Second-Order Variation: Guarantee of Stability

“Extremum is not enough to determine physics, stability is also needed.”

🎯 Why Second-Order Variation?

First-order variation $\delta S_{\text{gen}}=0$ gives Einstein’s equations, but this is only a necessary condition.

Key questions:

• Is this extremum a maximum or minimum?
• Is the solution stable?
• Will small perturbations cause divergence?

Answer: Need to check second-order variation!

Classical Analogy: Pendulum

Consider the potential energy of a pendulum:

$$V(\theta )=-mgl\cos \theta$$

Extremum points:

• $\theta =0$ (hanging down): $V^{\prime }=0$, and $V^{\prime \prime }>0$ (stable minimum)
• $\theta =\pi$ (inverted): $V^{\prime }=0$, but $V^{\prime \prime }<0$ (unstable maximum)

      ↑
     / \    Unstable (V'' < 0)
    /   \
   |  O  |
   |  |  |   Stable (V'' > 0)
   --------

Physical reality: Only extrema with $V^{\prime \prime }>0$ are physically realizable stable states!

IGVP Second-Order Condition

Similarly, IGVP requires:

1. First-order condition: $\delta S_{\text{gen}}=0$ → Einstein’s equations
2. Second-order condition: ${\delta }^2{S}_{\text{rel}}\ge 0$ → stability

graph TB
    I["IGVP Variational Principle"] --> F["First Order<br/>δS_gen = 0"]
    I --> S["Second Order<br/>δ²S_rel ≥ 0"]

    F --> E["Einstein Equations<br/>G_ab + Λg_ab = 8πGT_ab"]
    S --> H["Hollands-Wald<br/>Stability"]

    E --> P["Necessary Condition<br/>(Extremum)"]
    H --> C["Sufficient Condition<br/>(Stable Extremum)"]

    style I fill:#e1f5ff
    style F fill:#fff4e1
    style S fill:#ffe1e1,stroke:#ff6b6b,stroke-width:2px
    style E fill:#e1ffe1
    style H fill:#e1ffe1

📐 Relative Entropy

Definition

Relative entropy is a core concept in information geometry measuring the “distance” between two states.

For two quantum states $\rho$ and $\sigma$, define:

$$S(\rho ||\sigma ):=\text{tr}(\rho \ln \rho )-\text{tr}(\rho \ln \sigma )$$

Properties:

1. Non-negativity: $S(\rho ||\sigma )\ge 0$ (Klein inequality)
2. Zero if and only if: $S(\rho ||\sigma )=0\iff \rho =\sigma$
3. Monotonicity: For completely positive map $\Phi$, $S(\Phi (\rho )||\Phi (\sigma ))\le S(\rho ||\sigma )$

Geometric Interpretation

Relative entropy is the “distance squared” in information geometry:

$$S(\rho ||\sigma )\approx \frac{1}{2}{g}_{ij}\Delta {\theta }^i\Delta {\theta }^j$$

Where $g_{ij}$ is the Fisher information matrix (metric).

Second-order expansion:

$$S({\rho }_0+\delta \rho ||{\rho }_0)\approx \frac{1}{2}{\delta }^2{S}_{\text{rel}}$$

Where ${\delta }^2{S}_{\text{rel}}$ is the Hessian of relative entropy (second-order variation).

graph LR
    R0["Reference State ρ₀"] --> D["Perturbation<br/>ρ = ρ₀ + δρ"]
    D --> S["Relative Entropy<br/>S(ρ||ρ₀)"]

    S --> O["First Order: δS = 0<br/>(Extremum)"]
    S --> T["Second Order: δ²S ≥ 0<br/>(Stable)"]

    style R0 fill:#e1f5ff
    style T fill:#ffe1e1,stroke:#ff6b6b,stroke-width:2px

⚛️ Generalized Relative Entropy

Relative Entropy in IGVP

In the IGVP framework, consider generalized relative entropy:

$$S_{\text{rel}}:=S_{\text{gen}}(\text{perturbed state})-S_{\text{gen}}(\text{reference state})$$

Expanding:

$$S_{\text{rel}}=\left[\frac{A^{\prime }}{4G\hslash }+S_{\text{out}}^{\prime }({\rho }^{\prime })\right]-\left[\frac{A}{4G\hslash }+S_{\text{out}}(\rho )\right]$$

Second-Order Expansion

For small perturbations $\delta g_{ab}$ and $\delta \rho$:

$$S_{\text{rel}}=\underset{=0\text{ (first-order extremum)}}{\underbrace{\delta S_{\text{gen}}}}+\frac{1}{2}{\delta }^2{S}_{\text{rel}}+O({\delta }^3)$$

Stability requirement:

$$\boxed{{\delta }^2{S}_{\text{rel}}\ge 0}$$

Physical meaning: Perturbations cannot lower generalized entropy, providing a theoretical guarantee that the extremum is a stable minimum.

🔧 Hollands-Wald Canonical Energy

Definition

Hollands and Wald (2013), in studying linearized gravity stability, defined canonical energy:

$${\mathcal{E}}_{\text{can}}[h,h]:={\int }_{\Sigma }{\mathcal{E}}^{\mu \nu }{h}_{\mu \nu }\sqrt{-g}{d}^{d}x$$

Where:

• $h_{\mu \nu }$: metric perturbation
• $\Sigma$: Cauchy hypersurface
• ${\mathcal{E}}^{\mu \nu }$: canonical energy density (given by variation of gravitational Hamiltonian)

Properties:

1. Non-negativity: Under appropriate boundary conditions, ${\mathcal{E}}_{\text{can}}[h,h]\ge 0$
2. Conservation: Invariant along evolution (when field equations hold)
3. Gauge invariance: For pure gauge modes $h={\mathcal{L}}_{\xi }g$, ${\mathcal{E}}_{\text{can}}=0$

Physical Meaning

${\mathcal{E}}_{\text{can}}$ measures the energy of gravitational perturbations.

$${\mathcal{E}}_{\text{can}}\ge 0\iff \text{Gravitational waves carry positive energy}$$

Stability criterion: If ${\mathcal{E}}_{\text{can}}\ge 0$ for all allowed perturbations, then the background solution is generally considered linearly stable.

graph TB
    B["Background Solution<br/>g_ab"] --> P["Perturbation<br/>g_ab + h_ab"]

    P --> L["Linearized Einstein Equations<br/>δG_ab = 8πGδT_ab"]

    L --> E["Canonical Energy<br/>𝓔_can[h,h]"]

    E --> S["Non-Negativity<br/>𝓔_can ≥ 0"]

    S --> ST["Stable!"]

    style B fill:#e1f5ff
    style E fill:#fff4e1,stroke:#ff6b6b,stroke-width:2px
    style ST fill:#e1ffe1,stroke:#ff6b6b,stroke-width:3px

🔗 JLMS Equivalence

JLMS Relation

Jafferis, Lewkowycz, Maldacena, Suh (2016) proposed a profound equivalence relation:

Under appropriate conditions (spherical region, near vacuum state, fixed boundary conditions):

$$\boxed{{\delta }^2{S}_{\text{rel}}={\mathcal{E}}_{\text{can}}[h,h]}$$

Meaning: Second-order variation of relative entropy corresponds to Hollands-Wald canonical energy under specific conditions.

Proof Idea

Step 1: Modular Hamiltonian variation

$${\delta }^2{S}_{\text{out}}=\frac{{\delta }^2⟨K_{\chi }⟩}{T}$$

Where $K_{\chi }$ is the modular Hamiltonian.

Step 2: Relationship between $K_{\chi }$ and Hamiltonian

On small causal diamond, $K_{\chi }$ can be expressed as integral of boundary Hamiltonian.

Step 3: Boundary-bulk duality

Using AdS/CFT or holographic principle, boundary modular Hamiltonian corresponds to bulk canonical energy.

Step 4: Identification

$${\mathcal{F}}_Q:=\frac{{\delta }^2⟨K_{\chi }⟩}{T}={\mathcal{E}}_{\text{can}}[h,h]$$

Conditions

JLMS equivalence requires:

1. Code subspace: Perturbations satisfy $\delta M=\delta J=\delta P=0$ (conserved charges unchanged)
2. Boundary conditions: Dirichlet-type boundary conditions, fixed induced metric
3. No outward flux: Symplectic flux has no leakage ${\int }_{\partial \Sigma }{\iota }_n\omega =0$
4. Gauge fixing: Use Killing or covariant harmonic gauge

Under these conditions:

$${\delta }^2{S}_{\text{rel}}={\mathcal{F}}_Q={\mathcal{E}}_{\text{can}}[h,h]\ge 0$$

graph TB
    S["Relative Entropy Second Variation<br/>δ²S_rel"] --> M["Modular Hamiltonian Variation<br/>δ²⟨K_χ⟩"]

    M --> F["Quantum Fisher Information<br/>𝓕_Q"]

    F --> J["JLMS Identification<br/>𝓕_Q = 𝓔_can"]

    J --> E["Canonical Energy<br/>𝓔_can[h,h]"]

    E --> N["Non-Negativity<br/>𝓔_can ≥ 0"]

    style S fill:#e1f5ff
    style J fill:#fff4e1,stroke:#ff6b6b,stroke-width:3px
    style N fill:#e1ffe1,stroke:#ff6b6b,stroke-width:2px

🌊 QNEC: Alternative Criterion

Quantum Null Energy Condition

If JLMS equivalence is not used (e.g., when its conditions are not satisfied), Quantum Null Energy Condition (QNEC) can be used as an alternative.

QNEC (Bousso et al., 2016):

$$⟨T_{kk}(x){⟩}_{\psi }\ge \frac{\hslash }{2\pi }\frac{d^2{S}_{\text{out}}}{d{\lambda }^2}(x)$$

Where:

• $T_{kk}:=T_{ab}{k}^a{k}^b$: stress tensor along null direction
• $\lambda$: affine parameter of null geodesic
• $S_{\text{out}}$: entanglement entropy outside boundary

Second-Order Shape Derivative

Second-order shape derivative of QNEC gives non-negative quadratic form:

$$\boxed{{\mathcal{Q}}_{\text{QNEC}}[h,h]:={\int }_{\mathcal{H}}\frac{\hslash }{2\pi }\frac{{\partial }^2}{\partial {\lambda }^2}\left(\frac{{\delta }^2{S}_{\text{out}}}{A_{\perp }}\right)dA\ge 0}$$

Advantages:

• Does not depend on JLMS identification
• Holds in broader situations (strictly proven in CFT)
• Compatible with first-order chain

Disadvantages:

• High technical requirements for shape derivative calculation
• Consistency with ${\mathcal{E}}_{\text{can}}$ needs additional verification

📊 Two Independent Chains

Logical Structure

IGVP derivation is divided into two logically independent chains:

Chain A (Thermodynamics-Geometric Optics):

Generalized Entropy Variation δS_gen = 0
    ↓
Family Constraint ∫ λ(R_kk - 8πGT_kk) = 0
    ↓
Radon Closure
    ↓
Null-Direction Equation R_kk = 8πGT_kk
    ↓
Tensorization (Null Cone Characterization + Bianchi)
    ↓
Einstein Equations G_ab + Λg_ab = 8πGT_ab

Chain B (Entanglement-Relative Entropy):

Relative Entropy Non-Negative δ²S_rel ≥ 0
    ↓
JLMS Identification (or QNEC)
    ↓
Canonical Energy Non-Negative 𝓔_can ≥ 0
    ↓
Linear Stability

Key:

• Chain A gives field equations (first order)
• Chain B gives stability (second order)
• The two are logically independent but physically unified

graph TB
    subgraph "Chain A: Field Equations"
        A1["δS_gen = 0"] --> A2["Family Constraint"]
        A2 --> A3["R_kk = 8πGT_kk"]
        A3 --> A4["G_ab + Λg_ab = 8πGT_ab"]
    end

    subgraph "Chain B: Stability"
        B1["δ²S_rel ≥ 0"] --> B2["JLMS / QNEC"]
        B2 --> B3["𝓔_can ≥ 0"]
        B3 --> B4["Linear Stability"]
    end

    A4 --> U["Complete IGVP"]
    B4 --> U

    style A1 fill:#e1f5ff
    style B1 fill:#ffe1e1
    style A4 fill:#fff4e1
    style B4 fill:#e1ffe1
    style U fill:#e1ffe1,stroke:#ff6b6b,stroke-width:3px

💡 Physical Meaning

Extremum ≠ Physics

First-order extremum $\delta S_{\text{gen}}=0$ is only a necessary condition.

Physically realizable solutions are generally considered to be stable extrema:

$$\delta S_{\text{gen}}=0\text{and}{\delta }^2{S}_{\text{rel}}\ge 0$$

Analogy:

• Thermodynamics: Equilibrium is maximum of entropy ($\delta S=0$, ${\delta }^{2}S<0$)
• Mechanics: Stable equilibrium is minimum of potential ($\delta V=0$, ${\delta }^{2}V>0$)
• IGVP: Stable gravity is minimum of generalized entropy ($\delta S_{\text{gen}}=0$, ${\delta }^2{S}_{\text{rel}}\ge 0$)

Gravitational Waves Carry Positive Energy

${\mathcal{E}}_{\text{can}}[h,h]\ge 0$ implies:

Gravitational perturbations (gravitational waves) typically carry non-negative energy

This is a fundamental requirement for physical consistency.

Quantum Energy Condition

QNEC gives quantum-corrected energy condition:

$$T_{kk}\ge \frac{\hslash }{2\pi }\frac{d^2{S}_{\text{out}}}{d{\lambda }^2}$$

Meaning:

• Classical energy condition ($T_{kk}\ge 0$) can be violated by quantum effects
• But violation is bounded, bound given by change in entanglement entropy

📝 Key Theorems Summary

Theorem 5.1 (Conditional Version)

Assumptions:

1. Linearized Einstein equations hold (from Chain A)
2. Code subspace: $\delta M=\delta J=\delta P=0$
3. Dirichlet boundary conditions + no outward flux
4. Gauge fixing

Then (under JLMS identification):

$${\delta }^2{S}_{\text{rel}}={\mathcal{E}}_{\text{can}}[h,h]\ge 0$$

Conclusion: Solutions of Einstein’s equations are linearly stable.

Theorem 5.2 (No-Duality Version)

Assumptions:

1. Linearized Einstein equations hold
2. No outward flux at boundary

Then (using QNEC):

$${\mathcal{Q}}_{\text{QNEC}}[h,h]={\int }_{\mathcal{H}}\frac{\hslash }{2\pi }{\partial }_{\lambda }^2\left(\frac{{\delta }^2{S}_{\text{out}}}{A_{\perp }}\right)dA\ge 0$$

Conclusion: Provides universal stability criterion compatible with first-order chain.

🎓 Further Reading

• Hollands-Wald original paper: S. Hollands, R.M. Wald, “Stability of black holes and black branes” (CMP 321, 629, 2013)
• JLMS relation: D. Jafferis et al., “Relative entropy equals bulk relative entropy” (JHEP 06, 004, 2016)
• QNEC: R. Bousso et al., “Proof of the QNEC” (PRD 93, 024017, 2016)
• GLS complete derivation: igvp-einstein-complete.md
• Previous: 04-first-order-variation_en.md - First-Order Variation
• Next: 06-igvp-summary_en.md - IGVP Summary

🤔 Exercises

1. Conceptual understanding:

   • Why is first-order extremum insufficient to guarantee physical stability?
   • How to prove non-negativity of relative entropy (Klein inequality)?
   • What is the relationship between Hollands-Wald canonical energy and ADM energy?

2. Calculation exercises:

   • For Schwarzschild black hole, calculate linearized Einstein equations
   • Verify pure gauge modes $h={\mathcal{L}}_{\xi }g$ satisfy ${\mathcal{E}}_{\text{can}}[h,h]=0$
   • In flat spacetime, calculate right-hand side of QNEC

3. Physical applications:

   • How is QNEC verified in CFT?
   • How does black hole Hawking radiation satisfy ${\delta }^2{S}_{\text{rel}}\ge 0$?
   • What is the physical meaning of code subspace condition?

4. Advanced thinking:

   • What happens if ${\delta }^2{S}_{\text{rel}}<0$?
   • Under what circumstances might JLMS equivalence fail?
   • Can Einstein equations be derived directly from second-order variation (without first order)?

Next step: We have completed the core derivation of IGVP. Let’s review the complete picture in the summary!