Equivalence of the Einstein–Hilbert Action in Metric and Vielbein Variables

Derives Lemma A8.2. Both sides of Equation (A8.3) are, as functions of the four free indices \(i,j,l,m\), antisymmetric under \(i\leftrightarrow j\) and under \(l\leftrightarrow m\) separately: for the right side this is immediate; for the left side, swapping \(i\leftrightarrow j\) swaps the first two slots of \(\epsilon^{ij k_{3}\ldots}\), which flips its sign by total antisymmetry, and the swap \(l\leftrightarrow m\) acts the same way on the second factor. Hence both sides vanish whenever \(i=j\) or \(l=m\), and, by the double antisymmetry, it suffices to verify Equation (A8.3) for the case \(i=l\), \(j=m\), \(i\neq j\) (every other nonvanishing assignment of \(\set{i,j}=\set{l,m}\) then follows by the antisymmetries just established, and every assignment with \(\set{i,j}\neq\set{l,m}\) gives \(0=0\): the left side vanishes because no choice of \(k_{3},\ldots,k_{D}\) can make both \(\epsilon\) factors nonzero when \(\set{i,j}\ne\set{l,m}\), since each factor forces its own two free indices to be distinct from the shared \(k_{3},\ldots,k_{D}\) and from each other, and a nonzero product further forces the sets \(\set{i,j}\) and \(\set{l,m}\) to be identical, as both equal the complement of \(\set{k_{3},\ldots,k_{D}}\) in \(\set{1,\ldots,D}\)).

At \(i=l\), \(j=m\), \(i\ne j\): the right side of Equation (A8.3) is \((D-2)!\,(1-0)=(D-2)!\). For the left side, \(\epsilon^{ijk_{3}\ldots k_{D}}\epsilon_{ijk_{3}\ldots k_{D}}\) (no sum yet on \(i,j\)) is nonzero only when \((k_{3},\ldots,k_{D})\) is a permutation of the \(D-2\) elements of \(\set{1,\ldots,D}\setminus\set{i,j}\), and each such term contributes \(\left(\epsilon^{ijk_{3}\ldots k_{D}}\right)^{2}=1\); there are exactly \((D-2)!\) such permutations, so the sum equals \((D-2)!\). The two sides agree.

Proof of Theorem A8.1. Derives Theorem A8.1. Fix a point \(P\in M\). Expand the curvature two-form and the vielbeins in the coordinate basis,

so that, writing \(\Omega\) for the left side of Equation (A8.1),

Choice of point-adapted frame.

Both sides of Equation (A8.1) are built — with no further differentiation — from the algebraic data \(\left(g_{\mu\nu}(P),\,R^{a}{}_{b\mu\nu}(P),\,e^{a}{}_{\mu}(P)\right)\) and the numerical symbols \(\epsilon,\eta\); consequently, once verified in one admissible frame and coordinate system at \(P\), the identity holds in every frame and coordinate system at \(P\), by the tensorial (respectively, local-Lorentz-covariant) transformation laws of \(g\), \(R^{a}{}_{b\mu\nu}\) and \(e^{a}{}_{\mu}\) established in Section 17.5.2, Section 17.8 and Theorem 17.156 — both sides of Equation (A8.1) transform as the components of a \(D\)-form, i.e. by the same Jacobian determinant, so an equality of components in one frame is an equality of components in every frame. It therefore suffices to prove Equation (A8.1) at \(P\) in a coordinate system and frame chosen so that

Such a choice exists: pick any coordinates first, so that \(g_{\mu\nu}(P)\) is some fixed nondegenerate symmetric matrix of signature \((p,q)\); a constant linear change of coordinates diagonalizes it to \(\eta_{\mu\nu}\) at \(P\) (Sylvester's law of inertia, Algebraic Structures), after which the vielbein \(e^{a}{}_{\mu}=\delta^{a}_{\mu}\) satisfies \(\eta_{ab}\,\delta^{a}_{\mu}\delta^{b}_{\nu}=\eta_{\mu\nu}=g_{\mu\nu}(P)\) (Equation (17.264)), and every other vielbein compatible with \(g_{\mu\nu}(P)\) differs from this one by a local Lorentz transformation (Proposition 17.123), so this choice is always reachable.

Component computation at $P$.

Under Equation (A8.4), \(e^{a_{i}}{}_{\lambda_{i}}(P) =\delta^{a_{i}}_{\lambda_{i}}\) collapses each frame sum \(a_{3},\ldots,a_{D}\) against the epsilon symbol into the corresponding coordinate index, and \(R^{a_{1}a_{2}}{}_{\mu\nu}(P)\) becomes numerically the doubly-raised Riemann tensor \(g^{a_{1}\rho}(P)g^{a_{2}\sigma}(P)R_{\rho\sigma\mu\nu}(P)\), so that at \(P\),

where the \(\rho,\sigma\) sum has simply been relabelled from \(a_{1},a_{2}\). Reordering the wedge of coordinate differentials (Section 17.6.3), \(\dd x^{\mu}\wedge\dd x^{\nu}\wedge\dd x^{\lambda_{3}} \wedge\cdots\wedge\dd x^{\lambda_{D}} =\varepsilon^{\mu\nu\lambda_{3}\ldots\lambda_{D}}\, \dd x^{1}\wedge\cdots\wedge\dd x^{D}\), so

By Lemma A8.2 (with \(D-2\) contracted indices \(\lambda_{3},\ldots,\lambda_{D}\)), the bracket equals \((D-2)!\left(\delta^{\mu}_{\rho}\delta^{\nu}_{\sigma} -\delta^{\mu}_{\sigma}\delta^{\nu}_{\rho}\right)\), giving

Renaming the summed indices \(\rho\leftrightarrow\sigma\) in the second term and using the antisymmetry of the curvature two-form in its upper pair, \(R^{\rho\sigma}{}_{\mu\nu}=-R^{\sigma\rho}{}_{\mu\nu}\) (inherited from \(\omega^{ab}=-\omega^{ba}\), Equation (17.310)),

so the bracket is \(2\,R^{\rho\sigma}{}_{\rho\sigma}(P)\) and

Identification with the Ricci scalar.

It remains to recognize \(R^{\rho\sigma}{}_{\rho\sigma}(P)\). Lowering both upper indices with \(g^{-1}(P)\),

Contract the inner index pair first: by Definition 17.153, \(R_{\mu\nu}=R^{\lambda}{}_{\mu\lambda\nu}=g^{\lambda\kappa} R_{\kappa\mu\lambda\nu}\), i.e. contracting the first index of the (fully lowered) Riemann tensor with the third, via \(g^{-1}\), produces the Ricci tensor in its second and fourth slots. Applying this with \(\kappa\to\rho,\,\mu\to\beta,\,\lambda\to\alpha,\,\nu\to\sigma\) gives \(g^{\rho\alpha}R_{\alpha\beta\rho\sigma}=R_{\beta\sigma}\), so

the last equality being Equation (17.307). Hence \(R^{\rho\sigma}{}_{\rho\sigma}(P) = R(P)\), and Equation (A8.5) reads

Finally, under Equation (A8.4), \(g(P)=\det\eta_{\mu\nu} =(-1)^{q}\), so \(\sqrt{\abs{g(P)}}=1\) and \(\dd x^{1}\wedge\cdots\wedge\dd x^{D}=\sqrt{\abs{g(P)}}\,\dd^{D}x\) at \(P\). Substituting,

which is Equation (A8.1) at \(P\). Since \(P\in M\) was arbitrary and, by the covariance argument above, the identity transported to any frame at \(P\), Equation (A8.1) holds identically on \(M\). Dividing by \(2\kappa\,(D-2)!\) and integrating gives Equation (A8.2).