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Curvature And Tides — Examples

1 · Prerequisites

2 · Summary

These examples and counterexamples work the constructions of gr-curvature-and-tides on explicit components. All six items are mathematics: each is a complete conditional verification or refutation inside the stated model, and none reports a measurement or an empirical claim.

Two curvature computations bracket the theory. Locally in cylindrical coordinates, the flat metric $g = - (d x^{0})^{2} + (d x^{1})^{2} + (x^{1})^{2} (d x^{2})^{2} + (d x^{3})^{2}$ has the nowhere-vanishing Christoffel symbols $Γ^{1}_{22} = - x^{1}$ and $Γ^{2}_{12} = Γ^{2}_{21} = 1 / x^{1}$ , while the coordinate formula and a finite case check show that every curvature component vanishes (A flat metric with nowhere-zero Christoffel coefficients: flat space in cylindrical coordinates): nonvanishing connection coefficients are a chart artifact, not curvature. On the actual product $R^{2} \times S^{2}$ of flat time and space directions with a round unit two-sphere, the local chart $0 < x^{1} < π$ , $0 < x^{2} < 2 π$ has $g = - (d x^{0})^{2} + (d x^{1})^{2} + \sin^{2} (x^{1}) (d x^{2})^{2} + (d x^{3})^{2}$ . The same machinery gives $R_{1212} = \sin^{2} (x^{1})$ , $Ric = diag (0, 1, \sin^{2} (x^{1}), 0)$ , $S = 2$ and $G = diag (1, 0, 0, - 1)$ , and the contracted second Bianchi identity is checked term by term in this local chart (Curvature, Ricci, scalar and Einstein tensors of a metric with a round two-sphere factor).

The Jacobi theory is then exercised on both sides. In flat space the Jacobi fields along a geodesic are exactly the affine fields $J_{0} + t J_{1}$ with $D_{t} J (0) = J_{1}$ , and no geodesic has a conjugate instant (Jacobi fields in flat space are affine, and no geodesic has conjugate instants); on the equator of the round-sphere factor the field $\sin (s) \partial_{1}$ is a nonzero Jacobi field vanishing at $s = 0$ and $s = π$ , so $π$ is a first conjugate instant of multiplicity one, with a one-dimensional kernel of $d (\exp_{p})$ at $π v$ (An equator geodesic of the round-sphere factor: conjugate instant at distance pi with multiplicity one).

Finally the two counterexamples delimit the algebraic hypotheses. The constant tensor $T_{a b c d} = ω_{a b} ω_{c d}$ built from $ω = d x^{0} \land d x^{1} + d x^{2} \land d x^{3}$ satisfies the skew-symmetries and the pair exchange but has Bianchi sum $1$ at $(0, 1, 2, 3)$ , so the exchange symmetries do not imply the first Bianchi identity (The exchange symmetries of a curvature-like tensor do not imply the first Bianchi identity); and the algebraic curvature tensor $R_{a b c d} = h_{a c} h_{b d} - h_{a d} h_{b c}$ of a degenerate form $h$ has a tidal operator that vanishes for $\partial_{0}$ while $R (\partial_{0} + \frac{1}{2} \partial_{1}, \partial_{3}) (\partial_{0} + \frac{1}{2} \partial_{1}) = - \frac{1}{4} \partial_{3}$ , so the vanishing criterion cannot be weakened from "all timelike directions" to a single observer (A nonzero algebraic curvature tensor whose tidal operator vanishes for one timelike direction).

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

None yet.

5 · Examples, experiments and counterexamples

ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passOpen item page →

A flat metric with nowhere-zero Christoffel coefficients: flat space in cylindrical coordinates

Example

On the open set $Ω : = {x \in R^{4} : x^{1} > 0}$ with the identity chart $x = (x^{0}, x^{1}, x^{2}, x^{3})$ consider the metric $g : = - (d x^{0})^{2} + (d x^{1})^{2} + (x^{1})^{2} (d x^{2})^{2} + (d x^{3})^{2}, N : = \partial_{0},$ and let $\nabla$ be its metric-selected connection and $R$ its curvature (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, The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors). Writing $X : = x^{1} \cos x^{2}$ and $Y : = x^{1} \sin x^{2}$ , the chain rule gives $d X = \cos x^{2} d x^{1} - x^{1} \sin x^{2} d x^{2}$ and $d Y = \sin x^{2} d x^{1} + x^{1} \cos x^{2} d x^{2}$ , hence $d X^{2} + d Y^{2} = (d x^{1})^{2} + (x^{1})^{2} (d x^{2})^{2}$ : the displayed metric is the flat metric $- (d x^{0})^{2} + d X^{2} + d Y^{2} + (d x^{3})^{2}$ pulled back by the map $(x^{0}, x^{1}, x^{2}, x^{3}) \mapsto (x^{0}, X, Y, x^{3})$ . This map is a local diffeomorphism on all of $Ω$ , so the metric is locally the flat Minkowski metric. It is not globally injective because $x^{2} \in R$ is an unwrapped angular coordinate; on every restriction where $x^{2}$ ranges over an interval of length less than $2 π$ , it gives genuine cylindrical coordinates. The component verification below uses only the displayed metric.

The only nonzero Christoffel symbols of $\nabla$ in this chart are $Γ^{1}_{22} = - x^{1}, Γ^{2}_{12} = Γ^{2}_{21} = \frac{1}{x^{1}},$ so these three coefficients are nonzero at every point of $Ω$ , while all other Christoffel symbols vanish identically.
The curvature vanishes identically, $R = 0$ , even though the chart has nonvanishing Christoffel symbols everywhere.
Consequently $R (T, J) T = 0$ for every geodesic tangent $T$ and every field $J$ along it: for every smooth family of affinely parametrized geodesics the deviation equation reads $D_{s}^{2} J = 0$ , with no tidal term.

Facts & Assumptions

Given: The open set $Ω = {x^{1} > 0} \subseteq R^{4}$ with the identity chart, the metric $g = diag (- 1, 1, (x^{1})^{2}, 1)$ , its metric-selected connection $\nabla$ and curvature $R$ .

[L1]

The metric $g$ is a smooth Lorentzian metric of signature $(-, +, +, +)$ on $Ω$ with time orientation $N = \partial_{0}$ , and $\nabla$ exists, is unique, and has Christoffel symbols $Γ^{a}_{b c} = \frac{1}{2} g^{a d} (\partial_{b} g_{d c} + \partial_{c} g_{b d} - \partial_{d} g_{b c})$ in the identity chart, with $g^{a d}$ the matrix inverse of $g_{a b}$ (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 clause (a), Christoffel symbols of an affine connection, The inverse metric, musical isomorphisms and index raising).

[L2]

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

[L3]

The curvature is antisymmetric in the loop indices: $R^{a}_{b c d} = - R^{a}_{b d c}$ (The algebraic symmetries of the curvature tensor and the symmetry of the Ricci and Einstein tensors clause (a)).

[L4]

For a smooth family $F$ of affinely parametrized geodesics with $T = \partial_{t} F$ and $J = \partial_{r} F$ , the deviation equation $D_{t}^{2} J = R (T, J) T$ holds (The Jacobi equation, geodesic deviation and the initial-value theory of Jacobi fields clause (b), Covariant derivative along a curve, Geodesic of an affine connection).

Verification

technique · direct

1.1L1algebra

(Christoffel symbols.) In the identity chart only $g_{22} = (x^{1})^{2}$ is nonconstant, with $\partial_{1} g_{22} = 2 x^{1}$ , and $g^{22} = (x^{1})^{- 2}$ ; by [L1] the possible nonzero symbols are therefore $Γ^{1}_{22} = \frac{1}{2} g^{11} (- \partial_{1} g_{22}) = - x^{1}$ , $Γ^{2}_{12} = Γ^{2}_{21} = \frac{1}{2} g^{22} \partial_{1} g_{22} = 1 / x^{1}$ , and every other symbol has a factor $\partial_{a} g_{b c}$ with $a \neq 1$ or a constant metric component and vanishes. All displayed symbols are nonzero on all of $Ω$ because $x^{1} > 0$ there.

2.1L2step 1.1L3algebra

(Vanishing outside a finite list.) Let $a, b, c, d \in {0, 1, 2, 3}$ . Each of the four terms of [L2] contains a factor $Γ^{a}_{\cdot \cdot}$ , so $R^{a}_{b c d} = 0$ unless $a \in {1, 2}$ . If $c \notin {1, 2}$ then $c \in {0, 3}$ : the first term vanishes because $Γ^{a}_{d b}$ depends on $x^{1}$ alone, so $\partial_{0}$ and $\partial_{3}$ annihilate it; the second and fourth vanish because they contain $Γ^{a}_{c b}$ or $Γ^{e}_{c b}$ , whose lower index pair contains $c \notin {1, 2}$ ; the third vanishes because $Γ^{a}_{c e} = 0$ . If $d \notin {1, 2}$ : the first and third terms contain $Γ^{a}_{d b}$ or $Γ^{e}_{d b}$ with $d \notin {1, 2}$ and vanish, the second vanishes because $\partial_{d} Γ^{a}_{c b} = 0$ , and the fourth contains $Γ^{a}_{d e} = 0$ . Hence $R^{a}_{b c d}$ can be nonzero only when $a \in {1, 2}$ and $c, d \in {1, 2}$ ; by [L3] it then suffices to check the eight components $R^{a}_{b 12}$ with $a \in {1, 2}$ and $b \in {0, 1, 2, 3}$ .

2.2L2step 1.1algebra

(Components with a lower index outside the block.) Let $b \notin {1, 2}$ . In $R^{a}_{b 12}$ the first term is $\partial_{1} Γ^{a}_{2 b} = 0$ and the second is $\partial_{2} Γ^{a}_{1 b} = 0$ because both symbols have the lower index $b \notin {1, 2}$ ; the third term contains $Γ^{e}_{2 b} = 0$ and the fourth contains $Γ^{e}_{1 b} = 0$ for the same reason. Hence $R^{a}_{b 12} = 0$ for $b \in {0, 3}$ and $a \in {1, 2}$ .

3.1step 2.1step 2.2L2algebra

(The four remaining components.) By steps 2.1 and 2.2 only $R^{1}_{112}$ , $R^{1}_{212}$ , $R^{2}_{112}$ and $R^{2}_{212}$ remain. Term by term, $Γ^{1}_{1 e} = 0$ for all $e$ and $Γ^{e}_{11} = 0$ for all $e$ , so the middle products vanish in the surviving components and $R^{1}_{112} = 0$ : using step 1.1 and the fourth term $- Γ^{1}_{2 e} Γ^{e}_{12}$ with $e = 2$ , $R^{1}_{212} = \partial_{1} Γ^{1}_{22} - \partial_{2} Γ^{1}_{12} - 0 - Γ^{1}_{22} Γ^{2}_{12} = - 1 + (- x^{1}) (- \frac{1}{x^{1}}) = 0$ and $R^{2}_{112} = \partial_{1} Γ^{2}_{21} - \partial_{2} Γ^{2}_{11} + Γ^{2}_{12} Γ^{2}_{21} - 0 = - \frac{1}{(x^{1})^{2}} + \frac{1}{(x^{1})^{2}} = 0,$ while every term of $R^{1}_{112}$ and of $R^{2}_{212}$ contains a vanishing symbol, so those two components are $0$ as well.

4.1step 1.1step 3.1L4given∎

(Conclusion and consequence.) Steps 2.1, 2.2 and 3.1 show that every component $R^{a}_{b c d}$ vanishes at every point of $Ω$ , so $R = 0$ , although step 1.1 shows that the Christoffel symbols do not vanish anywhere. Nonvanishing connection coefficients in curvilinear coordinates are chart artifacts, not curvature: no coordinate-invariant conclusion can be drawn from them. By [L4], for every smooth family $F$ of affinely parametrized geodesics the variation field satisfies $D_{s}^{2} J = R (T, J) T = 0$ : the congruence has no tidal term, and its deviation is governed only by the affine initial data.

Remarks

The metric is flat and the identity chart on $Ω$ is not a point-normal chart at any point: its three displayed Christoffel coefficients are nonzero everywhere, while every other coefficient is zero. The vanishing of Christoffel symbols is therefore a property of a chart at a point and not a property of the geometry; only the curvature computed above, which is the second-order chart-independent information (Tensoriality of the curvature, its coordinate formula and its point-normal metric formula clause (a)), can detect the difference between flat space and a genuinely curved spacetime.

ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passOpen item page →

Curvature, Ricci, scalar and Einstein tensors of a metric with a round two-sphere factor

Example

Let $S^{2} : = {(X, Y, Z) \in R^{3} : X^{2} + Y^{2} + Z^{2} = 1}$ with the round metric induced by the Euclidean inner product, and let $M : = R^{2} \times S^{2}$ with the product Lorentzian metric whose first factor has line element $- (d x^{0})^{2} + (d x^{3})^{2}$ . Use the sphere chart $σ : (0, π) \times (0, 2 π) \to S^{2}$ given by $σ (θ, φ) = (\sin θ \cos φ, \sin θ \sin φ, \cos θ),$ and put $U : = R^{2} \times σ ((0, π) \times (0, 2 π)) \subset M$ . In the product chart with coordinate order $x = (x^{0}, x^{1}, x^{2}, x^{3}) = (x^{0}, θ, φ, x^{3})$ , the metric is $g : = - (d x^{0})^{2} + (d x^{1})^{2} + \sin^{2} (x^{1}) (d x^{2})^{2} + (d x^{3})^{2}, N : = \partial_{0} .$ This chart contains the equatorial geodesic segment used in An equator geodesic of the round-sphere factor: conjugate instant at distance pi with multiplicity one; the underlying manifold has an actual $S^{2}$ factor, not an unwrapped angular cover. The metric is smooth and Lorentzian on $M$ with time orientation $N$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model). Let $\nabla$ be its metric-selected connection and $R$ , $R_{a b c d}$ , ${Ric}_{a b}$ , $S$ and $G_{a b}$ its curvature data (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, The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors). Then:

The only nonzero Christoffel symbols in this chart are $Γ^{1}_{22} = - \sin x^{1} \cos x^{1}, Γ^{2}_{12} = Γ^{2}_{21} = \cot x^{1} .$
The curvature has exactly one independent nonzero lowered component up to the algebraic symmetries, namely $R_{1212} = \sin^{2} (x^{1});$ equivalently $R (\partial_{1}, \partial_{2}) \partial_{2} = \sin^{2} (x^{1}) \partial_{1}$ and $R (\partial_{1}, \partial_{2}) \partial_{1} = - \partial_{2}$ , and every component containing an index $0$ or $3$ vanishes.
The Ricci tensor, scalar curvature and Einstein tensor are ${Ric}_{11} = 1, {Ric}_{22} = \sin^{2} (x^{1}), S = 2, G = diag (1, 0, 0, - 1) .$
The contracted second Bianchi identity $\nabla^{a} G_{a b} = 0$ holds. In the displayed chart $U$ , $\nabla_{e} G_{a b} = 0$ identically, so the divergence vanishes term by term there.

Facts & Assumptions

Given: The product $M = R^{2} \times S^{2}$ with its product metric and time orientation $N$ , and its local chart $U$ with $0 < x^{1} < π$ and $0 < x^{2} < 2 π$ as specified above. Let $\nabla$ be the metric-selected connection and $R$ , $R_{a b c d}$ , $Ric$ , $S$ , and $G$ its curvature data.

[L1]

The product metric is smooth Lorentzian of signature $(-, +, +, +)$ on $M$ with time orientation $N = \partial_{0}$ ; its components on $U$ are $diag (- 1, 1, \sin^{2} (x^{1}), 1)$ . The metric-selected connection exists and is unique, and on $U$ its Christoffel symbols 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 $g_{a b}$ (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 clause (a), Christoffel symbols of an affine connection, The inverse metric, musical isomorphisms and index raising).

[L2]

The curvature components in a chart are $R^{a}_{b c d} = \partial_{c} Γ^{a}_{d b} - \partial_{d} Γ^{a}_{c b} + \sum_{e} (Γ^{a}_{c e} Γ^{e}_{d b} - Γ^{a}_{d e} Γ^{e}_{c b})$ , and $R$ is a tensor: its components transform between charts, while tensor identities and vanishing are chart-independent; the displayed nonzero component values below refer to the specified product chart (Tensoriality of the curvature, its coordinate formula and its point-normal metric formula clauses (a)-(b), The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors clause (b)).

[L3]

The lowered curvature satisfies $R_{a b c d} = - R_{b a c d} = - R_{a b d c} = R_{c d a b}$ and $R_{a b c d} + R_{a c d b} + R_{a d b c} = 0$ ; equivalently $R (X, Y) Z = - R (Y, X) Z$ and $g (R (X, Y) Z, W) = - g (R (X, Y) W, Z)$ (The algebraic symmetries of the curvature tensor and the symmetry of the Ricci and Einstein tensors clauses (a)-(c)).

[L4]

The Ricci tensor, scalar curvature and Einstein tensor are ${Ric}_{b d} = g^{a c} R_{a b c d}$ , $S = g^{b d} {Ric}_{b d}$ 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 clause (c)).

[L5]

The contracted second Bianchi identity reads $\nabla^{a} G_{a b} = 0$ , where $\nabla^{a} G_{a b} = g^{a e} \nabla_{e} G_{a b}$ and $\nabla$ acts on tensor components by the metric-compatible derivative (The second Bianchi identity and the vanishing divergence of the Einstein tensor clause (b), Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields clause (c)).

Verification

technique · direct

1.1L1algebra

(Christoffel symbols.) Only $g_{22} = \sin^{2} (x^{1})$ is nonconstant, with $\partial_{1} g_{22} = 2 \sin x^{1} \cos x^{1}$ and $g^{22} = \sin^{- 2} (x^{1})$ ; by [L1] the only possible nonzero symbols are $Γ^{1}_{22} = \frac{1}{2} g^{11} (- \partial_{1} g_{22}) = - \sin x^{1} \cos x^{1}$ and $Γ^{2}_{12} = Γ^{2}_{21} = \frac{1}{2} g^{22} \partial_{1} g_{22} = \cot x^{1}$ , every other symbol vanishing because it contains $\partial_{a} g_{b c}$ with $a \neq 1$ or a constant component.

2.1L2step 1.1L3algebra

(Candidate components.) In the coordinate formula of [L2] every term contains a factor $Γ^{a}_{\cdot \cdot}$ or a derivative of one; by step 1.1 a term can be nonzero only if all its symbol indices lie in ${1, 2}$ . Hence $R^{a}_{b c d} = 0$ whenever one of $a, b, c, d$ equals $0$ or $3$ : the derivative terms contain $Γ^{a}_{d b}$ or $Γ^{a}_{c b}$ with a forbidden index in the lower pair, or are annihilated by $\partial_{0}, \partial_{3}$ , and each product term contains a symbol with a forbidden index. Of the four remaining candidates $R^{1}_{112}$ , $R^{1}_{212}$ , $R^{2}_{112}$ , $R^{2}_{212}$ , the first and the last have a repeated index in the first lower pair, so their lowered components vanish by the skew-symmetry of [L3], and $R^{2}_{112}$ is the negative of $R^{2}_{121}$ ; it therefore suffices to check $R^{1}_{212}$ and $R^{2}_{121}$ , which are equal up to lowering and sign.

2.2step 1.1L2L3L4algebra

(The curvature component.) By [L2] and step 1.1, the products in $R^{1}_{212}$ contribute only through $- \sum_{e} Γ^{1}_{2 e} Γ^{e}_{12} = - Γ^{1}_{22} Γ^{2}_{12}$ , because $Γ^{1}_{1 e} = 0$ for every $e$ kills the first product and the only nonzero term of the second has $e = 2$ ; hence $R^{1}_{212} = \partial_{1} Γ^{1}_{22} - \partial_{2} Γ^{1}_{12} - Γ^{1}_{22} Γ^{2}_{12} = - \partial_{1} (\sin x^{1} \cos x^{1}) + \sin x^{1} \cos x^{1} \cot x^{1} .$ Substituting and using $\cos x^{1} \cot x^{1} = \cos^{2} x^{1} / \sin x^{1}$ , $R^{1}_{212} = - (\cos^{2} x^{1} - \sin^{2} x^{1}) + \cos^{2} x^{1} = \sin^{2} (x^{1})$ . Lowering with $g_{11} = 1$ gives $R_{1212} = \sin^{2} (x^{1})$ ; by [L3], $R_{2121} = R_{1212}$ and the skew-symmetry gives $R^{2}_{121} = \sin^{- 2} (x^{1}) R_{2121} = 1$ , and the vector-valued forms follow from $R (\partial_{c}, \partial_{d}) \partial_{b} = R^{a}_{b c d} \partial_{a}$ .

3.1step 2.1step 2.2L4algebra

(Ricci tensor.) By [L4] and step 2.1, the only components of $R_{a b c d}$ that can contribute to a contraction are those with all indices in ${1, 2}$ : ${Ric}_{11} = g^{22} R_{2121} = \sin^{- 2} (x^{1}) \sin^{2} (x^{1}) = 1$ and ${Ric}_{22} = g^{11} R_{1212} = \sin^{2} (x^{1})$ , while ${Ric}_{b d} = 0$ whenever $b$ or $d$ is $0$ or $3$ , because every term of the sum $g^{a c} R_{a b c d}$ then contains a component with a forbidden index.

4.1step 3.1L4algebra

(Scalar curvature and Einstein tensor.) By [L4] and step 3.1, $S = g^{11} {Ric}_{11} + g^{22} {Ric}_{22} = 1 + \sin^{- 2} (x^{1}) \sin^{2} (x^{1}) = 2$ . Hence $G_{11} = 1 - \frac{1}{2} \cdot 2 \cdot 1 = 0$ , $G_{22} = \sin^{2} (x^{1}) - \frac{1}{2} \cdot 2 \cdot \sin^{2} (x^{1}) = 0$ , $G_{00} = 0 - \frac{1}{2} \cdot 2 \cdot (- 1) = 1$ , $G_{33} = 0 - \frac{1}{2} \cdot 2 \cdot 1 = - 1$ , and all off-diagonal components vanish: $G = diag (1, 0, 0, - 1)$ .

5.1step 4.1L5algebra∎

(Divergence of the Einstein tensor in the chart.) By [L5] and the formula for the covariant derivative of a tensor, $\nabla_{e} G_{a b} = \partial_{e} G_{a b} - \sum_{d} (Γ^{d}_{e a} G_{d b} + Γ^{d}_{e b} G_{a d})$ on $U$ . Step 4.1 gives $\partial_{e} G_{a b} = 0$ ; and each product vanishes because a nonzero $Γ^{d}_{e a}$ has all three indices in ${1, 2}$ , while $G_{d b}$ or $G_{a d}$ is nonzero only when both its indices are $0$ or both are $3$ , so no index value is available for $d$ . Hence $\nabla_{e} G_{a b} = 0$ throughout $U$ , and the contracted divergence vanishes term by term there, in accordance with the globally valid identity [L5].

Remarks

What the example checks. The computation verifies on an explicit curved metric the definitions of the Ricci, scalar and Einstein tensors, the algebraic symmetries in the form $R_{1212} = R_{2121}$ and $R_{1212} + R_{1122} + R_{1221} = 0$ , and the contracted second Bianchi identity. The metric is not Ricci flat and is not flat: $R_{1212} = \sin^{2} (x^{1})$ is nonzero throughout the chart $U$ , so $Ric \neq 0$ by step 3.1.
Relation to the tidal page. On the equator $x^{1} = π / 2$ , the geodesic and field of An equator geodesic of the round-sphere factor: conjugate instant at distance pi with multiplicity one have $T = \partial_{2}$ and $J (s) = \sin (s) \partial_{1}$ . The component $R (\partial_{1}, \partial_{2}) \partial_{2} = \sin^{2} (x^{1}) \partial_{1}$ computed above, together with skew-symmetry, gives the curvature term $R (T, J) T = - \sin (s) \partial_{1}$ . That item verifies the Jacobi equation and the conjugate instant it entails.

ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passOpen item page →

Jacobi fields in flat space are affine, and no geodesic has conjugate instants

Example

Let $M = R^{4}$ carry the identity chart $x = (x^{0}, x^{1}, x^{2}, x^{3})$ , the flat Lorentzian metric $g = diag (- 1, 1, 1, 1)$ and the time orientation $N = \partial_{0}$ ; let $\nabla$ be its metric-selected connection and $R$ its curvature (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, Metric compatibility, torsion freedom and the metric-selected connection class). Then:

In the identity chart $\nabla$ is the coordinate connection, $Γ \equiv 0$ , and its geodesics are exactly the affine lines $t \mapsto p + t v$ ; moreover $\exp_{p} (v) = p + v$ and $d (\exp_{p})_{u} = id$ for every $u \in T_{p} M$ .
$R = 0$ : every curvature component vanishes, although the identity chart is global.
Along the geodesic $γ (t) = p + t v$ the covariant derivative is the componentwise derivative in the identity chart, and the Jacobi fields along $γ$ are exactly the affine fields $J (t) = J_{0} + t J_{1}, J_{0}, J_{1} \in R^{4},$ with $D_{t} J (0) = J_{1}$ ; the map $J \mapsto (J (0), D_{t} J (0))$ from this space onto $T_{p} M \oplus T_{p} M$ is the initial-value isomorphism of The Jacobi equation, geodesic deviation and the initial-value theory of Jacobi fields clause (c), and its inverse sends $(J_{0}, J_{1})$ to the field $t \mapsto J_{0} + t J_{1}$ .
No geodesic of this model has a conjugate instant: if $J (t_{0}) = J (t_{1}) = 0$ for $t_{0} < t_{1}$ , then $J = 0$ . Consistently $d (\exp_{p})_{u} = id$ is invertible for every $u \in T_{p} M$ , because $\exp_{p} (v) = p + v$ is a translation.
For the flat congruence $F (t, r) : = p + t v + r w$ with $p, v, w \in R^{4}$ , $v \neq 0$ , the variation field is the constant field $J = w$ and $D_{t}^{2} J = 0 = R (T, J) T$ : the deviation equation holds with no curvature term, and the relative acceleration of the congruence vanishes identically.

Facts & Assumptions

Given: The flat Lorentzian four-manifold $(R^{4}, g)$ with identity chart $x$ , $g = diag (- 1, 1, 1, 1)$ , and the geodesics $t \mapsto p + t v$ .

[L1]

The constant components $g_{a b} = diag (- 1, 1, 1, 1)$ in the global identity chart define a smooth Lorentzian metric of signature $(-, +, +, +)$ with time orientation $N = \partial_{0}$ , since $g (\partial_{0}, \partial_{0}) = - 1$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model).

[L2]

An affine connection is $g$ -compatible if and only if $\partial_{a} g_{b c} = \sum_{d} (Γ^{d}_{a b} g_{d c} + Γ^{d}_{a c} g_{b d})$ in every chart, and torsion-free if and only if $Γ^{a}_{b c} = Γ^{a}_{c b}$ ; the metric-selected connection exists, is unique, and is exactly the torsion-free $g$ -compatible connection (Torsion freedom and metric compatibility in chart components clauses 1-2, 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 clause (a)).

[L3]

In a chart, a geodesic with affine parameter satisfies ${\ddot{x}}^{k} + \sum_{i, j} Γ^{k}_{i j} {\dot{x}}^{i} {\dot{x}}^{j} = 0$ ; the geodesic initial-value problem has a unique maximal solution $γ_{p, v}$ , and $\exp_{p} (v) = γ_{p, v} (1)$ (Coordinate geodesic equation, Geodesic of an affine connection, The geodesic initial-value problem and the exponential map of the metric-selected connection).

[L4]

The curvature components in a chart are $R^{a}_{b c d} = \partial_{c} Γ^{a}_{d b} - \partial_{d} Γ^{a}_{c b} + \sum_{e} (Γ^{a}_{c e} Γ^{e}_{d b} - Γ^{a}_{d e} Γ^{e}_{c b})$ , and $R$ is a smooth tensor field (Tensoriality of the curvature, its coordinate formula and its point-normal metric formula clauses (a)-(b), The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors).

[L5]

Along a curve the covariant derivative has, in a chart, the component expression $D_{t} V^{a} = \partial_{t} V^{a} + \sum_{b, c} Γ^{a}_{b c} {\dot{γ}}^{b} V^{c}$ for a field $V = \sum_{a} V^{a} \partial_{a}$ (Covariant derivative along a curve).

[L6]

A Jacobi field along a geodesic $γ$ is a smooth field with $D_{t}^{2} J = R (γ^{'}, J) γ^{'}$ ; for a smooth family of affinely parametrized geodesics the variation field satisfies the same equation (Jacobi fields, geodesic variations and conjugate instants along a geodesic clauses (b)-(c), The Jacobi equation, geodesic deviation and the initial-value theory of Jacobi fields clause (b)).

[L7]

For a geodesic $γ (t) = \exp_{p} (t v)$ with $v \in T_{p} M$ and $s_{1} > 0$ with $s_{1} v$ in the exponential domain, $d (\exp_{p})_{s_{1} v}$ is injective if and only if no nonzero Jacobi field along $γ$ vanishes at $t = 0$ and at $t = s_{1}$ (The differential of the exponential map and conjugacy along a geodesic clause (b)).

Verification

technique · direct

1.1L1L2L3algebra

(The metric-selected connection and its geodesics.) For the coordinate connection with $Γ \equiv 0$ the symmetry $Γ^{a}_{b c} = Γ^{a}_{c b}$ is immediate and the compatibility identity of [L2] reads $\partial_{a} g_{b c} = 0$ , which holds because the metric components are constant; hence by [L2] this connection is the metric-selected connection and unique. By [L3] its geodesic equation is ${\ddot{x}}^{k} = 0$ , whose solutions are the affine lines, so the maximal geodesic with initial data $(p, v)$ is $t \mapsto p + t v$ defined on all of $R$ ; therefore $\exp_{p} (v) = p + v$ , and differentiating the chart expression of this translation gives $d (\exp_{p})_{u} = id$ for every $u$ .

1.2L4L5algebra

(Vanishing curvature and the componentwise derivative.) With $Γ \equiv 0$ every term of the coordinate formula [L4] contains a Christoffel symbol or a derivative of one, so $R^{a}_{b c d} = 0$ for all $a, b, c, d$ and $R = 0$ . By [L5], for a field $V = \sum_{a} V^{a} \partial_{a}$ along a curve, $D_{t} V = \sum_{a} (\partial_{t} V^{a}) \partial_{a}$ : the identity frame is parallel along every curve, and $D_{t}$ is the componentwise derivative.

2.1step 1.1step 1.2L6algebra

(Jacobi fields are affine.) Let $J = \sum_{a} J^{a} \partial_{a}$ be a smooth field along the geodesic $γ (t) = p + t v$ ; by step 1.1 $γ^{'} = v$ has constant components. By step 1.2, $D_{t}^{2} J = \sum_{a} (J^{a})^{''} \partial_{a}$ and $R (γ^{'}, J) γ^{'} = 0$ , so the Jacobi equation of [L6] is equivalent to $(J^{a})^{''} = 0$ for $a = 0, 1, 2, 3$ , whose solutions are exactly the affine functions $J^{a} (t) = J_{0}^{a} + t J_{1}^{a}$ . Hence the Jacobi fields are exactly the fields $J (t) = J_{0} + t J_{1}$ with $J_{0}, J_{1} \in R^{4}$ , and $D_{t} J (0) = J_{1}$ by step 1.2.

3.1step 2.1step 1.1L7algebra

(No conjugate instants.) Let $t_{0} < t_{1}$ and let $J (t) = J_{0} + t J_{1}$ be a Jacobi field as in step 2.1. If $J (t_{0}) = J (t_{1}) = 0$ , then subtracting the two equations gives $(t_{1} - t_{0}) J_{1} = 0$ , so $J_{1} = 0$ , and then $J_{0} = J (t_{0}) = 0$ : the only Jacobi field vanishing at two instants is the zero field. By [L7] applied to the geodesic $γ (t) = \exp_{p} (t v)$ , whose exponential data lie in the domain $T_{p} M$ by step 1.1, $d (\exp_{p})_{s_{1} v}$ is injective for every $s_{1} > 0$ and every $v$ : no geodesic of the flat model has a conjugate instant relative to any of its points.

4.1step 1.2step 1.1L6algebra∎

(The flat congruence.) Let $F (t, r) : = p + t v + r w$ and $T = \partial_{t} F = v$ , $J = \partial_{r} F = w$ , both with constant components. By step 1.2, $D_{t}^{2} J = 0$ and $R (T, J) T = 0$ , so $D_{t}^{2} J = R (T, J) T$ holds, in accordance with the deviation equation of [L6]; the variation field is parallel and the relative acceleration of the family vanishes everywhere.

Remarks

Flat versus curved. In the flat model the deviation of nearby geodesics is governed by affine initial data alone, and no conjugate instants occur. The contrast with the sphere-factor metric of Curvature, Ricci, scalar and Einstein tensors of a metric with a round two-sphere factor, where a Jacobi field vanishes again at parameter distance $π$ , is computed in An equator geodesic of the round-sphere factor: conjugate instant at distance pi with multiplicity one.
Scope. The statements are about the flat Lorentzian four-manifold with its metric-selected connection; no maximality of geodesics, completeness or causal claim beyond the displayed computations is made.

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An equator geodesic of the round-sphere factor: conjugate instant at distance pi with multiplicity one

Example

Let $M = R^{2} \times S^{2}$ carry the product metric of Curvature, Ricci, scalar and Einstein tensors of a metric with a round two-sphere factor. Use its local chart $U$ with $0 < θ : = x^{1} < π$ , $0 < φ : = x^{2} < 2 π$ and coordinate order $(x^{0}, θ, φ, x^{3})$ ; this chart contains the full geodesic segment below. The metric has time orientation $N = \partial_{0}$ , and its Christoffel symbols and curvature are computed in Curvature, Ricci, scalar and Einstein tensors of a metric with a round two-sphere factor. Let $\nabla$ be the metric-selected connection. Define $γ : [0, π] \to M$ and fix $p : = (0, \frac{π}{2}, \frac{π}{2}, 0), γ (s) : = (0, \frac{π}{2}, \frac{π}{2} + s, 0), T : = γ^{'} = \partial_{2} .$ Then:

$γ$ is a geodesic with affine parameter $s$ and unit spacelike tangent, $g (T, T) = \sin^{2} (π / 2) = 1$ , and every Christoffel symbol of the chart vanishes along $γ$ ; hence the coordinate frame $\partial_{0}, \partial_{1}, \partial_{2}, \partial_{3}$ is parallel along $γ$ .
The smooth field $J (s) : = \sin (s) \partial_{1} ∣_{γ (s)}$ along $γ$ is a nonzero Jacobi field with $J (0) = J (π) = 0$ . Therefore $π$ is a conjugate instant of $γ$ relative to $0$ (Jacobi fields, geodesic variations and conjugate instants along a geodesic clause (d)), and the events $p = γ (0)$ and $γ (π) = (0, \frac{π}{2}, \frac{3 π}{2}, 0)$ are distinct conjugate points of $M$ .
No $s_{1} \in (0, π)$ is a conjugate instant of $γ$ relative to $0$ : the first conjugate instant is exactly $π$ .
The conjugate instant $π$ has multiplicity $1$ , and consistently $\ker d (\exp_{p})_{π T (p)}$ is the one-dimensional subspace $π R \partial_{1} ∣_{p}$ of $T_{p} M$ (The differential of the exponential map and conjugacy along a geodesic clause (b)).

Facts & Assumptions

Given: The product $M = R^{2} \times S^{2}$ with the local chart and metric of Curvature, Ricci, scalar and Einstein tensors of a metric with a round two-sphere factor, its metric-selected connection $\nabla$ , the geodesic $γ : [0, π] \to M$ with coordinates $(0, π / 2, π / 2 + s, 0)$ , and $p = γ (0)$ .

[L1]

In this chart the only nonzero Christoffel symbols are $Γ^{1}_{22} = - \sin x^{1} \cos x^{1}$ and $Γ^{2}_{12} = Γ^{2}_{21} = \cot x^{1}$ ; the curvature satisfies $R_{1212} = \sin^{2} (x^{1})$ , $R (\partial_{1}, \partial_{2}) \partial_{1} = - \partial_{2}$ , $R (\partial_{1}, \partial_{2}) \partial_{2} = \sin^{2} (x^{1}) \partial_{1}$ , and every component containing an index $0$ or $3$ vanishes (Curvature, Ricci, scalar and Einstein tensors of a metric with a round two-sphere factor clauses 1-2).

[L2]

A geodesic with affine parameter satisfies $D_{s} γ^{'} = 0$ , and the covariant derivative along a curve is $D_{s} V^{a} = \partial_{s} V^{a} + \sum_{b, c} Γ^{a}_{b c} {\dot{γ}}^{b} V^{c}$ in a chart (Geodesic of an affine connection, Covariant derivative along a curve, Christoffel symbols of an affine connection).

[L3]

A Jacobi field along a geodesic is a smooth field with $D_{s}^{2} J = R (T, J) T$ , $T = γ^{'}$ ; $s_{1}$ is a conjugate instant relative to $s_{0}$ when a nonzero Jacobi field vanishes at $s_{0}$ and $s_{1}$ , and the multiplicity is the dimension of the space of such fields (Jacobi fields, geodesic variations and conjugate instants along a geodesic clauses (b)-(d), The Jacobi equation, geodesic deviation and the initial-value theory of Jacobi fields clause (b)).

[L4]

The curvature is antisymmetric in its first two arguments, $R (X, Y) Z = - R (Y, X) Z$ , and in its second pair, so in particular $R (\partial_{2}, \partial_{1}) \partial_{2} = - R (\partial_{1}, \partial_{2}) \partial_{2}$ (The algebraic symmetries of the curvature tensor and the symmetry of the Ricci and Einstein tensors clause (a), The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors).

[L5]

For $v \in T_{p} M$ and $s_{1} > 0$ with $s_{1} v$ in the exponential domain $E_{p}$ , $\ker d (\exp_{p})_{s_{1} v} = s_{1} {u \in T_{p} M : J_{u} (s_{1}) = 0}$ , where $J_{u}$ is the Jacobi field along $t \mapsto \exp_{p} (t v)$ with $J_{u} (0) = 0$ , $D_{t} J_{u} (0) = u$ ; this kernel is zero if and only if no nonzero Jacobi field along the geodesic vanishes at $0$ and $s_{1}$ (The differential of the exponential map and conjugacy along a geodesic clauses (a)-(b), The geodesic initial-value problem and the exponential map of the metric-selected connection).

Verification

technique · direct

1.1L1L2algebra

(The geodesic and its parallel frame.) Along $γ$ one has $x^{1} \equiv π / 2$ , so $Γ^{1}_{22} = - \sin (π / 2) \cos (π / 2) = 0$ and $Γ^{2}_{12} = \cot (π / 2) = 0$ ; by [L1] every Christoffel symbol vanishes along $γ$ . By [L2], $\nabla_{\partial_{2}} \partial_{2} = 0$ , so $D_{s} T = 0$ and $γ$ is a geodesic with affine parameter $s$ ; likewise $D_{s} \partial_{a} = 0$ for $a = 0, 1, 2, 3$ , so the coordinate frame is parallel along $γ$ and $D_{s}$ is the componentwise derivative for fields written in this frame. Finally $g (T, T) = \sin^{2} (π / 2) = 1$ .

2.1step 1.1L1L3L4algebra

(The Jacobi field.) Let $J (s) : = \sin (s) \partial_{1} ∣_{γ (s)}$ . By step 1.1, $D_{s} J = \cos (s) \partial_{1}$ and $D_{s}^{2} J = - \sin (s) \partial_{1}$ . By [L1] and [L4], $R (T, J) T = \sin (s) R (\partial_{2}, \partial_{1}) \partial_{2} = - \sin (s) R (\partial_{1}, \partial_{2}) \partial_{2} = - \sin (s) \sin^{2} (π / 2) \partial_{1} = - \sin (s) \partial_{1}$ . Hence $D_{s}^{2} J = R (T, J) T$ : $J$ is a Jacobi field by [L3]. It is nonzero (for instance $J (π / 2) = \partial_{1} \neq 0$ ) and $J (0) = J (π) = 0$ .

2.2step 1.1L1L3L4algebra

(The general Jacobi field and the first conjugate instant.) Let $K = a \partial_{0} + b \partial_{1} + c \partial_{2} + d \partial_{3}$ be a smooth field along $γ$ . By step 1.1, $D_{s}^{2} K = a^{''} \partial_{0} + b^{''} \partial_{1} + c^{''} \partial_{2} + d^{''} \partial_{3}$ , while $R (T, K) T = b R (\partial_{2}, \partial_{1}) \partial_{2} = - b \partial_{1}$ by [L1] and [L4]: the Jacobi equation is the system $a^{''} = c^{''} = d^{''} = 0$ , $b^{''} = - b$ , with general solution $a = a_{0} + a_{1} s$ , $b = b_{0} \cos s + b_{1} \sin s$ , $c = c_{0} + c_{1} s$ , $d = d_{0} + d_{1} s$ . If $K (0) = 0$ and $K (s_{1}) = 0$ for some $s_{1} \in (0, π)$ : the first condition gives $a_{0} = b_{0} = c_{0} = d_{0} = 0$ ; the second gives $a_{1} = c_{1} = d_{1} = 0$ and $b_{1} \sin s_{1} = 0$ , so $b_{1} = 0$ because $\sin s_{1} \neq 0$ . Hence $K = 0$ : no instant in $(0, π)$ is conjugate to $0$ , and the first conjugate instant of $γ$ is $π$ .

3.1step 2.1L3given

(Conjugacy at $π$ .) By step 2.1 and [L3], $π$ is a conjugate instant of $γ$ relative to $0$ with $p = γ (0)$ and $γ (π) = (0, π / 2, 3 π / 2, 0) \neq p$ : the two distinct points $p$ and $γ (π)$ of $M$ are conjugate along $γ$ .

4.1step 2.2L3L5algebra∎

(Multiplicity one and the kernel of the exponential differential.) By step 2.2, a Jacobi field with $K (0) = K (π) = 0$ has $a_{0} = b_{0} = c_{0} = d_{0} = a_{1} = c_{1} = d_{1} = 0$ , hence is $K = b_{1} \sin s \partial_{1}$ : the space of such fields is one-dimensional, so the multiplicity of the conjugate instant $π$ is $1$ by [L3]. For the kernel statement, let $v : = T (p) = \partial_{2} ∣_{p}$ and note that the geodesic $τ \mapsto \exp_{p} (τ π v) = γ (τ π)$ exists for $τ \in [0, 1]$ , so $π v \in E_{p}$ . By [L5], $\ker d (\exp_{p})_{π v} = π {u \in T_{p} M : J_{u} (π) = 0}$ ; solving the system of step 2.2 with initial data $J_{u} (0) = 0$ , $D_{s} J_{u} (0) = u$ gives $J_{u} (s) = u_{0} s \partial_{0} + u_{1} \sin s \partial_{1} + u_{2} s \partial_{2} + u_{3} s \partial_{3}$ for $u = \sum_{a} u_{a} \partial_{a} ∣_{p}$ , so $J_{u} (π) = 0$ if and only if $u_{0} = u_{2} = u_{3} = 0$ , that is $u \in R \partial_{1} ∣_{p}$ . Hence $\ker d (\exp_{p})_{π v} = π R \partial_{1} ∣_{p}$ is one-dimensional, in agreement with the multiplicity computed above.

Remarks

Relation to the index-form results. For every $s_{1} < π$ the segment $γ ∣_{[0, s_{1}]}$ has no conjugate instant, and $π$ is the first conjugate instant (step 2.2). This is the pattern of "no conjugate instant below a bound" that the index-form positivity and local-maximum results on the curvature page require in the timelike case; the example itself is spacelike, $g (T, T) = 1$ , so those statements, which assume $g (T, T) = - 1$ , are not applied to it.
Spacelike nature of the example. The geodesic $γ$ is spacelike ( $g (T, T) = 1$ ); the conjugacy and multiplicity statements are those of the general definitions, which do not require a timelike geodesic, and no proper-time maximum claim is made here.

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The exchange symmetries of a curvature-like tensor do not imply the first Bianchi identity

Statement refuted

Statement refuted: "Let $M$ be a smooth four-manifold with a global chart and let $T$ be a smooth $(0, 4)$ -tensor field on $M$ whose components satisfy, at every point, $T_{a b c d} = - T_{b a c d}, T_{a b c d} = - T_{a b d c}, T_{a b c d} = T_{c d a b}$ for all $a, b, c, d$ — the skew-symmetries and the exchange of the two index pairs of The algebraic symmetries of the curvature tensor and the symmetry of the Ricci and Einstein tensors clauses (a)-(b). Then $T$ also satisfies the first Bianchi identity $T_{a b c d} + T_{a c d b} + T_{a d b c} = 0$ of clause (c)."

Facts & Assumptions

Given: The smooth four-manifold $M = R^{4}$ with the identity chart $x = (x^{0}, x^{1}, x^{2}, x^{3})$ and the flat Lorentzian metric $g = diag (- 1, 1, 1, 1)$ with time orientation $N = \partial_{0}$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model).

[L1]

For the curvature tensor of a metric-selected connection the identities $R_{a b c d} = - R_{b a c d}$ , $R_{a b c d} = - R_{a b d c}$ , $R_{a b c d} = R_{c d a b}$ and $R_{a b c d} + R_{a c d b} + R_{a d b c} = 0$ all hold; the displayed implicational statement above asserts only the first three for an arbitrary tensor field (The algebraic symmetries of the curvature tensor and the symmetry of the Ricci and Einstein tensors clauses (a)-(c)).

[L2]

A smooth tensor field of type $(0, 4)$ on $M$ is determined by its component functions in the maximal atlas, and a family of component functions that is $C^{\infty}$ in every chart — in particular a family that is constant in the global identity chart — defines a smooth $(0, 4)$ -tensor field (Tensor fields of type (r,s) on a Lorentzian four-manifold and their components clauses (b)-(c)).

[L3]

An algebraic curvature tensor at a point is defined to satisfy the four identities of [L1] together, the Bianchi identity included (A curvature tensor vanishes if the tidal operator vanishes for every timelike direction).

Counterexample

In the identity chart define the constant component families $ω_{01} = ω_{23} : = 1, ω_{10} = ω_{32} : = - 1, ω_{a b} : = 0 otherwise,$ and $T_{a b c d} : = ω_{a b} ω_{c d} .$ By [L2] these constants define a smooth $(0, 2)$ -tensor field $ω$ and a smooth $(0, 4)$ -tensor field $T$ on $M$ .

1.1L2constructalgebra

(The skew-symmetries.) Since $ω_{a b} = - ω_{b a}$ for all $a, b$ , $T_{a b c d} = ω_{a b} ω_{c d} = - ω_{b a} ω_{c d} = - T_{b a c d}$ and $T_{a b c d} = - ω_{a b} ω_{d c} = - T_{a b d c}$ : the first two identities of the statement hold everywhere.

1.2L2algebra

(Exchange of the pairs.) Since multiplication of real numbers is commutative, $T_{c d a b} = ω_{c d} ω_{a b} = ω_{a b} ω_{c d} = T_{a b c d}$ : the third identity holds everywhere.

2.1L2step 1.2algebra

(The Bianchi sum does not vanish.) For $(a, b, c, d) = (0, 1, 2, 3)$ the sum of the statement is $T_{0123} + T_{0231} + T_{0312}$ . The components are $T_{0123} = ω_{01} ω_{23} = 1 \cdot 1 = 1$ , $T_{0231} = ω_{02} ω_{31} = 0 \cdot 0 = 0$ and $T_{0312} = ω_{03} ω_{12} = 0 \cdot 0 = 0$ . Hence $T_{0123} + T_{0231} + T_{0312} = 1 \neq 0$ at every point of $M$ : $T$ violates the first Bianchi identity.

3.1step 1.1step 1.2step 2.1L1L3algebra∎

(Conclusion.) Steps 1.1 and 1.2 show that $T$ satisfies the two skew-symmetries and the pair exchange, and step 2.1 shows that it does not satisfy the Bianchi identity. Therefore the displayed statement is refuted: the exchange symmetries alone do not imply the Bianchi identity, which is an independent condition and is imposed separately in the definition of an algebraic curvature tensor [L3]. In particular, by clause (c) of [L1] no metric-selected connection on $M$ has $T$ as its curvature tensor.

Remarks

Why the example works. The tensor is defined directly by $T_{a b c d} = ω_{a b} ω_{c d}$ for the two-form $ω = d x^{0} \land d x^{1} + d x^{2} \land d x^{3}$ ; no Kulkarni-Nomizu construction is involved. The three exchange identities are immediate from the skew-symmetry of $ω$ , while the Bianchi sum for $(0, 1, 2, 3)$ receives the single term $ω_{01} ω_{23}$ that the other two orderings of the indices miss. The example is the promised companion of the independence remark in The algebraic symmetries of the curvature tensor and the symmetry of the Ricci and Einstein tensors.
What is not claimed. The counterexample is algebraic: it refutes the implication, not the Bianchi identity for curvature tensors; that identity is a theorem of this page (The algebraic symmetries of the curvature tensor and the symmetry of the Ricci and Einstein tensors clause (c)). No claim is made that $T$ arises geometrically, and no count of independent components is asserted.

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A nonzero algebraic curvature tensor whose tidal operator vanishes for one timelike direction

Statement refuted

Statement refuted: "Let $R_{p}$ be an algebraic curvature tensor at a point $p$ of a time-oriented smooth Lorentzian four-manifold, with the four identities of A curvature tensor vanishes if the tidal operator vanishes for every timelike direction, and suppose there is one timelike vector $T \in T_{p} M$ such that the tidal operator $J \mapsto R (T, J) T$ vanishes for every $J \in T_{p} M$ . Then $R_{p} = 0$ ."

Facts & Assumptions

Given: The flat Lorentzian four-manifold $M = R^{4}$ with identity chart $x = (x^{0}, x^{1}, x^{2}, x^{3})$ , metric $g = diag (- 1, 1, 1, 1)$ and time orientation $N = \partial_{0}$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model).

[L1]

A vector $R (T, J) T$ associated with a $(0, 4)$ -tensor $R$ is defined by the pairing $g (R (T, J) T, W) = R_{a b c d} W^{a} T^{c} J^{d} T^{b}$ for all $W$ , the convention fixed in The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors clause (b) and A curvature tensor vanishes if the tidal operator vanishes for every timelike direction; $g$ is nondegenerate, so such a vector is zero if and only if the pairing vanishes for all $W$ .

[L2]

An algebraic curvature tensor at a point is a multilinear map whose components in a basis satisfy $R_{a b c d} = - R_{b a c d} = - R_{a b d c} = R_{c d a b}$ and $R_{a b c d} + R_{a c d b} + R_{a d b c} = 0$ (A curvature tensor vanishes if the tidal operator vanishes for every timelike direction).

[L3]

A family of component functions that is constant in the global identity chart defines a smooth tensor field (Tensor fields of type (r,s) on a Lorentzian four-manifold and their components clauses (b)-(c)).

Counterexample

Let $h$ be the smooth $(0, 2)$ -tensor field on $M$ with constant components in the identity chart $h_{00} = h_{0 i} = h_{i 0} : = 0, h_{i j} : = δ_{i j} (i, j \in {1, 2, 3}),$ the Euclidean metric on the spacelike coordinate three-plane extended by zero, and let $R$ be the smooth $(0, 4)$ -tensor field with constant components $R_{a b c d} : = h_{a c} h_{b d} - h_{a d} h_{b c} .$ Both exist by [L3]. For a future timelike vector $T$ write $R (T, J) T$ for the vector defined in [L1].

1.1L2L3algebra

(The four algebraic identities.) Since $h_{a b} = h_{b a}$ , one has $R_{b a c d} = h_{b c} h_{a d} - h_{b d} h_{a c} = - R_{a b c d}$ and $R_{a b d c} = h_{a d} h_{b c} - h_{a c} h_{b d} = - R_{a b c d}$ ; and $R_{c d a b} = h_{c a} h_{d b} - h_{c b} h_{d a} = h_{a c} h_{b d} - h_{a d} h_{b c} = R_{a b c d}$ . Moreover $R_{a b c d} + R_{a c d b} + R_{a d b c} = (h_{a c} h_{b d} - h_{a d} h_{b c}) + (h_{a d} h_{c b} - h_{a b} h_{c d}) + (h_{a b} h_{d c} - h_{a c} h_{d b}) = 0$ , each of the three pairs cancelling by $h_{x y} = h_{y x}$ . Hence $R$ satisfies the four identities of [L2] at every point.

1.2L3algebra

(Nonvanishing.) $R_{1212} = h_{11} h_{22} - h_{12} h_{21} = 1 \cdot 1 - 0 = 1 \neq 0$ , so $R \neq 0$ .

1.3L1L3algebra

(The tidal operator vanishes for $T = \partial_{0}$ .) Let $T : = \partial_{0}$ , a future timelike vector with $g (T, T) = - 1$ . For all $J, W$ , by [L1] and the definition of $R$ , $g (R (T, J) T, W) = R_{a b c d} W^{a} T^{c} J^{d} T^{b} = R_{a 00 d} W^{a} J^{d}$ and $R_{a 00 d} = h_{a 0} h_{0 d} - h_{a d} h_{00} = 0 - 0 = 0$ , because all components of $h$ with an index $0$ vanish. Hence the pairing vanishes for all $J, W$ and, by nondegeneracy [L1], $R (T, J) T = 0$ for every $J \in T_{p} M$ .

2.1L1step 1.3algebra

(A tilted timelike direction detects the tensor.) Let $T^{'} : = \partial_{0} + \frac{1}{2} \partial_{1}$ ; then $g (T^{'}, T^{'}) = - 1 + \frac{1}{4} = - \frac{3}{4} < 0$ , so $T^{'}$ is timelike. Expanding in the constant components, $g (R (T^{'}, \partial_{3}) T^{'}, W) = R_{a b c d} W^{a} T^{' c} (\partial_{3})^{d} T^{' b} = \frac{1}{4} R_{a 113} W^{a}$ , with the remaining three terms $R_{a 003}$ , $R_{a 013}$ and $R_{a 103}$ vanishing: each of their products contains a factor $h_{0 b}$ or $h_{b 0}$ for some $b$ , and all such components of $h$ are zero. Now $R_{a 113} = h_{a 1} h_{13} - h_{a 3} h_{11} = - h_{a 3}$ , so $g (R (T^{'}, \partial_{3}) T^{'}, W) = - \frac{1}{4} W^{3}$ and, by [L1], $R (T^{'}, \partial_{3}) T^{'} = - \frac{1}{4} \partial_{3} \neq 0$ : the tidal operator of $R$ does not vanish for every timelike direction.

3.1step 1.1step 1.2step 1.3step 2.1L2algebra∎

(Conclusion.) By steps 1.1 and 1.2, $R$ is a nonzero algebraic curvature tensor; by step 1.3 there is a timelike vector, namely $\partial_{0}$ , for which the tidal operator vanishes identically; by step 2.1 the operator is nonzero for the timelike direction $T^{'}$ . Hence the displayed statement is refuted: the vanishing of the tidal operator for a single timelike direction does not force $R = 0$ , and the hypothesis "for every timelike $T$ " in A curvature tensor vanishes if the tidal operator vanishes for every timelike direction cannot be weakened to a single observer.

Remarks

Where the single oblique direction matters. For $T = \partial_{0}$ every slot of $R$ built from $T$ contains a factor $h_{\cdot 0} = 0$ , so the whole tidal pairing collapses; the tilt $T^{'} = \partial_{0} + \frac{1}{2} \partial_{1}$ gives the slot $h_{11} = 1$ a nonzero weight and produces the component $R (T^{'}, \partial_{3}) T^{'} = - \frac{1}{4} \partial_{3}$ . The example thus separates the hypothesis "one timelike direction" from the hypothesis "all timelike directions" used by the criterion.
Scope. The counterexample is algebraic and concerns the stated implication only; no realization of $R$ as the curvature of a metric-selected connection is claimed, and no empirical statement is made.

Curvature And Tides — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

5 · Examples, experiments and counterexamples

A flat metric with nowhere-zero Christoffel coefficients: flat space in cylindrical coordinates

Example

Facts & Assumptions

Verification

Remarks

Curvature, Ricci, scalar and Einstein tensors of a metric with a round two-sphere factor

Example

Facts & Assumptions

Verification

Remarks

Jacobi fields in flat space are affine, and no geodesic has conjugate instants

Example

Facts & Assumptions

Verification

Remarks

An equator geodesic of the round-sphere factor: conjugate instant at distance pi with multiplicity one

Example

Facts & Assumptions

Verification

Remarks

The exchange symmetries of a curvature-like tensor do not imply the first Bianchi identity

Statement refuted

Facts & Assumptions

Counterexample

Remarks

A nonzero algebraic curvature tensor whose tidal operator vanishes for one timelike direction

Statement refuted

Facts & Assumptions

Counterexample

Remarks

Sources