Intervals determine the metric

A calibration turns coordinate directions into measured components. Its quadratic readout recovers every local Lorentz metric, and its complete fibre identifies precisely the frame freedom left unresolved.

3. Local measurements and metric recovery

Einstein’s Part A motivates a generally covariant description through local rods and clocks, universal free fall and the invariance of event coincidences. His Machian discussion is motivation, not a theorem establishing that distant matter determines every inertial property. The mathematical starting premise in §4 is local special relativity: the measured infinitesimal interval has form $\eta$, and local measured differentials depend linearly on general coordinate differentials. We inherit that premise and make its representation explicit.

On a patch $U\subset M$, let $E_p:T_pU\to\mathbb R^{3,1}$ be a smooth linear isomorphism. Set

$$\theta^a=E^a{}_{\mu}\mathrm{d}x^\mu,\qquad Q_E(p,v)=\eta(E_pv,E_pv),\qquad g=\pi(E)=E^{\mathsf T}\eta E. \tag{3.1}$$

Polarization gives $g(u,v)=[Q_E(u+v)-Q_E(u)-Q_E(v)]/2$. Symmetry and nondegeneracy follow immediately, and the invertible change $v\mapsto Ev$ proves the signature. In particular,

$$\det g=-(\det E)^2<0,\qquad q=E^{-1}\eta^{-1}E^{-\mathsf T},\qquad \nu=|\det E|. \tag{3.2}$$

Negativity of the determinant alone would not identify the chosen inertia. Equations (3.1)–(3.2) recover Einstein (1)–Einstein (4); $E=I$ gives the special-relativistic coordinate case. The one-forms $\theta^a$ need not be simultaneously exact on an open region.

At one event the ten independent metric components are recovered from four diagonal and six pair evaluations:

$$g_{\mu\mu}=Q_E(\partial_\mu),\qquad g_{\mu\nu}=\frac{Q_E(\partial_\mu+\partial_\nu)-Q_E(\partial_\mu)-Q_E(\partial_\nu)}2 \quad(\mu<\nu). \tag{3.3}$$

This is algebraic coverage of a mathematical readout, not a claim that a laboratory protocol implementing these ten probes has been supplied.

Theorem 3.1 (Local section and complete frame fibre)

Every smooth Lorentz metric of signature (2.5) has local calibration preimages. At fixed event coordinates two calibrations $E,\widetilde E$ have the same metric exactly when

$$\widetilde E=\Lambda E,\qquad \Lambda^{\mathsf T}\eta\Lambda=\eta. \tag{3.4}$$

Thus the fibre is precisely the smooth local $O(3,1)$ frame orbit. This is not a quotient by coordinate transformations or a uniqueness theorem for the native source.

Proof.

Choose a sufficiently small chart whose fourth coordinate vector is timelike, and write

$$g=\begin{pmatrix} C&b\\b^{\mathsf T}&a\end{pmatrix},\quad a>0, \qquad S=\frac{bb^{\mathsf T}}a-C,\quad R^{\mathsf T}R=S, \qquad \iota(g)=\begin{pmatrix}R&0\\b^{\mathsf T}/\sqrt a&\sqrt a\end{pmatrix}. \tag{3.5}$$

Completion of squares shows $S$ is positive definite: $a$ accounts for the sole positive direction of $g$. Its upper triangular Cholesky factor $R$ with positive diagonal is smooth. Explicitly $R_{11}=\sqrt{S_{11}}$, $R_{1j}=S_{1j}/R_{11}$, $R_{22}=\sqrt{S_{22}-R_{12}^2}$, $R_{23}=(S_{23}-R_{12}R_{13})/R_{22}$ and $R_{33}=\sqrt{S_{33}-R_{13}^2-R_{23}^2}$. All radicands are positive. Multiplication yields $\iota(g)^{\mathsf T}\eta\iota(g)=g$. Every Lorentz metric admits a timelike coordinate direction near any point, so this covers every target locally, not just metrics already in this chart form.

If the two metric readouts agree, put $\Lambda=\widetilde E E^{-1}$. Then $\Lambda^{\mathsf T}\eta\Lambda=E^{-\mathsf T}\widetilde E^{\mathsf T}\eta\widetilde E E^{-1}=\eta$. Conversely (3.4) preserves $g$. Smoothness follows from smooth matrix inversion. Generally $\iota\pi(E)\ne E$, but it is frame equivalent to $E$.

$\square$

Variations, coordinate changes and compatible patches

For an arbitrary calibration variation $H$,

$$D\pi_E[H]=H^{\mathsf T}\eta E+E^{\mathsf T}\eta H, \qquad H=\tfrac12E qh\quad\Longrightarrow\quad D\pi_E[H]=h \tag{3.6}$$

for every symmetric metric variation $h$. The kernel consists of $H=AE$ with $A^{\mathsf T}\eta+\eta A=0$, as follows by multiplying $D\pi_E[H]=0$ by $E^{-\mathsf T}$ and $E^{-1}$. Compact support is preserved, and sufficiently small compactly supported variations preserve invertibility. Six vertical frame directions account for the difference between sixteen calibration components and ten metric components; this is not a count of propagating gravitational degrees of freedom.

For $x=x(y)$, let $J=\partial x/\partial y$. Coordinate and frame operations obey

$$E_y=E_xJ,\qquad g_y=J^{\mathsf T}g_xJ,\qquad Q_{E_y}(v_y)=Q_{E_x}(Jv_y),\qquad E'=\Lambda E. \tag{3.7}$$

Jacobians compose in chain-rule order. Coordinate injectivity preserves event coincidences. Left frame multiplication and right coordinate multiplication commute with the necessary pullback of their coefficient fields, but are conceptually different operations.

On an atlas retain independent $E_i$ and overlap comparisons $L_{ji}$. Their raw discrepancies include

$$E_j-L_{ji}E_iJ_{ij},\qquad L_{ji}^{\mathsf T}\eta L_{ji}-\eta, \qquad L_{ki}-L_{kj}L_{ji}, \quad J_{ij}=\frac{\partial x_i}{\partial x_j}. \tag{3.8}$$

All values refer to the same event, and reciprocal and identity transition rules are retained. Zero discrepancies give a smooth metric by (3.7). Conversely local sections of any smooth global Lorentz metric induce $L_{ji}=E_j(E_iJ_{ij})^{-1}$. Theorem 3.1 makes these matrices Lorentz; the Jacobian chain rule proves their cocycle identity. This supplies atlas coverage without requiring a global coframe or a trivial frame bundle.

What a coarser readout would lose

The same event and null-direction records can fail to determine calibrated intervals. On one event domain take $E_1=I$ and $E_2=\Omega I$, $\Omega=1+(x^1)^2$. The null sets agree, but

$$Q_{E_2}=\Omega^2Q_{E_1},\qquad Q_{E_1}(\partial_4)=1, \qquad Q_{E_2}(\partial_4)=\Omega^2. \tag{3.9}$$

Fixed units prevent erasing this nonconstant difference by one unit conversion. Einstein’s measurement coincidences include rods and clock readings; they must not be reduced to an uncalibrated equality relation on events.

At the other extreme, demanding $\theta^a=\mathrm{d}X^a$ simultaneously throughout a neighborhood gives, by invertibility, coordinates $X$ with $g=\eta_{ab}\mathrm{d}X^a\mathrm{d}X^b$. This restricts the geometry to local flatness. Nonclosed calibration forms, however, do not themselves prove curvature.

For an explicit flat counterexample, set $N=1+ax>0$, $a\ne0$, and $E=\operatorname{diag}(1,1,1,N)$. The transformation

$$X=(x+a^{-1})\cosh(at)-a^{-1},\qquad T=(x+a^{-1})\sinh(at) \tag{3.10}$$

gives $-\mathrm{d}X^2+\mathrm{d}T^2=-\mathrm{d}x^2+N^2\mathrm{d}t^2$. The same interval readout gives $\mathrm{d}\tau=N\mathrm{d}t$ for stationary clocks and $\mathrm{d}x/\mathrm{d}t=\pm N$ for coordinate light speeds, while the local calibrated speed is one. Yet $\mathrm{d}\theta^4=a\mathrm{d}x\wedge\mathrm{d}t\ne0$. Connection and curvature, rather than this frame-dependent nonclosure alone, are needed to distinguish genuine curvature.

The rotating measurement example

Einstein’s rotating-clock and rod argument in §3 is recovered by

$$(\theta^1,\theta^2,\theta^3,\theta^4) =(\mathrm{d}r,r(\mathrm{d}\varphi+\Omega\mathrm{d}t),\mathrm{d}z,\mathrm{d}t), \tag{3.11}$$

whose metric is

$$\mathrm{d}s^2=(1-\Omega^2r^2)\mathrm{d}t^2-2\Omega r^2\mathrm{d}\varphi\mathrm{d}t -\mathrm{d}r^2-r^2\mathrm{d}\varphi^2-\mathrm{d}z^2. \tag{3.12}$$

The Cartesian map $X=r\cos(\varphi+\Omega t)$, $Y=r\sin(\varphi+\Omega t)$, $Z=z$, $T=t$ independently produces this metric. At fixed rotating spatial coordinates and $|\Omega r|<1$, $\mathrm{d}\tau=\sqrt{1-\Omega^2r^2}\mathrm{d}t$. For a local rest-space rod, orthogonality to that observer gives $g_{44}\mathrm{d}t+g_{4i}\mathrm{d}x^i=0$. Only after retaining the mixed term may one eliminate $\mathrm{d}t$, obtaining

$$\mathrm{d}\ell^2=\left(-g_{ij}+\frac{g_{i4}g_{j4}}{g_{44}}\right)\mathrm{d}x^i\mathrm{d}x^j =\mathrm{d}r^2+\mathrm{d}z^2+\frac{r^2}{1-\Omega^2r^2}\mathrm{d}\varphi^2. \tag{3.13}$$

Thus $C(R)=2\pi R/\sqrt{1-\Omega^2R^2}$ and $D(R)=2R$, giving $C/D>\pi$ for a nonzero subluminal rim speed. This is a sum of infinitesimal comoving rod comparisons, not a globally synchronized rotating hypersurface. The center uses a Cartesian patch; the polar frame is not assumed nonsingular at $r=0$.