First-Order Variation: From Entropy to Einstein’s Equations

“When entropy takes an extremum, Einstein’s equations naturally emerge.”

🎯 Goal

We now have all the tools:

• Generalized entropy: $S_{\text{gen}}=A/(4G\hslash )+S_{\text{out}}$
• Small causal diamond: Stage for variation ${\mathcal{D}}_{\ell }(p)$
• Raychaudhuri equation: $\delta A\approx -\int \lambda R_{kk}d\lambda dA$

Now, let’s complete the core derivation of IGVP:

$$\boxed{\delta S_{\text{gen}}=0\Rightarrow G_{ab}+\Lambda g_{ab}=8\pi G{T}_{ab}}$$

📐 Variation Setup

Basic Variation

On the small causal diamond ${\mathcal{D}}_{\ell }(p)$, generalized entropy is:

$$S_{\text{gen}}=\frac{A(S_{\ell })}{4G\hslash }+S_{\text{out}}^{\text{ren}}-\frac{\Lambda }{8\pi G}\frac{V(B_{\ell })}{T}$$

Constraints:

1. Fixed endpoints: $p^{-}$ and $p^{+}$ unchanged
2. Fixed volume: $\delta V(B_{\ell })=0$
3. Fixed temperature: $\delta T=0$ (at first-order level)

Variation functional:

$$\delta S_{\text{gen}}=\frac{\delta A}{4G\hslash }+\delta S_{\text{out}}^{\text{ren}}$$

IGVP principle:

$$\boxed{\delta S_{\text{gen}}=0}$$

graph TB
    S["Generalized Entropy<br/>S_gen = A/4Gℏ + S_out"] --> V["Variation (Fixed Volume)<br/>δS_gen"]

    V --> G["Geometric Contribution<br/>δA/4Gℏ"]
    V --> Q["Quantum Contribution<br/>δS_out"]

    G --> R["Raychaudhuri Equation<br/>δA ~ -∫ λR_kk dλ dA"]
    Q --> M["Modular Theory<br/>δS_out ~ ∫ λT_kk dλ dA"]

    R --> E["Variation Zero<br/>δS_gen = 0"]
    M --> E

    E --> F["Family Constraint<br/>∫ λ(R_kk - 8πGT_kk) = o(ℓ²)"]

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

🔧 Calculation of Area Variation

From Raychaudhuri to Area

Recall the integral form of Raychaudhuri equation (derived in previous article):

$$\frac{\delta A}{4G\hslash }=-\frac{1}{4G\hslash }{\int }_{\mathcal{H}}{\int }_0^{{\lambda }_{\ast }}\lambda R_{kk}(\lambda )d\lambda dA+O({\epsilon }^2)$$

Where:

• $\mathcal{H}$: waist $S_{\ell }$
• $\lambda$: affine parameter along null geodesics
• ${\lambda }_{\ast }\sim \ell$: upper limit
• $dA$: area element of waist
• $O({\epsilon }^2)$: higher-order contributions from shear and expansion

Key insight: Area change is directly related to the integral of curvature along null direction!

Error Control

In the small diamond limit $\ell \to 0$, error term is:

$$\left|\delta A+{\int }_{\mathcal{H}}{\int }_0^{{\lambda }_{\ast }}\lambda R_{kk}d\lambda dA\right|\le C_d{\epsilon }^3{\ell }^{d-2}$$

Where $C_d=C_d(C_R,C_{\nabla R},C_{\mathcal{C}};d)$ is a geometric constant.

Key points:

• Main term proportional to ${\ell }^{d-2}$ (area scale)
• Error is $O({\epsilon }^3)$ (third-order small quantity)
• $\epsilon =\ell /L_{\text{curv}}\ll 1$

⚛️ Quantum Field Entropy Variation

Result from Modular Theory

Under Hadamard state and approximate KMS conditions, quantum field entropy variation satisfies the first law:

$$\delta S_{\text{out}}^{\text{ren}}=\frac{\delta ⟨K_{\chi }⟩}{T}+O({\epsilon }^2)$$

Where:

• $K_{\chi }$: modular Hamiltonian
• $T=\hslash |{\kappa }_{\chi }|/(2\pi )$: Unruh temperature
• ${\kappa }_{\chi }$: surface gravity of approximate Killing field ${\chi }^a$

Localization of Modular Hamiltonian

On the small causal diamond, modular Hamiltonian can be localized as:

$$K_{\chi }={\int }_{\mathcal{H}}{\int }_0^{{\lambda }_{\ast }}2\pi \lambda T_{kk}(\lambda )d\lambda dA+O({\epsilon }^2)$$

Where:

• $T_{kk}:=T_{ab}{k}^a{k}^b$: component of stress tensor along null direction
• Weight $2\pi \lambda$: from Rindler geometry

Physical meaning: Modular Hamiltonian is a weighted integral of stress tensor near causal horizon!

graph LR
    K["Modular Hamiltonian<br/>K_χ"] --> T["Stress Tensor Integral<br/>∫ λT_kk dλ dA"]
    T --> S["Entropy Variation<br/>δS_out = δK_χ/T"]

    S --> F["First Law<br/>δS_out ~ ∫ λT_kk dλ dA"]

    style K fill:#e1f5ff
    style F fill:#ffe1e1,stroke:#ff6b6b,stroke-width:2px

Variation Formula

Therefore:

$$\delta S_{\text{out}}^{\text{ren}}=\frac{2\pi }{\hslash }{\int }_{\mathcal{H}}{\int }_0^{{\lambda }_{\ast }}\lambda T_{kk}(\lambda )d\lambda dA+O({\epsilon }^2)$$

Or simply:

$$\delta S_{\text{out}}=\frac{\delta Q}{T}$$

Where $\delta Q=2\pi \int \lambda T_{kk}d\lambda dA$ is the “heat” change.

⚖️ Combining Variations: Family Constraint

First-Order Extremum Condition

Combining the two terms:

$$\delta S_{\text{gen}}=\frac{\delta A}{4G\hslash }+\delta S_{\text{out}}^{\text{ren}}=0$$

Substituting explicit expressions:

$$-\frac{1}{4G\hslash }{\int }_{\mathcal{H}}{\int }_0^{{\lambda }_{\ast }}\lambda R_{kk}d\lambda dA+\frac{2\pi }{\hslash }{\int }_{\mathcal{H}}{\int }_0^{{\lambda }_{\ast }}\lambda T_{kk}d\lambda dA=O({\epsilon }^2)$$

Simplification (units $\hslash =1$, $4G\cdot 2\pi =8\pi G$):

$$\boxed{{\int }_{\mathcal{H}}{\int }_0^{{\lambda }_{\ast }}\lambda (R_{kk}-8\pi G{T}_{kk})d\lambda dA=o({\ell }^{d-2})}$$

This is the family constraint!

Meaning of Family Constraint

For all small causal diamonds ${\mathcal{D}}_{\ell }(p)$ (when $\ell$ is sufficiently small), the above integral is $o({\ell }^{d-2})$.

Question: How to derive the pointwise equation $R_{kk}=8\pi G{T}_{kk}$ holding at each point from this integral condition (holding for a family of diamonds)?

Answer: Radon-type closure!

graph TB
    F["Family Constraint<br/>∀ Small Diamonds: ∫ λ(R_kk - 8πGT_kk) = o(ℓ²)"] --> L["Localization<br/>Test Function φ ∈ C_c^∞(S_ℓ)"]

    L --> R["Weighted Ray Transform<br/>ℒ_λ[f](p,k̂) = ∫ λf dλ"]

    R --> I["Local Invertibility<br/>ℒ_λ[f] = 0 ⇒ f(p) = 0"]

    I --> P["Pointwise Equation<br/>R_kk = 8πGT_kk"]

    style F fill:#e1f5ff
    style R fill:#fff4e1,stroke:#ff6b6b,stroke-width:2px
    style P fill:#e1ffe1,stroke:#ff6b6b,stroke-width:3px

🔬 Radon-Type Closure: From Family to Point

Weighted Ray Transform

For function $f:{\mathcal{D}}_{\ell }\to \mathbb{R}$, define weighted ray transform:

$${\mathcal{L}}_{\lambda }[f](p,\hat{k}):={\int }_0^{{\lambda }_{\ast }}\lambda f({\gamma }_{p,\hat{k}}(\lambda ))d\lambda$$

Where ${\gamma }_{p,\hat{k}}$ is the null geodesic from $p$ along direction $\hat{k}$.

Physical meaning: Weighted average along light ray, weight $\lambda$ (dual to Rindler temperature).

Small Domain Expansion

In small diamond, Taylor expansion:

$${\mathcal{L}}_{\lambda }[f](p,\hat{k})=\frac{1}{2}{\lambda }_{\ast }^{2}f(p)+O({\lambda }_{\ast }^3|\nabla f{|}_{\infty })$$

Key: Leading term is proportional to $f(p)$!

Inverse problem: If ${\mathcal{L}}_{\lambda }[f]=o({\ell }^2)$ holds for all directions $\hat{k}$, can we derive $f(p)=0$?

Local Invertibility Theorem

Theorem (Local invertibility of null geodesic first moment):

In the normal neighborhood of $p$, if:

1. No conjugate points
2. Null geodesic bundle transverse space is smooth

Then weighted ray transform ${\mathcal{L}}_{\lambda }$ is locally invertible at point $p$:

$${\mathcal{L}}_{\lambda }[f](p,\hat{k})=o({\ell }^2)\forall \hat{k}\Rightarrow f(p)=0$$

Proof idea:

• Use Fubini theorem to separate space and “time” directions
• Approximate Dirac $\delta$ function with mollifier
• At small scales, ray transform is similar to first moment of Radon transform
• First moment data is sufficient to reconstruct value of $f$ at point $p$

Application to Family Constraint

Take $f=R_{kk}-8\pi G{T}_{kk}$, family constraint tells us:

$${\int }_{S_{\ell }}\phi (x){\int }_0^{{\lambda }_{\ast }}\lambda f({\gamma }_{p,\hat{k}}(\lambda ))d\lambda dA=o({\ell }^2)$$

Holds for all test functions $\phi \in C_c^{\infty }(S_{\ell })$.

Localization lemma guarantees this is equivalent to:

$${\mathcal{L}}_{\lambda }[f](p,\hat{k})=o({\ell }^2)\forall \hat{k}$$

By local invertibility:

$$f(p)=R_{kk}(p)-8\pi G{T}_{kk}(p)=0$$

Conclusion:

$$\boxed{R_{kk}=8\pi G{T}_{kk}\text{for all null directions }{k}^a}$$

This is the null-direction Einstein equation!

🎯 Tensorization: From Null Direction to Complete Equation

Null Cone Characterization Lemma

Lemma (requires $d\ge 3$):

Let $X_{ab}$ be a smooth symmetric tensor. If for all null vectors $k^a$:

$$X_{ab}{k}^a{k}^b=0$$

Then necessarily:

$$X_{ab}=\Phi g_{ab}$$

Where $\Phi$ is some scalar function.

Proof idea:

• In $d\ge 3$ dimensions, null cone spans entire tangent space
• Any symmetric tensor can be decomposed into trace and traceless parts
• Null cone constraint completely determines traceless part to be zero

Note: When $d=2$, this lemma does not hold, Einstein equations degenerate!

Applying Bianchi Identity

Define:

$$X_{ab}:=R_{ab}-8\pi G{T}_{ab}$$

We have proven $X_{ab}{k}^a{k}^b=0$ for all $k$, so:

$$X_{ab}=\Phi g_{ab}$$

Using Bianchi identity:

$${\nabla }^a{R}_{ab}=\frac{1}{2}{\nabla }_{b}R$$

Using energy-momentum conservation:

$${\nabla }^a{T}_{ab}=0$$

Therefore:

$${\nabla }^a{X}_{ab}={\nabla }^a{R}_{ab}-8\pi G{\nabla }^a{T}_{ab}=\frac{1}{2}{\nabla }_{b}R$$

But from $X_{ab}=\Phi g_{ab}$:

$${\nabla }^a{X}_{ab}={\nabla }^a(\Phi g_{ab})={\nabla }_b\Phi$$

Comparing the two:

$${\nabla }_b\Phi =\frac{1}{2}{\nabla }_{b}R$$

That is:

$${\nabla }_b\left(\Phi -\frac{1}{2}R\right)=0$$

Therefore $\Phi -\frac{1}{2}R$ is constant, denoted $-\Lambda$:

$$\Phi =\frac{1}{2}R-\Lambda$$

Einstein Equations

Substituting back $X_{ab}=\Phi g_{ab}$:

$$R_{ab}-8\pi G{T}_{ab}=\left(\frac{1}{2}R-\Lambda \right)g_{ab}$$

Rearranging:

$$R_{ab}-\frac{1}{2}R{g}_{ab}+\Lambda g_{ab}=8\pi G{T}_{ab}$$

That is:

$$\boxed{G_{ab}+\Lambda g_{ab}=8\pi G{T}_{ab}}$$

This is Einstein’s field equations with cosmological constant!

graph TB
    Z["Null-Direction Equation<br/>R_kk = 8πGT_kk ∀k"] --> L["Null Cone Characterization Lemma<br/>X_ab = Φg_ab"]

    L --> B["Bianchi Identity<br/>∇^a R_ab = ½∇_b R"]
    L --> C["Energy Conservation<br/>∇^a T_ab = 0"]

    B --> N["Derivative Constraint<br/>∇_b(Φ - ½R) = 0"]
    C --> N

    N --> LA["Cosmological Constant<br/>Λ := ½R - Φ"]

    LA --> E["Einstein Equations<br/>G_ab + Λg_ab = 8πGT_ab"]

    style Z fill:#e1f5ff
    style L fill:#fff4e1
    style E fill:#e1ffe1,stroke:#ff6b6b,stroke-width:3px

🌟 Complete Derivation Chain Summary

Let’s review the entire derivation process:

Step 1: Define Generalized Entropy

$$S_{\text{gen}}=\frac{A}{4G\hslash }+S_{\text{out}}$$

Step 2: Variation Setup

On small causal diamond, fix volume, set $\delta S_{\text{gen}}=0$.

Step 3: Calculate Area Variation

Using Raychaudhuri equation:

$$\frac{\delta A}{4G\hslash }=-\frac{1}{4G\hslash }\int \lambda R_{kk}d\lambda dA$$

Step 4: Calculate Field Entropy Variation

Using modular theory:

$$\delta S_{\text{out}}=\frac{2\pi }{\hslash }\int \lambda T_{kk}d\lambda dA$$

Step 5: Family Constraint

Combining:

$$\int \lambda (R_{kk}-8\pi G{T}_{kk})d\lambda dA=o({\ell }^{d-2})$$

Step 6: Radon-Type Closure

Using local invertibility of weighted ray transform:

$$R_{kk}=8\pi G{T}_{kk}\forall k$$

Step 7: Tensorization

Using null cone characterization lemma + Bianchi identity:

$$G_{ab}+\Lambda g_{ab}=8\pi G{T}_{ab}$$

Complete!

graph TB
    S1["Step 1: Generalized Entropy<br/>S_gen = A/4Gℏ + S_out"] --> S2["Step 2: Variation Principle<br/>δS_gen = 0 (Fixed V)"]

    S2 --> S3["Step 3: Raychaudhuri<br/>δA ~ -∫ λR_kk dλ dA"]
    S2 --> S4["Step 4: Modular Theory<br/>δS_out ~ ∫ λT_kk dλ dA"]

    S3 --> S5["Step 5: Family Constraint<br/>∫ λ(R_kk - 8πGT_kk) = o(ℓ²)"]
    S4 --> S5

    S5 --> S6["Step 6: Radon Closure<br/>R_kk = 8πGT_kk ∀k"]

    S6 --> S7["Step 7: Null Cone Characterization<br/>R_ab - 8πGT_ab = Φg_ab"]

    S7 --> S8["Step 8: Bianchi<br/>∇_b(Φ - ½R) = 0"]

    S8 --> S9["Step 9: Einstein Equations<br/>G_ab + Λg_ab = 8πGT_ab"]

    style S1 fill:#e1f5ff
    style S5 fill:#fff4e1,stroke:#ff6b6b,stroke-width:2px
    style S6 fill:#ffe1e1
    style S9 fill:#e1ffe1,stroke:#ff6b6b,stroke-width:3px

💡 Physical Insights

Entropy is Fundamental

Traditional perspective: Einstein equations are fundamental axioms → black hole entropy is derived result

IGVP perspective: Generalized entropy is fundamental functional → Einstein equations are variation result

$$\text{Entropy Extremum}\Rightarrow \text{Gravitational Field Equations}$$

Gravity as a Thermodynamic Phenomenon

Einstein’s equations in this framework can be understood as:

$$\underset{\text{Geometric Entropy Change}}{\underbrace{\frac{\delta A}{4G\hslash }}}+\underset{\text{First Law of Thermodynamics}}{\underbrace{\frac{\delta Q}{T}}}=0$$

This is similar to a thermal equilibrium condition.

Manifestation of Locality

Deriving pointwise field equations from local entropy extremum (small causal diamond) highlights the local nature of gravity.

Emergence of Cosmological Constant

$\Lambda$ is not assumed beforehand in this derivation, but naturally emerges as an integration constant from the variation process.

Its physical meaning can be interpreted as the dual variable of the volume constraint.

📝 Key Formulas Quick Reference

🎓 Further Reading

• Jacobson’s original derivation: T. Jacobson, “Thermodynamics of spacetime” (1995)
• Radon transform: S. Helgason, The Radon Transform (Birkhäuser, 1999)
• GLS complete proof: igvp-einstein-complete.md
• Previous: 03-raychaudhuri-equation_en.md - Raychaudhuri Equation
• Next: 05-second-order-variation_en.md - Second-Order Variation and Stability

🤔 Exercises

1. Conceptual understanding:

   • Why is the weight in family constraint $\lambda$ rather than constant?
   • What does “local invertibility” of Radon-type closure mean?
   • Why does null cone characterization lemma require $d\ge 3$?

2. Derivation exercises:

   • Verify that $X_{ab}=\Phi g_{ab}$ implies ${\nabla }^a{X}_{ab}={\nabla }_b\Phi$
   • Derive ${\nabla }^a{G}_{ab}=0$ from Bianchi identity
   • Check units of cosmological constant when $d=4$

3. Physical applications:

   • If $T_{ab}=0$ (vacuum), what do Einstein equations become?
   • How does Schwarzschild solution satisfy $\delta S_{\text{gen}}=0$?
   • Why do we say gravity is a “thermodynamic phenomenon”?

4. Advanced thinking:

   • If volume is not fixed, what equations would variation yield?
   • How to modify IGVP derivation for higher-order gravity theories (e.g., $f(R)$)?
   • Can IGVP derive Lovelock equations? (Hint: use Wald entropy)

Next step: First-order variation gives field equations, but how is stability guaranteed? Let’s enter the world of second-order variation!