Comparison recovers covariant differentiation

Tensor slots can be read out reversibly. Differentiation needs more: a compatible comparison rule. Its construction recovers the full metric connection while keeping raw torsion and nonmetricity visible.

4. Tensor operations and compatible comparison

Tensor readouts and their inverses

For a calibrated tensor $U$ of type $(r,s)$ define

$$(\Pi_EU)^{\mu_1\cdots\mu_r}{}_{\nu_1\cdots\nu_s} =(E^{-1})^{\mu_1}{}_{a_1}\cdots(E^{-1})^{\mu_r}{}_{a_r} E^{b_1}{}_{\nu_1}\cdots E^{b_s}{}_{\nu_s} U^{a_1\cdots a_r}{}_{b_1\cdots b_s}. \tag{4.1}$$

Its inverse inserts $E$ in every upper coordinate slot and $E^{-1}$ in every lower coordinate slot. In particular $v=E^{-1}u$ and $\alpha=E^{\mathsf T}a$ for calibrated vector $u$ and covector $a$.

Proposition 4.1 (Tensor-algebra recovery)

The maps (4.1) are smooth fibrewise linear isomorphisms. They preserve sums, products, index permutations, symmetrization, antisymmetrization and contraction. Raising or lowering with $\eta$ before readout agrees with raising or lowering with $g$ afterward.

Proof.

Inverse factors cancel in each slot. A product adds slots, a permutation reorders factors, and a contraction cancels one $E$ with one $E^{-1}$. For the metric operations, $gE^{-1}u=E^{\mathsf T}\eta u$ and $qE^{\mathsf T}a=E^{-1}\eta^{-1}a$. Tensor products extend these identities to all ranks. Every tensor is a finite sum of elementary tensor products in a local basis, so the proof is not restricted to decomposable tensors.

$\square$

This recovers the transformations and algebra in Einstein (5)–Einstein (17). The invariant-pairing test for tensor character follows from nondegeneracy of evaluation on arbitrary basis tuples. For symmetric bilinear arrays, polarization permits diagonal probes; without symmetry they miss the antisymmetric part. Completely antisymmetric tensors in four dimensions have dimensions $\binom4k$, in particular $6,4,1$ at ranks $2,3,4$, and vanish above rank four. Symmetric tensors do not have that cutoff; the printed qualification is discussed in Appendix A.

The volume density transforms as $\nu_y=|\det J|\nu_x$, so $\nu|\mathrm{d}^4x|$ is invariant. This recovers Einstein (18) and Einstein (18a) without assuming an orientation. Signed integration uses an orientation chosen in addition.

Why a comparison connection is necessary

An arbitrary matrix-valued one-form $\omega$ defines

$$D_\mu u=\partial_\mu u+\omega_\mu u,\qquad D_\mu a=\partial_\mu a-\omega_\mu^{\mathsf T}a. \tag{4.2}$$

The product rule and contraction extend it to tensors. Since $u'=\Lambda u$, ordinary differentiation produces the unwanted term $(\mathrm{d}\Lambda)u$. Requiring $D'(\Lambda u)=\Lambda Du$ for every $u$ forces

$$\omega'=\Lambda\omega\Lambda^{-1}-\mathrm{d}\Lambda\Lambda^{-1}. \tag{4.3}$$

The raw discrepancies are

$$M_\mu=\omega_\mu^{\mathsf T}\eta+\eta\omega_\mu, \qquad \Theta^a=\mathrm{d}\theta^a+\omega^a{}_b\wedge\theta^b. \tag{4.4}$$

They transform as $M'=\Lambda^{-\mathsf T}M\Lambda^{-1}$ and $\Theta'=\Lambda\Theta$. Their zero sector preserves the interval pairing and is torsion-free. This is the realization law corresponding to Einstein’s metric differential calculus, not a consequence of native admission.

Let $e_b=(E^{-1})^\mu{}_b\partial_\mu$ and define structure coefficients by

$$\mathrm{d}\theta^a=-\tfrac12c^a{}_{bc}\theta^b\wedge\theta^c, \qquad c^a{}_{bc}=-(E^{-1})^\mu{}_b(E^{-1})^\nu{}_c (\partial_\mu E^a{}_{\nu}-\partial_\nu E^a{}_{\mu}). \tag{4.5}$$

Lower the first index with $\eta$, and write $\omega^a{}_b=\omega^a{}_{bc}\theta^c$.

Theorem 4.2 (Unique compatible comparison)

For every smooth invertible calibration, the unique connection with $M=0$ and $\Theta=0$ is

$$\boxed{\omega_{abc}=\tfrac12(c_{cab}-c_{abc}-c_{bca}),\qquad \omega^a{}_{bc}=\eta^{ad}\omega_{dbc}.} \tag{4.6}$$

The last index is the differentiating frame direction.

Proof.

Antisymmetry of the final two $c$ indices gives $\omega_{abc}=-\omega_{bac}$ and $\omega_{acb}-\omega_{abc}=c_{abc}$. These are respectively $M=0$ and $\Theta=0$, because $\Theta^a(e_b,e_c)=-c^a{}_{bc}+\omega^a{}_{cb}-\omega^a{}_{bc}$. The difference $K$ of two solutions is antisymmetric in its first two indices and symmetric in its last two. Thus

$$K_{abc}=K_{acb}=-K_{cab}=-K_{cba}=K_{bca}=K_{bac}=-K_{abc},$$

so $K=0$. The displayed formula uses only $E,E^{-1}$ and first derivatives, and is smooth on the declared domain.

$\square$

Connection readout and complete defects

For arbitrary raw $\omega$, define

$$C_\mu=E^{-1}(\partial_\mu E+\omega_\mu E),\qquad \omega_\mu=EC_\mu E^{-1}-(\partial_\mu E)E^{-1}. \tag{4.7}$$

Direct substitution, without imposing compatibility, gives

$$\begin{aligned} \nabla^C_\mu g&=\partial_\mu g-C_\mu^{\mathsf T}g-gC_\mu =-E^{\mathsf T}M_\mu E,&&\text{(4.8)}\\ C^\rho{}_{\mu\nu}-C^\rho{}_{\nu\mu} &=(E^{-1})^\rho{}_a\Theta^a{}_{\mu\nu}. &&\text{(4.9)}\end{aligned}$$

The two zero conditions are therefore equivalent to metric compatibility and lower-index symmetry of $C$. Writing metric compatibility for three cyclic placements of indices, adding two equations and subtracting the third, yields

$$\boxed{C^\rho{}_{\mu\nu}=\tfrac12q^{\rho\lambda} (\partial_\mu g_{\lambda\nu}+\partial_\nu g_{\lambda\mu} -\partial_\lambda g_{\mu\nu}).} \tag{4.10}$$

This recovers Einstein’s brackets and braces in Einstein (21) and Einstein (23) from the source comparison, rather than using the completed Christoffel formula as its forward definition.

Internal frame changes leave $C$ fixed. Coordinate changes instead give

$$C^{(y)}_\alpha=J^{-1}\!\left(\sum_\mu J^\mu{}_{\alpha}C^{(x)}_\mu\right)J +J^{-1}\partial_\alpha J. \tag{4.11}$$

This follows by differentiating $E_xJ$. Equal metric readouts imply frame-related calibrations; the transformed $\omega$ satisfies the same zero conditions, so uniqueness in Theorem 4.2 implies equal coordinate differential operations. The same proof on overlaps gives compatible connection gluing without a global coframe.

Theorem 4.3 (Differentiation intertwines with readout)

Define the appended calibrated derivative slot by $D_b=(E^{-1})^\mu{}_bD_\mu$. For every tensor type, after reading out that slot as well,

$$\boxed{\Pi_E(DU)=\nabla^C(\Pi_EU).} \tag{4.12}$$

The identity holds on the raw comparison domain. On $M=\Theta=0$, the right side is Einstein’s metric covariant derivative.

Proof.

For vectors, $u=Ev$ gives $D_\mu(Ev)=E(\partial_\mu v+C_\mu v)$ by (4.7). For covectors, $\alpha=E^{\mathsf T}a$ gives $\partial_\mu\alpha-C_\mu^{\mathsf T}\alpha =E^{\mathsf T}(\partial_\mu a-\omega_\mu^{\mathsf T}a)$. The extra $E$ on the derivative slot cancels its $E^{-1}$. Scalars, products and contractions give all ranks by Proposition 4.1.

$\square$