QIQTH.Raychaudhuri

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Raychaudhuri · entries 564–574 of 1000

Definition 564 (covDeriv2Vec).  source ↗

Second covariant derivative of a vector field, ∇_μ ∇_ν V^ρ. Treating W^ρ_ν := ∇_ν V^ρ (= covDerivVec) as a (1,1) tensor: ∇_μ W^ρ_ν = ∂_μ W^ρ_ν + Γ^ρ_{μσ} W^σ_ν − Γ^σ_{μν} W^ρ_σ.

Used by ricci_identity, ricci_identity_contracted, covDeriv2Vec_trace, raychaudhuri_focusing, geodesic_leibniz, raychaudhuri_geodesic.

Lemma 565 (pd_covDerivVec).  source ↗

The partial derivative of ∇_ν V^ρ, expanded via the product rule: ∂_μ(∇_ν V^ρ) = ∂_μ∂_ν V^ρ + Σ_σ (∂_μ Γ^ρ_{νσ}) V^σ + Σ_σ Γ^ρ_{νσ} ∂_μ V^σ.

Proof. By PdiffAt, pd_add, PdiffAt_of_contDiff, mul, PdiffAt_sum, pd_sum, pd_mul, PdiffAt_pd. $\square$

Used by ricci_identity.

Lemma 566 (ricci_identity).  source ↗

The Ricci identity — the commutator of covariant derivatives is the Riemann curvature: (∇_μ ∇_ν − ∇_ν ∇_μ) V^ρ = R^ρ_{σμν} V^σ. The geometric heart of Raychaudhuri focusing.

Proof. By pd, pd_comm, christoffel_symm, covDerivVec, pd_covDerivVec. $\square$

Used by ricci_identity_contracted.

Lemma 567 (ricci_identity_contracted).  source ↗

The contracted Ricci identity — tracing the commutator on the upper index (ρ = μ, summed) turns the Riemann tensor into the Ricci tensor: ∑_μ (∇_μ∇_ν − ∇_ν∇_μ) V^μ = R_{σν} V^σ. This is exactly the step that introduces the R_{μν} focusing term into the expansion evolution.

Proof. By riemann, ricci_identity. $\square$

Used by raychaudhuri_focusing.

Definition 568 (expansion).  source ↗

The expansion θ = ∇_μ V^μ — the covariant divergence of a vector field.

Used by qiqt_gr_freefield_complete, qiqt_gr_freefield_complete_covCong, qiqt_gr_freefield_localized', qiqt_gr_freefield_nullEnergy, qiqt_gr_freefield_geom, qiqt_gr_freefield_gaussian, qiqt_gr_ppwave, qiqt_gr_freefield_thermo, and 8 more.

Lemma 569 (covDeriv2Vec_trace).  source ↗

Covariant derivative commutes with contraction (the geodesic-direction Γ terms cancel by torsion-freeness): the trace ∑_μ ∇_ν ∇_μ V^μ is just the ordinary derivative of the expansion, ∂_ν θ. This is what turns ∇_ν(∇_μ V^μ) into ∂_ν θ in the Raychaudhuri derivation.

Proof. By PdiffAt, PdiffAt_of_contDiff, mul, add, PdiffAt_sum, pd_sum, PdiffAt_pd, covDerivVec. $\square$

Used by raychaudhuri_focusing.

Lemma 570 (raychaudhuri_focusing).  source ↗

The Raychaudhuri focusing equation. Contracting the (contracted) Ricci identity with V gives the evolution of the expansion θ along V, with the Ricci focusing term −R_{σν}V^σV^ν made explicit:

V^ν ∂_ν θ = Σ_{μν} V^ν ∇_μ∇_ν V^μ − R_{σν} V^σ V^ν.

This is Jacobson’s focusing step (the geometry of his front half). For a geodesic V (V^σ∇_σV^μ=0) the first right-hand term equals −(∇_μV^ν)(∇_νV^μ) (the −½θ²−σ² shear part); that geodesic simplification is the remaining (Leibniz) polish. Holds for any vector field.

Proof. By ricci_identity_contracted, covDeriv2Vec_trace. $\square$

Used by raychaudhuri_geodesic.

Lemma 571 (geodesic_divergence_leibniz).  source ↗

Partial-Leibniz of the geodesic acceleration. For a geodesic vector field V (Σ_ν V^ν ∇_ν V^μ = 0 as a field), the divergence of the acceleration vanishes, expanded by the product rule: Σ_ν (∂_μ V^ν · ∇_ν V^μ + V^ν · ∂_μ(∇_ν V^μ)) = 0. The step that lets Σ V^ν∇_μ∇_νV^μ be rewritten as −(∇_μV^ν)(∇_νV^μ) (the −½θ²−σ² shear part).

Proof. By PdiffAt, pd_const, PdiffAt_of_contDiff, mul, add, PdiffAt_sum, pd_sum, pd_mul, PdiffAt_pd. $\square$

Used by geodesic_leibniz.

Lemma 572 (geodesic_leibniz).  source ↗

Geodesic Leibniz identity. For a geodesic field V, the Raychaudhuri second-derivative term is the shear/expansion quadratic: Σ_{νμ} V^ν ∇_μ∇_ν V^μ = − Σ_{μν} (∇_μ V^ν)(∇_ν V^μ).

Proof. By pd, geodesic_divergence_leibniz. $\square$

Used by raychaudhuri_geodesic.

Lemma 573 (raychaudhuri_geodesic).  source ↗

The Raychaudhuri equation (geodesic congruence), in Jacobson’s exact form:

V^ν ∂_ν θ = − (∇_μ V^ν)(∇_ν V^μ) − R_{σν} V^σ V^ν.

The expansion θ of a geodesic congruence focuses, driven by the shear/expansion quadratic −(∇V)(∇V) (Jacobson’s −½θ²−σ², the term he neglects near a stationary horizon) and the Ricci focusing term −R(V,V) (the term he uses). Assembled from raychaudhuri_focusing and geodesic_leibniz. The full geometry of Jacobson’s front half is now machine-checked, axiom-free.

Proof. By covDeriv2Vec, raychaudhuri_focusing, geodesic_leibniz. $\square$

Used by raychaudhuri_focusing_at_equilibrium.

Lemma 574 (raychaudhuri_focusing_at_equilibrium).  source ↗

Leading-order Raychaudhuri focusing at equilibrium — the geometric content of Jacobson’s hFocus. At a moment of local equilibrium (a stationary/bifurcation horizon, where the shear–expansion quadratic (∇_μV^ν)(∇_νV^μ) vanishes — θ = σ = ω = 0, the condition Jacobson imposes), the Raychaudhuri equation collapses to pure Ricci focusing:

V^ν ∂_ν θ = − R_{σν} V^σ V^ν (i.e. …

Proof. By raychaudhuri_geodesic. $\square$

Used by hFocus_of_raychaudhuri.

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