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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.

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

Statement

Facts & Assumptions

Proof

Remarks

Depends on

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