Keep the nonlinear curvature

The comparison commutator recovers both derivative and quadratic curvature terms. Local flatness, coordinate-volume choices and variation coverage are then different consequences of that complete operation.

6. Curvature, local flatness and variation coverage

The comparison curvature is defined by the commutator of the source derivatives:

$$\mathcal F=\mathrm{d}\omega+\omega\wedge\omega,\qquad \mathcal F_{\mu\nu}=\partial_\mu\omega_\nu-\partial_\nu\omega_\mu +[\omega_\mu,\omega_\nu]. \tag{6.1}$$

The ordered matrix product cannot be discarded.

Theorem 6.1 (Curvature recovery on the raw domain)

For every raw comparison pair $(E,\omega)$ and its readout (4.7),

$$\boxed{R_{\mu\nu}[C]=E^{-1}\mathcal F_{\mu\nu}[\omega]E.} \tag{6.2}$$

On the compatible torsion-free sector this is the Riemann tensor of the recovered metric.

Proof.

The vector derivative identity is the operator equality $(\partial_\mu+\omega_\mu)E=E(\partial_\mu+C_\mu)$. Take its commutator in $\mu,\nu$. Derivatives of the arbitrary test vector cancel, leaving $\mathcal F_{\mu\nu}E=ER_{\mu\nu}$. Expansion yields all four terms in (2.6), including both quadratic products. No zero residual or field equation has been used.

$\square$

Under a frame change $\mathcal F'=\Lambda\mathcal F\Lambda^{-1}$, so its coordinate readout is fixed. Coordinate changes act on the two direction slots and the vector/covector slots, proving tensor character. Einstein appends derivative indices, so his $A_{\mu\sigma\tau}$ means $\nabla_\tau\nabla_\sigma A_\mu$, including the derivative-slot connection. Subtraction gives

$$A_{\mu\sigma\tau}-A_{\mu\tau\sigma} =R^\rho{}_{\mu\sigma\tau}A_\rho +T^\lambda{}_{\sigma\tau}\nabla_\lambda A_\mu. \tag{6.3}$$

On zero torsion this is Einstein (42)–Einstein (43). Einstein contracts the upper index with the final lower index, giving $B_{\mu\nu}=R^\rho{}_{\mu\nu\rho}=-\operatorname{Ric}_{\mu\nu}$. In particular,

$$\begin{aligned} B_{\mu\nu}&=W_{\mu\nu}+S_{\mu\nu},&&\text{(6.4)}\\ W_{\mu\nu}&=-\partial_\alpha C^\alpha{}_{\mu\nu} +\operatorname{tr}(C_\mu C_\nu),\qquad S_{\mu\nu}=\partial_\mu c_\nu-C^\alpha{}_{\mu\nu}c_\alpha. &&\text{(6.5)}\end{aligned}$$

This is Einstein (44), including its original two-part decomposition. The first term $W$ is not the unrestricted Ricci tensor; the volume defect $S$ is part of the identity.

Flatness and coordinate-volume specialization

If a metric is constant on a chart, $C=0$ there, so $R=0$. Conversely, on a sufficiently small simply connected neighborhood, $\mathcal F=0$ is the integrability condition for $\mathrm{d}\Lambda=\Lambda\omega$ with $\Lambda(p)=I$. One may solve along paths by the linear transport ODE. Differentiating the transport with respect to an endpoint-fixed path deformation gives an integral of conjugated curvature, hence zero. The local simply connected domain makes the result path independent. Interval compatibility keeps $\Lambda$ Lorentz. Equation (4.3) gives $\omega'=0$, and zero torsion then gives $\mathrm{d}(\Lambda\theta)=0$. On a star-shaped subpatch, integration of closed one-forms gives $\Lambda\theta=\mathrm{d}X$; invertibility makes $X$ coordinates. This proves the local flatness converse mentioned in Einstein’s footnote, not global constant coordinates or trivial holonomy on an arbitrary manifold.

Likewise, any positive smooth coordinate density can be made one locally. Set

$$y^1=\int_{x_0^1}^{x^1}\nu(s,x^2,x^3,x^4)\mathrm{d}s, \qquad y^j=x^j\quad(j=2,3,4). \tag{6.6}$$

Its Jacobian determinant is $\nu>0$, so the inverse function theorem gives $\nu_y=1$. Orientation-preserving transitions between such charts have determinant one, recovering Einstein (19). This coordinate choice makes $S=0$ in (6.4); it is neither a restriction to flat geometry nor an internal Lorentz-frame change.

Variations of the differential operation

A smooth metric family lifts through the local section of Theorem 3.1, and its connection is then constructed by (4.6). Differentiating the two comparison conditions gives

$$\delta\omega^{\mathsf T}\eta+\eta\delta\omega=0,\qquad \mathrm{d}(\delta\theta)+\delta\omega\wedge\theta+\omega\wedge\delta\theta=0. \tag{6.7}$$

The algebraic operator on $\delta\omega$ is the same invertible torsion system used in Theorem 4.2. Every $\delta E$ therefore induces a unique comparison response. Projecting the differentiated compatibility and symmetry identities gives

$$\boxed{\delta C^\rho{}_{\mu\nu} =\tfrac12q^{\rho\lambda} (\nabla_\mu h_{\lambda\nu}+\nabla_\nu h_{\lambda\mu} -\nabla_\lambda h_{\mu\nu}),\qquad h=\delta g.} \tag{6.8}$$

The proof is the same three-index addition and subtraction that gave (4.10), now linearized. Compact support is preserved because the construction uses only finitely many derivatives. This is variation coverage, not a separate independent-connection action principle.

Controls on the nonlinear comparison

For $E=\operatorname{diag}(1,1,1,N(x))$, setting $\omega=0$ preserves $M=0$ but gives $\Theta\ne0$ when $N'\ne0$. The bare connection has $C^4{}_{14}=N'/N$, $C^4{}_{41}=0$ and zero matrix curvature. The compatible construction instead gives

$$\omega^1{}_4=\omega^4{}_1=N'\mathrm{d}t,\quad C^1{}_{44}=NN',\quad C^4{}_{14}=C^4{}_{41}=N'/N, \quad R^1{}_{414}=NN''. \tag{6.9}$$

Linear $N$ gives the flat example (3.10); $N=e^{kx}$ gives nonzero curvature. Matching the metric alone does not license an arbitrary comparison derivative.

A lapse-only example does not test every quadratic term. With $E=\operatorname{diag}(a,a,a,1)$, $a=e^{Ht}$ and $H\ne0$, the compatible connection satisfies $\omega^i{}_4=\omega^4{}_i=H\theta^i$. Then

$$\mathcal F^1{}_2=H^2\theta^1\wedge\theta^2=H^2a^2\mathrm{d}x\wedge\mathrm{d}y, \qquad \mathrm{d}\omega^1{}_2=0. \tag{6.10}$$

Deleting $\omega\wedge\omega$ loses this component. The full calculation gives $\operatorname{Ric}=-3H^2g$ and $\mathcal R=-12H^2$ in the declared convention. This is a designed test geometry, not a vacuum solution or a new spacetime discovery.