Brown-York Quasi-Local Energy: Generator of Boundary Time

“In curved spacetime, energy is considered not to be at points, but on boundaries.”

🎯 Core Problems

Problem 1: How to define “energy” in curved spacetime?

Traditional Difficulties:

• No global time translation symmetry (Killing vector)
• Energy density $T_{00}$ coordinate-dependent
• Cannot integrate to get “total energy”

Brown-York Solution: Define quasi-local energy on boundary!

Problem 2: What is this “quasi-local energy” related to?

Answer: It is proposed to be the generator of boundary time evolution.

💡 Intuitive Image: “Weight” of a Region

Analogy: Weighing a Room

Traditional Method (Fails):

• Place a scale at each point inside room
• But scale readings depend on “how to place”
• Cannot simply add

Brown-York Method (Succeeds):

• Only weigh the walls!
• “Tension” of walls tells you total energy of room
• This is natural, well-defined

graph LR
    ROOM["Room (Spacetime Region)"] --> TRAD["Traditional Method"]
    ROOM --> BY["Brown-York Method"]

    TRAD --> FAIL["✗ Internal Energy Density<br/>Coordinate-Dependent"]
    BY --> SUCCESS["✓ Boundary Tension<br/>Well-Defined"]

    style FAIL fill:#ffe1e1
    style SUCCESS fill:#e1ffe1

Key Insight:

• Energy is viewed as not “something in volume”
• But “property of boundary”
• Boundary tells you how much energy is inside

📜 From GHY to Brown-York

Review of GHY Boundary Term

From previous article:

$$S_{\mathrm{total}}=S_{\mathrm{EH}}+S_{\mathrm{GHY}}$$

$$S_{\mathrm{GHY}}=\frac{\epsilon }{8\pi G}{\int }_{\partial \mathcal{M}}\sqrt{|h|}K{\mathrm{d}}^{3}x$$

Variation gives:

$$\delta S_{\mathrm{total}}=\frac{1}{16\pi G}{\int }_{\mathcal{M}}\sqrt{-g}{G}_{\mu \nu }\delta g^{\mu \nu }+{\int }_{\partial \mathcal{M}}\sqrt{|h|}{\Pi }^{ab}\delta h_{ab}$$

where:

$$\boxed{{\Pi }^{ab}=\frac{1}{8\pi G}(K^{ab}-K{h}^{ab})}$$

This is the canonical momentum!

Hamiltonian Form

In $(3+1)$ decomposition, for spacelike hypersurface $\Sigma$ with induced metric $h_{ij}$, its conjugate momentum is exactly:

$${\pi }^{ij}=\frac{\partial \mathcal{L}}{\partial {\dot{h}}_{ij}}=\frac{\sqrt{|h|}}{16\pi G}(K^{ij}-K{h}^{ij})$$

Canonical Pair: $(h_{ij},{\pi }^{ij})$

Hamiltonian:

$$H[\xi ]={\int }_{\Sigma }\left[N\mathcal{H}+N^i{\mathcal{H}}_i\right]{\mathrm{d}}^{3}x+{\oint }_{\partial \Sigma }\left(\text{boundary term}\right){\mathrm{d}}^{2}x$$

where $\mathcal{H},{\mathcal{H}}_i$ are constraints (zero on-shell).

Boundary term is exactly the source of Brown-York energy!

⭐ Brown-York Surface Stress Tensor

Definition

Brown-York Surface Stress Tensor:

$$\boxed{T_{\mathrm{BY}}^{ab}=\frac{2}{\sqrt{|h|}}\frac{\delta S_{\mathrm{GHY}}}{\delta h_{ab}}=\frac{1}{8\pi G}(K^{ab}-K{h}^{ab})}$$

Physical Meaning:

• $T_{\mathrm{BY}}^{ab}$ is “stress” on boundary
• Symmetric tensor: $T_{\mathrm{BY}}^{ab}=T_{\mathrm{BY}}^{ba}$
• Depends on extrinsic curvature $K^{ab}$

graph TB
    GHY["GHY Boundary Term<br/>S_GHY = ∫√|h| K"] --> VAR["Variation<br/>δS_GHY/δh_ab"]
    VAR --> TBY["Brown-York Stress<br/>T^ab_BY"]

    TBY --> SYMM["Symmetry<br/>T^ab = T^ba"]
    TBY --> CURV["Depends on K^ab<br/>Extrinsic Curvature"]

    style TBY fill:#fff4e1,stroke:#ff6b6b,stroke-width:3px

Component Decomposition

On boundary $\partial \Sigma$, choose:

• Time-like unit vector: $u^a$ (along boundary time direction)
• Space-like normal vector: $n^a$ (perpendicular to $\partial \Sigma$ in $\Sigma$)

2D induced metric: $${\sigma }_{ab}=h_{ab}+u_a{u}_b$$

Energy Density: $$\epsilon :=u_a{u}_b{T}_{\mathrm{BY}}^{ab}$$

Momentum Density: $$j_a:=-{\sigma }_a{}^b{u}_c{T}_{\mathrm{BY}}^{bc}$$

Stress Tensor: $${\tau }_{ab}:={\sigma }_a{}^c{\sigma }_b{}^d{T}_{\mathrm{BY}}^{cd}$$

🌟 Brown-York Quasi-Local Energy

Definition

For 2D cross-section $\mathcal{S}$ of boundary $\partial \Sigma$:

$$\boxed{E_{\mathrm{BY}}={\int }_{\mathcal{S}}\sqrt{\sigma }\epsilon {\mathrm{d}}^{2}x={\int }_{\mathcal{S}}\sqrt{\sigma }{u}_a{u}_b{T}_{\mathrm{BY}}^{ab}{\mathrm{d}}^{2}x}$$

Expanded:

$$E_{\mathrm{BY}}=\frac{1}{8\pi G}{\int }_{\mathcal{S}}\sqrt{\sigma }{u}_a{u}_b\left(K^{ab}-K{h}^{ab}\right){\mathrm{d}}^{2}x$$

Physical Meaning:

• $E_{\mathrm{BY}}$: energy of region as seen by boundary observer
• Depends on choice of boundary (quasi-locality)
• Depends on choice of time direction ($u^a$)

graph TB
    REGION["Spacetime Region<br/>M"] --> BOUND["Boundary<br/>∂Σ"]
    BOUND --> SLICE["2D Cross-Section<br/>S"]

    SLICE --> ENERGY["Quasi-Local Energy<br/>E_BY = ∫√σ ε"]

    ENERGY --> DEP1["Depends on Boundary"]
    ENERGY --> DEP2["Depends on Time Vector u^a"]

    style ENERGY fill:#fff4e1,stroke:#ff6b6b,stroke-width:3px

Reference Subtraction

Problem: Directly calculated $E_{\mathrm{BY}}$ usually diverges (at large $R$)!

Solution: Subtract reference background contribution

$$\boxed{E_{\mathrm{BY},\mathrm{ren}}=E_{\mathrm{BY}}-E_{\mathrm{ref}}}$$

Usually choose:

• Asymptotically Flat: Reference is Minkowski space
• Asymptotically AdS: Reference is pure AdS space

Renormalized Energy:

$$E_{\mathrm{BY},\mathrm{ren}}=\frac{1}{8\pi G}{\int }_{\mathcal{S}}\sqrt{\sigma }{u}_a{u}_b\left[(K^{ab}-K{h}^{ab})-(K_0^{ab}-K_0{h}_{ab})\right]{\mathrm{d}}^{2}x$$

where $K_0^{ab}$ is extrinsic curvature of reference background.

🔢 Example: Schwarzschild Spacetime

Setup

Schwarzschild metric:

$$\mathrm{d}{s}^2=-f(r)\mathrm{d}{t}^2+f(r{)}^{-1}\mathrm{d}{r}^2+r^2\mathrm{d}{\Omega }_2^2$$

where $f(r)=1-2M/r$.

Take boundary as sphere at $r=R$, time vector:

$$u^a=\frac{1}{\sqrt{f(R)}}({\partial }_t{)}^a$$

Extrinsic Curvature

From previous article:

$$K=\frac{2\sqrt{f(R)}}{R}+\frac{f^{\prime }(R)}{2\sqrt{f(R)}}$$

For spherical symmetry, $K^{ab}$ diagonal, key components:

$$K_{tt}=0,K_{\theta \theta }=R^2\sqrt{f(R)},K_{\varphi \varphi }=R^2{\sin }^2\theta \sqrt{f(R)}$$

Brown-York Stress

$$T_{\mathrm{BY}}^{ab}=\frac{1}{8\pi G}\left(K^{ab}-K{h}^{ab}\right)$$

Energy density:

$$\epsilon =u_a{u}_b{T}_{\mathrm{BY}}^{ab}=\frac{1}{8\pi G}f(R{)}^{-1}\left(K_{tt}-K{h}_{tt}\right)$$

$$=\frac{1}{8\pi G}f(R{)}^{-1}\left(0+Kf(R)\right)=\frac{K}{8\pi G}$$

Quasi-Local Energy

$$E_{\mathrm{BY}}={\int }_{\mathcal{S}}\sqrt{\sigma }\epsilon {\mathrm{d}}^{2}x=\frac{1}{8\pi G}{\int }_0^{2\pi }\mathrm{d}\varphi {\int }_0^{\pi }\mathrm{d}\theta R^2\sin \theta \cdot K$$

$$=\frac{1}{8\pi G}\cdot 4\pi R^2\cdot K=\frac{R^{2}K}{2G}$$

Substitute $K$:

$$E_{\mathrm{BY}}(R)=\frac{R^2}{2G}\left[\frac{2\sqrt{f(R)}}{R}+\frac{f^{\prime }(R)}{2\sqrt{f(R)}}\right]$$

$$=\frac{R}{G}\sqrt{f(R)}+\frac{R^2{f}^{\prime }(R)}{4G\sqrt{f(R)}}$$

For $f=1-2M/r$:

$$f^{\prime }(r)=\frac{2M}{r^2}$$

$$E_{\mathrm{BY}}(R)=\frac{R}{G}\sqrt{1-\frac{2M}{R}}+\frac{R^2}{4G\sqrt{1-2M/R}}\cdot \frac{2M}{R^2}$$

$$=\frac{R}{G}\sqrt{1-\frac{2M}{R}}+\frac{M}{2G\sqrt{1-2M/R}}$$

Asymptotic Behavior

As $R\to \infty$:

$$\sqrt{1-\frac{2M}{R}}=1-\frac{M}{R}+O(R^{-2})$$

$$E_{\mathrm{BY}}(R)=\frac{R}{G}\left(1-\frac{M}{R}\right)+\frac{M}{2G}\left(1+\frac{M}{R}\right)+O(R^{-1})$$

$$=\frac{R}{G}-M+\frac{M}{2G}+O(R^{-1})=\frac{R}{G}-\frac{M}{2G}+O(R^{-1})$$

Diverges! Need reference subtraction.

Reference Subtraction

Minkowski space: $f_0=1$, $K_0=2/R$

$$E_0=\frac{R^2}{2G}\cdot \frac{2}{R}=\frac{R}{G}$$

Renormalized Energy:

$$\boxed{E_{\mathrm{BY},\mathrm{ren}}(R)=E_{\mathrm{BY}}(R)-E_0(R)=M}$$

Perfect! Converges to ADM mass $M$!

graph TB
    SCHW["Schwarzschild<br/>r=R Boundary"] --> RAW["Raw E_BY(R)<br/>~ R/G + ..."]
    REF["Reference E_0(R)<br/>~ R/G"] --> SUB["Subtract"]

    RAW --> SUB
    SUB --> REN["Renormalized<br/>E_BY,ren = M"]

    REN --> ADM["✓ Converges to<br/>ADM Mass"]

    style RAW fill:#ffe1e1
    style REN fill:#e1ffe1
    style ADM fill:#e1ffe1

📊 Comparison of Three Mass Concepts

Relation:

In asymptotically flat spacetime, after appropriate renormalization:

$$\underset{R\to \infty }{\lim }{E}_{\mathrm{BY},\mathrm{ren}}(R)=M_{\mathrm{ADM}}$$

graph LR
    BY["Brown-York<br/>Quasi-Local"] --> ADM["ADM<br/>Spatial Infinity"]
    BY --> BONDI["Bondi<br/>Null Infinity"]

    ADM -.->|"Asymptotic Limit"| BY
    BONDI -.->|"Null Plane Limit"| BY

    style BY fill:#fff4e1,stroke:#ff6b6b,stroke-width:2px

🔗 Connection to Boundary Time Generator

Boundary Part of Hamiltonian

In canonical form, Hamiltonian is:

$$H_{\xi }={\int }_{\Sigma }(N\mathcal{H}+N^i{\mathcal{H}}_i){\mathrm{d}}^{3}x+H_{\partial \Sigma }[\xi ]$$

where boundary part:

$$H_{\partial \Sigma }[\xi ]={\int }_{\partial \Sigma }\sqrt{\sigma }{\xi }^a{j}_a{\mathrm{d}}^{2}x$$

$j_a$ is component of Brown-York stress!

When $\xi ={\partial }_t$ (time translation Killing vector):

$$H_{\partial \Sigma }[{\partial }_t]=E_{\mathrm{BY}}$$

Physical Meaning: $$\boxed{\text{Brown-York Energy}\cong \text{Generator of Boundary Time Translation}}$$

Connection to Unified Time Scale

Recall Time Scale Identity from Unified Time chapter:

$$\kappa (\omega )=\frac{{\phi }^{\prime }(\omega )}{\pi }={\rho }_{\mathrm{rel}}(\omega )=\frac{1}{2\pi }\mathrm{tr}Q(\omega )$$

Now we see: This unified scale is considered to be realized in the gravitational end by Brown-York energy.

Boundary Trinity:

graph TB
    UNITY["Unified Time Scale<br/>κ(ω)"] --> SCATTER["Scattering End<br/>tr Q/(2π)"]
    UNITY --> MOD["Algebraic End<br/>Modular Hamiltonian K_ω"]
    UNITY --> GRAV["Gravitational End<br/>Brown-York E_BY"]

    SCATTER -.->|"Same Object"| MOD
    MOD -.->|"Same Object"| GRAV
    GRAV -.->|"Same Object"| SCATTER

    style UNITY fill:#fff4e1,stroke:#ff6b6b,stroke-width:4px

🌌 Generalization: Non-Asymptotically Flat Cases

AdS Spacetime

For asymptotically AdS spacetime, need:

1. Counterterms:

$$S_{\mathrm{ct}}=\frac{1}{8\pi G}{\int }_{\partial \mathcal{M}}\sqrt{|h|}\left(\frac{2}{L}+\frac{L}{2}\hat{\mathcal{R}}\right){\mathrm{d}}^{3}x$$

where $L$ is AdS curvature radius, $\hat{\mathcal{R}}$ is intrinsic Ricci scalar of boundary.

2. Renormalized Stress Tensor:

$$T_{\mathrm{BY},\mathrm{ren}}^{ab}=T_{\mathrm{BY}}^{ab}+T_{\mathrm{ct}}^{ab}-T_{\mathrm{ref}}^{ab}$$

$$T_{\mathrm{ct}}^{ab}=\frac{2}{\sqrt{|h|}}\frac{\delta S_{\mathrm{ct}}}{\delta h_{ab}}$$

de Sitter Universe

For de Sitter spacetime, horizon is null surface, need to use null Brown-York energy:

$$T^A{}_B{|}_{\mathcal{N}}=-\frac{1}{8\pi G}\left(W^A{}_B-\theta {\delta }^A{}_B\right)$$

where $W^A{}_B$ is shape operator, $\theta$ is expansion.

🎓 Conservation Laws and First Law

Energy Conservation

In time-independent case (existence of Killing vector ${\xi }^a=({\partial }_t{)}^a$):

$$\frac{\mathrm{d}{E}_{\mathrm{BY}}}{\mathrm{d}t}=0$$

Proof Outline:

• Hamiltonian evolution: $\frac{\mathrm{d}{h}_{ab}}{\mathrm{d}t}=\{\dots ,H\}$
• On-shell (when Einstein equations satisfied): bulk constraints $\mathcal{H}={\mathcal{H}}_i=0$
• Boundary term unchanged (because $\xi$ is Killing vector)

First Law of Black Holes

For static black holes, define:

• $M$: ADM mass ($=E_{\mathrm{BY},\mathrm{ren}}$ at infinity)
• $J$: angular momentum
• ${\mathcal{A}}_{\mathcal{H}}$: horizon area
• ${\kappa }_{\mathcal{H}}$: surface gravity

First Law:

$$\delta M=\frac{{\kappa }_{\mathcal{H}}}{8\pi G}\delta {\mathcal{A}}_{\mathcal{H}}+{\Omega }_{\mathcal{H}}\delta J$$

where ${\Omega }_{\mathcal{H}}$ is horizon angular velocity.

Thermodynamic Analogy:

$$\mathrm{d}E=T\mathrm{d}S+\text{work}$$

Identify:

• $T={\kappa }_{\mathcal{H}}/(2\pi )$: Hawking temperature
• $S={\mathcal{A}}_{\mathcal{H}}/(4G)$: Bekenstein-Hawking entropy

graph LR
    FIRST["First Law<br/>δM = ..."] --> THERMO["Thermodynamics<br/>dE = T dS"]

    TEMP["Temperature<br/>T = κ/(2π)"] -.-> FIRST
    ENT["Entropy<br/>S = A/(4G)"] -.-> FIRST

    TEMP -.-> THERMO
    ENT -.-> THERMO

    style FIRST fill:#fff4e1,stroke:#ff6b6b,stroke-width:2px
    style THERMO fill:#e1ffe1

💎 Deep Understanding of Physical Meaning

Why “Quasi-Local”?

Local:

• Defined at a spacetime point
• Example: energy density $T_{00}(x)$

Global:

• Needs entire spacetime
• Example: ADM mass (spatial infinity)

Quasi-Local:

• Defined on finite boundary
• Can “move” boundary to get different values
• Brown-York energy is exactly this type

Why Depends on Boundary?

Answer: Because energy is essentially considered a property of the boundary.

Deep Reasons:

1. General Covariance: No preferred coordinate system, cannot define “same moment”
2. Equivalence Principle: Locally can always eliminate gravitational field, energy density coordinate-dependent
3. Boundary Observation: Experiments always on some boundary, quasi-local energy is natural observable

Why Converges to ADM Mass?

Physical Image:

• Larger boundary, farther from gravitational source
• Spacetime tends to flat at infinity
• Extrinsic curvature $K\to K_0+M/R^2$ (only difference from mass)
• After integration, converges to total mass

🤔 Exercises

1. Conceptual Understanding

Question: Why is Brown-York energy zero in Minkowski space (after reference subtraction)?

Hint: Minkowski space itself is reference, $K=K_0$.

2. Calculation Exercise

Question: Calculate Brown-York energy for Reissner-Nordström black hole (charged).

Hint: $$f(r)=1-\frac{2M}{r}+\frac{Q^2}{r^2}$$

3. Physical Application

Question: How does Hawking radiation change Brown-York energy?

Hint: Bondi mass monotonically decreases along null infinity, relates to time dependence of Brown-York energy.

4. Philosophical Reflection

Question: Is Brown-York energy “subjective” (depends on boundary choice) or “objective” (physical reality)?

Hint: Like velocity depends on reference frame but is still physical quantity, quasi-local energy depends on boundary but has physical meaning.

📝 Chapter Summary

Core Definition

Brown-York Surface Stress Tensor:

$$T_{\mathrm{BY}}^{ab}=\frac{1}{8\pi G}(K^{ab}-K{h}^{ab})$$

Quasi-Local Energy:

$$E_{\mathrm{BY}}={\int }_{\mathcal{S}}\sqrt{\sigma }{u}_a{u}_b{T}_{\mathrm{BY}}^{ab}{\mathrm{d}}^{2}x$$

Core Properties

1. Well-Defined: Can calculate on any boundary
2. Quasi-Local: Depends on boundary choice
3. Convergence: Asymptotic limit gives ADM/Bondi mass
4. Generator: Is Hamiltonian of boundary time translation
5. Conservation: Conserved in Killing case

Connection to Unified Framework

Boundary Trinity:

$$\text{Scattering Time Delay}⟺\text{Modular Flow Parameter}⟺\text{Brown-York Energy}$$

All are different realizations of unified time scale $\kappa (\omega )$!

Physical Meaning

• Energy in curved spacetime is property of boundary
• Quasi-local energy is natural observable
• Generator of boundary time evolution
• Foundation of black hole thermodynamics

Next Step: We’ve seen boundary data, GHY term, Brown-York energy, now it’s time to unify boundary observer perspectives!

Navigation:

• Previous: 03-ghy-boundary-term_en.md
• Next: 05-boundary-observables_en.md
• Overview: 00-boundary-overview_en.md