The Einstein Field Equations

Derives Proposition 44.4. Tracing Equation (44.2) with \(g^{\mu\nu}\) in \(D=4\) gives \(R-2R=\kappa T\), so \(R=-\kappa T\). Substituting back, \(R_{\mu\nu}=\kappa T_{\mu\nu}+\tfrac12 g_{\mu\nu}R =\kappa(T_{\mu\nu}-\tfrac12 g_{\mu\nu}T)\), and the two forms determine each other. With \(T_{\mu\nu}=0\) both give \(R_{\mu\nu}=0\), while the Riemann tensor need not vanish — the Schwarzschild exterior (Section 44.10) is the standing example. This trace-reversed form is the one Einstein printed on 25 November [Einstein:1915b], and it is the form the Newtonian limit reads most easily (Section 44.4.2).

Derives Lemma 44.7. Varying \(g^{\mu\alpha}g_{\alpha\nu}=\delta^{\mu}_{\nu}\) gives \(\delta g^{\mu\alpha}\,g_{\alpha\nu} =-g^{\mu\alpha}\,\delta g_{\alpha\nu}\); contracting with \(g^{\nu\beta}\) yields Equation (44.6). For the determinant, Jacobi's formula Equation (5.21) applied to the matrix \(g_{\mu\nu}\) along any one-parameter family gives \(\delta(\det g)=(\det g)\,g^{\mu\nu}\delta g_{\mu\nu}\), and since \(\abs{g}=-\det g\) here, \(\delta\sqrt{\abs{g}} =\tfrac12\abs{g}^{-1/2}\,\delta\abs{g} =\tfrac12\sqrt{\abs{g}}\,g^{\mu\nu}\delta g_{\mu\nu}\). The second equality in Equation (44.7) follows from Equation (44.6).

Derives Lemma 44.8. Step 1: Equation (44.8). Vary the Christoffel formula of Theorem 13.150, \(\Gamma^{\lambda}{}_{\mu\nu}=\tfrac12 g^{\lambda\rho} (\pp_{\mu}g_{\rho\nu}+\pp_{\nu}g_{\rho\mu}-\pp_{\rho}g_{\mu\nu})\). The \(\delta g^{\lambda\rho}\) term contributes, by Equation (44.6) and the same formula read backwards, \(-g^{\lambda\alpha}\,\Gamma^{\sigma}{}_{\mu\nu}\,\delta g_{\alpha\sigma}\). In the \(\delta(\pp g)\) terms replace each \(\pp\,\delta g\) by \(\nabla\,\delta g\) plus its two connection terms; summing the three combinations, four of the six connection terms cancel in pairs by the symmetry of \(\Gamma\), and the surviving two give \(+g^{\lambda\rho}\,\Gamma^{\sigma}{}_{\mu\nu}\,\delta g_{\rho\sigma}\), which cancels the \(\delta g^{\lambda\rho}\) contribution exactly. What remains is Equation (44.8); being built covariantly from the tensor \(\delta g_{\mu\nu}\), it is a tensor — as the difference of two connections must be.

Step 2: Equation (44.9). Vary Equation (13.306):

On the other hand, treating \(\delta\Gamma\) as the \((1,2)\) tensor it is,

the terms in \(\Gamma^{\sigma}{}_{\mu\nu}\delta\Gamma^{\lambda}{}_{\sigma\rho}\) cancelling between the two covariant derivatives because \(\Gamma^{\sigma}{}_{\mu\nu}\) is symmetric. The right-hand sides agree term by term, so \(\delta R^{\lambda}{}_{\rho\mu\nu} =\nabla_{\mu}\delta\Gamma^{\lambda}{}_{\nu\rho} -\nabla_{\nu}\delta\Gamma^{\lambda}{}_{\mu\rho}\); contracting \(\lambda\) with \(\mu\) gives Equation (44.9).

Step 3: Equation (44.10). Contract Equation (44.9) with \(g^{\mu\nu}\) and pull the metric through \(\nabla\) by metricity (Definition 13.149); the first expression for \(v^{\mu}\) results. Substituting Equation (44.8): in the first slot, \(g^{\alpha\beta}\delta\Gamma^{\mu}{}_{\alpha\beta} =g^{\mu\rho}g^{\alpha\beta}(\nabla_{\alpha}\delta g_{\rho\beta} -\tfrac12\nabla_{\rho}\delta g_{\alpha\beta})\); in the second, \(\delta\Gamma^{\lambda}{}_{\lambda\beta} =\tfrac12 g^{\lambda\rho}\nabla_{\beta}\delta g_{\rho\lambda}\) because the first and third terms of Equation (44.8) cancel under the \(\lambda\rho\) contraction. Combining the two, the half-terms add up and give Equation (44.11).

Derives Theorem 44.9. Write \(\sqrt{\abs{g}}\,R =\sqrt{\abs{g}}\,g^{\mu\nu}R_{\mu\nu}\) and vary the three factors:

By Equation (44.7) the first term is \(-\tfrac12\sqrt{\abs{g}}\,g_{\mu\nu}R\,\delta g^{\mu\nu}\), which combines with the second into \(\sqrt{\abs{g}}\,G_{\mu\nu}\,\delta g^{\mu\nu}\); by Equation (44.10) the third is \(\sqrt{\abs{g}}\,\nabla_{\mu}v^{\mu}\). That is Equation (44.12).

For the divergence term, note the identity \(\Gamma^{\lambda}{}_{\lambda\mu} =\tfrac12 g^{\lambda\rho}\pp_{\mu}g_{\lambda\rho} =\pp_{\mu}\ln\sqrt{\abs{g}}\) — Jacobi's formula Equation (5.21) again — whence \(\sqrt{\abs{g}}\,\nabla_{\mu}v^{\mu} =\pp_{\mu}(\sqrt{\abs{g}}\,v^{\mu})\): a coordinate divergence. If \(\delta g_{\mu\nu}\), and with it \(v^{\mu}\), vanishes outside a compact subset of the interior, the integral of \(\pp_{\mu}(\sqrt{\abs{g}}\,v^{\mu})\) vanishes by the fundamental theorem of calculus applied coordinate by coordinate (Theorem 7.133 is its three-dimensional integral form). Adding \(\delta S_{\mathrm{m}}\) from Equation (44.5),

and since \(\delta g^{\mu\nu}\) is an arbitrary symmetric compactly supported tensor, the fundamental lemma of the calculus of variations in its several-variable form (Lemma 16.18, as used in Theorem 16.36) gives Equation (44.13). The vacuum case and the equivalence \(G_{\mu\nu}=0\Leftrightarrow R_{\mu\nu}=0\) are Proposition 44.4.

Derives Lemma 44.10. As in Theorem 44.9, \(\sqrt{\abs{g}}\,\nabla_{\mu}v^{\mu} =\pp_{\mu}(\sqrt{\abs{g}}\,v^{\mu})\), and the flat divergence theorem (Theorem 7.133, in its four-dimensional form, which is proved by the identical slicing argument) turns the left side into the flux of \(\sqrt{\abs{g}}\,v^{\mu}\) through \(\pp\Omega\). It remains to identify the flux element. On a boundary face given by \(x^{1}=\text{const}\) (outward direction of increasing \(x^{1}\), the other faces being identical), the flux is \(\int\dd^{3}y\,\sqrt{\abs{g}}\,v^{1}\). By the adjugate formula for the inverse matrix (Proposition 5.49), \(g^{11}\) equals the \((1,1)\) cofactor of \(g_{\mu\nu}\) divided by \(\det g\); the cofactor is the determinant of the \(3\times3\) block of \(g\) on the coordinates \((x^{0},x^{2},x^{3})\) of the face, which is exactly \(\det h\). Hence \(\abs{g}\,\abs{g^{11}}=\abs{h}\), so \(\sqrt{\abs{g}}=\sqrt{\abs{h}}/\sqrt{\abs{g^{11}}}\) — finite precisely because the face is not null, \(g^{11}\neq0\). The outward unit conormal is \(n_{\mu}=\pp_{\mu}x^{1}/\sqrt{\abs{g^{11}}}\), with \(n^{\mu}n_{\mu}=g^{11}/\abs{g^{11}}=\epsilon\), and \(n_{\mu}v^{\mu}=v^{1}/\sqrt{\abs{g^{11}}}\). Therefore \(\sqrt{\abs{g}}\,v^{1}=\sqrt{\abs{h}}\;n_{\mu}v^{\mu}\), which is Equation (44.14).

Derives Proposition 44.11. On \(\pp\Omega\) every term of \(v^{\mu}\) in Equation (44.11) may be written with partial derivatives, since \(\nabla\delta g=\pp\,\delta g-\Gamma\,\delta g\) and \(\delta g=0\) there. Decompose \(g^{\alpha\beta}=h^{\alpha\beta}+\epsilon^{-1}n^{\alpha}n^{\beta}\) along the boundary. In \(n_{\mu}v^{\mu}=g^{\alpha\beta}n^{\rho} (\pp_{\alpha}\delta g_{\rho\beta}-\pp_{\rho}\delta g_{\alpha\beta})\) the two \(n^{\alpha}n^{\beta}\) contributions are equal and cancel between the two terms; in the \(h^{\alpha\beta}\) part, the first term is a tangential derivative of \(\delta g\), which vanishes because \(\delta g\) vanishes identically along \(\pp\Omega\). That is the first equality of Equation (44.15).

For the second, vary \(K=h^{\alpha\beta}\nabla_{\alpha}n_{\beta}\) at fixed boundary metric. Since \(\delta g_{\mu\nu}=0\) on \(\pp\Omega\), the unit conormal \(n_{\mu}\propto\pp_{\mu}(\text{boundary function})\) and its normalization are unvaried there, and so is \(h^{\alpha\beta}\); moreover \(\delta n_{\beta}\) vanishes identically along \(\pp\Omega\), so its \(h\)-projected (tangential) derivatives vanish too, and the only surviving variation is through the connection, \(\delta(\nabla_{\alpha}n_{\beta}) =-\delta\Gamma^{\sigma}{}_{\alpha\beta}\,n_{\sigma}\). With Equation (44.8), again dropping tangential derivatives of \(\delta g\),

By Lemma 44.10 the offending term of Equation (44.12) is \((2\kappa)^{-1}\oint\sqrt{\abs{h}}\,n_{\mu}v^{\mu} =-\kappa^{-1}\oint\sqrt{\abs{h}}\,\delta K =-\,\delta S_{\mathrm{GHY}}\), using once more that \(\sqrt{\abs{h}}\) is unvaried on the boundary. Adding \(S_{\mathrm{GHY}}\) therefore cancels it identically.

Derives Proposition 44.13. The \(\omega\) equation. From the second structure equation Equation (13.312), \(\delta R^{ab}=\dd\,\delta\omega^{ab} +\delta\omega^{a}{}_{c}\wedge\omega^{cb} +\omega^{a}{}_{c}\wedge\delta\omega^{cb} =\mathrm{D}\,\delta\omega^{ab}\). The frame Levi-Civita symbol is \(\mathrm{D}\)-constant: \(\mathrm{D}\epsilon_{abcd}\) is totally antisymmetric in \(abcd\), hence proportional to \(\epsilon_{abcd}\), with coefficient the trace \(\omega^{e}{}_{e}=\eta_{ef}\,\omega^{ef}=0\) by the antisymmetry Equation (13.310). The graded Leibniz rule for \(\mathrm{D}\), applied to the one-form \(\delta\omega^{ab}\), therefore gives

where \(\dd\) replaces \(\mathrm{D}\) on the full contraction because it is a Lorentz scalar. The exact term integrates to zero, so \(\delta_{\omega}S=(4\kappa)^{-1}\!\int \epsilon_{abcd}\,\delta\omega^{ab}\wedge \mathrm{D}(e^{c}\wedge e^{d})\). With \(\mathrm{D}(e^{c}\wedge e^{d})=T^{c}\wedge e^{d}-e^{c}\wedge T^{d}\) (first structure equation, Equation (13.311)) and the relabelling symmetry of \(\epsilon_{abcd}\), the integrand equals \(2\,\epsilon_{abcd}\,\delta\omega^{ab}\wedge T^{c}\wedge e^{d}\), so the field equation is \(\epsilon_{abcd}\,T^{c}\wedge e^{d}=0\) for every pair \(ab\).

To see that this forces \(T^{a}=0\), expand \(T^{c}=\tfrac12 T^{c}{}_{ef}\,e^{e}\wedge e^{f}\) and wedge the equation with \(e^{g}\); using \(e^{e}\wedge e^{f}\wedge e^{d}\wedge e^{g} =\epsilon^{efdg}\,e^{0}\wedge e^{1}\wedge e^{2}\wedge e^{3}\) with the numerical symbol, the content is \(\epsilon_{abcd}\,\epsilon^{defg}\,T^{c}{}_{ef}=0\). The numerical contraction identity \(\epsilon^{defg}\epsilon_{dabc}=\delta^{efg}_{abc}\) (the \(3\times3\) determinant of Kronecker deltas; Lemma A.50 is its twice-contracted consequence) reduces this, after the six contractions, to

Tracing \(g\) with \(b\) gives \(T^{c}{}_{ac}=4T^{c}{}_{ac}-T^{c}{}_{ac}\), so the trace vanishes, and with it the whole tensor: \(T^{a}=0\).

The \(e\) equation. On the surface \(T^{a}=0\) the connection is the unique torsion-free \(\omega(e)\) of Proposition 13.158, and substituting it back gives the second-order functional \(S[e,\omega(e)]\), which by Theorem 43.3 equals \(S_{\mathrm{EH}}[g]\) with \(g_{\mu\nu}=\eta_{ab}e^{a}{}_{\mu}e^{b}{}_{\nu}\). Now vary \(e\) in the first-order action at \(\omega=\omega(e)\): the chain rule splits the variation into the explicit \(\delta_{e}\) at fixed \(\omega\) and the induced \(\delta_{\omega}\) evaluated on \(\delta\omega(e)\) — and the latter vanishes, because \(\omega(e)\) is precisely the point where \(\delta_{\omega}S=0\). So the first-order \(e\) equation coincides with the total variation of \(S_{\mathrm{EH}}[g(e)]\), which by Theorem 44.9 and \(\delta g_{\mu\nu} =\eta_{ab}(e^{a}{}_{\mu}\,\delta e^{b}{}_{\nu} +e^{a}{}_{\nu}\,\delta e^{b}{}_{\mu})\) is \(\kappa^{-1}\!\int\sqrt{\abs{g}}\;G^{\mu\nu}\, \eta_{ab}e^{b}{}_{\nu}\,\delta e^{a}{}_{\mu}\,\dd^{4}x\). Since the vielbein is invertible, stationarity for all \(\delta e^{a}{}_{\mu}\) is equivalent to \(G^{\mu\nu}=0\).

Derives Proposition 44.17. \(F_{\mu\nu}\), defined from the potential with lower indices, does not involve the metric; all metric dependence of Equation (44.20) sits in the two raised indices and the volume factor. Varying,

and inserting Equation (44.7) for the first term,

which by Equation (44.5) is Equation (44.21). The trace is \(\mu_{0}^{-1}(F^{\nu\alpha}F_{\nu\alpha} -\tfrac14\cdot4\,F_{\alpha\beta}F^{\alpha\beta})=0\), a special feature of \(D=4\). For the energy density, work in a local orthonormal frame: \(F_{0}{}^{\alpha}F_{0\alpha}=g^{ij}F_{0i}F_{0j}=E^{2}/c^{2}\) and \(F_{\alpha\beta}F^{\alpha\beta}=2(B^{2}-E^{2}/c^{2})\), so

using \(\epsilon_{0}\mu_{0}=1/c^{2}\); both intermediate quantities are squares, so the identification is independent of the sign convention relating \(F_{0i}\) to \(E_{i}\).

Derives Theorem 44.19. Let \(\xi^{\mu}\) be an arbitrary compactly supported vector field and drag all fields along its flow. Diffeomorphism invariance says the action does not change: \(0=\delta_{\xi}S_{\mathrm{m}} =\int(\delta S_{\mathrm{m}}/\delta\psi)\,\delta_{\xi}\psi +\int(\delta S_{\mathrm{m}}/\delta g^{\mu\nu})\, \delta_{\xi}g^{\mu\nu}\). On shell the first integral vanishes. For the second, the Lie derivative of the metric is \(\delta_{\xi}g_{\mu\nu} =\nabla_{\mu}\xi_{\nu}+\nabla_{\nu}\xi_{\mu}\) (Equation (13.275) with \(\pp\to\nabla\) by metricity, as in the proof of Killing's equation Equation (13.290)), so by Equations (44.5) and (44.6)

using the symmetry of \(T^{\mu\nu}\). Integrate by parts with Lemma 44.10 (no boundary term, compact support): \(0=-\int\sqrt{\abs{g}}\,(\nabla_{\mu}T^{\mu\nu})\,\xi_{\nu}\), and since \(\xi_{\nu}\) is arbitrary, Equation (44.22) follows. This is Noether's second theorem (Theorem 16.93) in action: the symmetry group depends on four arbitrary functions \(\xi^{\mu}\), and what it yields is not four conserved charges but four differential identities [Noether:1918].

Derives Proposition 44.20. Insert Equation (44.19) into \(\nabla_{\mu}T^{\mu\lambda}=0\):

Contract with \(u_{\lambda}\): since \(u_{\lambda}u^{\lambda}=-c^{2}\) is constant, \(u_{\lambda}\nabla_{\mu}u^{\lambda}=0\), so the middle term dies, the first gives \(-c^{2}\nabla_{\mu}[(\rho+p/c^{2})u^{\mu}]\), and the last gives \(u^{\mu}\pp_{\mu}p\). Expanding \(\nabla_{\mu}[(p/c^{2})u^{\mu}] =(p/c^{2})\nabla_{\mu}u^{\mu}+(u^{\mu}/c^{2})\pp_{\mu}p\), the two pressure-gradient terms cancel and what remains, divided by \(-c^{2}\), is exactly Equation (44.23). Projecting instead with \(h^{\lambda}{}_{\sigma} =\delta^{\lambda}{}_{\sigma}+u^{\lambda}u_{\sigma}/c^{2}\), which annihilates \(u^{\lambda}\), kills the first term and produces Equation (44.24). In the nonrelativistic limit \(u^{\mu}\to(c,\vect{v})\), \(\nabla\to\pp\), \(\rho+p/c^{2}\to\rho\) and the projector becomes spatial; Equation (44.23) becomes \(\pp_{t}\rho+\nabla\cdot(\rho\vect{v})=0\) and Equation (44.24) becomes \(\rho(\pp_{t}+\vect{v}\cdot\nabla)\vect{v}=-\nabla p\), the equations of Fluid Dynamics.

Derives Corollary 44.21. Set \(p=0\) in Equation (44.24); the right side vanishes and the left is \(\rho\) times the geodesic equation Equation (43.1) in its velocity-field form.

Derives Theorem 44.23. Write the second Bianchi identity of Equation (13.308) with its cycle explicit,

Contract \(\lambda\) with \(\mu\) (legitimate under \(\nabla\) by metricity): the first term gives \(\nabla_{\sigma}R_{\rho\nu}\); the third, by antisymmetry in the last index pair, \(-\nabla_{\nu}R_{\rho\sigma}\); the second stays \(\nabla_{\lambda}R^{\lambda}{}_{\rho\nu\sigma}\):

Now contract with \(g^{\rho\sigma}\). The first term gives \(\nabla^{\rho}R_{\rho\nu}\) and the second \(-\pp_{\nu}R\). In the third, lower the index, \(g^{\rho\sigma}R^{\lambda}{}_{\rho\nu\sigma} =g^{\rho\sigma}g^{\lambda\alpha}R_{\alpha\rho\nu\sigma}\), and use the pair-exchange symmetry of Equation (13.308) followed by antisymmetry in each pair:

the last step being the Ricci contraction Equation (13.307) of the first and third slots. So the third term is \(\nabla_{\lambda}(g^{\lambda\alpha}R_{\nu\alpha}) =\nabla^{\rho}R_{\rho\nu}\). Altogether \(2\nabla^{\rho}R_{\rho\nu}=\pp_{\nu}R\), and

Derives Corollary 44.24. Apply \(\nabla^{\mu}\) to both sides of Equation (44.2) and use Equation (44.25).

Derives Proposition 44.26. For slow motion \(\dd x^{i}/\dd\tau\ll\dd x^{0}/\dd\tau\) and \(\dd x^{0}/\dd\tau=c\,\dd t/\dd\tau\approx c\), so among the \(\Gamma^{i}{}_{\mu\nu}\,\dot{x}^{\mu}\dot{x}^{\nu}\) terms of Equation (43.1) only \(\Gamma^{i}{}_{00}(\dot{x}^{0})^{2}\) survives at leading order. For a static metric,

where the spatial perturbation in \(g^{ij}\) multiplies a quantity already \(O(\Phi)\) and drops. Hence \(\dd^{2}x^{i}/\dd\tau^{2}=-c^{2}\,\Gamma^{i}{}_{00}=-\pp_{i}\Phi\) and, since \(\dd\tau\approx\dd t\) at this order, Equation (44.28). Note that only \(g_{00}\) enters: the identification of the potential is insensitive to the spatial part of Equation (44.27). The same \(g_{00}\) is what the gravitational-redshift measurements read directly (Phenomenon 42.5).

Derives Theorem 44.27. Work in the trace-reversed form Equation (44.3). For static dust at rest, Equation (44.18) gives \(T_{00}=\rho c^{2}\), the other components vanishing, and \(T=g^{\mu\nu}T_{\mu\nu}=-\rho c^{2}\) to leading order. Hence

On the geometric side, the Ricci contraction of Equation (13.306) is \(R_{00}=\pp_{\lambda}\Gamma^{\lambda}{}_{00} -\pp_{0}\Gamma^{\lambda}{}_{\lambda0}+O(\Gamma^{2})\); the \(\Gamma^{2}\) terms are \(O(\Phi^{2}/c^{4})\) and every time derivative vanishes, so with \(\Gamma^{i}{}_{00}=\pp_{i}\Phi/c^{2}\) from Proposition 44.26,

Equating the two expressions gives Equation (44.29). The Newtonian side is the field form of the inverse-square law: for a point mass, \(\Phi=-Gm/\abs{\vect{x}}\) (Central Forces and Statics), and superposing over a continuous density, \(\Phi(\vect{x})=-G\int\rho(\vect{x}')\, \abs{\vect{x}-\vect{x}'}^{-1}\dd^{3}x'\), the fundamental-solution property of \(-1/4\pi\abs{\vect{x}}\) (Proposition 10.67) gives \(\nabla^{2}\Phi=4\pi G\rho\). Matching coefficients, \(\kappa c^{4}/2=4\pi G\), which is Equation (44.30); the numerical value uses the CODATA \(G\) of Appendix B, whose laboratory determination is Experiment: The Cavendish Torsion Balance.

Derives Proposition 44.30. Divergence-freedom is \(\nabla^{\mu}(\Lambda g_{\mu\nu})=\Lambda\nabla^{\mu}g_{\mu\nu}=0\) together with Equation (44.25). For the action route, the extra term is \(-(\Lambda/\kappa)\int\dd^{4}x\sqrt{\abs{g}}\), whose variation by Equation (44.7) is \(+(\Lambda/2\kappa)\int\sqrt{\abs{g}}\,g_{\mu\nu}\delta g^{\mu\nu}\); added to Equation (44.12) it shifts \(G_{\mu\nu}\to G_{\mu\nu}+\Lambda g_{\mu\nu}\) in Equation (44.13).

Derives Proposition 44.31. By Theorem 13.160 a maximally symmetric metric has \(R_{\lambda\rho\mu\nu} =\tfrac{R}{12}(g_{\lambda\mu}g_{\rho\nu} -g_{\lambda\nu}g_{\rho\mu})\) with \(R\) constant, whence \(R_{\mu\nu}=\tfrac{R}{4}g_{\mu\nu}\). Insert into Equation (44.34) with \(T=0\): \(\tfrac{R}{4}g_{\mu\nu}-\tfrac{R}{2}g_{\mu\nu} +\Lambda g_{\mu\nu}=0\), so \(R=4\Lambda\), and back-substitution gives Equation (44.35). The curvature constant of Equation (13.318) is \(R/12=\Lambda/3\), giving \(\ell=(3/\Lambda)^{1/2}\); existence is the explicit quadric construction of Section 13.13. The case \(\Lambda<0\) gives anti-de Sitter space the same way.

Derives Corollary 44.35. Immediate from Theorem 44.33 applied to the left-hand side, plus the two calibrations cited.

Derives Proposition 44.37. Write the observer as \(w^{\mu}=\gamma(u^{\mu}+v^{\mu})\) with \(u_{\mu}v^{\mu}=0\), \(v^{\mu}\) spacelike of norm \(v^{2}<c^{2}\); normalization forces \(\gamma^{2}(c^{2}-v^{2})=c^{2}\), the Lorentz factor of the relative motion, and \(u_{\mu}w^{\mu}=-\gamma c^{2}\). Then from Equation (44.19),

Nonnegativity for all \(\gamma\ge1\) is equivalent to the two endpoint conditions \(\gamma=1\) (\(\rho\ge0\)) and \(\gamma\to\infty\) (\(\rho c^{2}+p\ge0\)), since the expression is affine in \(\gamma^{2}\): that is WEC. For a null \(k\), \(T_{\mu\nu}k^{\mu}k^{\nu}=(\rho+p/c^{2})(u_{\mu}k^{\mu})^{2}\) with \(u_{\mu}k^{\mu}\neq0\), giving NEC. For SEC, the trace is \(T=-\rho c^{2}+3p\), and

again affine in \(\gamma^{2}\): nonnegative for all \(\gamma\) iff the slope \(\rho c^{2}+p\ge0\) and the value at \(\gamma=1\), \(\tfrac12(\rho c^{2}+3p)\), is nonnegative. For DEC, compute the flux:

Causality of \(f\) for every \(v^{2}<c^{2}\) is equivalent to \(p^{2}c^{2}\le\rho^{2}c^{6}\), i.e. \(\abs{p}\le\rho c^{2}\), and future-direction follows from \(f^{\mu}u_{\mu}=-\gamma\rho c^{4}\le0\) when \(\rho\ge0\).

Derives Proposition 44.38. WEC: for any observer, evaluate in that observer's local orthonormal frame; by Proposition 44.17, \(T_{\mu\nu}w^{\mu}w^{\nu}/c^{2}=T_{\hat0\hat0} =\epsilon_{0}E^{2}/2+B^{2}/2\mu_{0}\ge0\), with \(E\), \(B\) the fields that observer measures. NEC follows by continuity. SEC: the tensor is traceless, so the SEC quadratic form equals the WEC form. DEC: in the observer's frame the flux components are \(f^{\hat0}=c\,T^{\hat0\hat0}=cu\) and \(f^{\hat\imath}=c\,T^{\hat\imath\hat0}=S^{\hat\imath}\), where \(u\) is the energy density above and \(\vect{S}=\vect{E}\times\vect{B}/\mu_{0}\) is the Poynting flux (The Maxwell Equations); causality of \(f\) is \(c^{2}u^{2}\ge\abs{\vect{S}}^{2}\). With \(a=\epsilon_{0}E^{2}\), \(b=B^{2}/\mu_{0}\),

using \(\abs{\vect{E}\times\vect{B}}\le\abs{\vect{E}}\abs{\vect{B}}\) and \(\epsilon_{0}/\mu_{0}\cdot E^{2}B^{2}=ab\). Future-direction is \(f^{\hat0}=cu\ge0\).

Derives Proposition 44.39. Trace-reverse Equation (44.34) as in Proposition 44.4: tracing gives \(-R+4\Lambda=\kappa T\), so \(R_{\mu\nu}=\kappa(T_{\mu\nu}-\tfrac12 g_{\mu\nu}T) +\Lambda g_{\mu\nu}\), and contracting twice with \(w\) (using \(g_{\mu\nu}w^{\mu}w^{\nu}=-c^{2}\)) gives Equation (44.38). For a null vector the \(g_{\mu\nu}\) terms contract to zero, leaving \(R_{\mu\nu}k^{\mu}k^{\nu}=\kappa T_{\mu\nu}k^{\mu}k^{\nu}\).

Derives Proposition 44.41. The contracted Bianchi identity Equation (44.25), written out,

holds identically in the metric. Every term on the right contains at most second \(x^{0}\)-derivatives of \(g_{\mu\nu}\) (\(G\) is second order and \(\pp_{i}\), \(\Gamma\) add no time derivative). If \(G^{0\nu}\) contained a second time derivative, say a term \(A\,\pp_{0}^{2}g_{\alpha\beta}\) with \(A\) built from lower-order quantities, the left side would contain the third derivative \(A\,\pp_{0}^{3}g_{\alpha\beta}\), which nothing on the right can supply; since the identity holds for every metric, the coefficient \(A\) vanishes identically.

Derives Theorem 44.42. The geometric identities are the Gauss and Codazzi equations, proved for arbitrary signature in Appendix A.52 with the normalization \(\epsilon=n^{\mu}n_{\mu}\) carried as a symbol; in this chapter's signature a spacelike surface has \(\epsilon=-1\) and the induced metric \(\gamma_{\mu\nu}=g_{\mu\nu}-\epsilon n_{\mu}n_{\nu}\) used there is our positive-definite \(h_{\mu\nu}\).

Hamiltonian constraint. Contract the Gauss equation Equation (A.985) over its first and third indices, then trace with \(h^{\nu\mu}\). The intrinsic side gives \({}^{(3)}\!R\). The quadratic terms give \(-\epsilon(K_{\mu\nu}K^{\mu\nu}-K^{2})\). The ambient side is the double \(h\)-trace of the projected Riemann tensor; expanding \(h^{\mu\nu}=g^{\mu\nu}+n^{\mu}n^{\nu}\) in both traces and using the Riemann symmetries, the \(nnnn\) term vanishes by antisymmetry, and

Hence, at \(\epsilon=-1\), \({}^{(3)}\!R =R+2R_{\mu\nu}n^{\mu}n^{\nu} +K_{\mu\nu}K^{\mu\nu}-K^{2}\). But \(2G_{\mu\nu}n^{\mu}n^{\nu} =2R_{\mu\nu}n^{\mu}n^{\nu}+R\) since \(g_{\mu\nu}n^{\mu}n^{\nu}=-1\), so

and the field equations replace \(G_{nn}\) by \(\kappa T_{nn}\), giving Equation (44.39).

Momentum constraint. The contracted Codazzi equation Equation (A.990) is \(D_{\nu}(K^{\nu}{}_{\mu}-K\,h^{\nu}{}_{\mu}) =h^{\alpha}{}_{\mu}R_{\alpha\kappa}n^{\kappa}\), with no factor of \(\epsilon\). Since \(h^{\alpha}{}_{\mu}g_{\alpha\kappa}n^{\kappa} =h^{\alpha}{}_{\mu}n_{\alpha}=0\), the Ricci tensor may be replaced by the Einstein tensor, and the field equations by \(\kappa T\); restricting the free index to \(\Sigma\) gives Equation (44.40).

That these are the four equations of Proposition 44.41 follows because \(G_{\mu\nu}n^{\nu}\) and \(G^{0\nu}\) span the same four combinations: \(n_{\mu}\propto\pp_{\mu}x^{0}\) on the surface. Neither Equation (44.39) nor Equation (44.40) contains a time derivative of \(K_{ij}\), i.e. a second time derivative of \(h_{ij}\): they restrict the initial data.

Derives Proposition 44.44. Only the terms of \(R_{\mu\nu}\) carrying second derivatives need tracking; everything else is by definition collected in \(Q_{\mu\nu}\). From the Ricci contraction of Equation (13.306), \(R_{\mu\nu}=\pp_{\lambda}\Gamma^{\lambda}{}_{\nu\mu} -\pp_{\nu}\Gamma^{\lambda}{}_{\lambda\mu}+\Gamma\Gamma\), and the Christoffel formula, the second-derivative content is

\(\simeq\) denoting equality modulo first-derivative terms. On the other hand \(\Gamma_{\nu}\simeq g^{\lambda\sigma}(\pp_{\lambda}g_{\sigma\nu} -\tfrac12\pp_{\nu}g_{\lambda\sigma})\) up to products \((\pp g)(g)\), so

Subtracting reproduces exactly the missing \(-\tfrac12 g^{\lambda\sigma}\pp_{\lambda}\pp_{\sigma}g_{\mu\nu}\), which is Equation (44.41). A coordinate function \(x^{\alpha}\) satisfies \(\Box_{g}x^{\alpha} =g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}x^{\alpha} =-g^{\mu\nu}\Gamma^{\alpha}{}_{\mu\nu}=-\Gamma^{\alpha}\), so harmonicity of all four coordinates is \(\Gamma^{\alpha}=0\), and Equation (44.42) follows.