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Curvature And Tides

1 · Prerequisites

2 · Summary

This page builds the curvature of the metric-selected connection on gr-connections-and-free-fall and its reading as the tidal field. It has eleven mathematical items: two definitions, eight proved results, and one worked example, all within a time-oriented smooth Lorentzian four-manifold. Its two physical items are one model-conditional theorem and one scope note. No item reports a measurement or asserts that the model describes nature.

Curvature and its identities. The curvature operator $R (X, Y) Z = \nabla_{X} \nabla_{Y} Z - \nabla_{Y} \nabla_{X} Z - \nabla_{[X, Y]} Z$ and the derived lowered Riemann, Ricci, scalar and Einstein tensors are defined with a fixed sign and index order (The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors); the tensoriality and the chart and point-normal coordinate formulas are proved in Tensoriality of the curvature, its coordinate formula and its point-normal metric formula, and the algebraic symmetries together with the first Bianchi identity and the Ricci and Einstein symmetries in The algebraic symmetries of the curvature tensor and the symmetry of the Ricci and Einstein tensors. The second Bianchi identity and its two contractions give the divergence-free Einstein tensor $\nabla^{a} G_{a b} = 0$ (The second Bianchi identity and the vanishing divergence of the Einstein tensor).

Geodesic deviation, conjugacy and the local maximum. Jacobi fields along a geodesic are defined by $D_{s}^{2} J = R (T, J) T$ , with conjugate instants, multiplicity and the normal bundle (Jacobi fields, geodesic variations and conjugate instants along a geodesic); the commutator identity $[D_{t}, D_{r}] V = R (T, J) V$ , the geodesic deviation equation and the eight-dimensional initial-value theory are proved in The Jacobi equation, geodesic deviation and the initial-value theory of Jacobi fields. Conjugacy is then characterized as non-injectivity of the exponential differential with kernel dimension equal to the multiplicity (The differential of the exponential map and conjugacy along a geodesic), the normal index form is factored as a sum of squares before the first conjugate instant by a matrix Riccati computation (The normal index form before the first conjugate instant is positive definite), and the local proper-time maximum is proved for every future timelike segment with no conjugate instant through its endpoint; no claim is made about maximality at or after a conjugate instant (Conjugacy, the exponential differential and the local proper-time maximum before the first conjugate instant).

Tidal reading. The algebraic vanishing criterion (A curvature tensor vanishes if the tidal operator vanishes for every timelike direction) is the mathematical lemma behind the model deduction that free-fall congruence deviation is exactly the curvature term, vanishes identically when the curvature does, and contains no rest mass or internal-structure parameter (Free-fall congruences, tidal (curvature) relative acceleration and the flat comparison); the scope note Conventions, units and scope for curvature and tidal quantities fixes units and conventions and separates chart artifacts from curvature. The in-page example Curvature survives a first-order normal chart: an explicit metric, its normal form and its deviation field shows explicitly that a first-order normal chart with vanishing Christoffel symbols at a point can still carry nonzero curvature. The companion page gr-curvature-and-tides-examples computes all of this on flat space in cylindrical coordinates, on a product with a round sphere, and on two counterexamples separating the algebraic hypotheses.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

DefinitionDefinition: AI-adaptedProof: Not applicableOpen item page →

The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors

Definition

Let $(M, g, N)$ be a time-oriented smooth Lorentzian four-manifold (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model) with metric-selected connection $\nabla$ (Metric compatibility, torsion freedom and the metric-selected connection class), whose existence and uniqueness are supplied by Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields; that theorem also extends $\nabla$ to a derivative on tensor fields commuting with contractions and annihilating $g$ and $g^{- 1}$ . Let $U \subseteq M$ be open and let $X, Y, Z$ be smooth vector fields on $U$ ; brackets are those of The Lie bracket of smooth vector fields, and contractions those of The contraction of a mixed tensor.

(a) The curvature operator. The curvature operator of $\nabla$ is the assignment

$R (X, Y) Z : = \nabla_{X} \nabla_{Y} Z - \nabla_{Y} \nabla_{X} Z - \nabla_{[X, Y]} Z .$

The right-hand side is a smooth vector field on $U$ : $\nabla_{Y} Z$ is smooth because $\nabla$ is an affine connection (Affine connection on a smooth manifold), $[X, Y]$ is smooth, and $\nabla_{X} (\cdot)$ and $\nabla_{[X, Y]} (\cdot)$ preserve smoothness. The assignment is defined for the given smooth fields; that its value at a point depends only on the values of $X, Y, Z$ at that point, so that $R$ is a smooth tensor field of type $(1, 3)$ , is the content of Tensoriality of the curvature, its coordinate formula and its point-normal metric formula ↗ clause (a), and likewise the algebraic symmetries used below are proved in The algebraic symmetries of the curvature tensor and the symmetry of the Ricci and Einstein tensors ↗.

(b) Component functions. Let $(V, x)$ be a smooth chart of $M$ and let $Γ^{a}_{b c}$ be the Christoffel symbols of $\nabla$ in this chart, characterized by $\nabla_{\partial_{b}} \partial_{c} = \sum_{a} Γ^{a}_{b c} \partial_{a}$ (Christoffel symbols of an affine connection); the coordinate fields commute, so $\nabla_{\partial_{c}} \partial_{d}$ and $\nabla_{\partial_{d}} \partial_{c}$ are obtained from the same symbols. The components of the curvature in the chart are the smooth functions $R^{a}_{b c d}$ on $V$ defined by

$R (\partial_{c}, \partial_{d}) \partial_{b} = \sum_{a} R^{a}_{b c d} \partial_{a} .$

The first index is the output slot, the second is the differentiated slot and the last two are the two directions of the loop; indices are $a, b, c, d = 0, 1, 2, 3$ and the Einstein summation convention of Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model applies.

(c) Lowered tensor, Ricci tensor, scalar curvature, Einstein tensor. On the chart domain define, using the inverse metric $g^{a b}$ of The inverse metric, musical isomorphisms and index raising,

$R_{a b c d} : = \sum_{e} g_{a e} R^{e}_{b c d}, {Ric}_{b d} : = \sum_{a, c} g^{a c} R_{a b c d}, S : = \sum_{b, d} g^{b d} {Ric}_{b d}, G_{a b} : = {Ric}_{a b} - \frac{1}{2} S g_{a b} .$

The fields $R_{a b c d}$ , ${Ric}_{a b}$ , $S$ and $G_{a b}$ are called the Riemann tensor with lowered indices, the Ricci tensor, the scalar curvature and the Einstein tensor of the metric-selected connection. Tensoriality of $R$ is proved in Tensoriality of the curvature, its coordinate formula and its point-normal metric formula ↗ clause (a). Lowering its output slot with the smooth metric and contracting with the smooth inverse metric are intrinsic tensor operations (Tensor fields of type (r,s) on a Lorentzian four-manifold and their components clauses (b)-(c), The contraction of a mixed tensor). Thus these formulas give smooth fields of types $(0, 4)$ , $(0, 2)$ , $(0, 0)$ and $(0, 2)$ , respectively, independent of the chart. Symmetry of $Ric$ and $G$ is proved separately in The algebraic symmetries of the curvature tensor and the symmetry of the Ricci and Einstein tensors ↗ clause (d).

Conventions and scope

Sign convention. The signs of the curvature operator and of the component order are fixed by the displays of clauses (a) and (b) and are those used by every curvature item on this page and by the curvature consumers on later pages: with these conventions $R (X, Y) Z = \nabla_{X} \nabla_{Y} Z - \nabla_{Y} \nabla_{X} Z - \nabla_{[X, Y]} Z$ and $R (\partial_{c}, \partial_{d}) \partial_{b} = R^{a}_{b c d} \partial_{a}$ . Other sources order the loop indices differently; no formula of another source is used without checking its convention against this one.
What is defined where. The operator and components of clauses (a)-(b) are defined from the connection alone; the lowered tensor, Ricci tensor, scalar curvature and Einstein tensor of clause (c) require the metric, and in clause (c) everything is defined chartwise from the components, with the chart-independence supplied by curvature tensoriality and intrinsic metric contractions, as specified there. The Christoffel symbols $Γ^{a}_{b c}$ are chart functions and not tensor components (Christoffel symbols of an affine connection); no tensority of $Γ$ is asserted here or later.
Scope. This definition asserts no relation between curvature and any physical quantity, no equation satisfied by $Ric$ or $G$ , and no vanishing criterion: in particular $Γ^{a}_{b c} (p) = 0$ at a point does not imply $R^{a}_{b c d} (p) = 0$ , since curvature is second-order metric information in the precise sense of Tensoriality of the curvature, its coordinate formula and its point-normal metric formula ↗ clause (c). The physical reading of $G_{a b}$ as one side of a field equation is a separate postulate of the framework and is not part of this definition.

TheoremStatement: AI-adaptedProof: AI-generatedprecheck passOpen item page →

Tensoriality of the curvature, its coordinate formula and its point-normal metric formula

Statement

(a) Tensoriality. For all smooth functions $f$ on $M$ and smooth vector fields $X, Y, Z$ on a common domain, $R (X, Y) Z = - R (Y, X) Z$ and $R (f X, Y) Z = f R (X, Y) Z, R (X, f Y) Z = f R (X, Y) Z, R (X, Y) (f Z) = f R (X, Y) Z .$ Consequently, for every $p \in M$ , the value $R (X, Y) Z ∣_{p}$ depends only on $X_{p}, Y_{p}, Z_{p}$ , and $R$ is a smooth tensor field of type $(1, 3)$ on $M$ whose components in every smooth chart are the functions $R^{a}_{b c d}$ of The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors clause (b).

(b) Coordinate formula. In every smooth chart $(U, x)$ with Christoffel symbols $Γ^{a}_{b c}$ of $\nabla$ (Christoffel symbols of an affine connection), $R^{a}_{b c d} = \partial_{c} Γ^{a}_{d b} - \partial_{d} Γ^{a}_{c b} + \sum_{e} (Γ^{a}_{c e} Γ^{e}_{d b} - Γ^{a}_{d e} Γ^{e}_{c b})$ on $U$ , for all $a, b, c, d \in {0, 1, 2, 3}$ .

(c) Point-normal formula. If the chart is point-normal at $p$ , that is $g_{a b} (p) = η_{a b}$ , $\partial_{a} g_{b c} (p) = 0$ and $Γ^{a}_{b c} (p) = 0$ (Point-normal and exponential coordinate charts of the metric-selected connection), then, writing $g_{a d, b c} : = \partial_{b} \partial_{c} g_{a d}$ , $R_{a b c d} (p) = \frac{1}{2} (g_{a d, b c} + g_{b c, a d} - g_{b d, a c} - g_{a c, b d}) (p) .$ Equivalently, $R^{a}_{b c d} (p) = \frac{1}{2} g^{a e} (p) (\partial_{b} \partial_{c} g_{e d} + \partial_{e} \partial_{d} g_{b c} - \partial_{e} \partial_{c} g_{b d} - \partial_{b} \partial_{d} g_{e c}) (p)$ .

Facts & Assumptions

Given: A time-oriented smooth Lorentzian four-manifold $(M, g, N)$ with metric-selected connection $\nabla$ , its curvature operator $R$ , a chart $(U, x)$ , and a point $p$ with a point-normal chart as in the claims.

All field calculations below are made on a common open domain $U$ , using the restriction of the connection to local fields there. The function-linearity identities are proved for every $f \in C^{\infty} (U)$ ; the Statement's global smooth functions are their restrictions, and step 3.1 uses these same local identities for its chart coefficient functions.

[L1]

The curvature operator is $R (X, Y) Z = \nabla_{X} \nabla_{Y} Z - \nabla_{Y} \nabla_{X} Z - \nabla_{[X, Y]} Z$ for smooth fields, is a smooth field, and its components in a chart are defined by $R (\partial_{c}, \partial_{d}) \partial_{b} = \sum_{a} R^{a}_{b c d} \partial_{a}$ (The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors).

[L2]

On every common open domain $U$ , the restricted affine connection is real bilinear on local fields, function-linear in its first argument, satisfies $\nabla_{X} (f Y) = X (f) Y + f \nabla_{X} Y$ for $f \in C^{\infty} (U)$ , and takes smooth local fields to smooth local fields (Affine connection on a smooth manifold).

[L3]

On every common open domain $U$ , the Lie bracket satisfies $[f X, Y] = f [X, Y] - Y (f) X$ and $[X, f Y] = X (f) Y + f [X, Y]$ for $f \in C^{\infty} (U)$ , $[Y, X] = - [X, Y]$ , and $[\partial_{c}, \partial_{d}] = 0$ for coordinate fields (The Lie bracket of smooth vector fields, Coordinate formula for the Lie bracket).

[L4]

In a chart, $\nabla_{\partial_{b}} \partial_{c} = \sum_{a} Γ^{a}_{b c} \partial_{a}$ , and torsion freedom of $\nabla$ gives $Γ^{a}_{b c} = Γ^{a}_{c b}$ (Christoffel symbols of an affine connection, Torsion freedom and metric compatibility in chart components clauses 1, Metric compatibility, torsion freedom and the metric-selected connection class).

[L5]

The Christoffel symbols of the metric-selected connection satisfy $Γ^{a}_{b c} = \frac{1}{2} g^{a d} (\partial_{b} g_{d c} + \partial_{c} g_{b d} - \partial_{d} g_{b c})$ in every chart, and $g_{a e} g^{e b} = δ_{a}^{b}$ (Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields clause (a), The inverse metric, musical isomorphisms and index raising).

[L6]

For every $p$ there is a smooth chart with $g_{a b} (p) = η_{a b}$ , $\partial_{a} g_{b c} (p) = 0$ and $Γ^{a}_{b c} (p) = 0$ (Point-normal and exponential coordinate charts of the metric-selected connection clause (a)).

[L7]

A smooth vector field on a chart domain is a finite sum $\sum_{a} f^{a} \partial_{a}$ with smooth coefficients $f^{a}$ , and $\sum_{a} f^{a} \partial_{a}$ vanishes at a point whenever every $f^{a}$ does (Coordinate derivations form a basis of the tangent space).

[L8]

Mixed partial derivatives of a smooth function commute (Clairaut--Schwarz theorem for continuous second partial derivatives).

[L9]

A smooth tensor field of type $(1, 3)$ is a family of multilinear maps with $C^{\infty}$ components in every chart (Tensor fields of type (r,s) on a Lorentzian four-manifold and their components clauses (a)-(b)).

Proof

technique · direct

1.1L1L2L3algebra

(Function-linearity in the first slot.) Let $f$ be smooth. Using [L3] for the bracket and [L2] for the connection, $R (f X, Y) Z = \nabla_{f X} \nabla_{Y} Z - \nabla_{Y} \nabla_{f X} Z - \nabla_{[f X, Y]} Z = f \nabla_{X} \nabla_{Y} Z - \nabla_{Y} (f \nabla_{X} Z) - \nabla_{f [X, Y] - Y (f) X} Z = f \nabla_{X} \nabla_{Y} Z - Y (f) \nabla_{X} Z - f \nabla_{Y} \nabla_{X} Z - f \nabla_{[X, Y]} Z + Y (f) \nabla_{X} Z = f R (X, Y) Z$ , the two $Y (f) \nabla_{X} Z$ terms cancelling.

1.2L1L2L3algebra

(Function-linearity in the third slot.) Expanding with [L2] and [L3], $\nabla_{X} \nabla_{Y} (f Z) = \nabla_{X} (Y (f) Z + f \nabla_{Y} Z) = X (Y (f)) Z + Y (f) \nabla_{X} Z + X (f) \nabla_{Y} Z + f \nabla_{X} \nabla_{Y} Z$ , while $\nabla_{Y} \nabla_{X} (f Z) = Y (X (f)) Z + X (f) \nabla_{Y} Z + Y (f) \nabla_{X} Z + f \nabla_{Y} \nabla_{X} Z$ , and $\nabla_{[X, Y]} (f Z) = [X, Y] (f) Z + f \nabla_{[X, Y]} Z$ . Subtracting the second and third displays from the first, the $X (f)$ , $Y (f)$ and $Z$ -coefficient terms cancel in pairs and the remaining scalar coefficient is $X (Y (f)) - Y (X (f)) - [X, Y] (f) = 0$ by definition of the bracket, so $R (X, Y) (f Z) = f R (X, Y) Z$ .

2.1step 1.1L1L3algebra

(Skew-symmetry and the second slot.) Directly from [L1] and $[Y, X] = - [X, Y]$ , $R (Y, X) Z = - R (X, Y) Z$ ; combining this with step 1.1, $R (X, f Y) Z = - R (f Y, X) Z = - f R (Y, X) Z = f R (X, Y) Z$ .

3.1step 1.1step 2.1step 1.2L7algebra

(Pointwise dependence.) Let $p \in M$ and let $(V, x)$ be a chart around $p$ . By [L7], fields $X^{'}, Y^{'}, Z^{'}$ with $X_{p}^{'} = X_{p}$ , $Y_{p}^{'} = Y_{p}$ , $Z_{p}^{'} = Z_{p}$ have the form $X^{'} = X + \sum_{a} f^{a} \partial_{a}$ , $Y^{'} = Y + \sum_{b} g^{b} \partial_{b}$ , $Z^{'} = Z + \sum_{c} h^{c} \partial_{c}$ on a possibly smaller chart neighborhood, with $f^{a} (p) = g^{b} (p) = h^{c} (p) = 0$ . By additivity of $R$ in each field argument and the three function-linearity steps, expanding $R (X^{'}, Y^{'}, Z^{'})$ yields $R (X, Y, Z)$ plus a finite sum of terms each carrying at least one of the smooth coefficients $f^{a}, g^{b}, h^{c}$ (a term with $k$ such factors expands, by steps 1.1, 2.1 and 1.2, into $k$ -fold products of coefficients times $R$ evaluated on coordinate fields). Every such term vanishes at $p$ because all coefficients do, so $R (X^{'}, Y^{'}, Z^{'}) ∣_{p} = R (X, Y, Z) ∣_{p}$ : the value depends only on the pointwise data.

4.1step 3.1L1L9

(Smooth tensor field.) By step 3.1 the assignment descends to a map $R_{p} : T_{p} M \times T_{p} M \times T_{p} M \to T_{p} M$ , $(v, w, u) \mapsto R (X, Y) Z ∣_{p}$ for any smooth local extensions, which is $R$ -multilinear because the field-level assignment is additive over $R$ and function-linear. In a chart, $R (\partial_{c}, \partial_{d}) \partial_{b}$ is a smooth field by [L1], so its coefficients $R^{a}_{b c d}$ in the coordinate frame are $C^{\infty}$ functions on the chart domain. By [L9] these data constitute a smooth tensor field of type $(1, 3)$ whose components are exactly $R^{a}_{b c d}$ . This proves (a).

5.1step 4.1L1L2L3L4algebra

(Coordinate formula.) In the chart, [L3] gives $[\partial_{c}, \partial_{d}] = 0$ , so [L1] gives $R (\partial_{c}, \partial_{d}) \partial_{b} = \nabla_{\partial_{c}} \nabla_{\partial_{d}} \partial_{b} - \nabla_{\partial_{d}} \nabla_{\partial_{c}} \partial_{b}$ . By [L4], $\nabla_{\partial_{d}} \partial_{b} = \sum_{e} Γ^{e}_{d b} \partial_{e}$ , so [L2] gives $\nabla_{\partial_{c}} (\sum_{e} Γ^{e}_{d b} \partial_{e}) = \sum_{e} (\partial_{c} Γ^{e}_{d b}) \partial_{e} + \sum_{e, a} Γ^{e}_{d b} Γ^{a}_{c e} \partial_{a}$ ; exchanging $c, d$ and subtracting yields $R^{a}_{b c d} = \partial_{c} Γ^{a}_{d b} - \partial_{d} Γ^{a}_{c b} + \sum_{e} (Γ^{a}_{c e} Γ^{e}_{d b} - Γ^{a}_{d e} Γ^{e}_{c b})$ . This proves (b).

6.1step 5.1L5L6L8algebra

(Point-normal derivative of the Christoffel symbols.) In the point-normal chart of [L6], differentiate the Christoffel formula of [L5]. The derivative of the factor $g^{a e}$ multiplies the bracket $(\partial_{b} g_{e d} + \partial_{d} g_{e b} - \partial_{e} g_{d b})$ , which vanishes at $p$ because $\partial g (p) = 0$ , so only the bracket's derivative survives: $\partial_{c} Γ^{a}_{d b} (p) = \frac{1}{2} g^{a e} (p) (\partial_{c} \partial_{b} g_{e d} + \partial_{c} \partial_{d} g_{e b} - \partial_{c} \partial_{e} g_{d b}) (p)$ , and likewise $\partial_{d} Γ^{a}_{c b} (p) = \frac{1}{2} g^{a e} (p) (\partial_{d} \partial_{b} g_{e c} + \partial_{d} \partial_{c} g_{e b} - \partial_{d} \partial_{e} g_{c b}) (p)$ . Since $Γ (p) = 0$ , formula (b) of step 5.1 reduces at $p$ to the difference of these two derivatives, in which the terms $\partial_{c} \partial_{d} g_{e b}$ and $\partial_{d} \partial_{c} g_{e b}$ agree by [L8] and cancel: $R^{a}_{b c d} (p) = \frac{1}{2} g^{a e} (p) (\partial_{b} \partial_{c} g_{e d} - \partial_{c} \partial_{e} g_{d b} - \partial_{b} \partial_{d} g_{e c} + \partial_{d} \partial_{e} g_{c b}) (p)$ .

7.1step 6.1L5L8algebra∎

(Lowering and final form.) Lowering the first index with [L5], $R_{a b c d} (p) = g_{a e} (p) R^{e}_{b c d} (p) = \frac{1}{2} (\partial_{b} \partial_{c} g_{a d} - \partial_{c} \partial_{a} g_{d b} - \partial_{b} \partial_{d} g_{a c} + \partial_{d} \partial_{a} g_{c b}) (p)$ . Using [L8] to commute the mixed partials and the symmetry $g_{d b} = g_{b d}$ , $g_{c b} = g_{b c}$ of the metric, this is $\frac{1}{2} (g_{a d, b c} + g_{b c, a d} - g_{b d, a c} - g_{a c, b d}) (p)$ , and raising back with $g^{a e} (p)$ recovers the equivalent raised display. This proves (c) and completes the proof.

Remarks

What is used where. Clause (a) uses only the connection rules and the bracket identity; the coordinate formula (b) uses in addition the symmetry of the Christoffel symbols, which is torsion freedom; the point-normal formula (c) uses the Christoffel formula, that is metric compatibility and nondegeneracy. No positive-definite or Riemannian hypothesis is used anywhere. The result is stated for a Lorentzian metric only because that is the ambient framework; the computation is signature-independent.
Tensoriality of $Γ$ is not claimed. The formula of (b) contains the non-tensorial chart functions $Γ^{a}_{b c}$ , but its left-hand side is tensorial by (a); this is exactly why (b) cannot be used to conclude that vanishing of $Γ$ at a point implies vanishing of curvature. Clause (c) shows that curvature is precisely the second-order metric information that a point-normal chart cannot remove.

TheoremStatement: AI-adaptedProof: AI-generatedprecheck passOpen item page →

The algebraic symmetries of the curvature tensor and the symmetry of the Ricci and Einstein tensors

Statement

(a) Skew-symmetries. $R_{a b c d} = - R_{b a c d}$ and $R_{a b c d} = - R_{a b d c}$ . Equivalently, for smooth vector fields, $g (R (X, Y) Z, W) = - g (R (X, Y) W, Z) and g (R (X, Y) Z, W) = - g (R (Y, X) Z, W) .$

(b) Exchange of the index pairs. $R_{a b c d} = R_{c d a b}$ , equivalently $g (R (X, Y) Z, W) = g (R (Z, W) X, Y)$ .

(c) First Bianchi identity. $R_{a [b c d]} = 0$ , that is, $R_{a b c d} + R_{a c d b} + R_{a d b c} = 0.$

(d) Ricci symmetry. ${Ric}_{a b} = {Ric}_{b a}$ ; consequently the Einstein tensor is symmetric, $G_{a b} = G_{b a}$ .

Facts & Assumptions

Given: A time-oriented smooth Lorentzian four-manifold $(M, g, N)$ with metric-selected connection, its Riemann and Ricci tensors, and a point $p$ .

[L1]

The components, the lowered tensor, the Ricci tensor, the scalar curvature and the Einstein tensor are defined by $R (\partial_{c}, \partial_{d}) \partial_{b} = \sum_{a} R^{a}_{b c d} \partial_{a}$ , $R_{a b c d} = g_{a e} R^{e}_{b c d}$ , ${Ric}_{b d} = g^{a c} R_{a b c d}$ , $S = g^{b d} {Ric}_{b d}$ and $G_{a b} = {Ric}_{a b} - \frac{1}{2} S g_{a b}$ (The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors).

[L2]

$R$ is a smooth $(1, 3)$ -tensor, so its components in any chart and the tensors built from them by contraction with $g$ and $g^{- 1}$ are components of tensor fields; the components in a chart are related to those in any other chart by the tensor transformation law (Tensoriality of the curvature, its coordinate formula and its point-normal metric formula clause (a), Tensor fields of type (r,s) on a Lorentzian four-manifold and their components clauses (b)-(c)).

[L3]

For every $p$ there is a point-normal chart with $g_{a b} (p) = η_{a b}$ , $\partial_{a} g_{b c} (p) = 0$ , $Γ^{a}_{b c} (p) = 0$ , and in it the lowered components at $p$ are given by the four-term formula $R_{a b c d} (p) = \frac{1}{2} (g_{a d, b c} + g_{b c, a d} - g_{b d, a c} - g_{a c, b d}) (p)$ , with $g_{a d, b c} = \partial_{b} \partial_{c} g_{a d}$ (Point-normal and exponential coordinate charts of the metric-selected connection clause (a), Tensoriality of the curvature, its coordinate formula and its point-normal metric formula clause (c)).

[L4]

Mixed partial derivatives commute and the metric components are symmetric: $g_{a d, b c} = g_{a d, c b}$ and $g_{a b} = g_{b a}$ (Clairaut--Schwarz theorem for continuous second partial derivatives, Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model clause (a)).

[L5]

$g^{a b}$ is the inverse matrix of $g_{a b}$ , so $g^{a c} g_{c b} = δ_{b}^{a}$ , and relabelling a summed index pair does not change a sum (The inverse metric, musical isomorphisms and index raising, Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model clause (d)).

Proof

technique · direct

1.1L3L2given

(Point-normal reduction.) Fix $p$ and choose a point-normal chart at $p$ as in [L3]. By [L3], $R_{a b c d} (p) = \frac{1}{2} (g_{a d, b c} + g_{b c, a d} - g_{b d, a c} - g_{a c, b d}) (p)$ . Use this second-derivative formula only in the chosen point-normal chart. By [L2] the lowered curvature is a tensor; each skew-symmetry, pair-exchange and cyclic-sum identity below says that a fixed sum of slot permutations of that tensor vanishes. If its components vanish in the chosen basis, multilinearity makes that tensor zero at $p$ , hence its components vanish in every chart. The same applies to $Ric - {Ric}^{T}$ and $G - G^{T}$ , since metric contraction gives tensors by [L2]. Thus it suffices to verify these particular identities in the point-normal chart at each $p$ ; no tensor transformation law for metric second derivatives is asserted.

2.1step 1.1L1L2algebra

(Skew-symmetries.) Interchanging $a$ and $b$ in the display of step 1.1 negates it, because each of its four terms is a second partial derivative with one of $a, b$ in a fixed slot: $R_{b a c d} (p) = \frac{1}{2} (g_{b d, a c} + g_{a c, b d} - g_{a d, b c} - g_{b c, a d}) (p) = - R_{a b c d} (p)$ . Interchanging $c$ and $d$ likewise negates it. This proves (a), the two operator forms following by expanding $g (R (X, Y) Z, W) = R_{a b c d} W^{a} X^{c} Y^{d} Z^{b}$ in the same components.

2.2step 1.1L1L2L4algebra

(Pair exchange.) Substituting $(a, b, c, d) \mapsto (c, d, a, b)$ into the display of step 1.1 turns its four terms into $g_{c b, d a}$ , $g_{d a, c b}$ , $- g_{d b, c a}$ and $- g_{c a, d b}$ ; by the symmetry of $g$ and [L4] these are $g_{b c, a d}$ , $g_{a d, b c}$ , $- g_{b d, a c}$ and $- g_{a c, b d}$ , that is, the four terms of the original display in permuted order. Hence $R_{c d a b} (p) = R_{a b c d} (p)$ , which proves (b).

2.3step 1.1L4algebra

(First Bianchi identity.) By step 1.1, $2 (R_{a b c d} + R_{a c d b} + R_{a d b c}) (p)$ equals the sum of the twelve terms obtained by substituting the four-term formula into the three summands. The terms pair off with opposite signs: $g_{a d, b c}$ cancels $g_{a d, c b}$ , $g_{b c, a d}$ cancels $g_{c b, a d}$ , $- g_{b d, a c}$ cancels $+ g_{b d, a c}$ , $- g_{a c, b d}$ cancels $+ g_{a c, b d}$ , $g_{a b, c d}$ cancels $- g_{a b, c d}$ , and $g_{c d, a b}$ cancels $- g_{c d, a b}$ , each cancellation using [L4]. Hence $R_{a b c d} + R_{a c d b} + R_{a d b c} = 0$ at $p$ , and since $p$ was arbitrary this proves (c).

2.4step 1.1L1L4L5algebra

(Ricci symmetry.) By [L1] and step 1.1, ${Ric}_{b d} (p) = \frac{1}{2} g^{a c} (p) (g_{a d, b c} + g_{b c, a d} - g_{b d, a c} - g_{a c, b d}) (p)$ and, writing out the same expression with $b$ and $d$ interchanged, ${Ric}_{d b} (p) = \frac{1}{2} g^{a c} (p) (g_{a b, d c} + g_{d c, a b} - g_{d b, a c} - g_{a c, d b}) (p)$ . Relabelling the summed pair $a, c$ as $c, a$ in the second expression and using $g^{a c} = g^{c a}$ [L5], its first term becomes $g^{a c} g_{c b, d a} = g^{a c} g_{b c, a d}$ by [L4], which is the second term of ${Ric}_{b d}$ ; its second term becomes $g^{a c} g_{d a, c b} = g^{a c} g_{a d, b c}$ , the first term of ${Ric}_{b d}$ ; its third term is $g^{a c} g_{b d, a c}$ , the third term of ${Ric}_{b d}$ ; and its fourth term is $g^{a c} g_{a c, b d}$ , the fourth term. Hence ${Ric}_{d b} (p) = {Ric}_{b d} (p)$ .

3.1step 2.1step 2.2step 2.3step 2.4L1L4algebra∎

(Einstein symmetry and conclusion.) By [L1], $G_{a b} (p) = {Ric}_{a b} (p) - \frac{1}{2} S (p) g_{a b} (p)$ and $G_{b a} (p) = {Ric}_{b a} (p) - \frac{1}{2} S (p) g_{b a} (p)$ ; step 2.4 and the symmetry of $g$ [L4] give $G_{a b} (p) = G_{b a} (p)$ . Since $p$ was arbitrary, (d) holds, and with steps 2.1, 2.2 and 2.3 this completes the proof.

Remarks

What the identities say and do not say. The four algebraic symmetries constrain the components of $R$ at a point and, in four dimensions, reduce its naively $4^{4}$ component functions to $20$ independent ones; the derivation above uses no counting of independent components and claims none. The identities are algebraic and hold pointwise; they are not differential equations, and they do not constrain how curvature varies from point to point. The differential identity satisfied by $\nabla R$ is a separate result (The second Bianchi identity and the vanishing divergence of the Einstein tensor).
Independence of the identities. The exchange symmetries (a) and (b) do not by themselves imply the Bianchi identity (c); a counterexample is given on the companion examples page. Every use of (c) below therefore cites it as an independent ingredient.

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The second Bianchi identity and the vanishing divergence of the Einstein tensor

Statement

Let $(M, g, N)$ be a time-oriented smooth Lorentzian four-manifold with metric-selected connection $\nabla$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, Metric compatibility, torsion freedom and the metric-selected connection class, Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields), and let $R$ , $R_{a b c d}$ , ${Ric}_{a b}$ , $S$ and $G_{a b}$ be as in The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors. Write $; e$ for the induced covariant derivative and raise indices with $g^{- 1}$ (Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields clause (c), The inverse metric, musical isomorphisms and index raising).

(a) Second Bianchi identity. For all indices $a, b, c, d, e$ , $R^{a}_{b c d; e} + R^{a}_{b d e; c} + R^{a}_{b e c; d} = 0, equivalently R_{a b c d; e} + R_{a b d e; c} + R_{a b e c; d} = 0.$ Equivalently, the antisymmetrization over the last three slots vanishes: $R^{a}_{b [c d; e]} = 0$ .

(b) Contracted identities. $\nabla^{a} {Ric}_{a b} = \frac{1}{2} \nabla_{b} S$ , and consequently $\nabla^{a} G_{a b} = 0 for every b,$ where $\nabla^{a} G_{a b} : = g^{a e} \nabla_{e} G_{a b}$ . The conclusion is the vanishing of the contracted divergence for each second index; it does not assert that all components $\nabla_{c} G_{a b}$ vanish.

Facts & Assumptions

Given: A time-oriented smooth Lorentzian four-manifold with metric-selected connection, its curvature tensors as in the definition, a point $p$ , and a point-normal chart at $p$ .

[L1]

$R^{a}_{b c d}$ , $R_{a b c d}$ , ${Ric}_{b d}$ , $S$ and $G_{a b}$ are defined by the component, lowering, contraction and Einstein formulas of The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors, and ${Ric}_{a b} = {Ric}_{b a}$ (The algebraic symmetries of the curvature tensor and the symmetry of the Ricci and Einstein tensors clause (d)).

[L2]

In every chart, $R^{a}_{b c d} = \partial_{c} Γ^{a}_{d b} - \partial_{d} Γ^{a}_{c b} + Γ^{a}_{c e} Γ^{e}_{d b} - Γ^{a}_{d e} Γ^{e}_{c b}$ (Tensoriality of the curvature, its coordinate formula and its point-normal metric formula clause (b)).

[L3]

For a smooth tensor field the induced covariant derivative has, in a chart, the component formula $R^{a}_{b c d; e} = \partial_{e} R^{a}_{b c d} + Γ^{a}_{f e} R^{f}_{b c d} - Γ^{f}_{b e} R^{a}_{f c d} - Γ^{f}_{c e} R^{a}_{b f d} - Γ^{f}_{d e} R^{a}_{b c f}$ ; it commutes with contractions and satisfies $g_{a b; c} = 0 = g^{a b}_{; c}$ (Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields clause (c)).

[L4]

For every $p$ there is a point-normal chart with $g_{a b} (p) = η_{a b}$ , $\partial_{a} g_{b c} (p) = 0$ and $Γ^{a}_{b c} (p) = 0$ (Point-normal and exponential coordinate charts of the metric-selected connection clause (a), Metric compatibility, torsion freedom and the metric-selected connection class).

[L5]

Mixed partial derivatives commute (Clairaut--Schwarz theorem for continuous second partial derivatives), and $g^{a b} g_{b c} = δ_{c}^{a}$ (The inverse metric, musical isomorphisms and index raising).

[L6]

The components of a tensor field in a chart determine the field, so an identity between tensor components proved at every point in a point-normal chart holds everywhere (Tensor fields of type (r,s) on a Lorentzian four-manifold and their components clause (b), Tensoriality of the curvature, its coordinate formula and its point-normal metric formula clause (a)).

[L7]

The Riemann tensor satisfies $R_{a b c d} = - R_{a b d c}$ and $R_{a b c d} = R_{c d a b}$ , and hence $R_{a b c d} = - R_{b a c d}$ by applying the exchange symmetry, the last-pair skew-symmetry and the exchange symmetry again (The algebraic symmetries of the curvature tensor and the symmetry of the Ricci and Einstein tensors clauses (a)-(b)).

Proof

technique · direct

1.1L2L3L4given

(Reduction to a point-normal chart.) Fix $p$ and take the point-normal chart of [L4]. By [L3], $R^{a}_{b c d; e} (p) = \partial_{e} R^{a}_{b c d} (p)$ , because every term of the derivative formula carrying a Christoffel symbol vanishes at $p$ . By [L2], $R^{a}_{b c d} (p) = \partial_{c} Γ^{a}_{d b} (p) - \partial_{d} Γ^{a}_{c b} (p)$ , since the quadratic terms carry a factor $Γ (p) = 0$ .

2.1step 1.1L2L3L4algebra

(Differentiate on the chart, then evaluate.) The full coordinate identity [L2] holds as an equality of smooth functions throughout the chart. Apply $\partial_{e}$ to that identity before evaluating at $p$ . Each derivative of a quadratic term is a sum of terms $\partial_{e} (Γ^{a}_{c f} Γ^{f}_{d b}) = (\partial_{e} Γ^{a}_{c f}) Γ^{f}_{d b} + Γ^{a}_{c f} (\partial_{e} Γ^{f}_{d b})$ , and similarly for the second product. Every such term contains an undifferentiated Christoffel symbol, which is zero at $p$ by [L4]. Since [L3] also gives $R^{a}_{b c d; e} (p) = \partial_{e} R^{a}_{b c d} (p)$ , this yields $R^{a}_{b c d; e} (p) = \partial_{e} \partial_{c} Γ^{a}_{d b} (p) - \partial_{e} \partial_{d} Γ^{a}_{c b} (p)$ , and cyclically $R^{a}_{b d e; c} (p) = \partial_{c} \partial_{d} Γ^{a}_{e b} (p) - \partial_{c} \partial_{e} Γ^{a}_{d b} (p)$ and $R^{a}_{b e c; d} (p) = \partial_{d} \partial_{e} Γ^{a}_{c b} (p) - \partial_{d} \partial_{c} Γ^{a}_{e b} (p)$ .

3.1step 2.1L3L5L6algebra

(Cyclic cancellation.) Adding the three displays of step 2.1, the six second partial derivatives of Christoffel symbols cancel in pairs by [L5]: $\partial_{e} \partial_{c} Γ^{a}_{d b}$ cancels $- \partial_{c} \partial_{e} Γ^{a}_{d b}$ , $- \partial_{e} \partial_{d} Γ^{a}_{c b}$ cancels $+ \partial_{d} \partial_{e} Γ^{a}_{c b}$ , and $+ \partial_{c} \partial_{d} Γ^{a}_{e b}$ cancels $- \partial_{d} \partial_{c} Γ^{a}_{e b}$ . Hence $R^{a}_{b c d; e} + R^{a}_{b d e; c} + R^{a}_{b e c; d} = 0$ at $p$ ; by [L6] this tensor identity holds everywhere. Lowering the first index using [L3] ( $g_{a f; e} = 0$ ) gives $R_{a b c d; e} + R_{a b d e; c} + R_{a b e c; d} = 0$ , and the antisymmetrized form is the same display rewritten. This proves (a).

4.1step 3.1L1L3L7L5algebra

(First contraction.) Contract the identity of step 3.1 with $g^{a c}$ , using [L3] to move the metric inside the derivatives: $g^{a c} R_{a b c d; e} = \nabla_{e} (g^{a c} R_{a b c d}) = \nabla_{e} {Ric}_{b d}$ ; $g^{a c} R_{a b d e; c} = \nabla^{a} R_{a b d e}$ ; and $g^{a c} R_{a b e c; d} = \nabla_{d} (g^{a c} R_{a b e c})$ . For the last, $R_{a b e c} = - R_{a b c e}$ by [L7], so $g^{a c} R_{a b e c} = - g^{a c} R_{a b c e} = - {Ric}_{b e}$ by [L1] and the Einstein summation convention. Hence $\nabla_{e} {Ric}_{b d} + \nabla^{a} R_{a b d e} - \nabla_{d} {Ric}_{b e} = 0$ .

5.1step 4.1L1L3L5L7algebra

(Second contraction and the Einstein divergence.) Contract the display of step 4.1 with $g^{b d}$ , again moving the metric inside derivatives with [L3]. The first term gives $\nabla_{e} (g^{b d} {Ric}_{b d}) = \nabla_{e} S$ . For the second, renaming the summed pair $(b, d)$ as $(d, b)$ and using the skew-symmetry $R_{a d b e} = - R_{d a b e}$ of [L7], $g^{b d} R_{a b d e} = g^{d b} R_{a d b e} = - g^{b d} R_{d a b e} = - {Ric}_{a e}$ , because the defining contraction of [L1] reads ${Ric}_{a e} = g^{b d} R_{d a b e}$ with its dummy pair renamed. Hence $\nabla^{a} (g^{b d} R_{a b d e}) = - \nabla^{a} {Ric}_{a e}$ . The third term gives $- g^{b d} \nabla_{d} {Ric}_{b e} = - \nabla^{b} {Ric}_{b e} = - \nabla^{a} {Ric}_{a e}$ after renaming. Collecting, $\nabla_{e} S - 2 \nabla^{a} {Ric}_{a e} = 0$ , that is $\nabla^{a} {Ric}_{a b} = \frac{1}{2} \nabla_{b} S$ after renaming $e \to b$ .

6.1step 5.1L1L3L5algebra∎

(Vanishing divergence.) By [L1], $G_{a e} = {Ric}_{a e} - \frac{1}{2} S g_{a e}$ , so by [L3] and step 5.1, $\nabla^{a} G_{a e} = \nabla^{a} {Ric}_{a e} - \frac{1}{2} g^{a b} g_{a e} \nabla_{b} S = \frac{1}{2} \nabla_{e} S - \frac{1}{2} \nabla_{e} S = 0$ . Since $e$ is arbitrary this is $\nabla^{a} G_{a b} = 0$ for every $b$ , and the proof is complete.

Remarks

The point-normal chart is the only place torsion freedom enters. Steps 1.1-2.1 evaluate the derivative of curvature in a chart with $Γ (p) = 0$ ; the existence of such a chart is part of the point-normal construction for the torsion-free metric-selected connection. The cyclic cancellation itself is then only the commutativity of mixed partials. The displayed identities are tensor identities and therefore independent of the chart in which they were derived.
What is not claimed. The identity $\nabla^{a} G_{a b} = 0$ is a purely geometric consequence of the definitions; it is not an equation of motion, it asserts nothing about any matter field, and it does not say that $\nabla_{c} G_{a b} = 0$ in all slots. Its use in the framework's field-law discussion is recorded on the Einstein-and-matter page, where the field equation is separately adopted as a postulate.

DefinitionDefinition: AI-adaptedProof: Not applicableOpen item page →

Jacobi fields, geodesic variations and conjugate instants along a geodesic

Definition

Let $(M, g, N)$ be a time-oriented smooth Lorentzian four-manifold with metric-selected connection $\nabla$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, Metric compatibility, torsion freedom and the metric-selected connection class, Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields), and let $R$ be its curvature (The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors). Let $I \subseteq R$ be an interval with nonempty interior (Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length) and let $γ : I \to M$ be an affinely parametrized geodesic with $D_{s} γ^{'} = 0$ , written $T : = γ^{'}$ (Geodesic of an affine connection); the parameter is the affinely parametrized one of that definition and need not be proper time.

(a) Fields along $γ$ and the derivative $D_{s}$ . A field along $γ$ is a section $J$ of the pullback bundle $γ^{*} T M$ , and $D_{s}$ denotes the covariant derivative along $γ$ of Covariant derivative along a curve. A field is smooth when its coefficients in a smooth local frame of $M$ around each point of $γ$ are $C^{\infty}$ on the corresponding parameter interval. At an included endpoint of $I$ all derivatives below are one-sided.

(b) Jacobi field. A Jacobi field along $γ$ is a smooth field $J$ along $γ$ satisfying the Jacobi equation $D_{s}^{2} J = R (T, J) T (s \in I),$ that is $D_{s}^{2} J = R (γ^{'}, J) γ^{'}$ . The assignment $J \mapsto R (T, J) T$ , which associates with a field along $γ$ another field along $γ$ and is $R$ -linear, is the Jacobi operator of $γ$ . Whether the field is smooth only piecewise is stated whenever a statement uses piecewise fields; the unqualified term means globally smooth.

(c) Geodesic variations. A geodesic variation with affine parameter $s$ is a smooth map $F : J_{0} \times I \to M$ , where $J_{0} \subseteq R$ is an open interval containing $0$ , such that for every $r \in J_{0}$ the curve $s \mapsto F (r, s)$ is a geodesic with affine parameter $s$ . Its variation field is the field $s \mapsto \partial_{r} F (0, s)$ along the curve $s \mapsto F (0, s)$ . The variation field of a geodesic variation is a Jacobi field along that curve (The Jacobi equation, geodesic deviation and the initial-value theory of Jacobi fields ↗ clause (b)).

(d) Conjugate instants and multiplicity. Let $s_{0} < s_{1}$ be points of $I$ . The event $γ (s_{1})$ is conjugate to $γ (s_{0})$ along $γ$ when there is a nonzero Jacobi field $J$ along $γ$ with $J (s_{0}) = J (s_{1}) = 0$ . In that case $s_{1}$ is called a conjugate instant of $γ$ (with respect to $s_{0}$ , when the base point needs to be named), and the multiplicity of the conjugate instant is the dimension of the real vector space of Jacobi fields vanishing at $s_{0}$ and $s_{1}$ ; the fields vanishing at $s_{0}$ form a $4$ -dimensional space and the multiplicity is at most $4$ , since a Jacobi field is determined by its value and derivative at $s_{0}$ (The Jacobi equation, geodesic deviation and the initial-value theory of Jacobi fields ↗ clause (c)). For a geodesic with $γ (s_{0}) = p$ and $γ^{'} (s_{0}) = v$ , the corresponding possibility of conjugacy is measured by the derivative of the exponential map at $(s_{1} - s_{0}) v$ , provided this vector lies in its domain (The differential of the exponential map and conjugacy along a geodesic ↗ applied to $t \mapsto γ (s_{0} + t)$ on $[0, s_{1} - s_{0}]$ ); that equivalence is proved, not assumed here. No conjugate instant is asserted to exist.

(e) The normal bundle. If $γ$ is future timelike with $g (T, T) = - 1$ , define its normal bundle to be the vector subbundle $N_{γ} : = ⨆_{s \in I} N_{γ} (s) \subseteq γ^{*} T M, N_{γ} (s) : = {v \in T_{γ (s)} M : g_{γ (s)} (v, T (s)) = 0} .$ A normal field is a smooth section $J$ of $N_{γ}$ , equivalently a smooth field along $γ$ satisfying $g (J, T) = 0$ . Each fibre $N_{γ} (s)$ has dimension three, the restriction of $g_{γ (s)}$ to it is positive definite, and $D_{s}$ maps normal fields to normal fields, since $\frac{d}{d s} g (J, T) = g (D_{s} J, T) + g (J, D_{s} T) = g (D_{s} J, T)$ and $D_{s} T = 0$ ; these facts are proved in The normal index form before the first conjugate instant is positive definite ↗ clause (b). The norm of a normal field is $∣ J (s) ∣ : = \sqrt{g_{γ (s)} (J (s), J (s))}$ . For any field $V$ along $γ$ , its normal component is the field $V^{⊥} (s) : = V (s) + g_{γ (s)} (V (s), T (s)) T (s) .$ Indeed $g (T, T) = - 1$ implies $g (V^{⊥}, T) = 0$ and $V = V^{⊥} - g (V, T) T$ .

Remarks

What is and is not fixed. This definition fixes the vocabulary of the deviation and conjugate-point results on this page: Jacobi field, variation field, conjugate instant and multiplicity. It asserts nothing about the existence of Jacobi fields with prescribed initial data, about conjugacy for any particular geodesic, or about maximality; those are theorems proved separately. The sign convention of the Jacobi operator is the one induced by the curvature sign convention of The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors: $D_{s}^{2} J = R (T, J) T$ , not its negative. Sources using the opposite sign or the opposite curvature convention state the equation with a minus sign; no formula is taken from another source without converting its convention.
Relation to proper time. When $γ$ is future timelike with $g (T, T) = - c^{2}$ and the affine parameter is proper time $τ$ , the equation above is the same as $D_{τ}^{2} J = R (U, J) U$ for the four-velocity $U$ ; the conversion factor is a positive constant and does not change which fields vanish at which parameter values (Proper time is well defined and reparametrization invariant; the four-velocity is its proper-time tangent).

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The Jacobi equation, geodesic deviation and the initial-value theory of Jacobi fields

Statement

Let $(M, g, N)$ be a time-oriented smooth Lorentzian four-manifold with metric-selected connection $\nabla$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, Metric compatibility, torsion freedom and the metric-selected connection class, Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields), let $R$ be its curvature (The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors), let $I, J_{0} \subseteq R$ be intervals with nonempty interior (Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length), and let $F : J_{0} \times I \to M$ be a smooth map. Write $T : = \partial_{t} F$ and $J^{F} : = \partial_{r} F$ ; a field along $F$ means a smooth section of $F^{*} T M$ over $J_{0} \times I$ , and $D_{r}, D_{t}$ are the covariant derivatives in the two parameters (Covariant derivative along a curve).

(a) Curvature as the commutator of the two derivatives. For every smooth field $V$ along $F$ , $(D_{t} D_{r} - D_{r} D_{t}) V = R (T, J^{F}) V .$

(b) Geodesic deviation. Suppose in addition that for every $r \in J_{0}$ the curve $t \mapsto F (r, t)$ is a geodesic with affine parameter $t$ , that is $D_{t} T = 0$ . Then $D_{t} J^{F} = D_{r} T$ and $D_{t}^{2} J^{F} = R (T, J^{F}) T .$ Consequently, for every $r$ the field $t \mapsto J^{F} (r, t)$ is a Jacobi field along the geodesic $t \mapsto F (r, t)$ (Jacobi fields, geodesic variations and conjugate instants along a geodesic clause (b)), and for a geodesic variation the variation field is a Jacobi field.

(c) Initial data. Let $γ : I \to M$ be a smooth geodesic and let $s_{0} \in I$ . For every pair $J_{0}, J_{1} \in T_{γ (s_{0})} M$ there exists exactly one smooth Jacobi field $J$ along $γ$ with $J (s_{0}) = J_{0}, D_{s} J (s_{0}) = J_{1} .$ The set of all Jacobi fields along $γ$ is a real vector space of dimension $8$ , and $J \mapsto (J (s_{0}), D_{s} J (s_{0}))$ is a linear isomorphism onto $T_{γ (s_{0})} M \oplus T_{γ (s_{0})} M$ .

(d) The tangential component. Let $γ$ be a smooth geodesic and $J$ a smooth Jacobi field along it. Then $s \mapsto g (J, γ^{'}) (s)$ is affine, and if $γ$ is future timelike with $g (γ^{'}, γ^{'}) = - 1$ , the normal component $J^{⊥}$ of Jacobi fields, geodesic variations and conjugate instants along a geodesic clause (e), explicitly $J^{⊥} = J + g (J, γ^{'}) γ^{'}$ , is again a Jacobi field along $γ$ .

Facts & Assumptions

Given: A time-oriented smooth Lorentzian four-manifold with metric-selected connection and curvature, a smooth two-parameter map $F$ , a geodesic $γ$ , and the data of the claims.

[L1]

The curvature operator is $R (X, Y) Z = \nabla_{X} \nabla_{Y} Z - \nabla_{Y} \nabla_{X} Z - \nabla_{[X, Y]} Z$ and is a smooth $(1, 3)$ -tensor, whose components satisfy $R (\partial_{c}, \partial_{d}) \partial_{b} = \sum_{a} R^{a}_{b c d} \partial_{a}$ and $g (R (X, Y) Z, W) = R_{a b c d} W^{a} X^{c} Y^{d} Z^{b}$ (The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors, Tensoriality of the curvature, its coordinate formula and its point-normal metric formula clause (a)).

[L2]

In a chart, $\nabla_{\partial_{b}} \partial_{c} = \sum_{a} Γ^{a}_{b c} \partial_{a}$ with $Γ^{a}_{b c} = Γ^{a}_{c b}$ because $\nabla$ is torsion-free; the coordinate fields commute and mixed partials commute (Christoffel symbols of an affine connection, Torsion freedom and metric compatibility in chart components clause 1, Clairaut--Schwarz theorem for continuous second partial derivatives).

[L3]

For a smooth curve $σ$ and a field $V$ along it, the covariant derivative along the curve has, in a chart, the component expression $D_{s} V^{a} = \partial_{s} V^{a} + \sum_{b, c} Γ^{a}_{b c} \partial_{s} σ^{b} V^{c}$ , and satisfies $D_{s} (f V) = f^{'} V + f D_{s} V$ (Covariant derivative along a curve, Christoffel symbols of an affine connection, Vector field and section along a smooth curve).

[L4]

Along a geodesic $D_{t} T = 0$ . Metric compatibility gives $\frac{d}{d t} g (V, W) = g (D_{t} V, W) + g (V, D_{t} W)$ for fields along it (Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields clause (c), Covariant derivative along a curve). The curvature satisfies $R (X, Y) Z = - R (Y, X) Z$ and $g (R (X, Y) Z, W) = - g (R (X, Y) W, Z)$ (Geodesic of an affine connection, The algebraic symmetries of the curvature tensor and the symmetry of the Ricci and Einstein tensors clause (a)).

[L5]

For a smooth curve there is a parallel smooth field with any prescribed initial value; parallel transport is linear and preserves the metric, so the parallel transports of a $g$ -orthonormal basis form a smooth orthonormal frame along the curve (Existence and uniqueness of parallel sections, Parallel section along a curve, Covariant differentiation along curves: metric pairing, parallel transport and geodesic speed clause 2).

[L6]

On a compact interval, a linear first-order system $Y^{'} = A (s) Y$ with continuous coefficient matrix has a unique solution with prescribed initial value (Linear matrix ODEs have unique global solutions on a fixed interval).

[L7]

A Jacobi field along a geodesic is by definition a smooth field with $D_{t}^{2} J = R (γ^{'}, J) γ^{'}$ ; the Jacobi operator is $J \mapsto R (γ^{'}, J) γ^{'}$ (Jacobi fields, geodesic variations and conjugate instants along a geodesic clause (b)).

Proof

technique · direct

1.1L2L3algebra

(Chart computation of the commutator.) Let $V$ be a smooth field along $F$ , fix a parameter pair $(r, t)$ and a chart around $F (r, t)$ . By [L3] applied to the curves with the other parameter fixed, $D_{r} V^{a} = \partial_{r} V^{a} + \sum_{b, c} Γ^{a}_{b c} \partial_{r} F^{b} V^{c}$ and $D_{t} V^{a} = \partial_{t} V^{a} + \sum_{b, c} Γ^{a}_{b c} \partial_{t} F^{b} V^{c}$ (indices here are chart components of the fields). Applying $D_{t}$ to the first expression and $D_{r}$ to the second and subtracting, the second-order terms $\partial_{t} \partial_{r} V^{a}$ and $\partial_{r} \partial_{t} V^{a}$ cancel by [L2], and the remaining terms are exactly $\sum_{b, c, d} (\partial_{c} Γ^{a}_{d b} - \partial_{d} Γ^{a}_{c b} + Γ^{a}_{c e} Γ^{e}_{d b} - Γ^{a}_{d e} Γ^{e}_{c b}) T^{c} (J^{F})^{d} V^{b}$ , with the Christoffel-product terms collected by relabelling summed indices; this is the coordinate expression whose coefficient is the curvature component of [L1].

1.2L2L3algebra

(The two derivatives commute on the tangent.) If $D_{t} T = 0$ , then by the component expressions of [L3] with $V = T$ and by [L2], $D_{t} J^{F} - D_{r} T$ has components $\partial_{t} \partial_{r} F^{a} + Γ^{a}_{b c} T^{b} (J^{F})^{c} - \partial_{r} \partial_{t} F^{a} - Γ^{a}_{b c} (J^{F})^{b} T^{c}$ , which vanishes because the mixed partials commute and $Γ^{a}_{b c}$ is symmetric. Hence $D_{t} J^{F} = D_{r} T$ .

1.3L5L1L3algebra

(Jacobi fields with prescribed initial data.) Let $p : = γ (s_{0})$ and extend a basis $E_{1} (s_{0}), \dots, E_{4} (s_{0})$ of $T_{p} M$ to a parallel frame $E_{1}, \dots, E_{4}$ along $γ$ by [L5]. Writing a field along $γ$ as $J = \sum_{i} J^{i} E_{i}$ , [L3] and $D_{s} E_{i} = 0$ give $D_{s}^{2} J = \sum_{i} (J^{i})^{''} E_{i}$ . For each $j$ , define the smooth coefficient functions $C^{i}_{j}$ by the unique frame expansion $R (γ^{'}, E_{j}) γ^{'} = \sum_{i} C^{i}_{j} E_{i}$ ; then [L1] gives $R (γ^{'}, J) γ^{'} = \sum_{i, j} J^{j} C^{i}_{j} E_{i}$ . The Jacobi equation is therefore equivalent to the second-order linear system $(J^{i})^{''} = \sum_{j} C^{i}_{j} J^{j}$ for the coefficient functions.

2.1step 1.1L1algebra

(Identification with the curvature and conclusion of (a).) The coefficient of $T^{c} (J^{F})^{d} V^{b}$ in step 1.1 is $\partial_{c} Γ^{a}_{d b} - \partial_{d} Γ^{a}_{c b} + \sum_{e} (Γ^{a}_{c e} Γ^{e}_{d b} - Γ^{a}_{d e} Γ^{e}_{c b})$ , which is the coordinate formula for $R^{a}_{b c d}$ of [L1]. Therefore $(D_{t} D_{r} - D_{r} D_{t}) V = R (T, J^{F}) V$ at the chosen point; the point was arbitrary and the identity is between smooth fields, so (a) holds.

2.2step 1.3L5L6algebra

(Solution of the linear system.) The system of step 1.3 is equivalent to the first-order system $Y^{'} = A (s) Y$ for $Y = (J^{1}, \dots, J^{4}, (J^{1})^{'}, \dots, (J^{4})^{'})$ and the continuous $8 \times 8$ block matrix $A = (\begin{matrix} 0 & I \\ C & 0 \end{matrix})$ . By [L6], on each compact subinterval of $I$ there is a unique solution with prescribed initial value; two such solutions agree on the overlap of their intervals, so they patch to a unique solution on all of $I$ . Every pair $(J_{0}, J_{1}) = (J^{i} (s_{0}), (J^{i})^{'} (s_{0}))$ is an admissible initial value, and since the frame is a basis, prescribing $(J_{0}, J_{1}) \in T_{p} M \oplus T_{p} M$ prescribes exactly these coefficients.

3.1step 2.1step 1.2L4L7algebra

(Deviation equation.) By step 1.2 and $D_{t} T = 0$ , $D_{t}^{2} J^{F} = D_{t} D_{r} T$ , and by step 2.1 with $V = T$ , $D_{t} D_{r} T = D_{r} D_{t} T + R (T, J^{F}) T = 0 + R (T, J^{F}) T$ . Hence $D_{t}^{2} J^{F} = R (T, J^{F}) T$ ; by [L7] the field $t \mapsto J^{F} (r, t)$ is a Jacobi field along the geodesic $t \mapsto F (r, t)$ for each $r$ , which proves (b).

3.2step 2.2algebra

(Vector space and dimension.) By step 2.2 the map $J \mapsto (J (s_{0}), D_{s} J (s_{0}))$ from the set of smooth Jacobi fields along $γ$ to $T_{p} M \oplus T_{p} M$ is well defined, linear over $R$ , injective (two Jacobi fields with the same data agree by uniqueness of step 2.2) and surjective (existence in step 2.2). It is therefore a linear isomorphism, the space of Jacobi fields has dimension $\dim (T_{p} M \oplus T_{p} M) = 8$ , and (c) holds.

4.1step 3.1step 3.2L4L7algebra∎

(Tangential component.) For a Jacobi field $J$ along a geodesic, put $a (s) : = g (J, γ^{'})$ . Metric compatibility and $D_{s} γ^{'} = 0$ give $a^{''} = g (D_{s}^{2} J, γ^{'}) = g (R (γ^{'}, J) γ^{'}, γ^{'}) = 0$ by [L4], so $a$ is affine. If in addition $g (γ^{'}, γ^{'}) = - 1$ , the normal component defined in Jacobi fields, geodesic variations and conjugate instants along a geodesic is $J^{⊥} = J + a γ^{'}$ . Therefore $D_{s}^{2} J^{⊥} = D_{s}^{2} J + a^{''} γ^{'} + 2 a^{'} D_{s} γ^{'} + a D_{s}^{2} γ^{'} = R (γ^{'}, J) γ^{'}$ . Since $J = J^{⊥} - a γ^{'}$ and $R (γ^{'}, γ^{'}) γ^{'} = 0$ by the first-slot skew-symmetry of [L4], this is $R (γ^{'}, J^{⊥}) γ^{'}$ . Thus $J^{⊥}$ is a Jacobi field, proving (d).

Remarks

The deviation equation is exact for a geodesic family. No smallness of the separation or of the family is assumed: the equation of claim (b) holds exactly for every smooth family of affinely parametrized geodesics and every parameter value, and holds even where the family map is not an immersion. Smallness enters only in physical applications, where a family of test bodies models an infinitesimal separation.
What determines a Jacobi field. Claim (c) states the initial-value theory: initial value and initial derivative at one instant determine a Jacobi field on the whole parameter interval, and the space is eight-dimensional for a four-manifold. Nothing here asserts the existence of conjugate instants; conjugacy is the existence of a nonzero Jacobi field with two zeros, characterized separately in The differential of the exponential map and conjugacy along a geodesic.

LemmaStatement: AI-adaptedProof: AI-generatedprecheck passOpen item page →

The differential of the exponential map and conjugacy along a geodesic

Statement

Let $(M, g, N)$ be a time-oriented smooth Lorentzian four-manifold with metric-selected connection $\nabla$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, Metric compatibility, torsion freedom and the metric-selected connection class, Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields), let $p \in M$ , let $E_{p} \subseteq T_{p} M$ be the exponential domain and $\exp_{p} : E_{p} \to M$ the exponential map of The geodesic initial-value problem and the exponential map of the metric-selected connection, and identify $T_{h} (T_{p} M)$ with $T_{p} M$ by the canonical isomorphism for every $h \in T_{p} M$ (The differential of a smooth map).

(a) The differential in terms of Jacobi fields. Let $v \in T_{p} M$ , let $s_{1} > 0$ satisfy $s_{1} v \in E_{p}$ , and let $u \in T_{p} M$ . Let $γ : [0, s_{1}] \to M$ , $γ (t) : = \exp_{p} (t v)$ (well defined by star-shapedness of $E_{p}$ ), and let $J_{u}$ be the unique Jacobi field along $γ$ with $J_{u} (0) = 0$ and $D_{t} J_{u} (0) = u$ (The Jacobi equation, geodesic deviation and the initial-value theory of Jacobi fields clause (c), Jacobi fields, geodesic variations and conjugate instants along a geodesic clause (b)). Then $d (\exp_{p})_{t v} (t u) = J_{u} (t) (0 \leq t \leq s_{1});$ in particular $d (\exp_{p})_{s_{1} v} (s_{1} u) = J_{u} (s_{1})$ and, in the special case $s_{1} = 1$ , $v \in E_{p}$ , $d (\exp_{p})_{v} (u) = J_{u} (1)$ .

(b) Conjugacy as a kernel condition. With the notation of (a), $d (\exp_{p})_{s_{1} v}$ is injective if and only if no nonzero Jacobi field along $γ$ vanishes at $t = 0$ and at $t = s_{1}$ . Moreover $\ker d (\exp_{p})_{s_{1} v} = s_{1} {u \in T_{p} M : J_{u} (s_{1}) = 0}$ , so $\dim \ker d (\exp_{p})_{s_{1} v}$ equals the multiplicity of the conjugate instant $s_{1}$ whenever $γ (s_{1})$ is conjugate to $p$ along $γ$ (Jacobi fields, geodesic variations and conjugate instants along a geodesic clause (d)).

(c) Conjugate instants of a geodesic. Let $γ : I \to M$ be a smooth geodesic with $γ (0) = p$ , and let $s_{1} > 0$ with $s_{1} \in I$ and $s_{1} γ^{'} (0) \in E_{p}$ . Then $γ (s_{1})$ is conjugate to $γ (0) = p$ along $γ$ if and only if $d (\exp_{p})_{s_{1} γ^{'} (0)}$ is not injective.

Facts & Assumptions

Given: A time-oriented smooth Lorentzian four-manifold with metric-selected connection, a point $p$ , the exponential domain $E_{p}$ and map $\exp_{p}$ , a vector $v \in E_{p}$ and a vector $w \in T_{p} M$ .

[L1]

$E_{p}$ is open and star-shaped about $0$ ; for every $h \in E_{p}$ the curve $t \mapsto \exp_{p} (t h)$ on an interval containing $[0, 1]$ is the geodesic $γ_{p, h}$ with $γ_{p, h} (0) = p$ , $γ_{p, h}^{'} (0) = h$ , and $\exp_{p} (t h) = γ_{p, h} (t)$ ; the map $(t, p, h) \mapsto γ_{p, h} (t)$ is smooth on its domain (The geodesic initial-value problem and the exponential map of the metric-selected connection clauses (a)-(d)).

[L2]

For a smooth two-parameter map $Φ (r, s)$ , the variation field $\partial_{r} Φ$ of a family of geodesics is a Jacobi field along each longitudinal geodesic, the two covariant derivatives satisfy $D_{s} \partial_{r} Φ = D_{r} \partial_{s} Φ$ whenever the longitudinal curves are geodesics, and the differential of a smooth map is computed by the chain rule (The Jacobi equation, geodesic deviation and the initial-value theory of Jacobi fields clauses (a)-(b), The differential of a smooth map, The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ).

[L3]

A Jacobi field along a geodesic is determined by its value and covariant derivative at one parameter; the initial-value map $J \mapsto (J (s_{0}), D_{s} J (s_{0}))$ is a linear isomorphism onto $T_{γ (s_{0})} M \oplus T_{γ (s_{0})} M$ (The Jacobi equation, geodesic deviation and the initial-value theory of Jacobi fields clause (c)).

[L4]

A field $J$ along a geodesic is a Jacobi field when $D_{t}^{2} J = R (γ^{'}, J) γ^{'}$ ; conjugacy at a pair of instants means the existence of a nonzero Jacobi field vanishing there, and the multiplicity is the dimension of the space of such fields (Jacobi fields, geodesic variations and conjugate instants along a geodesic clauses (b) and (d)).

[L5]

The canonical identification $T_{h} (T_{p} M) ≅ T_{p} M$ of a vector space with its tangent space at a point, and the linearity of the differential in its second argument (The differential of a smooth map).

Proof

technique · direct

1.1L1given

(The two-parameter family.) Fix $v \in T_{p} M$ , $u \in T_{p} M$ and $s_{1} > 0$ with $s_{1} v \in E_{p}$ , and put $Φ (r, s) : = \exp_{p} (s (v + r u))$ for $s \in [0, s_{1}]$ and $∣ r ∣$ small. This is well defined and smooth: $E_{p}$ is open and star-shaped by [L1], for small $∣ r ∣$ the vector $v + r u$ is close to $v$ , hence $s_{1} (v + r u) \in E_{p}$ , and then $s (v + r u) = \frac{s}{s_{1}} (s_{1} (v + r u)) \in E_{p}$ for $0 \leq s \leq s_{1}$ . For each $r$ , $s \mapsto Φ (r, s)$ is the geodesic with initial tangent $v + r u$ at $s = 0$ by [L1]; its tangent field is $T : = \partial_{s} Φ$ , and $T (0, r) = v + r u$ .

2.1step 1.1L1L2algebra

(The variation field is Jacobi.) By [L2], $J (s) : = \partial_{r} Φ (0, s)$ is a Jacobi field along $s \mapsto Φ (0, s) = \exp_{p} (s v) = γ (s)$ , and $D_{s} J = D_{r} T$ . Its initial data are $J (0) = \partial_{r} Φ (0, 0) = 0$ because $Φ (r, 0) = \exp_{p} (0) = p$ for all $r$ , and $D_{s} J (0) = D_{r} T (0, 0) = \partial_{r} (v + r u) ∣_{r = 0} = u$ , because the curve $r \mapsto Φ (r, 0)$ is constant and hence $D_{r}$ at $s = 0$ is the ordinary derivative.

3.1step 2.1L2L5algebra

(Value at the parameter $t$ .) For $0 \leq t \leq s_{1}$ , by [L2] and the chain rule, $J (t) = \partial_{r} (\exp_{p} (t (v + r u))) ∣_{r = 0} = d (\exp_{p})_{t v} (t u)$ , the derivative of $r \mapsto t (v + r u)$ at $r = 0$ being $t u$ .

4.1step 3.1L3algebra

(Identification with $J_{u}$ .) By step 2.1, $J$ is a Jacobi field along $γ$ with initial data $(0, u)$ ; by the uniqueness and linearity of the initial-data correspondence [L3], $J = J_{u}$ . Combining with step 3.1, $d (\exp_{p})_{t v} (t u) = J_{u} (t)$ for every $t \in [0, s_{1}]$ ; at $t = 0$ both sides vanish. The cases $t = s_{1}$ and $s_{1} = 1$ are the stated special cases, so (a) holds.

5.1step 4.1L3L4algebra

(Kernel and conjugacy.) By step 4.1 with $t = s_{1}$ , $d (\exp_{p})_{s_{1} v} (s_{1} u) = J_{u} (s_{1})$ ; since $u \mapsto s_{1} u$ is a linear isomorphism, $\ker d (\exp_{p})_{s_{1} v} = {s_{1} u : J_{u} (s_{1}) = 0}$ . By [L3], $u \mapsto J_{u}$ is a linear isomorphism from $T_{p} M$ onto the Jacobi fields along $γ$ vanishing at $0$ (those with initial value $0$ ), so the kernel is nonzero exactly when some nonzero Jacobi field vanishes at $0$ and $s_{1}$ , and its dimension, namely $\dim {u : J_{u} (s_{1}) = 0}$ , is the multiplicity by [L4]. This proves (b).

6.1step 5.1L1L4algebra∎

(Geodesics starting at $p$ .) Let $γ : I \to M$ be a geodesic with $γ (0) = p$ and let $s_{1} γ^{'} (0) \in E_{p}$ . By [L1], $γ (s) = \exp_{p} (s γ^{'} (0))$ for those $s \in I$ with $s γ^{'} (0) \in E_{p}$ , and $s_{1}$ is such a value; so the claim is the special case of step 5.1 with $v = γ^{'} (0)$ . This proves (c) and completes the proof.

Remarks

No Riemannian positivity is used. The statement and proof use only the smooth exponential map, the geodesic-variation equation and the initial-value theory of Jacobi fields. Positivity of the metric on a normal bundle enters only in the separate index-form lemma The normal index form before the first conjugate instant is positive definite.
The base point. Formula (a) compares the differential at $t v$ with a Jacobi field along the geodesic from $p$ ; the factor $t$ in the argument is essential and is what makes the kernel computation of (b) exact. The convention $v \in E_{p}$ ensures that the entire radial segment $[0, v]$ is in the exponential domain, so no completeness of $(M, g)$ is assumed.

LemmaStatement: AI-adaptedProof: AI-generatedprecheck passOpen item page →

The normal index form before the first conjugate instant is positive definite

Statement

Let $(M, g, N)$ be a time-oriented smooth Lorentzian four-manifold with metric-selected connection $\nabla$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, Metric compatibility, torsion freedom and the metric-selected connection class, Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields), let $a < b$ and let $γ : [a, b] \to M$ be a smooth future timelike geodesic with $g (T, T) = - 1$ , $T : = γ^{'}$ , such that no instant $s \in (a, b]$ is conjugate to $a$ along $γ$ (Jacobi fields, geodesic variations and conjugate instants along a geodesic). Let $N_{γ}$ be the normal bundle of $γ$ (clause (e) there) with the positive definite fibre metric $g$ , and for continuous piecewise- $C^{1}$ normal fields $J, K$ vanishing at $a$ and $b$ define $I (J, K) : = \int_{a}^{b} (g (D_{s} J, D_{s} K) + g (R (T, J) T, K)) d s,$ the integral being over the smooth pieces of $J$ and $K$ ; write $I (J) : = I (J, J)$ and $∣ J ∣ : = \sqrt{g (J, J)}$ .

(a) The Jacobi operator on the normal bundle. For every normal field $J$ the field $C_{T} (J) : = R (T, J) T$ is again normal, and $C_{T}$ is $g$ -selfadjoint on each fibre: $g (C_{T} (J), K) = g (J, C_{T} (K))$ for normal fields $J, K$ .

(b) Matrix form. Let $E_{1}, E_{2}, E_{3}$ be a parallel $g$ -orthonormal frame of $N_{γ}$ (which exists), and write $J = \sum_{i} J^{i} E_{i}$ , $K = \sum_{i} K^{i} E_{i}$ with smooth coefficients on the smooth pieces. Then $g (D_{s} J, D_{s} K) = \sum_{i} (J^{i})^{'} (K^{i})^{'}$ and $g (R (T, J) T, K) = \sum_{i, j} J^{j} C^{i}_{j} K^{i}$ for the smooth symmetric coefficient functions $C^{i}_{j} : = g (C_{T} (E_{j}), E_{i})$ , so that the matrix $C (s) = (C^{i}_{j} (s))_{i, j}$ is symmetric for every $s$ and $I (J, K) = \int_{a}^{b} (\sum_{i} (J^{i})^{'} (K^{i})^{'} + J^{T} C K) d s .$

(c) Factorization. Let $A : [a, b] \to M_{3} (R)$ be the unique solution of the initial-value problem $A^{''} = C A, A (a) = 0, A^{'} (a) = I_{3} .$ Then $A (s)$ is invertible for every $s \in (a, b]$ , the field $Θ (s) : = A^{'} (s) A (s)^{- 1}$ is symmetric on $(a, b]$ with $Θ^{'} = C - Θ^{2},$ and for every continuous piecewise- $C^{1}$ normal field $J$ vanishing at $a$ and $b$ , $I (J) = \int_{a}^{b} ∣ J^{'} - Θ J ∣^{2} d s,$ where $∣ J^{'} - Θ J ∣^{2} : = \sum_{i} ((J^{i})^{'} - \sum_{j} Θ^{i}_{j} J^{j})^{2}$ in the frame of (b). Consequently $I (J) \geq 0$ , and $I (J) = 0$ if and only if $J = 0$ ; the index form is positive definite on the space of such fields.

Facts & Assumptions

Given: A smooth future unit timelike geodesic $γ : [a, b] \to M$ with no conjugate instant in $(a, b]$ , its normal bundle with a parallel orthonormal frame, and piecewise- $C^{1}$ normal fields vanishing at $a$ and $b$ .

[L1]

$T = γ^{'}$ satisfies $D_{s} T = 0$ and $g (T, T) = - 1$ ; the fibres of $N_{γ}$ are $N_{γ} (s) = {v \in T_{γ (s)} M : g (v, T (s)) = 0}$ , their smooth sections are the normal fields, $D_{s}$ preserves them, and their fibre metric is positive definite. The normal component is $V^{⊥} = V + g (V, T) T$ (Jacobi fields, geodesic variations and conjugate instants along a geodesic clause (e), Observers, rest spaces and the spatial projection at an event clause (b), The observer rest-space decomposition and the spatial projection).

[L2]

The curvature satisfies $R_{a b c d} = - R_{b a c d}$ , $R_{a b c d} = - R_{a b d c}$ and $R_{a b c d} = R_{c d a b}$ , and $g (R (X, Y) Z, W) = R_{a b c d} W^{a} X^{c} Y^{d} Z^{b}$ (The algebraic symmetries of the curvature tensor and the symmetry of the Ricci and Einstein tensors clauses (a)-(b), Tensoriality of the curvature, its coordinate formula and its point-normal metric formula clause (a)).

[L3]

A Jacobi field is a field with $D_{s}^{2} J = R (T, J) T$ ; for a Jacobi field $J$ along $γ$ , $g (J, T)$ is affine and $J^{⊥}$ is again a Jacobi field (Jacobi fields, geodesic variations and conjugate instants along a geodesic clauses (b), (c) and (e), The Jacobi equation, geodesic deviation and the initial-value theory of Jacobi fields clause (d)).

[L4]

The parallel transport of a $g$ -orthonormal basis is a smooth $g$ -orthonormal frame with $D_{s} E_{i} = 0$ , so for $J = \sum_{i} J^{i} E_{i}$ one has $D_{s} J = \sum_{i} (J^{i})^{'} E_{i}$ and $g (D_{s} J, D_{s} K) = \sum_{i} (J^{i})^{'} (K^{i})^{'}$ (Existence and uniqueness of parallel sections, Parallel section along a curve, Covariant differentiation along curves: metric pairing, parallel transport and geodesic speed clause 2).

[L5]

On $[a, b]$ the linear matrix initial-value problem $Y^{'} = B Y$ has a unique solution for continuous $B$ ; in particular the problem of claim (c) has a unique solution $A$ on $[a, b]$ (Linear matrix ODEs have unique global solutions on a fixed interval).

[L6]

The coefficient matrix $C$ of claim (c) is smooth, so the solution $A$ of $A^{''} = C A$ , $A (a) = 0$ , $A^{'} (a) = I_{3}$ is $C^{2}$ and satisfies $A (s) = (s - a) I_{3} + o (s - a)$ and $A^{'} (s) = I_{3} + o (1)$ as $s ↓ a$ . Write $A (s) = (s - a) B (s)$ , so $B (s) \to I_{3}$ ; the cofactor formula and $\det B (s) \to 1$ give $B (s)^{- 1} \to I_{3}$ . Thus $A (s)^{- 1} = (s - a)^{- 1} (I_{3} + o (1))$ and $Θ (s) = A^{'} (s) A (s)^{- 1} = (s - a)^{- 1} (I_{3} + o (1))$ .

[L7]

For $C^{1}$ functions on $[a, b]$ the fundamental theorem of calculus holds, and continuous functions on compact intervals are integrable; continuous piecewise- $C^{1}$ fields have continuous integrands on each piece. A continuous nonnegative function on a nondegenerate compact interval with zero integral vanishes identically (The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ , A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion, A continuous $f \geq 0$ on $[a, b]$ with $\int_{a}^{b} f = 0$ is identically $0$ , The lower and upper Darboux integrals of a bounded $f$ on $[a, b]$ as $\sup_{P} L (f, P)$ and $\inf_{P} U (f, P)$ , Darboux integrability as their equality, and the notation $\int_{a}^{b} f$ ).

Proof

technique · direct

1.1L1L2algebra

(The Jacobi operator preserves the normal bundle and is selfadjoint.) Let $J, K$ be normal fields. For the bilinear form $R (X, Y, Z, W) : = g (R (X, Y) Z, W)$ the symmetries of [L2] read $R (X, Y, Z, W) = - R (Y, X, Z, W) = - R (X, Y, W, Z) = R (Z, W, X, Y)$ . Hence $g (C_{T} (J), T) = R (T, J, T, T) = - R (T, J, T, T) = 0$ , so $C_{T} (J)$ is again normal by [L1]. For selfadjointness, $g (C_{T} (J), K) = R (T, J, T, K) = - R (T, J, K, T) = - R (K, T, T, J) = R (T, K, T, J) = g (C_{T} (K), J)$ , using the last-pair skew, pair exchange and first-pair skew in that order; by symmetry of $g$ this is $g (J, C_{T} (K))$ , so $C_{T}$ is $g$ -selfadjoint.

2.1step 1.1L1L4algebra

(Matrix form in a parallel frame.) Let $E_{1}, E_{2}, E_{3}$ be a parallel orthonormal frame of $N_{γ}$ by [L4] and write $J = \sum_{i} J^{i} E_{i}$ , $K = \sum_{i} K^{i} E_{i}$ . Then $D_{s} J = \sum_{i} (J^{i})^{'} E_{i}$ and $g (D_{s} J, D_{s} K) = \sum_{i} (J^{i})^{'} (K^{i})^{'}$ by orthonormality; and $C_{T} (J) = \sum_{j} J^{j} C_{T} (E_{j})$ , so $g (C_{T} (J), K) = \sum_{i, j} J^{j} C^{i}_{j} K^{i}$ with $C^{i}_{j} = g (C_{T} (E_{j}), E_{i})$ , the coefficient functions being smooth along $γ$ . By step 1.1 the matrix $C (s)$ is symmetric for every $s$ .

3.1step 2.1L2L3L5algebra

(The fundamental solution.) By [L5] applied to the first-order system for $(A, A^{'})$ with coefficient matrix $(\begin{matrix} 0 & I \\ C & 0 \end{matrix})$ , the problem $A^{''} = C A$ , $A (a) = 0$ , $A^{'} (a) = I_{3}$ has a unique solution on $[a, b]$ . For $v \in R^{3}$ the field $s \mapsto A (s) v$ in the frame is the Jacobi field along $γ$ with initial data $(0, v)$ : its coefficient vector satisfies $(A v)^{''} = C A v$ and the initial data are $(A v) (a) = 0$ , $(A v)^{'} (a) = v$ , which is the Jacobi initial-value problem of [L3] in the frame of step 2.1. Hence $A (s) v = 0$ for a nonzero $v$ would exhibit a nonzero Jacobi field vanishing at $a$ and $s$ , contradicting the absence of conjugate instants; so $A (s)$ is invertible for every $s \in (a, b]$ .

4.1step 2.1step 3.1algebra

(The Wronskian and $Θ$ .) Define $W (s) : = A (s)^{T} A^{'} (s) - A^{'} (s)^{T} A (s)$ , a $3 \times 3$ matrix function on $[a, b]$ . Then $W^{'} = A^{' T} A^{'} + A^{T} A^{''} - A^{'' T} A - A^{' T} A^{'} = A^{T} C A - (C A)^{T} A = 0$ because $C$ is symmetric by step 2.1, and $W (a) = 0$ ; hence $W = 0$ on $[a, b]$ . On $(a, b]$ set $Θ : = A^{'} A^{- 1}$ , which is well defined by step 3.1. The identity $W = 0$ gives $A^{T} A^{'} = A^{' T} A$ , and multiplying on the left by $(A^{T})^{- 1}$ and on the right by $A^{- 1}$ gives $Θ^{T} = Θ$ ; differentiability of $A$ and $A^{- 1}$ gives $Θ^{'} = A^{''} A^{- 1} - A^{'} A^{- 1} A^{'} A^{- 1} = C - Θ^{2}$ .

5.1step 3.1step 4.1L6L7algebrastep 2.1

(The factorization identity.) Let $J$ be a continuous piecewise- $C^{1}$ normal field vanishing at $a$ and $b$ , with coefficient vector $j (s)$ in the frame of step 2.1. On each smooth piece, $∣ j^{'} - Θ j ∣^{2} = ∣ j^{'} ∣^{2} - 2 j^{' T} Θ j + j^{T} Θ^{2} j$ , because $Θ$ is symmetric. For $ε > 0$ sufficiently small, integrate by parts on the smooth pieces of $[a + ε, b]$ ; the interior endpoint terms cancel because $j$ and $Θ$ are continuous. The identity $(j^{T} Θ j)^{'} = 2 j^{' T} Θ j + j^{T} Θ^{'} j$ gives $\int_{a + ε}^{b} - 2 j^{' T} Θ j d s = - [j^{T} Θ j]_{a + ε}^{b} + \int_{a + ε}^{b} j^{T} Θ^{'} j d s .$ At $b$ the boundary term vanishes because $j (b) = 0$ . At $a$ , one-sided $C^{1}$ regularity gives $j (s) = (s - a) j^{'} (a) + o (s - a)$ and $j^{'} (s) = j^{'} (a) + o (1)$ . By [L6], $j^{T} Θ j = O (s - a) \to 0$ and $Θ (s) j (s) = j^{'} (a) + o (1)$ , so $j^{'} (s) - Θ (s) j (s) \to 0$ on the first smooth piece as $s ↓ a$ . Define the factorized integrand at $a$ to be this limit squared, namely $0$ ; it is then piecewise continuous on $[a, b]$ . Using $Θ^{'} = C - Θ^{2}$ from step 4.1 and letting $ε ↓ 0$ in the integrated identity yields $I (J) = \int_{a}^{b} (∣ j^{'} ∣^{2} + j^{T} C j) d s = \int_{a}^{b} ∣ j^{'} - Θ j ∣^{2} d s .$ This is the factorization in (c), with its right-hand integrand continuously extended at $a$ .

6.1step 5.1step 3.1step 4.1L7algebrastep 1.1∎

(Positivity.) By step 5.1, $I (J) = \int_{a}^{b} ∣ J^{'} - Θ J ∣^{2} d s \geq 0$ . If $I (J) = 0$ , nonnegativity gives zero integral on every compact subinterval inside a smooth piece; [L7] makes the continuous integrand zero there, so $J^{'} = Θ J$ on each open piece. Then $(A^{- 1} J)^{'} = A^{- 1} J^{'} - A^{- 1} A^{'} A^{- 1} J = A^{- 1} (J^{'} - Θ J) = 0$ on $(a, b]$ , so $A^{- 1} J$ is constant on each open piece; its continuity across the finitely many joints makes this one constant $v$ on $(a, b]$ , giving $J (s) = A (s) v$ ; since $J (b) = 0$ and $A (b)$ is invertible by step 3.1, $v = 0$ and $J = 0$ . Conversely $J = 0$ gives $I (J) = 0$ . This proves the positivity statement, and with steps 1.1, 2.1, 3.1 and 5.1 the lemma.

Remarks

Where positivity is used. Positive definiteness enters only through the fibre metric on the normal bundle of the timelike geodesic (clause (e) of Jacobi fields, geodesic variations and conjugate instants along a geodesic); no positive-definite metric on all of $T M$ and no Riemannian index lemma is used. On the full tangent space the quadratic form $g (D_{s} J, D_{s} J) + g (R (T, J) T, J)$ is not positive definite, because the tangential directions contribute negatively.
The meaning of the bound. The lemma is conditional on the absence of conjugate instants in $(a, b]$ ; this is exactly the hypothesis under which the local maximum of Conjugacy, the exponential differential and the local proper-time maximum before the first conjugate instant applies. The factorization is the finite-dimensional Lorentzian analogue of the classical index-form argument, with the operator $C_{T}$ acting on the positive definite normal three-space.

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Conjugacy, the exponential differential and the local proper-time maximum before the first conjugate instant

Statement

(a) Conjugacy and the exponential differential. For every $s \in (0, b]$ , $γ (s)$ is conjugate to $p$ along $γ$ if and only if $d (\exp_{p})_{s T (0)}$ is not injective; consequently $d (\exp_{p})_{s T (0)}$ is invertible for every $s \in [0, b]$ .

(b) Local proper-time maximum. Fix a finite family of charts $(U_{1}, x_{1}), \dots, (U_{m}, x_{m})$ whose domains cover $γ ([0, b])$ and a partition $0 = s_{0} < s_{1} < \dots < s_{m} = b$ with $γ ([s_{i - 1}, s_{i}]) \subseteq U_{i}$ for each $i$ . For $δ > 0$ call a piecewise- $C^{1}$ curve $α : [0, b] \to M$ $C^{1}$ - $δ$ -close to $γ$ relative to this data when $α ([s_{i - 1}, s_{i}]) \subseteq U_{i}$ for every $i$ and, on the common refinement of the admissible partitions of $α$ and $γ$ , the coordinate representatives satisfy $\sup ∥ x_{i} \circ α - x_{i} \circ γ ∥ < δ, \sup ∥ (x_{i} \circ α)^{'} - (x_{i} \circ γ)^{'} ∥ < δ$ on each common open piece contained in $(s_{i - 1}, s_{i})$ . Then there exists $δ > 0$ such that every piecewise- $C^{1}$ future timelike curve $α : [0, b] \to M$ with $α (0) = p$ and $α (b) = γ (b)$ that is $C^{1}$ - $δ$ -close to $γ$ relative to the fixed data satisfies $L_{g} (α) \leq b = L_{g} (γ),$ with equality if and only if there is a piecewise- $C^{1}$ strictly increasing surjection $ρ : [0, b] \to [0, b]$ with $α = γ \circ ρ$ ; that is, $α$ is a strictly monotone reparametrization of $γ$ .

(c) The normal index form. The normal index form of The normal index form before the first conjugate instant is positive definite on the space of piecewise- $C^{1}$ normal fields along $γ$ vanishing at $0$ and $b$ is positive definite.

Facts & Assumptions

Given: A smooth future unit timelike geodesic $γ : [0, b] \to M$ with $p = γ (0)$ , no conjugate instant in $(0, b]$ , a chart-and-partition cover of $γ ([0, b])$ , and a piecewise- $C^{1}$ future timelike curve $α$ with the same endpoints that is $C^{1}$ -close to $γ$ .

[L1]

For $s > 0$ with $s T (0)$ in the exponential domain, $γ (s)$ is conjugate to $p$ along $γ$ if and only if $d (\exp_{p})_{s T (0)}$ is not injective; and $d (\exp_{p})_{t v} (t u) = J_{u} (t)$ for the Jacobi field $J_{u}$ with $J_{u} (0) = 0$ , $D_{t} J_{u} (0) = u$ (The differential of the exponential map and conjugacy along a geodesic clauses (a) and (c)).

[L2]

The normal index form is positive definite under the present hypotheses (The normal index form before the first conjugate instant is positive definite clause (c)).

[L3]

$E_{p}$ is open and star-shaped, $\exp_{p}$ is smooth with $d (\exp_{p})_{0} = {id}_{T_{p} M}$ , and $t \mapsto \exp_{p} (t v)$ is the geodesic with initial tangent $v$ ; in particular $γ (t) = \exp_{p} (t T (0))$ for $t \in [0, b]$ and the exponential map is a local diffeomorphism wherever its differential is invertible (The geodesic initial-value problem and the exponential map of the metric-selected connection clauses (a)-(d), Point-normal and exponential coordinate charts of the metric-selected connection).

[L4]

Gauss lemma: $g_{\exp_{p} v} (d (\exp_{p})_{v} (v), d (\exp_{p})_{v} (w)) = g_{p} (v, w)$ (The Lorentzian Gauss lemma and the radial maximum of proper time clause (a)).

[L5]

If $n$ is a future unit timelike vector and $w$ a future timelike vector at the same point, then $- g (n, w) \geq \sqrt{- g (w, w)}$ , with equality exactly when $w$ is a positive multiple of $n$ (Pairing of an observer with future causal vectors, Observers, rest spaces and the spatial projection at an event).

[L6]

Length is invariant under increasing piecewise- $C^{1}$ reparametrization whose derivative is strictly positive on every piece, including its one-sided endpoint derivatives, comparing the actual image interval. For a future timelike geodesic $γ : [0, b] \to M$ with $g (T, T) = - 1$ , one has $L_{g} (γ) = b$ (Proper time is well defined and reparametrization invariant; the four-velocity is its proper-time tangent clause 2, Timelike worldlines, proper time, length and four-velocity on a Lorentzian four-manifold clause (d)).

[L7]

The inverse function theorem: if $f$ is smooth on an open subset of $R^{4}$ and $d f$ is invertible at a point, then $f$ is a diffeomorphism from some open neighborhood of that point onto its image (The Euclidean inverse function theorem, The differential of a smooth map).

[L8]

Continuous functions on a compact interval are Darboux integrable, their integral is monotone, and the fundamental theorem of calculus holds for $C^{1}$ functions on each piece; a nonnegative continuous function with zero integral on an interval has a zero value at every point of that interval (The lower and upper Darboux integrals of a bounded $f$ on $[a, b]$ as $\sup_{P} L (f, P)$ and $\inf_{P} U (f, P)$ , Darboux integrability as their equality, and the notation $\int_{a}^{b} f$ , If $f \leq g$ on $[a, b]$ and both are integrable then $\int_{a}^{b} f \leq \int_{a}^{b} g$ ; and $m (b - a) \leq \int_{a}^{b} f \leq M (b - a)$ , A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion, A continuous $f \geq 0$ on $[a, b]$ with $\int_{a}^{b} f = 0$ is identically $0$ , The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ , Continuity of $f : A \to R$ at a point of $A$ and on $A$ : the $ε$ - $δ$ condition, its agreement with $\lim_{x \to c} f (x) = f (c)$ at a limit point, and continuity at an isolated point, Partition of $[a, b]$ as a finite strictly increasing list $a = t_{0} < t_{1} < \dots < t_{n} = b$ , its subintervals and their lengths, its mesh, refinement, and the common refinement of two partitions).

Proof

technique · direct

1.1L1L2L3given

(Invertibility of the exponential differential along the geodesic.) By [L1] and the hypothesis, for every $s \in (0, b]$ there is no nonzero Jacobi field vanishing at $0$ and $s$ , so $d (\exp_{p})_{s T (0)}$ is injective for every $s \in (0, b]$ ; at $s = 0$ the differential is the identity by [L3] and hence injective. This proves (a), and together with [L2] proves (c).

1.2L3L4L8givenalgebra

(Gauss identity throughout the exponential domain.) The identity in [L4] holds at each lift value even if it is outside the small injective normal domain of that supplier's Statement. Indeed fix $v \in E_{p}$ and $w \in T_{p} M$ . Openness and star-shapedness of $E_{p}$ let us form $F (t, r) : = \exp_{p} (t (v + r w))$ for $0 \leq t \leq 1$ and all sufficiently small $r$ . Put $A : = \partial_{t} F$ and $J : = \partial_{r} F$ . By [L3], $D_{t} A = 0$ and $g (A, A) = g_{p} (v + r w, v + r w)$ . Torsion freedom gives $D_{t} J = D_{r} A$ by the symmetric coordinate connection formula, and metric compatibility gives $\partial_{t} g (A, J) = g (A, D_{r} A) = \frac{1}{2} \partial_{r} g (A, A) = g_{p} (v + r w, w) .$ Since $J (0, r) = 0$ , integrating in $t$ and taking $r = 0$ yields $g_{\exp_{p} v} (d (\exp_{p})_{v} (v), d (\exp_{p})_{v} (w)) = g_{p} (v, w)$ , with no injectivity hypothesis. The metric compatibility and torsion identities used here are supplied by Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields and Metric compatibility, torsion freedom and the metric-selected connection class.

2.1step 1.1L3L7

(Finite local-inverse data.) For each $s \in [0, b]$ , $d (\exp_{p})_{s T (0)}$ is invertible, so by [L7] there is an open $D_{s} \subseteq T_{p} M$ containing $s T (0)$ such that $\exp_{p} ∣_{D_{s}}$ is a diffeomorphism onto $V_{s} : = \exp_{p} (D_{s})$ . In particular, $d (\exp_{p})_{v}$ is invertible for every $v \in D_{s}$ . Since ${s T (0) : s \in [0, b]}$ is compact, finitely many of these sets $D_{s_{1}}, \dots, D_{s_{N}}$ cover the segment; write $ψ_{j} : V_{j} \to D_{j}$ for the inverse of each diffeomorphism. Thus every selected $D_{j}$ lies in the locus where $d (\exp_{p})$ is invertible.

3.1step 2.1L3L7given

(Compatible local inverse branches and the lift.) Refine the fixed chart partition of (b) by a finite partition $0 = t_{0} < \dots < t_{M} = b$ so that each radial subsegment ${s T (0) : t_{k - 1} \leq s \leq t_{k}}$ is contained in one $D_{j (k)}$ . Such a refinement exists by compactness and a Lebesgue number for the open cover of the parameter interval induced by the $D_{j}$ . Write $D_{k}^{*} : = D_{j (k)}$ , $V_{k}^{*} : = V_{j (k)}$ , and $ψ_{k}^{*} : = ψ_{j (k)}$ . At a junction $t_{k}$ , the point $z_{k} : = t_{k} T (0)$ lies in both $D_{k}^{*}$ and $D_{k + 1}^{*}$ , and both inverses send $γ (t_{k})$ to $z_{k}$ . Therefore $O_{k} : = V_{k}^{*} \cap V_{k + 1}^{*} \cap (ψ_{k}^{*})^{- 1} (D_{k}^{*} \cap D_{k + 1}^{*}) \cap (ψ_{k + 1}^{*})^{- 1} (D_{k}^{*} \cap D_{k + 1}^{*})$ is an open neighborhood of $γ (t_{k})$ . On $O_{k}$ both inverse values belong to $D_{k}^{*}$ and have the same exponential image, so injectivity on $D_{k}^{*}$ makes the inverses agree. Choose the tolerance $δ$ so small that $α ([t_{k - 1}, t_{k}]) \subseteq V_{k}^{*}$ for each $k$ , and $α (t_{k}) \in O_{k}$ at every junction. These requirements follow from $C^{0}$ closeness in the fixed charts: each reference subsegment is compact in its corresponding open target, each junction lies in its open $O_{k}$ , and there are only finitely many such conditions. Define $v (s) : = ψ_{k}^{*} (α (s))$ on $[t_{k - 1}, t_{k}]$ . The definitions agree at junctions, giving a continuous piecewise- $C^{1}$ lift with $\exp_{p} (v (s)) = α (s)$ . Injectivity of the initial and final branches gives $v (0) = 0$ and $v (b) = b T (0)$ . This construction allows distinct inverse branches at a repeated spacetime event; compatibility is imposed only at adjacent parameter junctions.

4.1step 3.1L3L7L8algebrastep 2.1

(Uniform timelike control of the lift.) On every refined interval the inverse branch and its derivative are continuous near the compact reference image, so the fixed-chart $C^{1}$ closeness implies uniform closeness of $v$ to $s T (0)$ and, on each smooth piece, of $v^{'}$ to $T (0)$ ; the chain rule gives $v^{'} = d ψ_{k}^{*}_{α (s)} (α^{'} (s))$ . To control the endpoint $s = 0$ uniformly, choose a convex norm ball $B$ about $T (0)$ whose closure is inside the future timelike cone of $T_{p} M$ , and first shrink the tolerance so that $v^{'} (s) \in B$ throughout the initial refined interval, including one-sided values. Since $v (0) = 0$ , piecewise integration gives $v (s) / s = s^{- 1} \int_{0}^{s} v^{'} (t) d t \in B$ there; hence $v (s)$ is future timelike for every positive $s$ in this interval. On the remaining compact radial subsegment, bounded away from $0$ , uniform $C^{0}$ control keeps $v (s)$ in the open future timelike cone. Thus one tolerance works for all $s \in (0, b]$ . Every $v (s)$ lies in its $D_{k}^{*}$ , so $d (\exp_{p})_{v (s)}$ is invertible by step 2.1.

5.1step 4.1step 1.2L3L5L8algebra

(Length inequality.) Define $R (s) : = \sqrt{- g_{p} (v (s), v (s))} > 0$ on $(0, b]$ . On each smooth piece, step 1.2 gives $R^{'} = - g_{p} (v, v^{'}) / R = - g_{α} (N, α^{'})$ , where $N : = d (\exp_{p})_{v} (v) / R$ and $α^{'} = d (\exp_{p})_{v} (v^{'})$ . The radial tangent has constant squared norm $g_{p} (v, v) = - R^{2}$ by metric compatibility along its geodesic. It starts future timelike and remains so by continuity of its pairing with the time-orientation field; hence $N$ is a future unit timelike vector. By [L5], $R^{'} \geq \sqrt{- g (α^{'}, α^{'})}$ . Integrating over the pieces from $ε > 0$ to $b$ and letting $ε ↓ 0$ , using $R \to 0$ and bounded piecewise-continuous length density, gives $L_{g} (α) \leq R (b) = \sqrt{- g_{p} (b T (0), b T (0))} = b$ .

6.1step 5.1L5L8algebrastep 4.1

(Equality case.) If equality holds in step 5.1, the nonnegative function $R^{'} - \sqrt{- g (α^{'}, α^{'})}$ has total limiting integral zero. Its integral over every compact subinterval of a smooth piece inside $(0, b]$ is therefore zero, since it is bounded above by that total integral. It is continuous on each such compact subinterval, so [L8] makes it vanish there. These subintervals cover the smooth interiors, and by the equality case of [L5] the vector $α^{'} (s)$ is a positive multiple of $N (s)$ at every interior parameter value; since $α^{'} = d (\exp_{p})_{v} (v^{'})$ and $N = d (\exp_{p})_{v} (v) / R$ , and $d (\exp_{p})_{v (s)}$ is injective by step 4.1, $v^{'} (s)$ is a positive multiple of $v (s)$ : $v^{'} (s) = λ (s) v (s)$ with $λ (s) > 0$ , piecewise continuous.

7.1step 6.1step 5.1L3L8algebra

(Monotone reparametrization.) Under equality, step 6.1 gives $v^{'} = λ v$ with $λ > 0$ on the smooth pieces. Differentiating $R$ gives $R^{'} = λ R$ , so $(v / R)^{'} = 0$ there. Continuity across the finitely many corners makes $v / R$ one constant vector on $(0, b]$ , and its terminal value is $v (b) / R (b) = T (0)$ . Thus $v (s) = R (s) T (0)$ and $α (s) = γ (R (s))$ . Put $ρ (s) : = R (s)$ for $s > 0$ and $ρ (0) : = 0$ . One has $ρ (b) = b$ and $ρ^{'} = \sqrt{- g (α^{'}, α^{'})} > 0$ on every smooth piece, so $ρ$ is strictly increasing and surjects onto $[0, b]$ . At $0$ , $v (s) = s α^{'} (0) + o (s)$ and $v^{'} (s) \to α^{'} (0)$ from the initial piece. Consequently $R (s) / s \to \sqrt{- g_{p} (α^{'} (0), α^{'} (0))}$ and $R^{'} (s) = - g_{p} (v, v^{'}) / R$ has the same limit, proving the required one-sided $C^{1}$ extension. Hence $ρ$ is piecewise $C^{1}$ on $[0, b]$ and gives the equality characterization.

8.1step 5.1step 7.1L6algebrastep 1.1∎

(Converse and conclusion.) Conversely, if $α = γ \circ ρ$ with $ρ$ a piecewise- $C^{1}$ strictly increasing surjection of $[0, b]$ onto itself, and $α$ belongs to the comparison class of (b), then $α^{'} = ρ^{'} T \circ ρ$ on every piece and at its one-sided endpoints. Since $α$ is future timelike and $T$ is future timelike, these tangents force $ρ^{'} > 0$ there. Its image interval is $[0, b]$ , so [L6] gives $L_{g} (α) = L_{g} (γ) = b$ and equality holds in the stated sense. Steps 5.1 and 7.1 give the inequality and the forward equality characterization for every $α$ in the chosen neighborhood, and steps 1.1 and 2.1 give (a) and (c); the proof is complete.

Remarks

What "local" means. The maximum is asserted only among curves with the same endpoints that are $C^{1}$ -close to $γ$ in the fixed finite chart cover; it implies no global maximality of $γ$ , no statement about curves leaving that neighborhood or wrapping around $M$ , and no assertion that $γ$ stops being a maximum at or after the first conjugate instant. The neighborhood is nonempty and depends on $γ$ through the fixed chart-and-partition data and through the branch choices of step 2.1.
No Riemannian index lemma is consumed. The proof uses the Gauss lemma, the reverse Cauchy-Schwarz inequality and the inverse function theorem; the positivity of the index form in (c) is proved separately in The normal index form before the first conjugate instant is positive definite within the positive definite normal three-space, not by a positive-definite hypothesis on $T M$ .

LemmaStatement: AI-adaptedProof: AI-generatedprecheck passOpen item page →

A curvature tensor vanishes if the tidal operator vanishes for every timelike direction

Statement

Let $(M, g, N)$ be a time-oriented smooth Lorentzian four-manifold (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model) and let $p \in M$ . Let $R_{p} : (T_{p} M)^{4} \to R$ be an algebraic curvature tensor at $p$ : an $R$ -multilinear map whose components $R_{a b c d}$ in a basis (Tensor fields of type (r,s) on a Lorentzian four-manifold and their components) satisfy $R_{a b c d} = - R_{b a c d} = - R_{a b d c} = R_{c d a b}, R_{a b c d} + R_{a c d b} + R_{a d b c} = 0$ for all $a, b, c, d$ . Suppose that for every timelike $T \in T_{p} M$ , every $J \in T_{p} M$ and every $W \in T_{p} M$ , $R_{a b c d} W^{a} T^{b} T^{c} J^{d} = 0.$ Equivalently (by nondegeneracy of $g_{p}$ ), the tidal operator $J \mapsto R (T, J) T$ is zero for every timelike $T$ , where $R (T, J) T$ is the vector with $g_{p} (R (T, J) T, W) = R_{a b c d} W^{a} T^{b} T^{c} J^{d}$ for all $W$ . Then $R_{p} = 0$ .

Facts & Assumptions

Given: A point $p$ of a time-oriented smooth Lorentzian four-manifold and an algebraic curvature tensor $R_{p}$ at $p$ whose tidal operator vanishes for every timelike direction.

[L1]

The timelike set ${T \in T_{p} M : g_{p} (T, T) < 0}$ has exactly two connected components, each nonempty, open and convex; in particular there is a timelike $T_{0}$ and the timelike set contains a nonempty open set (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model clause (b), Timelike, null, spacelike and causal vectors of a Lorentz vector space).

[L2]

In a basis of $T_{p} M$ the components of a tensor are real numbers and a tensor vanishes if and only if all its components in one — equivalently every — basis vanish; indices are raised and lowered with $g_{p}$ and its inverse (Tensor fields of type (r,s) on a Lorentzian four-manifold and their components clauses (a)-(c), Tensoriality of the curvature, its coordinate formula and its point-normal metric formula clause (a)).

[L3]

For fixed $a, d$ and fixed $v \in T_{p} M$ , the function $λ \mapsto R_{a b c d} (v + λ T_{0})^{b} (v + λ T_{0})^{c}$ is a polynomial in $λ$ of degree at most $2$ , a finite sum of products of at most two affine functions of $λ$ (Tensor fields of type (r,s) on a Lorentzian four-manifold and their components clause (a), Continuity of $f : A \to R$ at a point of $A$ and on $A$ : the $ε$ - $δ$ condition, its agreement with $\lim_{x \to c} f (x) = f (c)$ at a limit point, and continuity at an isolated point).

[L4]

A nonzero real polynomial of degree at most $2$ has at most $2$ distinct real roots (A nonzero real polynomial of degree $n$ has no more than $n$ distinct real roots).

[L5]

If $g_{p} (T_{0}, T_{0}) < 0$ , then $g_{p} (v + λ T_{0}, v + λ T_{0}) = λ^{2} g_{p} (T_{0}, T_{0}) + 2 λ g_{p} (v, T_{0}) + g_{p} (v, v)$ tends to $- \infty$ as $∣ λ ∣ \to \infty$ ; hence $v + λ T_{0}$ is timelike for all sufficiently large $∣ λ ∣$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, Timelike, null, spacelike and causal vectors of a Lorentz vector space, algebra).

Proof

technique · direct

1.1givenL2algebra

(The quadratic contraction vanishes on all timelike vectors.) Fix indices $a, d$ and put $Q (v) : = R_{a b c d} v^{b} v^{c}$ for $v \in T_{p} M$ . Taking $W$ and $J$ to be basis vectors in the hypothesis gives $R_{a b c d} T^{b} T^{c} = 0$ for every timelike $T$ ; that is, $Q (T) = 0$ for every timelike $T$ .

2.1step 1.1L1L3L4L5algebra

(Extension to all vectors.) Fix $v \in T_{p} M$ and a timelike $T_{0}$ , which exists by [L1]. By [L3], $λ \mapsto Q (v + λ T_{0})$ is a polynomial of degree at most $2$ , and by [L5] the argument $v + λ T_{0}$ is timelike for all sufficiently large $∣ λ ∣$ , so by step 1.1 this polynomial has infinitely many roots. By [L4] the polynomial is identically zero, and evaluating at $λ = 0$ gives $Q (v) = 0$ . Since $v$ was arbitrary, $Q \equiv 0$ for the fixed pair $(a, d)$ ; as $a, d$ were arbitrary, $R_{a b c d} v^{b} v^{c} = 0$ for all $a, b, c, d$ and all $v \in T_{p} M$ .

3.1step 2.1L2algebra

(Polarization gives antisymmetry in the middle pair.) For arbitrary $u, w \in T_{p} M$ , step 2.1 applied to $u + w$ , $u$ and $w$ gives $0 = Q (u + w) - Q (u) - Q (w) = R_{a b c d} u^{b} w^{c} + R_{a b c d} w^{b} u^{c}$ . Relabelling the summed pair $(b, c)$ in the second term turns it into $R_{a c b d} u^{b} w^{c}$ , so $(R_{a b c d} + R_{a c b d}) u^{b} w^{c} = 0$ for all $u, w$ and all $a, d$ . Choosing $u, w$ to be basis vectors gives $R_{a b c d} = - R_{a c b d}$ for all indices, that is, $R$ is antisymmetric in its second and third slots.

4.1step 3.1givenL2algebra∎

(Bianchi cancellation.) Using step 3.1 with the given last-pair skew, $R_{a c d b} = - R_{a c b d} = R_{a b c d}$ : the first equality is the skew-symmetry in the last two slots, the second is step 3.1. Similarly, $R_{a d c b} = - R_{a c d b}$ by step 3.1 applied to the slots $(b, c)$ of $R_{a d c b}$ , so $R_{a d b c} = - R_{a d c b} = R_{a c d b} = R_{a b c d}$ , where the first equality is the last-pair skew. The algebraic Bianchi identity of the statement therefore gives $0 = R_{a b c d} + R_{a c d b} + R_{a d b c} = 3 R_{a b c d}$ ; since $R$ has characteristic $0$ , $R_{a b c d} = 0$ for all $a, b, c, d$ , and by [L2] $R_{p} = 0$ .

Remarks

The hypothesis must hold for every timelike direction. A single observer with vanishing tidal operator does not force $R_{p} = 0$ ; a nonzero algebraic curvature tensor supported on a spacelike three-space has zero tidal operator for a suitable timelike direction. The passage from one direction to all directions is what step 2.1 exploits, through the fact that the timelike set is open and the contraction is polynomial in the direction.
Geometric reading. For the curvature tensor of the metric-selected connection the symmetry hypotheses of the statement hold by The algebraic symmetries of the curvature tensor and the symmetry of the Ricci and Einstein tensors, and by The Jacobi equation, geodesic deviation and the initial-value theory of Jacobi fields the tidal operator is exactly the relative-acceleration operator of a family of geodesics. The lemma is the algebraic core of the statement that curvature is precisely what free-fall relative acceleration detects; the physical comparison is a separate item.

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Free-fall congruences, tidal (curvature) relative acceleration and the flat comparison

Physical scope

Smooth test-field regime of the classical general-relativistic framework: a smooth time-oriented Lorentzian four-manifold with metric-selected connection, ideal freely falling structureless test bodies whose worldlines are future timelike geodesics parametrized by proper time, and smooth one-parameter families of candidate geodesic histories. Their physical interpretation applies when body histories realize the family on a common interval; no existence of bodies with arbitrary release data is assumed or deduced. Excludes bodies with non-negligible self-gravity, self-force, spin, finite size or internal structure, non-geodesic motion, non-smooth or distributional matter and fields, quantum effects, and any comparison with measured data. Every conclusion is a conditional model deduction from the adopted postulates, not an observational claim.

Statement

Adopt the postulates Adopted spacetime model and classical domain of the gravitational framework and Adopted clock, free-fall and light-ray assumptions of the classical gravitational framework of the classical gravitational framework, and let $(M, g, N)$ be the model's smooth time-oriented Lorentzian four-manifold with metric-selected connection $\nabla$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, Metric compatibility, torsion freedom and the metric-selected connection class, Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields) and curvature $R$ (The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors). All statements below are deductions within the model so specified; they assert nothing about nature.

At an event $q$ , a frame means an ordered basis $(E_{0}, E_{1}, E_{2}, E_{3})$ of $T_{q} M$ . If an orthonormal frame is intended, it means a tetrad satisfying $g_{q} (E_{A}, E_{B}) = diag (- 1, 1, 1, 1)_{A B}$ ; these terms and their distinction from coordinate frames are defined in Local frame fields, coordinate frames and tetrad fields. Such a frame is pointwise basis data, not a chart on an open set or an observer congruence, which is a smooth future unit timelike vector field on an open set (Observer congruences, their observer curves and adapted charts).

(a) Free-fall congruences. A free-fall congruence of the model is a smooth map $γ : (- ε, ε) \times I \to M$ , with $ε > 0$ and $I \subseteq R$ an interval with nonempty interior, whose values we write $γ (r, s)$ . For each $r \in (- ε, ε)$ the curve $γ_{r} : I \to M$ is defined by $γ_{r} (s) : = γ (r, s)$ and is a future timelike geodesic parametrized by proper time, with tangent field $U_{r} (s) : = \partial_{s} γ (r, s)$ satisfying $g (U_{r}, U_{r}) = - c^{2}, D_{s} U_{r} = 0,$ where $D_{s}$ is the covariant derivative along $γ_{r}$ . Here this term denotes a mathematical family of candidate free-fall histories. Clause (ii) of Adopted clock, free-fall and light-ray assumptions of the classical gravitational framework requires any ideal freely falling massive test body's history to be such a geodesic on its actual proper-time interval; it does not guarantee bodies realizing arbitrary release data or an entire geodesic domain. If a smooth family of these body histories is realized on a common interval $I$ , it is a free-fall congruence in the sense just defined. Conversely each individual such body history can be included mathematically in a constant family $γ (r, s) = γ_{0} (s)$ , without asserting the existence of additional bodies. No injectivity or immersion condition is imposed on the family map.

(b) Deviation equation (model deduction). Let $γ$ be a free-fall congruence and set $J : = \partial_{r} γ$ . Then, along each $γ_{r}$ , the separation field satisfies $D_{s}^{2} J = R (U_{r}, J) U_{r},$ where $D_{s}$ is the covariant derivative along $γ_{r}$ . In particular, the relative four-acceleration encoded by the family’s infinitesimal separation field $J$ is exactly the curvature term: at each event, $J \mapsto R (U_{r}, J) U_{r}$ is an $R$ -linear map on the event's tangent space, vanishes at $J = 0$ , is tensorial and independent of the frame or chart used for its components, and contains no rest mass, charge or internal-structure parameter of the bodies. When the family is realized by ideal test-body histories, this is their infinitesimal relative acceleration within the adopted model.

(c) Flat comparison and the vanishing criterion. If the curvature vanishes on an open subset $O \subseteq M$ , then every free-fall congruence with worldlines in $O$ has $D_{s}^{2} J = 0$ there: the model predicts no tidal relative acceleration of freely falling test bodies in that region. Conversely, if at an event $q$ the operator $J \mapsto R (U, J) U$ vanishes for every future timelike $U$ and every $J \in T_{q} M$ , then $R_{q} = 0$ ; hence within the model the tidal response of all free-fall congruences detects exactly the curvature when all mathematical candidate families of (a) are considered, and $R$ is the model's complete tidal field at every event. Every normalized future timelike $U$ and every separation value $J$ are realized by such a local mathematical family. This does not assert the existence of test bodies realizing those families; inference from only physically realized families would additionally require that they supply all these directions and values.

Status. Clauses (a)-(c) are conditional consequences of the stated postulates and the cited mathematics. No observation, apparatus, measured input or statistical inference is used; the item contains no empirical premises, and the word "tidal" denotes the curvature term in the model's deviation equation, not a measured force on any particular body.

Facts & Assumptions

Given: The adopted framework postulates, a model spacetime with metric-selected connection, a free-fall congruence $γ$ , and an event $q$ .

[A1]

The framework model has a smooth time-oriented Lorentzian four-manifold $(M, g, N)$ with metric-selected connection $\nabla$ and curvature $R$ , all mathematics as in the cited definitions (Adopted spacetime model and classical domain of the gravitational framework clauses (i) and (iv), Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors).

[A2]

An ideal freely falling massive test body of the model has as worldline a future timelike geodesic with four-velocity $U$ , $g (U, U) = - c^{2}$ , and proper time an affine parameter. Its history is a restriction of the unique maximal geodesic with its release event and four-velocity; no force, charge or rest mass enters that evolution rule. This constrains a body's history when realized, without guaranteeing a body for each initial datum or requiring its history to have the maximal geodesic's domain (Adopted clock, free-fall and light-ray assumptions of the classical gravitational framework clause (ii), The free-fall worldline of an ideal test body is determined by its initial event and four-velocity, Proper time is well defined and reparametrization invariant; the four-velocity is its proper-time tangent clause 3).

[L1]

For a smooth family of geodesics parametrized by an affine parameter $σ$ with $g (T, T) = - 1$ , the variation field satisfies $D_{σ}^{2} J = R (T, J) T$ ; for a geodesic with $g (T, T) = - 1$ the proper time is $τ = σ / c$ , and the physical four-velocity is $U = c T$ (The Jacobi equation, geodesic deviation and the initial-value theory of Jacobi fields clause (b), Proper time is well defined and reparametrization invariant; the four-velocity is its proper-time tangent clauses 2-3, Timelike worldlines, proper time, length and four-velocity on a Lorentzian four-manifold clauses (d)-(e)).

[L2]

If an algebraic curvature tensor at a point of a Lorentzian four-manifold satisfies $R (U, J) U = 0$ for every timelike $U$ and every $J$ , then it vanishes at that point (A curvature tensor vanishes if the tidal operator vanishes for every timelike direction).

[L3]

The curvature tensor is a smooth $(1, 3)$ -tensor; its lowered components satisfy the symmetries and the pairing identity $g (R (X, Y) Z, W) = R_{a b c d} W^{a} X^{c} Y^{d} Z^{b}$ (Tensoriality of the curvature, its coordinate formula and its point-normal metric formula clause (a), The algebraic symmetries of the curvature tensor and the symmetry of the Ricci and Einstein tensors clauses (a)-(b)).

[L4]

The geodesic evaluation map is smooth on an open neighborhood of each initial position-velocity datum and time zero (The geodesic initial-value problem and the exponential map of the metric-selected connection clause (b)). Metric compatibility gives $\frac{d}{d s} g (γ', γ') = 2 g (D_{s} γ', γ') = 0$ along each geodesic (Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields), so a future timelike geodesic with initial squared speed $- c^{2}$ has proper-time parameter $s$ up to an additive constant (Proper time is well defined and reparametrization invariant; the four-velocity is its proper-time tangent clause 3). These are mathematical conclusions, not body-existence assertions.

Proof

technique · direct

1.1L1given

(From proper time to the unit affine parameter.) Let $γ$ be a free-fall congruence and fix $r$ . By definition, $t \mapsto γ_{r} (t)$ is a future timelike geodesic with $U_{r} = \partial_{t} γ$ and $g (U_{r}, U_{r}) = - c^{2}$ ; by [L1] the rescaled map $Γ_{r} (σ) : = γ_{r} (σ / c)$ is a geodesic with unit tangent $T_{r} = U_{r} / c$ and $g (T_{r}, T_{r}) = - 1$ , so $σ$ is an affine parameter for it.

2.1step 1.1L1L3algebra

(Deviation equation in the model.) Put $c I : = {c t : t \in I}$ and define the smooth family $Γ : (- ε, ε) \times c I \to M$ by $Γ (r, σ) : = γ (r, σ / c)$ . For each fixed $r$ , its variation field is ${\tilde{J}}_{r} (σ) : = \partial_{r} Γ (r, σ) = J (r, σ / c)$ , so [L1] gives $D_{σ}^{2} {\tilde{J}}_{r} = R (T_{r}, {\tilde{J}}_{r}) T_{r}$ . Since $D_{t} = c D_{σ}$ along $γ_{r}$ by the chain rule for the covariant derivative along a curve, $D_{t}^{2} J = c^{2} D_{σ}^{2} {\tilde{J}}_{r} = c^{2} R (T_{r}, J) T_{r} = R (c T_{r}, J) (c T_{r}) = R (U_{r}, J) U_{r}$ by $R$ -bilinearity of $R$ . This proves (b).

3.1step 2.1A2given

(No mass or structure parameter.) The right-hand side of the equation of step 2.1 is built from the worldlines (through $U_{r}$ and $J$ ) and the geometry only; no rest mass, charge or internal-structure parameter occurs in it, and by [A2] the worldlines themselves do not depend on the rest mass of the body. Hence two congruences of the model that agree in the worldlines of their test bodies have identical relative-acceleration fields, whether or not the bodies would be assigned different masses; this is a comparison inside the model, not a statement about any real bodies.

3.2step 2.1L2L3algebra

(Flat comparison and vanishing criterion.) If $R = 0$ on an open set $O$ , then step 2.1 gives $D_{t}^{2} J = 0$ for every free-fall congruence with worldlines in $O$ : the model predicts no tidal relative acceleration there. Conversely, suppose at the event $q$ that $R (U, J) U = 0$ for every future timelike $U$ and every $J$ . If $W$ is timelike, then either $W$ is future timelike or $- W$ is future timelike; in the second case $R (W, J) W = R (- W, J) (- W) = 0$ as well. Thus the tidal operator vanishes for every timelike direction at $q$ , and [L2] gives $R_{q} = 0$ .

3.3A2L4step 2.1L3constructalgebra

(Mathematical realization of arbitrary separation.) Fix $q \in M$ , a future timelike $U \in T_{q} M$ with $g (U, U) = - c^{2}$ , and $J \in T_{q} M$ . In a chart $x$ around $q$ choose $q (r) = x^{- 1} (x (q) + r d x_{q} J)$ for sufficiently small $∣ r ∣$ , and let $W (r)$ have the constant chart components of $U$ . By continuity, $W (r)$ remains future timelike after shrinking the interval. Set $V (r) = c W (r) / \sqrt{- g_{q (r)} (W (r), W (r))}$ ; this is smooth, future timelike, has squared norm $- c^{2}$ , and $V (0) = U$ . The open smooth evaluation domain of [L4] supplies a smooth geodesic family $γ (r, s)$ on a common open rectangle with $γ (r, 0) = q (r)$ and $\partial_{s} γ (r, 0) = V (r)$ . Shrink the rectangle so its tangents remain future-directed. By [L4] their squared speed is $- c^{2}$ , making $s$ proper time up to a constant. Thus this is a mathematical family of (a), without any body-existence inference from [A2]. At $(0, 0)$ its separation is $\partial_{r} γ = J$ and its velocity is $U$ , hence step 2.1 realizes every $R (U, J) U$ with this normalization. Every other future timelike $\tilde{U}$ is a positive multiple of such a $U$ , and tensor bilinearity gives $R (\tilde{U}, J) \tilde{U}$ as that positive square multiple. This supplies the mathematical family interpretation of (c); if body histories realize a restriction of the family on a common interval, [A2] and step 2.1 give its physical interpretation there.

4.1step 2.1step 3.1step 3.2step 3.3A1A2given∎

(Conditional status and conclusion.) Steps 2.1, 3.1, 3.2 and 3.3 derive clauses (b) and (c) from [A1]-[A2] and the cited mathematical results, using no observation, measurement or statistical model; the statements are therefore conditional deductions within the adopted model, and the item carries no empirical premises. Together with the definitions of clause (a) this completes the proof.

Remarks

What the deduction does and does not establish. The theorem shows that, if the framework's spacetime, free-fall and clock postulates are adopted, then free-fall relative acceleration is governed by curvature and is detected by the tidal operator for every observer direction. It does not establish the postulates, does not show that any real body is exactly freely falling, and does not compare the prediction with any measured tidal effect; such a comparison would require a specified apparatus model, empirical premises with uncertainties and a statistical analysis, none of which is supplied here.
Frame independence. The operator $J \mapsto R (U, J) U$ is a tensor contraction at an event; changing the ordered basis of that tangent space or changing coordinates alters its components but not the operator, its vanishing, or the geometric content of the equation. Neither this pointwise basis nor a chart is the smooth family of worldlines defined in clause (a). The absence of a zeroth-order relative acceleration term is the model's precise formulation of the statement that free fall is "locally inertial", and the curvature term is the model's tidal field; the equivalence-principle reading of these facts is motivational and is not asserted as a separate claim.

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Conventions, units and scope for curvature and tidal quantities

Physical scope

Bookkeeping remark for the curvature-and-tides page: it fixes the sign, index and unit conventions of the curvature items, distinguishes chart artifacts from curvature, and delimits the model status of the word 'tidal'. It makes no prediction, uses no empirical premise, and asserts nothing about nature beyond the adopted framework postulates it cites.

Conventions, units and scope

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: $R (X, Y) Z = \nabla_{X} \nabla_{Y} Z - \nabla_{Y} \nabla_{X} Z - \nabla_{[X, Y]} Z$ and $R (\partial_{c}, \partial_{d}) \partial_{b} = R^{a}_{b c d} \partial_{a}$ , 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 $\nabla^{a} G_{a b} = 0$ 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 $x^{0} = c t$ 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 $Γ^{a}_{b c}$ have units of inverse length and the components $R^{a}_{b c d}$ , $R_{a b c d}$ , ${Ric}_{a b}$ 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 $κ T_{a b}$ 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: $Ric = 0$ 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 $γ : [0, b] \to M$ , $b > 0$ , with $g (γ^{'}, γ^{'}) = - 1$ and no conjugate instant in $(0, b]$ relative to $0$ : no nonzero smooth field $J$ along $γ$ satisfying $D_{s}^{2} J = R (γ^{'}, J) γ^{'}$ vanishes both at $0$ and at such an instant. Fix a finite chart cover $(U_{i}, x_{i})$ and a partition $0 = s_{0} < \dots < s_{m} = b$ with $γ ([s_{i - 1}, s_{i}]) \subseteq U_{i}$ . The comparison curves $α : [0, b] \to M$ are continuous piecewise- $C^{1}$ future timelike curves with the same endpoints. For some $δ > 0$ , each comparison curve must remain in the corresponding $U_{i}$ and satisfy $\sup ∥ x_{i} \circ α - x_{i} \circ γ ∥ < δ$ and $\sup ∥ (x_{i} \circ α)^{'} - (x_{i} \circ γ)^{'} ∥ < δ$ on every common smooth partition piece, with Euclidean norms as in that clause. Only in this comparison class is $L_{g} (α) \leq b = L_{g} (γ)$ 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 $ψ_{k} : V_{k} \to D_{k} \subseteq T_{p} M$ , $p = γ (0)$ , inverse to $\exp_{p} ∣_{D_{k}} : D_{k} \to V_{k}$ , where $D_{k}, V_{k}$ are open and the restricted exponential map is a diffeomorphism. A finite parameter subdivision puts each radial subsegment $s γ^{'} (0)$ 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.

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Curvature survives a first-order normal chart: an explicit metric, its normal form and its deviation field

Example

On $M = R^{4}$ consider the identity chart $x = (x^{0}, x^{1}, x^{2}, x^{3})$ , the metric with components $g_{00} = - 1, g_{11} = 1 + (x^{0})^{2}, g_{22} = g_{33} = 1, g_{a b} = 0 for a \neq b,$ and the time orientation $N : = \partial_{0}$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model). Let $\nabla$ be its metric-selected connection and $R$ its curvature (The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors). Then:

The identity chart is point-normal at the origin: $g_{a b} (0) = η_{a b}$ , $\partial_{a} g_{b c} (0) = 0$ and $Γ^{a}_{b c} (0) = 0$ ; nevertheless the curvature does not vanish at the origin, since $R_{0101} (0) = R_{1010} (0) = - 1$ and $R_{0110} (0) = 1$ .
The only nonzero Christoffel symbols of the metric-selected connection are $Γ^{1}_{10} = Γ^{1}_{01} = \frac{x^{0}}{1 + (x^{0})^{2}}, Γ^{0}_{11} = x^{0} .$
The deviation equation is verified exactly on the explicit family of geodesics $Φ (s, r) = (s, r, 0, 0)$ : with $T = \partial_{s} Φ$ and $J = \partial_{r} Φ$ one has $D_{s}^{2} J = R (T, J) T$ for every $s$ , and the relative acceleration does not vanish at the event where all connection symbols of the chart vanish.

Facts & Assumptions

Given: The metric $g = diag (- 1, 1 + (x^{0})^{2}, 1, 1)$ on $R^{4}$ with its metric-selected connection $\nabla$ and curvature $R$ .

[L1]

The metric-selected connection exists, is unique, and its Christoffel symbols in the identity chart are $Γ^{a}_{b c} = \frac{1}{2} g^{a d} (\partial_{b} g_{d c} + \partial_{c} g_{b d} - \partial_{d} g_{b c})$ , with $g^{a d}$ the matrix inverse of $g_{a b}$ (Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields clause (a), Christoffel symbols of an affine connection, The inverse metric, musical isomorphisms and index raising).

[L2]

In a point-normal chart the lowered curvature at the point is given by $R_{a b c d} = \frac{1}{2} (g_{a d, b c} + g_{b c, a d} - g_{b d, a c} - g_{a c, b d})$ , and the components satisfy $R^{a}_{b c d} = g^{a e} R_{e b c d}$ (Tensoriality of the curvature, its coordinate formula and its point-normal metric formula clause (c), The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors clause (c)).

[L3]

A chart is point-normal at $p$ when $g_{a b} (p) = η_{a b}$ , $\partial_{a} g_{b c} (p) = 0$ and $Γ^{a}_{b c} (p) = 0$ (Point-normal and exponential coordinate charts of the metric-selected connection clause (a)).

[L4]

For a smooth family of affinely parametrized geodesics the variation field satisfies $D_{s}^{2} J = R (T, J) T$ , where $T = \partial_{s} Φ$ and $J = \partial_{r} Φ$ (The Jacobi equation, geodesic deviation and the initial-value theory of Jacobi fields clause (b), Jacobi fields, geodesic variations and conjugate instants along a geodesic clause (b)).

[L5]

For a smooth curve with chart coordinates $x^{a} (s)$ , $(D_{s} γ^{'})^{a} = (x^{a})^{''} + \sum_{b, c} Γ^{a}_{b c} (x (s)) (x^{b})^{'} (x^{c})^{'}$ . In particular, for the coordinate curve $s \mapsto (s, r, 0, 0)$ , $D_{s} γ^{'} = \sum_{a} Γ^{a}_{00} \partial_{a}$ , so it is a geodesic with affine parameter $s$ whenever these symbols vanish along it (Covariant derivative along a curve, Geodesic of an affine connection, Christoffel symbols of an affine connection).

[L6]

The components of the curvature in a chart are given by $R^{a}_{b c d} = \partial_{c} Γ^{a}_{d b} - \partial_{d} Γ^{a}_{c b} + \sum_{e} (Γ^{a}_{c e} Γ^{e}_{d b} - Γ^{a}_{d e} Γ^{e}_{c b})$ (Tensoriality of the curvature, its coordinate formula and its point-normal metric formula clause (b)).

Verification

technique · direct

1.1L1L3given

(First-order normal form.) With $g_{11} = 1 + (x^{0})^{2}$ and all other components constant, $g (0) = η$ and $\partial_{a} g_{b c} (0) = 0$ because $\partial_{0} g_{11} = 2 x^{0}$ vanishes at $0$ and no other component depends on the coordinates. By [L1] the Christoffel symbols at $0$ are linear combinations of the first derivatives $\partial g (0)$ , so they vanish as well; the identity chart is point-normal at $0$ by [L3].

1.2L1algebra

(Christoffel symbols.) By [L1], the only nonzero first derivative of a metric component is $\partial_{0} g_{11} = 2 x^{0}$ , so a symbol can be nonzero only if one of its indices is a derivative index $0$ and another index refers to the component $g_{11}$ . Checking the three slots: $Γ^{1}_{10} = Γ^{1}_{01} = \frac{1}{2} g^{11} \partial_{0} g_{11} = \frac{1}{2} (1 + (x^{0})^{2})^{- 1} (2 x^{0}) = \frac{x^{0}}{1 + (x^{0})^{2}}$ , and $Γ^{0}_{11} = \frac{1}{2} g^{00} (- \partial_{0} g_{11}) = \frac{1}{2} (- 1) (- 2 x^{0}) = x^{0}$ ; all remaining symbols have a factor $\partial_{a} g_{b c}$ with $a \neq 0$ or $g_{b c}$ one of the constant components and vanish.

2.1step 1.1step 1.2L2algebra

(Curvature at the origin.) By steps 1.1 and 1.2 the chart is point-normal at $0$ , so [L2] applies at that point. For $(a, b, c, d) = (0, 1, 0, 1)$ the four terms are $g_{01, 10} + g_{10, 01} - g_{11, 00} - g_{00, 11}$ , of which only $- g_{11, 00} (0) = - \partial_{0} \partial_{0} (1 + (x^{0})^{2}) ∣_{0} = - 2$ survives, so $R_{0101} (0) = - 1$ ; interchanging the pairs gives $R_{1010} (0) = - 1$ as well, and the last-pair skew gives $R_{0110} (0) = + 1$ . In particular $R (0) \neq 0$ , although all Christoffel symbols vanish at $0$ .

2.2step 1.2L5given

(An explicit geodesic family.) Let $Φ (s, r) : = (s, r, 0, 0)$ . By step 1.2, $Γ^{a}_{00} = 0$ identically, so by [L5] each curve $s \mapsto Φ (s, r)$ is a geodesic with affine parameter $s$ ; its tangent $T = \partial_{0}$ satisfies $g (T, T) = - 1$ . The variation field is $J = \partial_{1}$ .

2.3step 1.2L6algebra

(The curvature term.) By [L6] and step 1.2, $R^{1}_{001} = \partial_{0} Γ^{1}_{10} - \partial_{1} Γ^{1}_{00} + Γ^{1}_{0 e} Γ^{e}_{01} - Γ^{1}_{1 e} Γ^{e}_{00} = \frac{1 - s^{2}}{(1 + s^{2})^{2}} + 0 + \frac{s^{2}}{(1 + s^{2})^{2}} - 0 = \frac{1}{(1 + s^{2})^{2}}$ , all other components $R^{a}_{001}$ vanishing because their Christoffel combinations vanish. Hence $R (T, J) T = R^{1}_{001} \partial_{1} = \frac{1}{(1 + s^{2})^{2}} \partial_{1}$ .

3.1step 2.2L1algebra

(The variation field and its derivative.) By [L1], $D_{s} J = \nabla_{\partial_{0}} \partial_{1} = \sum_{a} Γ^{a}_{10} \partial_{a} = \frac{x^{0}}{1 + (x^{0})^{2}} \partial_{1}$ , and differentiating along the curve, $D_{s}^{2} J = \frac{d}{d s} (\frac{s}{1 + s^{2}}) \partial_{1} + \frac{s}{1 + s^{2}} \nabla_{\partial_{0}} \partial_{1} = \frac{1 - s^{2}}{(1 + s^{2})^{2}} \partial_{1} + \frac{s^{2}}{(1 + s^{2})^{2}} \partial_{1} = \frac{1}{(1 + s^{2})^{2}} \partial_{1}$ .

4.1step 3.1step 2.3L4algebra∎

(Verification of the deviation equation.) Steps 3.1 and 2.3 give $D_{s}^{2} J = R (T, J) T$ for every $s$ , in accordance with [L4]. At $s = 0$ the relative acceleration is $\partial_{1} \neq 0$ , while every Christoffel symbol of the chart vanishes there: the first-order normal form of the metric cannot remove the tidal (curvature) effect, and the deviation equation detects it at the very event where the connection symbols are zero. This completes the verification.

5 · Examples, experiments and counterexamples

None yet.