{\alpha\tau\lambda\sigma}=R_{\alpha\lambda}n^{\alpha}n^{\lambda}\): the \(g^{\tau\sigma}\) part is again a Ricci contraction on the second and fourth slots, and the \(n^{\tau}n^{\sigma}\) part vanishes because \(R_{\alpha\tau\lambda\sigma}n^{\alpha}n^{\tau}=0\). The quadratic terms give \(\mathcal{K}_{\mu\nu}\mathcal{K}^{\mu\nu}-\mathcal{K}^{2}\). Hence
Step 3: eliminate the normal–normal Ricci term. Substitute Equation (A52.28) into Equation (A52.36) and solve for \(R\):
which is Equation (A52.32): the quadratic term appears twice with opposite weight and what survives is \(-\epsilon\left(\mathcal{K}\mathcal{K}-\mathcal{K}^{2}\right)\). It is worth noticing that the sign of the quadratic term in the final answer is opposite to the sign it carries in the Gauss equation, and that the flip is entirely the work of Lemma A52.9; this is the step at which a derivation that quotes the Ricci equation instead of proving it can go wrong without leaving a trace.
Step 4: substitute. By Lemma A52.1, \(\epsilon=+1\); by Lemma A52.5, \(R[\gamma]=-{}^{(3)}\!R\), \(\mathcal{K}_{\mu\nu}\mathcal{K}^{\mu\nu}=K_{ij}K^{ij}\) and \(\mathcal{K}=K\). So
Multiplying by \(\sqrt{\abs{g}}=N\sqrt{h}\) and using the standard identity \(\sqrt{\abs{g}}\,\nabla_{\mu}V^{\mu} =\pp_{\mu}\left(\sqrt{\abs{g}}V^{\mu}\right)\) for the divergence of a vector field — which follows from \(\Gamma^{\mu}{}_{\mu\nu}=\pp_{\nu}\ln\sqrt{\abs{g}}\), itself a contraction of Equation (17.299) — gives Equation (A52.33).
Step 5: the velocities. \({}^{(3)}\!R\) is built from \(h_{ij}\) and its spatial derivatives alone. \(K_{ij}\) is given by Equation (30.79), in which \(\pp_{0}\) acts on \(h_{ij}\) and on nothing else: \(N\) and \(N^{i}\) enter algebraically and through spatial derivatives \(D_{i}N_{j}\) only. The divergence term contains \(\pp_{0}\) of \(\sqrt{\abs{g}}\left(a^{0}-Kn^{0}\right)\), hence of \(N\), \(N^{i}\) and \(h_{ij}\); but it is a total derivative and is removed by the boundary term of The boundary term, so no time derivative of the lapse or the shift survives in the bulk Lagrangian.
∎Equation (A52.33) carries an overall minus sign that Equation (30.80) does not display, and it is worth saying plainly where it comes from and what it costs, because this is exactly the bookkeeping the chapter's prose defers to this appendix.
The source is isolated in Lemma A52.5 and is a single asymmetry: of the four quantities in the dictionary, three are built with an even number of inverse metrics and are blind to \(\gamma_{ij}=-h_{ij}\), while the curvature scalar carries exactly one and changes sign. So with the curvature conventions of Theorem 17.152 and Definition 17.153 — the conventions for which the round two-sphere has \(R=2/r^{2}\), checked in Remark A52.7 — and the mostly-minus signature of Minkowski Space and Its Symmetries, the four-scalar that equals \(N\sqrt{h}\left({}^{(3)}\!R+K_{ij}K^{ij}-K^{2}\right)\) up to divergences is \(-\sqrt{\abs{g}}\,R\), and not \(+\sqrt{\abs{g}}\,R\).
Nothing physical turns on this, and three things should be said about it. First, the gravitational Lagrangian density in this signature is therefore \(-\sqrt{\abs{g}}\,R/2\kappa\); that is the sign carried, for precisely this reason, by every mostly-minus treatment of general relativity, and it is the combination whose ADM form is \(+N\sqrt{h}\left({}^{(3)}\!R+K_{ij}K^{ij}-K^{2}\right)/2\kappa\). Second, that is exactly the Lagrangian \(L\) used in the proof of Theorem 30.42 — read the first display of that proof — so the momentum Equation (30.82), the constraint densities Equations (30.84) and (30.85) and the degree-of-freedom count all stand unchanged. Third, and this is why Proposition 30.41 can be used as it is written, neither of the two facts the chapter draws from it is touched: \(\sqrt{\abs{g}} =N\sqrt{h}\) is Equation (A52.5), and the absence of \(\pp_{0}N\) and \(\pp_{0}N^{i}\) is Step 5 above, and both are independent of the overall sign. A reader working in a mostly-plus convention sets \(\epsilon=-1\) and \(\gamma_{ij}=+h_{ij}\) in Equation (A52.32) and recovers the familiar \(R={}^{(3)}\!R+K_{ij}K^{ij}-K^{2} -2\nabla_{\mu}\left(a^{\mu}-Kn^{\mu}\right)\) with no minus sign in front, which is the form printed in the standard references [Arnowitt:1962] [Misner:1973] [Wald:1984] and the reason the sign is so easy to import unexamined.
The boundary term
The divergence in Equation (A52.33) is a genuine coordinate divergence, so over a region \(\mathcal{R}\) bounded by two leaves it contributes only a surface integral. Evaluate it. The acceleration is tangential, so \(n_{\mu}a^{\mu}=Na^{0}=0\) and \(a^{0}=0\): the acceleration has no flux through a leaf. And \(\sqrt{\abs{g}}\,n^{0}=N\sqrt{h}\cdot N^{-1}=\sqrt{h}\) by Equations (A52.4) and (A52.5). Hence the \(\mu=0\) component of the divergence's argument is
and, integrating Equation (A52.33) over \(\mathcal{R}\),
the sign of the last term following the outward orientation of \(\pp\mathcal{R}\) once that orientation is fixed. Dividing by \(2\kappa\), the surface term is \(\kappa^{-1}\oint\sqrt{h}\,K\), which is the Gibbons–Hawking–York boundary action [Gibbons:1977] [York:1972].
Two statements are made about it here and only the first is proved. The first is Equation (A52.40) itself: the bulk ADM Lagrangian and the gravitational action differ by exactly that surface integral, which is why Proposition 30.41 may write “plus total derivatives” and Theorem 30.42 may drop them. The second is quoted: adding the Gibbons–Hawking–York term to the action makes the variational problem well posed with \(h_{ij}\) — and only \(h_{ij}\) — held fixed on \(\pp\mathcal{R}\), because the unwanted normal derivatives of the metric variation that \(\sqrt{\abs{g}}\,R\) produces on the boundary are precisely what it cancels. That is proved in [Gibbons:1977] and not here; the constrained-dynamics consequence of the same term, that the numerical value of the Hamiltonian of an isolated system is carried entirely by a surface integral, is Remark 30.45.
Two imports, and one debt of Differentiable Manifolds, Tensors, and Curvature.
Quoted. That the Gibbons–Hawking–York term makes the variational problem well posed, as stated in the paragraph above; it is used nowhere in Constrained Hamiltonian Systems: the Dirac–Bergmann Formalism except as the justification for discarding total derivatives, which Equation (A52.40) establishes independently.
Owed by Differentiable Manifolds, Tensors, and Curvature. A general-signature Gauss–Codazzi theory of hypersurfaces. That chapter carries the classical version Equation (17.92), but it is the theory of a surface in flat Euclidean \(\R^{3}\), built from a position vector \(\vect{x}\) and its derivatives, and it is not usable for a spacelike leaf of a Lorentzian spacetime: there is no position vector, the ambient is curved, and the normal's square is \(\epsilon\) rather than \(+1\). The chapter's later treatment of a quadric in a flat ambient of signature \((p,q)\) carries the right sign structure but is again an embedding into a flat space given by an explicit position vector. Lemmas A52.6, A52.8 and A52.9 are therefore proved here, from Theorem 17.152 and the projector alone, and the general statement belongs in the manifolds chapter. This is recorded as a Part II debt, in the same terms as Remarks 30.4 and 30.18 record theirs.
Everything else — the inverse metric, the determinant, the extrinsic curvature in coordinates, the dictionary, the two contractions of the Gauss equation and the elimination of the normal–normal Ricci term — is carried out above from the Levi-Civita connection of Theorem 17.150 and the Riemann tensor of Theorem 17.152.
The Gauss–Codazzi Form of the Einstein–Hilbert Lagrangian discharges the derivation owed at Proposition 30.41 of Section 30.6.5 in Constrained Hamiltonian Systems: the Dirac–Bergmann Formalism. What the chapter takes from it is narrow and is now established: \(\sqrt{\abs{g}}=N\sqrt{h}\) (Equation (A52.5)), and that the Einstein–Hilbert Lagrangian contains no time derivative of the lapse or the shift (Step 5 of Theorem A52.10). Those two facts are what make \(\pi_{N}\) and \(\pi_{i}\) vanish identically in Equation (30.81), hence what make \(N\) and \(N^{i}\) Lagrange multipliers rather than dynamical fields, hence what make \(\Ham_{\perp}\) and \(\Ham_{i}\) secondary constraints by outcome 2 of Proposition 30.10 — so the whole of Theorem 30.42, and with it the count of two degrees of freedom per point, rests on this section. The overall sign of Equation (30.80) is settled in Remark A52.11 and changes none of that. The brackets of the constraints so obtained are computed in The Hypersurface-Deformation Algebra.