From stationary intervals to differential identities

The same calibrated interval supplies geodesic motion, Hessians, volume derivatives and divergence identities. Boundary terms and tensor symmetries identify where each recovered operation applies.

5. Geodesic variation and differential identities

Fixed-endpoint interval variation

For a regular timelike curve $\gamma(\lambda)$ set

$$L=\sqrt{\eta(E\dot\gamma,E\dot\gamma)} =\sqrt{g_{\mu\nu}\dot x^\mu\dot x^\nu},\qquad S[\gamma]=\int L\mathrm{d}\lambda. \tag{5.1}$$

Here $E$ is a background during curve variation. Stationarity of the interval is Einstein’s inherited premise Einstein (20), not a native-source theorem. For variation $\xi$,

$$\delta S=\left[\frac{g_{\sigma\nu}\dot x^\nu}{L}\xi^\sigma\right]_{\lambda_1}^{\lambda_2} -\int r_\sigma\xi^\sigma\mathrm{d}\lambda, \qquad r_\sigma=\frac{\mathrm{d}}{\mathrm{d}\lambda}\frac{g_{\sigma\nu}\dot x^\nu}{L} -\frac{\partial_\sigma g_{\mu\nu}\dot x^\mu\dot x^\nu}{2L}. \tag{5.2}$$

The displayed boundary term vanishes for fixed endpoints, not automatically for moving endpoints. Substituting (4.10) gives

$$r_\sigma=\frac{g_{\sigma\rho}}L \left(\ddot x^\rho+C^\rho{}_{\mu\nu}\dot x^\mu\dot x^\nu -\frac{\dot L}{L}\dot x^\rho\right). \tag{5.3}$$

All curve variations lift as the same event-curve variations. With proper-time parameter $s$, $L=1$, and the equivalent source equation is

$$\boxed{D_su=0,\qquad u=E\frac{\mathrm{d}x}{\mathrm{d}s},\qquad \frac{\mathrm{d}^2x^\rho}{\mathrm{d}s^2}+C^\rho{}_{\mu\nu} \frac{\mathrm{d}x^\mu}{\mathrm{d}s}\frac{\mathrm{d}x^\nu}{\mathrm{d}s}=0.} \tag{5.4}$$

This recovers Einstein (20a)–Einstein (22). Metric compatibility preserves $\eta(u,u)$ along the source equation. Spacelike curves can be treated with the explicit real convention $L=\sqrt{|g(\dot\gamma,\dot\gamma)|}$ and a fixed sign; the same unparameterized equation follows. The division by $L$ excludes null curves. A separate optical or Hamiltonian operation is used in Section 16.

The original scalar-to-tensor progression

For a scalar $\phi$, $a_b=e_b\phi$ reads out as $\partial_\mu\phi$, recovering Einstein (24). On an affine geodesic,

$$\frac{\mathrm{d}^2\phi}{\mathrm{d}s^2} =\left(\partial_\mu\partial_\nu\phi-C^\rho{}_{\mu\nu}\partial_\rho\phi\right) \frac{\mathrm{d}x^\mu}{\mathrm{d}s}\frac{\mathrm{d}x^\nu}{\mathrm{d}s}. \tag{5.5}$$

The Hessian is the readout of $D(\mathrm{d}\phi)$ and is symmetric on the torsion-free sector, recovering Einstein (25). Timelike probes already form an open set, so a polynomial identity verified for all such probes determines its coefficients without an improper null normalization.

Locally every covector is $\sum_i f_i\mathrm{d}h_i$ by taking fixed coordinate functions $h_i$ and coefficient functions $f_i$. Under subsequent redescription they remain the same scalar functions. The product rule therefore extends the derivative from gradients to all covectors, recovering Einstein (26). Expanding a rank-two tensor in covector products gives

$$\nabla_\sigma A_{\mu\nu}=\partial_\sigma A_{\mu\nu} -C^\rho{}_{\sigma\mu}A_{\rho\nu}-C^\rho{}_{\sigma\nu}A_{\mu\rho}, \tag{5.6}$$

which is Einstein (27). Contravariant slots contribute plus signs and covariant slots minus signs, giving Einstein (38) and Einstein (39). This reproduces the original dependency argument, not only an isolated coordinate formula.

Determinants, divergences and antisymmetric derivatives

Differentiation of $qg=I$ and the determinant cofactor formula give

$$\begin{aligned} \partial_\lambda q&=-q(\partial_\lambda g)q,\qquad \partial_\lambda\det g=(\det g)q^{\mu\nu}\partial_\lambda g_{\mu\nu},&&\text{(5.7)}\\ c_\lambda:=\partial_\lambda\log\nu &=\tfrac12q^{\mu\nu}\partial_\lambda g_{\mu\nu} =-\tfrac12g_{\mu\nu}\partial_\lambda q^{\mu\nu} =C^\rho{}_{\lambda\rho}. &&\text{(5.8)}\end{aligned}$$

The last equality also follows from (4.7), because $\operatorname{tr}\omega=0$. Expanding compatibility yields

$$\partial_\lambda g_{\mu\nu} =C^\rho{}_{\lambda\mu}g_{\rho\nu}+C^\rho{}_{\lambda\nu}g_{\mu\rho}, \qquad \partial_\lambda q^{\mu\nu} =-C^\mu{}_{\lambda\rho}q^{\rho\nu}-C^\nu{}_{\lambda\rho}q^{\mu\rho}. \tag{5.9}$$

These supply Einstein (28)–Einstein (34), including Einstein (29a).

The corresponding contractions and antisymmetrizations are

$$\begin{aligned} \nabla_\mu V^\mu&=\nu^{-1}\partial_\mu(\nu V^\mu),&&\text{(5.10)}\\ A_{\mu;\nu}-A_{\nu;\mu}&=\partial_\nu A_\mu-\partial_\mu A_\nu,&&\text{(5.11)}\\ A_{\mu\nu;\sigma}+A_{\nu\sigma;\mu}+A_{\sigma\mu;\nu} &=\partial_\sigma A_{\mu\nu}+\partial_\mu A_{\nu\sigma}+\partial_\nu A_{\sigma\mu},&&\text{(5.12)}\\ \nabla_\nu A^{\mu\nu}&=\nu^{-1}\partial_\nu(\nu A^{\mu\nu}) \quad(A^{\mu\nu}=-A^{\nu\mu}). &&\text{(5.13)}\end{aligned}$$

Equation (5.12) assumes antisymmetric $A$ as well. Symmetric connection terms cancel pairwise, and their remaining trace becomes (5.8). These recover Einstein (35)–Einstein (37) and Einstein (40); Einstein’s curl in (5.11) is minus the usual exterior-derivative component in the order $(\mu,\nu)$.

For a mixed tensor the full identity is

$$\nabla_\nu T^\nu{}_{\mu}=\nu^{-1}\partial_\nu(\nu T^\nu{}_{\mu}) -C^\rho{}_{\nu\mu}T^\nu{}_{\rho}. \tag{5.14}$$

Only when $T^{\alpha\beta}$ is symmetric does it reduce to

$$\nabla_\nu T^\nu{}_{\mu} =\nu^{-1}\partial_\nu(\nu T^\nu{}_{\mu}) -\tfrac12\partial_\mu g_{\alpha\beta}T^{\alpha\beta} =\nu^{-1}\partial_\nu(\nu T^\nu{}_{\mu}) +\tfrac12\partial_\mu q^{\alpha\beta}T_{\alpha\beta}. \tag{5.15}$$

To prove it, lower the upper index of (4.10), contract with symmetric $T$, and exchange the two summed indices in its first and third terms. They cancel; the remaining term is the half metric derivative. Differentiating $qg=I$ supplies the last form. This retains the hypotheses of Einstein (41), Einstein (41a) and Einstein (41b).