Conventions, units and scope for curvature and tidal quantities

Signs and index order. The curvature operator, the component order and the lowered, Ricci, scalar and Einstein contractions are fixed once in The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors and used unchanged throughout this page: $R(X,Y)Z={\mathrm{\nabla }}_X{\mathrm{\nabla }}_{Y}Z-{\mathrm{\nabla }}_Y{\mathrm{\nabla }}_{X}Z-{\mathrm{\nabla }}_{[X,Y]}Z$ and $R({\mathrm{\partial }}_c,{\mathrm{\partial }}_d){\mathrm{\partial }}_b=R^a{}_{bcd}{\mathrm{\partial }}_a$, so that the algebraic symmetries of The algebraic symmetries of the curvature tensor and the symmetry of the Ricci and Einstein tensors and the divergence identity ${\mathrm{\nabla }}^a{G}_{ab}=0$ of The second Bianchi identity and the vanishing divergence of the Einstein tensor hold in exactly the displayed form. Sources that order the loop indices differently, or that use the opposite overall sign, state the same mathematics in another convention; no formula is imported from a source without converting it to this one.

Units. In length coordinates, in which $x^0=ct$ and all $x^a$ carry metres and the metric components are dimensionless (SI unit and coordinate conventions of the classical gravitational framework), the Christoffel symbols ${\mathrm{\Gamma }}^a{}_{bc}$ have units of inverse length and the components $R^a{}_{bcd}$, $R_{abcd}$, ${\mathrm{Ric}}_{ab}$ and $S$ have units of inverse length squared. The Einstein tensor has the same units as the Ricci tensor, and the field-equation coupling of the framework is normalized so that $\kappa T_{ab}$ has the same units; no numerical value of any constant is asserted by these conventions. In other coordinates, such as charts with an angular coordinate, individual component units change with the Jacobian and carry no universal dimension; only invariant statements, proper times and observer-relative quantities keep their physical reading.

Connection artifacts versus curvature. The Christoffel symbols are chart functions and not tensor components; a chart can make all of them vanish at a point (Point-normal and exponential coordinate charts of the metric-selected connection) without making the curvature vanish there: Tensoriality of the curvature, its coordinate formula and its point-normal metric formula clause (c) shows that curvature at a point is exactly the second-order metric information that no coordinate choice can remove. Conversely a chart may have nonzero Christoffel symbols everywhere while the curvature vanishes identically, as for flat space in curvilinear coordinates. Curvature is also not a trace of itself: $\mathrm{Ric}=0$ is a weaker condition than $R=0$, and the converse implications are not asserted anywhere on this page.

"Tidal" is a model term. In Free-fall congruences, tidal (curvature) relative acceleration and the flat comparison the word tidal refers to the curvature term $R(U,J)U$ in the deviation equation satisfied by smooth free-fall congruences of the adopted model; the deduction shows that this term is exactly the infinitesimal relative four-acceleration when ideal test-body histories realize the mathematical family on a common interval. Its vanishing for all timelike directions and all separation values at an event is equivalent to the vanishing of curvature there. All these values are attained by local mathematical candidate families, without guaranteeing bodies realizing them. It does not assert that any real body is freely falling, that any particular measured effect is of this form, or that agreement of any such comparison would establish the postulates; a comparison with data would require apparatus models, empirical premises with uncertainties and statistical inference, none of which is part of this page.

Scope of the maximum and conjugacy results. The exact maximum statement is Conjugacy, the exponential differential and the local proper-time maximum before the first conjugate instant clause (b). Its hypotheses are a smooth future timelike geodesic $\gamma :[0,b]\to M$, $b>0$, with $g({\gamma }^{\mathrm{\prime }},{\gamma }^{\mathrm{\prime }})=-1$ and no conjugate instant in $(0,b]$ relative to $0$: no nonzero smooth field $J$ along $\gamma$ satisfying $D_s^{2}J=R({\gamma }^{\mathrm{\prime }},J){\gamma }^{\mathrm{\prime }}$ vanishes both at $0$ and at such an instant. Fix a finite chart cover $(U_i,x_i)$ and a partition $0=s_0<\cdots <s_m=b$ with $\gamma ([s_{i-1},s_i])\subseteq U_i$. The comparison curves $\alpha :[0,b]\to M$ are continuous piecewise-$C^1$ future timelike curves with the same endpoints. For some $\delta >0$, each comparison curve must remain in the corresponding $U_i$ and satisfy $\sup \mathrm{\parallel }{x}_i\circ \alpha -x_i\circ \gamma \mathrm{\parallel }<\delta$ and $\sup \mathrm{\parallel }(x_i\circ \alpha {)}^{\mathrm{\prime }}-(x_i\circ \gamma {)}^{\mathrm{\prime }}\mathrm{\parallel }<\delta$ on every common smooth partition piece, with Euclidean norms as in that clause. Only in this comparison class is $L_g(\alpha )\le b=L_g(\gamma )$ asserted; equality means a piecewise-$C^1$ strictly increasing reparametrization onto $[0,b]$.

The local inverse branches used in that theorem's proof steps 2.1-3.1 are smooth maps ${\psi }_k:V_k\to D_k\subseteq T_{p}M$, $p=\gamma (0)$, inverse to ${\exp }_p{\mathrm{\mid }}_{D_k}:D_k\to V_k$, where $D_k,V_k$ are open and the restricted exponential map is a diffeomorphism. A finite parameter subdivision puts each radial subsegment $s{\gamma }^{\mathrm{\prime }}(0)$ in its corresponding $D_k$. Compatibility means adjacent branches agree on a neighborhood of the shared endpoint event and send that event to the same radial vector. Branches at nonadjacent occurrences of a repeated event need not agree; one injective exponential chart containing the whole segment is not assumed. These are local assertions, not global maximality, existence of conjugate instants, or a classification of where global maximality first fails. Global causal, completeness and singularity statements are reserved for later pages and are not prerequisites of the results here.