Raychaudhuri Equation: Convergence of Light

“Curvature causes light rays to converge, and area changes accordingly—this is the essence of the Raychaudhuri equation.”

🎯 Core Question

Varying generalized entropy on the small causal diamond:

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

Key question: How is area change $\delta A$ related to spacetime curvature $R_{ab}$?

Answer: The Raychaudhuri equation!

💡 Intuitive Image: Focusing Light Beam

Everyday Analogy

Imagine sunlight passing through a magnifying glass:

   ∥ ∥ ∥ ∥     ← Parallel beams
    ∥∥ ∥∥
     ∥ ∥       ← Convergence
      ∥
     Focus

Questions:

• Why do parallel light rays converge?
• What is the relationship between convergence rate and lens curvature?

Answers:

• Optics: law of refraction
• Gravity: Raychaudhuri equation

Gravitational Lensing

In curved spacetime, a bundle of light rays (null geodesics) will:

1. Expand (expansion) $\theta >0$: beam spreads out
2. Contract (contraction) $\theta <0$: beam converges
3. Remain constant: $\theta =0$

The Raychaudhuri equation describes how $\theta$ evolves with time, especially how curvature causes convergence!

graph TB
    L["Light Beam (Null Geodesic Bundle)"] --> E["Expansion Rate θ"]
    E --> R["Raychaudhuri Equation<br/>θ' = -θ²/(d-2) - σ² - R_kk"]

    R --> C1["Curvature Term -R_kk<br/>Causes Convergence"]
    R --> C2["Expansion Squared -θ²/(d-2)<br/>Self-Convergence"]
    R --> C3["Shear -σ²<br/>Enhances Convergence"]

    C1 --> A["Area Change<br/>dA/dλ = θA"]

    style R fill:#fff4e1,stroke:#ff6b6b,stroke-width:3px
    style A fill:#e1ffe1,stroke:#ff6b6b,stroke-width:2px

📐 Mathematical Framework of Null Geodesics

Null Geodesic Bundle

Consider a null geodesic bundle emitted from waist $S_{\ell }$:

Tangent vector: $k^a$, satisfying:

• $k^a{k}_a=0$ (null vector)
• $k^b{\nabla }_b{k}^a=0$ (geodesic, affinely parameterized)

Transverse space: At each point $\gamma (\lambda )$, define a $(d-2)$-dimensional space orthogonal to $k^a$, called “screen space”.

Choose orthonormal basis $\{e_A^a\}$ ($A=1,\dots ,d-2$), satisfying:

• $k^a{e}_{Aa}=0$ (orthogonal to $k^a$)
• $e_A^a{e}_{Ba}={\delta }_{AB}$ (normalized)

Deformation Tensor of Light Beam

Beam deformation is characterized by projected derivative:

$$B_{AB}:=e_A^a{\nabla }_a{k}_b\cdot e_B^b$$

Decomposition (no torsion case ${\omega }_{AB}=0$):

$$B_{AB}=\frac{1}{d-2}\theta {\delta }_{AB}+{\sigma }_{AB}$$

Where:

• Expansion: $\theta :=B_A{}^A={\nabla }_a{k}^a$ (trace)
• Shear: ${\sigma }_{AB}:=B_{AB}-\frac{1}{d-2}\theta {\delta }_{AB}$ (traceless part)

Physical meaning:

🌊 Raychaudhuri Equation

Equation Form

Along null geodesics, expansion rate evolution satisfies:

$$\boxed{\frac{d\theta }{d\lambda }=-\frac{1}{d-2}{\theta }^2-{\sigma }^2-R_{kk}}$$

Where:

• $\lambda$: affine parameter
• $R_{kk}:=R_{ab}{k}^a{k}^b$: contraction of Ricci curvature along null direction
• $d$: spacetime dimension

Assumption: Torsion ${\omega }_{AB}=0$ (holds when null geodesic bundle hypersurface is orthogonal)

Derivation Sketch

From the definition of Riemann curvature tensor:

$$[{\nabla }_a,{\nabla }_b]k^c=R_{abc}{}^d{k}^d$$

Take divergence of ${\nabla }_a{k}^b$, project onto transverse space, use geodesic condition and Ricci identity, and after a series of calculations obtain the Raychaudhuri equation.

Detailed derivation see standard GR textbooks (e.g., Wald, Carroll).

📊 Physical Meaning of Each Term

1. Curvature Term: $-R_{kk}$

$$R_{kk}=R_{ab}{k}^a{k}^b$$

Positive curvature $R_{kk}>0$:

• Represents gravitational attraction
• Causes ${\theta }^{\prime }<0$ (accelerated convergence)
• Light rays are “pulled together”

Example: Light deflection near Earth

graph LR
    M["Matter<br/>T_ab > 0"] --> |"Einstein Equation"| R["Positive Curvature<br/>R_kk > 0"]
    R --> C["Light Convergence<br/>θ' < 0"]

    style M fill:#e1f5ff
    style R fill:#fff4e1
    style C fill:#ffe1e1

2. Expansion Squared Term: $-\frac{1}{d-2}{\theta }^2$

Self-convergence:

• Even without curvature ($R_{kk}=0$)
• If the beam is already contracting ($\theta <0$)
• Contraction will self-accelerate

Example: Inertial convergence, similar to “snowball effect”

3. Shear Term: $-{\sigma }^2$

$${\sigma }^2={\sigma }_{AB}{\sigma }^{AB}\ge 0$$

Always non-negative:

• Shear always enhances convergence
• Represents “twisting deformation” of beam

Example: Deformation caused by tidal forces

🧮 Area Evolution Formula

From Expansion Rate to Area

Consider a small area element $d{A}_0$ on waist $S_{\ell }$, propagating along null geodesic to parameter $\lambda$, area becomes $dA(\lambda )$.

Area evolution:

$$\frac{1}{A}\frac{dA}{d\lambda }=\theta$$

Proof:

• Area element $dA$ is proportional to cross product of transverse direction vectors
• $\theta ={\nabla }_a{k}^a$ is exactly the logarithmic derivative of this “transverse volume”

Differential form:

$$\frac{dA}{d\lambda }=\theta A$$

Integral Form

From waist ($\lambda =0$, $\theta (0)=0$) to $\lambda ={\lambda }_{\ast }$:

$$A(\lambda )=A(0)\exp \left({\int }_0^{\lambda }\theta ({\lambda }^{\prime })d{\lambda }^{\prime }\right)$$

In small diamond limit $|\theta |\ll 1$, Taylor expansion:

$$A(\lambda )\approx A(0)\left[1+{\int }_0^{\lambda }\theta ({\lambda }^{\prime })d{\lambda }^{\prime }+O({\theta }^2)\right]$$

Area change:

$$\delta A=A(\lambda )-A(0)=A(0){\int }_0^{\lambda }\theta ({\lambda }^{\prime })d{\lambda }^{\prime }$$

🔍 Connection with Curvature

Integrating Raychaudhuri Equation

From Raychaudhuri equation:

$${\theta }^{\prime }=-\frac{1}{d-2}{\theta }^2-{\sigma }^2-R_{kk}$$

Multiply both sides by $\lambda$ and integrate (key technique):

$${\int }_0^{{\lambda }_{\ast }}\lambda {\theta }^{\prime }d\lambda =-{\int }_0^{{\lambda }_{\ast }}\lambda \left[\frac{1}{d-2}{\theta }^2+{\sigma }^2+R_{kk}\right]d\lambda$$

Integration by parts on left side:

$${\int }_0^{{\lambda }_{\ast }}\lambda {\theta }^{\prime }d\lambda =[\lambda \theta {]}_0^{{\lambda }_{\ast }}-{\int }_0^{{\lambda }_{\ast }}\theta d\lambda$$

In small diamond limit, ${\lambda }_{\ast }\sim \ell \to 0$, $\theta ({\lambda }_{\ast })=O(\epsilon )$, so ${\lambda }_{\ast }\theta ({\lambda }_{\ast })=o(\ell )$.

Also $\theta (0)=0$ (waist is maximum volume cross-section), we get:

$${\int }_0^{{\lambda }_{\ast }}\lambda {\theta }^{\prime }d\lambda =-{\int }_0^{{\lambda }_{\ast }}\theta d\lambda +o(\ell )$$

Area-Curvature Identity

Combining with $\delta A=A(0){\int }_0^{{\lambda }_{\ast }}\theta d\lambda$, we get:

$$\boxed{\delta A+{\int }_0^{{\lambda }_{\ast }}\lambda R_{kk}d\lambda \cdot A(0)=-{\int }_0^{{\lambda }_{\ast }}\lambda \left[\frac{1}{d-2}{\theta }^2+{\sigma }^2\right]d\lambda \cdot A(0)}$$

Right side is higher-order small quantity ($O({\epsilon }^2)$ or higher), ignored in first-order variation!

Key result:

$$\frac{\delta A}{4G\hslash }\approx -\frac{1}{4G\hslash }{\int }_0^{{\lambda }_{\ast }}\lambda R_{kk}d\lambda dA$$

This establishes the precise connection between area change and curvature.

graph TB
    R["Raychaudhuri Equation<br/>θ' = -θ²/(d-2) - σ² - R_kk"] --> I["Integrate (multiply by λ)"]
    I --> P["Integration by Parts"]
    P --> F["Area-Curvature Identity<br/>δA ≈ -∫ λ R_kk dλ dA"]

    T["Initial: θ(0)=0<br/>Waist is Maximum Cross-Section"] -.-> P

    F --> IG["Used in IGVP<br/>δS_gen = 0"]

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

📈 Precise Control of Small Diamond Limit

Error Estimation

In IGVP derivation, precise control of all error terms is needed:

Geometric constants:

• $C_R:={\sup }_{{\mathcal{D}}_{\ell }}|R_{kk}|$ (curvature upper bound)
• $C_{\nabla R}:={\sup }_{{\mathcal{D}}_{\ell }}|{\nabla }_k{R}_{kk}|$ (curvature gradient upper bound)
• $C_{\mathcal{C}}:={\sup }_{{\mathcal{D}}_{\ell }}|{\mathcal{C}}_{AB}|$ (Weyl curvature projection)
• $C_{\sigma ,0}:={\sup }_{S_{\ell }}|\sigma (0)|$ (initial shear)

Shear control (variable coefficient Grönwall):

$$|\sigma (\lambda )|\le (C_{\sigma ,0}+C_{\mathcal{C}}|\lambda |)\exp \left(\frac{2}{d-2}{\int }_0^{|\lambda |}|\theta |ds\right)$$

In small limit $|\theta |{\lambda }_{\ast }\ll 1$:

$$\underset{\lambda \in [0,{\lambda }_{\ast }]}{\sup }|\sigma (\lambda )|\le C_{\sigma }:=C_{\sigma ,0}+C_{\mathcal{C}}{\lambda }_{\ast }$$

Expansion control:

From Raychaudhuri equation:

$$|\theta (\lambda )|\le C_R|\lambda |+\frac{1}{d-2}{\int }_0^{|\lambda |}{\theta }^{2}ds+{\int }_0^{|\lambda |}{\sigma }^{2}ds$$

Using Grönwall inequality again, we get:

$$|\theta (\lambda )|=O(\epsilon \ell )$$

Where $\epsilon =\ell /L_{\text{curv}}\ll 1$.

Dominating Function

Define dominating function:

$${\tilde{M}}_{\text{dom}}(\lambda ):=\frac{1}{2}{C}_{\nabla R}{\lambda }^2+(C_{\sigma ,0}+C_{\mathcal{C}}{\lambda }_0{)}^2|\lambda |+\frac{4}{3(d-2)}{C}_R^2{\lambda }_0^3$$

Satisfying:

$$\left|\theta (\lambda )+\lambda R_{kk}\right|\le {\tilde{M}}_{\text{dom}}(\lambda )$$

And: ${\tilde{M}}_{\text{dom}}\in L^1([0,{\lambda }_0])$ (integrable!)

Importance: This guarantees that dominated convergence theorem applies, allowing exchange of limit and integral order!

🎨 Example in Minkowski Spacetime

Flat Spacetime

In flat spacetime $R_{ab}=0$, Raychaudhuri equation simplifies to:

$${\theta }^{\prime }=-\frac{1}{d-2}{\theta }^2-{\sigma }^2$$

Initial values $\theta (0)=0$, $\sigma (0)=0$ (symmetric configuration):

Solution: $\theta (\lambda )\equiv 0$, $\sigma (\lambda )\equiv 0$

Area: $A(\lambda )=A(0)=\text{constant}$

This is as expected: in flat spacetime, parallel beams remain parallel!

Adding Perturbation

If $\sigma (0)\ne 0$ (initial shear):

$${\theta }^{\prime }=-{\sigma }^2<0$$

Even if $\theta (0)=0$, $\theta$ becomes negative (convergence), area decreases!

Physical meaning: Initial “twist” leads to subsequent convergence.

🌌 Example in Schwarzschild Spacetime

Radial Null Geodesics

In Schwarzschild metric:

$$d{s}^2=-\left(1-\frac{2M}{r}\right)d{t}^2+{\left(1-\frac{2M}{r}\right)}^{-1}d{r}^2+r^{2}d{\Omega }^2$$

Radial outward null geodesic ($\theta =0,\varphi =\text{const}$):

$$k^a{\partial }_a={\partial }_t+\left(1-\frac{2M}{r}\right){\partial }_r$$

Curvature:

$$R_{kk}=\frac{4M}{r^3}(1-\frac{2M}{r})>0$$

Raychaudhuri:

$${\theta }^{\prime }=-{\sigma }^2-\frac{4M}{r^3}(1-\frac{2M}{r})$$

At $r\gg 2M$ (far from horizon):

$${\theta }^{\prime }\approx -\frac{4M}{r^3}<0$$

Conclusion: Even when light propagates outward, positive curvature causes expansion rate to decrease (convergence tendency)!

📝 Key Formulas Summary

🎓 Further Reading

• Original paper: A.K. Raychaudhuri, “Relativistic cosmology” (Phys. Rev. 98, 1123, 1955)
• Modern treatment: R.M. Wald, General Relativity (University of Chicago Press, 1984), §9.2
• Singularity theorems: S.W. Hawking, R. Penrose, “The singularities of gravitational collapse” (Proc. Roy. Soc. A 314, 529, 1970)
• GLS application: igvp-einstein-complete.md
• Previous: 02-causal-diamond_en.md - Small Causal Diamond
• Next: 04-first-order-variation_en.md - First-Order Variation and Einstein’s Equations

🤔 Exercises

1. Conceptual understanding:

   • Why is expansion rate $\theta$ defined as ${\nabla }_a{k}^a$?
   • Why are all terms in Raychaudhuri equation negative (except ${\omega }^2$)?
   • What is a “focal point”? What is its relationship with $\theta \to -\infty$?

2. Calculation exercises:

   • In two-dimensional $(t,x)$ spacetime, what is the form of Raychaudhuri equation? (Hint: $d=2$)
   • Verify that $\theta \equiv 0$ is a solution of Raychaudhuri equation in flat spacetime
   • In FRW universe, calculate $R_{kk}$ for radial null geodesics

3. Physical applications:

   • How to explain gravitational lensing using Raychaudhuri equation?
   • Why do we say “matter always causes convergence”? (Hint: energy conditions)
   • How does expansion rate $\theta$ of photons evolve in cosmic expansion?

4. Advanced thinking:

   • How does Raychaudhuri equation lead to Penrose-Hawking singularity theorems?
   • If ${\omega }_{AB}\ne 0$ (with torsion), how does the equation change?
   • Why multiply by weight $\lambda$ in area-curvature identity? (Hint: integration by parts)

Next step: After mastering Raychaudhuri equation, we will see how to derive Einstein’s equations from $\delta S_{\text{gen}}=0$!