GHY Boundary Term: Theoretical Proposal for Differentiability of Gravitational Action

$$\boxed{\text{Differentiability of Gravitational Action}\leftarrow \text{Determined by Boundary Term}}$$

“Boundary terms are viewed as requirements of completeness, not merely corrections.”

🎯 Core Problem

Question: Why does Einstein-Hilbert action need a boundary term?

Short Answer: Because the bulk action alone is typically ill-defined for variations fixing boundary metric!

Goals of This Article:

1. Understand why GHY boundary term is needed
2. Complete derivation of GHY term form
3. Verify cancellation mechanism of boundary terms
4. Generalize to piecewise boundaries and null boundaries

💡 Intuitive Image: Necessity of Integration by Parts

Analogy: Painting a Room

Imagine you want to paint a room:

Only Volume (Bulk Action):

• Calculate how much paint needed
• Formula: volume × thickness
• But when varying… walls “bleed”!

Add Walls (Boundary Term):

• Walls absorb “bleeding”
• Boundary conditions become natural
• Variation well-defined

graph TB
    PAINT["Painting Room"] --> VOL["Volume Integral<br/>S_EH"]
    PAINT --> WALL["Wall Treatment<br/>S_GHY"]

    VOL --> VAR1["Variation"]
    VAR1 --> BLEED["✗ Boundary 'Bleeding'<br/>Contains n·∇δg"]

    WALL --> VAR2["Compensation"]
    VAR2 --> ABSORB["✓ Absorb Bleeding<br/>Cancel n·∇δg"]

    BLEED -.->|"Needs"| ABSORB

    style BLEED fill:#ffe1e1
    style ABSORB fill:#e1ffe1

Mathematical Essence:

• Einstein equations are second-order differential equations
• Action contains square of first derivatives ($\Gamma \cdot \Gamma \sim (\partial g{)}^2$)
• Variation with integration by parts produces boundary terms
• Without boundary term, boundary has “uncontrollable” derivative terms

📜 Variation of Einstein-Hilbert Action

Original Action

$$S_{\mathrm{EH}}=\frac{1}{16\pi G}{\int }_{\mathcal{M}}\sqrt{-g}R{\mathrm{d}}^{4}x$$

where:

• $g=\det (g_{\mu \nu })$
• $R=g^{\mu \nu }{R}_{\mu \nu }$: Ricci scalar
• $G$: Newton’s gravitational constant

Three Steps of Variation

Step 1: Variation of Metric Determinant

$$\delta \sqrt{-g}=-\frac{1}{2}\sqrt{-g}{g}_{\mu \nu }\delta g^{\mu \nu }$$

Derivation: $$\delta g=g{g}^{\mu \nu }\delta g_{\mu \nu }=-g{g}_{\mu \nu }\delta g^{\mu \nu }$$

Step 2: Variation of Ricci Scalar

This is key! Ricci scalar contains Christoffel symbols:

$$R=g^{\mu \nu }{R}_{\mu \nu }=g^{\mu \nu }\left({\partial }_{\rho }{\Gamma }_{\mu \nu }^{\rho }-{\partial }_{\nu }{\Gamma }_{\mu \rho }^{\rho }+{\Gamma }_{\rho \sigma }^{\rho }{\Gamma }_{\mu \nu }^{\sigma }-{\Gamma }_{\nu \sigma }^{\rho }{\Gamma }_{\mu \rho }^{\sigma }\right)$$

Variation gives:

$$\delta R=R_{\mu \nu }\delta g^{\mu \nu }+g^{\mu \nu }\delta R_{\mu \nu }$$

Palatini Identity:

$$g^{\mu \nu }\delta R_{\mu \nu }={\nabla }_{\mu }\left(g^{\alpha \beta }\delta {\Gamma }_{\alpha \beta }^{\mu }-g^{\mu \alpha }\delta {\Gamma }_{\alpha \beta }^{\beta }\right)$$

This is a total divergence!

Step 3: Total Variation

$$\delta S_{\mathrm{EH}}=\frac{1}{16\pi G}{\int }_{\mathcal{M}}\sqrt{-g}\left[R_{\mu \nu }-\frac{1}{2}R{g}_{\mu \nu }\right]\delta g^{\mu \nu }{\mathrm{d}}^{4}x+\boxed{\text{Boundary Term}}$$

Bulk Term gives Einstein tensor $G_{\mu \nu }=R_{\mu \nu }-\frac{1}{2}R{g}_{\mu \nu }$, which is good!

Problem: What is the boundary term?

Explicit Form of Boundary Term

Using Stokes’ theorem:

$${\int }_{\mathcal{M}}{\nabla }_{\mu }{V}^{\mu }\sqrt{-g}{\mathrm{d}}^{4}x={\int }_{\partial \mathcal{M}}{V}^{\mu }{n}_{\mu }\sqrt{|h|}{\mathrm{d}}^{3}x$$

where $n^{\mu }$ is unit normal vector, $h$ is determinant of induced metric.

Boundary term becomes:

$$\delta S_{\mathrm{EH}}^{\mathrm{boundary}}=\frac{1}{16\pi G}{\int }_{\partial \mathcal{M}}\sqrt{|h|}{n}^{\mu }{\left(g^{\alpha \beta }\delta {\Gamma }_{\alpha \beta }^{\rho }-g^{\rho \alpha }\delta {\Gamma }_{\alpha \beta }^{\beta }\right)}_{\rho =\mu }{\mathrm{d}}^{3}x$$

graph TB
    EH["Einstein-Hilbert<br/>S_EH = ∫√(-g) R"] --> VAR["Variation δS_EH"]

    VAR --> BULK["Bulk Term<br/>∫√(-g) G_μν δg^μν"]
    VAR --> BOUND["Boundary Term<br/>∫√|h| (...)"]

    BULK --> GOOD["✓ Einstein Equations"]
    BOUND --> BAD["✗ Contains n·∇δg"]

    style BULK fill:#e1ffe1
    style BOUND fill:#ffe1e1

🔍 Detailed Analysis of Boundary Term

Projection onto Tangential and Normal

Decompose boundary term into tangential and normal:

$$h_{\mu }{}^{\nu }={\delta }_{\mu }{}^{\nu }-\epsilon n_{\mu }{n}^{\nu },\epsilon =n^{\mu }{n}_{\mu }\in \{\pm 1\}$$

• $\epsilon =-1$: spacelike boundary (initial/final time slices)
• $\epsilon =+1$: timelike boundary (spatial boundary)

After tedious index manipulation (see Appendix A), boundary term can be written as:

$$\delta S_{\mathrm{EH}}^{\mathrm{boundary}}=\frac{1}{16\pi G}{\int }_{\partial \mathcal{M}}\sqrt{|h|}\left[{\Pi }^{ab}\delta h_{ab}+n^{\rho }{h}^{\mu \alpha }{h}^{\nu \beta }{\nabla }_{\rho }\delta g_{\alpha \beta }\right]{\mathrm{d}}^{3}x$$

where:

• ${\Pi }^{ab}=K^{ab}-K{h}^{ab}$ (“momentum” related to extrinsic curvature)
• Second term is uncontrollable normal derivative term!

Essence of Ill-Definedness

Problem: When fixing induced metric $h_{ab}$, $\delta h_{ab}=0$, but:

$$n^{\rho }{h}^{\mu \alpha }{h}^{\nu \beta }{\nabla }_{\rho }\delta g_{\alpha \beta }\ne 0$$

This means:

1. Need to fix $n^{\rho }{\nabla }_{\rho }{g}_{\alpha \beta }$ (normal derivative)
2. This is unnatural boundary condition
3. Hamiltonian not differentiable

graph LR
    FIX["Fix Boundary Condition"] --> H["Fix h_ab"]
    FIX --> DERIV["Need to Fix<br/>n·∇g ?"]

    H --> NAT["✓ Natural"]
    DERIV --> UNNAT["✗ Unnatural"]

    style NAT fill:#e1ffe1
    style UNNAT fill:#ffe1e1

⭐ GHY Boundary Term: Perfect Solution

Gibbons-Hawking-York Term

Definition:

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

where:

• $K=h^{ab}{K}_{ab}$: trace of extrinsic curvature
• $K_{ab}=h_a{}^{\mu }{h}_b{}^{\nu }{\nabla }_{\mu }{n}_{\nu }$: extrinsic curvature
• $\epsilon =n^{\mu }{n}_{\mu }$: orientation factor

Physical Meaning:

• $K$ measures how boundary “curves” in bulk
• $K>0$: boundary convex outward
• $K<0$: boundary concave inward

graph TB
    CURV["Extrinsic Curvature K"] --> EMBED["Boundary Embedding<br/>in Bulk"]

    EMBED --> CONV["K > 0<br/>Convex Outward"]
    EMBED --> CONC["K < 0<br/>Concave Inward"]
    EMBED --> FLAT["K = 0<br/>Flat"]

    K_AB["K_ab = h^μ_a h^ν_b ∇_μ n_ν"]

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

Variation of GHY Term

Key Calculation:

$$\delta (\sqrt{|h|}K)=\sqrt{|h|}\left(\delta K+\frac{1}{2}K{h}^{ab}\delta h_{ab}\right)$$

where:

$$\delta K=h^{ab}\delta K_{ab}-K^{ab}\delta h_{ab}$$

and:

$$\delta K_{ab}=h_a{}^{\mu }{h}_b{}^{\nu }\left({\nabla }_{\mu }\delta n_{\nu }+\delta {\Gamma }_{\mu \nu }^{\rho }{n}_{\rho }\right)$$

Unit Normal Gauge: Fix embedding, vary only metric, then:

$$\boxed{\delta n_{\mu }=\frac{1}{2}\epsilon n_{\mu }{n}^{\alpha }{n}^{\beta }\delta g_{\alpha \beta }}$$

Something magical happens:

Substituting this into $\delta K_{ab}$, the ${\nabla }_{\mu }\delta n_{\nu }$ term exactly produces:

$$-\epsilon n^{\rho }{h}^{\mu \alpha }{h}^{\nu \beta }{\nabla }_{\rho }\delta g_{\alpha \beta }$$

This exactly cancels the ill-defined term in $\delta S_{\mathrm{EH}}$!

✨ Complete Proof of Cancellation Mechanism

Proposition (GHY Cancellation Mechanism)

For variation families fixing induced metric $\delta h_{ab}=0$:

$$\delta (S_{\mathrm{EH}}+S_{\mathrm{GHY}})=\frac{1}{16\pi G}{\int }_{\mathcal{M}}\sqrt{-g}{G}_{\mu \nu }\delta g^{\mu \nu }{\mathrm{d}}^{4}x$$

Boundary terms completely cancel!

Proof Skeleton

Step 1: Boundary term of $\delta S_{\mathrm{EH}}$

$$\delta S_{\mathrm{EH}}^{\mathrm{bdy}}=\frac{1}{16\pi G}{\int }_{\partial \mathcal{M}}\sqrt{|h|}\left[{\Pi }^{ab}\delta h_{ab}+n^{\rho }{h}^{\mu \alpha }{h}^{\nu \beta }{\nabla }_{\rho }\delta g_{\alpha \beta }\right]{\mathrm{d}}^{3}x$$

Step 2: Calculation of $\delta S_{\mathrm{GHY}}$

$$\delta S_{\mathrm{GHY}}=\frac{\epsilon }{8\pi G}{\int }_{\partial \mathcal{M}}\sqrt{|h|}\left(\delta K+\frac{1}{2}K{h}^{ab}\delta h_{ab}\right){\mathrm{d}}^{3}x$$

$$=\frac{\epsilon }{8\pi G}{\int }_{\partial \mathcal{M}}\sqrt{|h|}\left(h^{ab}\delta K_{ab}-K^{ab}\delta h_{ab}+\frac{1}{2}K{h}^{ab}\delta h_{ab}\right){\mathrm{d}}^{3}x$$

$$=\frac{\epsilon }{8\pi G}{\int }_{\partial \mathcal{M}}\sqrt{|h|}\left[{\Pi }^{ab}\delta h_{ab}+h^{ab}{\nabla }_a\delta n_b\right]{\mathrm{d}}^{3}x$$

Step 3: Substitute $\delta n_{\mu }$

$$h^{ab}{\nabla }_a\delta n_b=\frac{\epsilon }{2}{h}^{ab}{\nabla }_a(n_b{n}^{\alpha }{n}^{\beta }\delta g_{\alpha \beta })$$

Using projection relations and variation of Christoffel symbols, this term gives:

$$-\frac{\epsilon }{2}{n}^{\rho }{h}^{\mu \alpha }{h}^{\nu \beta }{\nabla }_{\rho }\delta g_{\alpha \beta }$$

Step 4: Sum

$$\delta S_{\mathrm{EH}}^{\mathrm{bdy}}+\delta S_{\mathrm{GHY}}=\frac{1}{16\pi G}{\int }_{\partial \mathcal{M}}\sqrt{|h|}2{\Pi }^{ab}\delta h_{ab}{\mathrm{d}}^{3}x$$

When $\delta h_{ab}=0$, boundary term is zero!

graph TB
    EH_BDY["δS_EH Boundary Term"] --> TERM1["Π^ab δh_ab"]
    EH_BDY --> TERM2["+ n·∇δg"]

    GHY["δS_GHY"] --> COMP1["Π^ab δh_ab"]
    GHY --> COMP2["- n·∇δg"]

    SUM["Sum"] --> TERM1
    SUM --> TERM2
    SUM --> COMP1
    SUM --> COMP2

    SUM --> CANCEL1["✓ n·∇δg<br/>Completely Canceled"]
    SUM --> REMAIN["2Π^ab δh_ab"]

    REMAIN --> FIXED["Fix h_ab<br/>⇒ δh_ab = 0"]
    FIXED --> ZERO["✓ Boundary Term = 0"]

    style CANCEL1 fill:#e1ffe1
    style ZERO fill:#e1ffe1

🔢 Concrete Example: Spherical Boundary

Setup

Consider Schwarzschild spacetime truncated at $r=R$:

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

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

Boundary $\mathcal{B}$ is timelike hypersurface at $r=R$.

Normal Vector

Outward unit normal:

$$n^{\mu }=\left(0,\sqrt{f(R)},0,0\right)$$

$$n_{\mu }=\left(0,\frac{1}{\sqrt{f(R)}},0,0\right)$$

$$\epsilon =n^{\mu }{n}_{\mu }=+1$$

(timelike)

Induced Metric

$$h_{ab}\mathrm{d}{x}^a\mathrm{d}{x}^b=-f(R)\mathrm{d}{t}^2+R^2\mathrm{d}{\Omega }_2^2$$

$$\sqrt{|h|}=R^2\sqrt{f(R)}\sin \theta$$

Extrinsic Curvature

Calculate $K_{ab}=h_a{}^{\mu }{h}_b{}^{\nu }{\nabla }_{\mu }{n}_{\nu }$:

Time-Time Component: $$K_{tt}=-f(R){\nabla }_r{n}_t=0$$

(by symmetry)

Angular Components: $$K_{\theta \theta }=R^2{\nabla }_r{n}_{\theta }=R\sqrt{f(R)}$$

$$K_{\varphi \varphi }=R^2{\sin }^2\theta {\nabla }_r{n}_{\varphi }=R\sqrt{f(R)}{\sin }^2\theta$$

Trace: $$K=h^{ab}{K}_{ab}=\frac{2\sqrt{f(R)}}{R}+\frac{f^{\prime }(R)}{2\sqrt{f(R)}}$$

where we used: $${\nabla }_r{n}_r=\frac{1}{2}{\partial }_r\ln f(R)=\frac{f^{\prime }(R)}{2f(R)}$$

GHY Term

$$S_{\mathrm{GHY}}=\frac{1}{8\pi G}{\int }_{\mathcal{B}}\sqrt{|h|}K{\mathrm{d}}^{3}x$$

$$=\frac{1}{8\pi G}\int \mathrm{d}t\mathrm{d}{\Omega }_2{R}^2\sqrt{f(R)}\left[\frac{2\sqrt{f(R)}}{R}+\frac{f^{\prime }(R)}{2\sqrt{f(R)}}\right]$$

$$=\frac{1}{8\pi G}\cdot 4\pi R^2\cdot T\left[2f(R)+\frac{R{f}^{\prime }(R)}{2}\right]$$

For large $R$ ($f\to 1$): $$K\to \frac{2}{R}+O(M/R^2)$$

Physical Meaning:

• $2/R$ term: intrinsic curvature of sphere
• $M/R^2$ term: correction from gravitational field

🧩 Piecewise Boundaries: Necessity of Corner Terms

Problem: Boundaries Have “Corners”

When boundary is piecewise, e.g., initial/final spacelike slices + timelike sides:

$$\partial \mathcal{M}={\mathcal{B}}_{\text{initial}}\cup {\mathcal{B}}_{\text{side}}\cup {\mathcal{B}}_{\text{final}}$$

At intersections (corners/joints) $\mathcal{C}$, GHY term is insufficient!

graph TB
    BOUND["Piecewise Boundary"] --> INIT["Initial<br/>Spacelike Piece"]
    BOUND --> SIDE["Side<br/>Timelike Piece"]
    BOUND --> FINAL["Final<br/>Spacelike Piece"]

    INIT --> CORNER1["Corner C₁"]
    SIDE --> CORNER1
    SIDE --> CORNER2["Corner C₂"]
    FINAL --> CORNER2

    CORNER1 -.->|"Need"| TERM["Corner Term"]
    CORNER2 -.->|"Need"| TERM

    style CORNER1 fill:#ffe1e1
    style CORNER2 fill:#ffe1e1

Form of Corner Term

For joints ${\mathcal{C}}_{ij}$ of non-null boundaries:

$$S_{\mathrm{corner}}=\frac{1}{8\pi G}{\int }_{\mathcal{C}}\sqrt{\sigma }\eta {\mathrm{d}}^{2}x$$

where $\eta$ is the angle:

• Two Spacelike Pieces: $\eta =\mathrm{arccosh}(-n_1\cdot n_2)$
• Two Timelike Pieces: $\eta =\arccos (n_1\cdot n_2)$
• Mixed: $\eta =\mathrm{arcsinh}(n_T\cdot n_S)$

Physical Meaning:

• $\eta$ measures “angle” between two boundary pieces
• Corner term compensates jump of GHY term at joints

Additivity Theorem

Proposition: After adding corner terms, action satisfies additivity:

$$S[{\mathcal{M}}_1{\cup }_{\Sigma }{\mathcal{M}}_2]=S[{\mathcal{M}}_1]+S[{\mathcal{M}}_2]$$

where $\Sigma$ is common boundary.

Proof Outline:

• Two regions glued at $\Sigma$
• GHY terms on both sides of $\Sigma$ have opposite signs, but don’t completely cancel (because normals opposite)
• Corner term exactly compensates this difference

🌌 Null Boundaries: $(\theta +\kappa )$ Structure

Special Nature of Null Boundaries

When boundary is null surface (e.g., horizon), $n^2=0$, above formulas fail!

New Metric Structure:

Null boundary generated by null generating vector ${\ell }^{\mu }$ ($\ell \cdot \ell =0$), with auxiliary vector $k^{\mu }$ (satisfying $\ell \cdot k=-1$).

Transverse 2D metric: $${\gamma }_{AB}\mathrm{d}{x}^A\mathrm{d}{x}^B$$

Null Boundary Term

Lehner-Myers-Poisson-Sorkin Formula:

$$\boxed{S_{\mathcal{N}}=\frac{1}{8\pi G}{\int }_{\mathcal{N}}\sqrt{\gamma }(\theta +\kappa )\mathrm{d}\lambda {\mathrm{d}}^{2}x}$$

where:

• $\theta ={\gamma }^{AB}{W}_{AB}$: expansion
• $W_{AB}={\gamma }_A{}^{\mu }{\gamma }_B{}^{\nu }{\nabla }_{\mu }{\ell }_{\nu }$: shape operator
• $\kappa =-k_{\mu }{\ell }^{\nu }{\nabla }_{\nu }{\ell }^{\mu }$: surface gravity
• $\lambda$: affine parameter along $\ell$

Physical Meaning:

• $\theta$: expansion rate of null geodesic bundle
• $\kappa$: “acceleration” of horizon

Rescaling Invariance

Key Property: Under constant rescaling $\ell \to e^{\alpha }\ell$, $k\to e^{-\alpha }k$:

$$\theta \to e^{\alpha }\theta ,\kappa \to e^{\alpha }\kappa$$

$$\int \sqrt{\gamma }(\theta +\kappa )\mathrm{d}\lambda \to \int \sqrt{\gamma }{e}^{\alpha }(\theta +\kappa )e^{-\alpha }\mathrm{d}{\lambda }^{\prime }=\text{invariant}$$

This ensures physical gauge invariance!

📊 Unification of Three Boundary Types

Unified formula:

$$S_{\mathrm{total}}=S_{\mathrm{EH}}+\sum _i{S}_{\mathrm{boundary}}^{(i)}+\sum _{ij}{S}_{\mathrm{corner}}^{(ij)}$$

graph TB
    ACTION["Total Action<br/>S_total"] --> EH["Einstein-Hilbert<br/>∫√(-g) R"]
    ACTION --> BDY["Boundary Terms"]
    ACTION --> CORNER["Corner Terms"]

    BDY --> SPACE["Spacelike<br/>-∫√|h| K"]
    BDY --> TIME["Timelike<br/>+∫√|h| K"]
    BDY --> NULL["Null<br/>∫√γ (θ+κ)dλ"]

    CORNER --> NN["Non-Null-Non-Null<br/>∫√σ η"]
    CORNER --> NL["Non-Null-Null<br/>∫√σ a"]
    CORNER --> LL["Null-Null<br/>∫√σ a"]

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

🎓 Chapter Summary

Core Conclusion

GHY Boundary Term is Considered Necessary:

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

So that: $$\delta (S_{\mathrm{EH}}+S_{\mathrm{GHY}})=\frac{1}{16\pi G}{\int }_{\mathcal{M}}\sqrt{-g}{G}_{\mu \nu }\delta g^{\mu \nu }$$

Boundary terms completely cancel!

Three Levels of Boundaries

1. Non-Null Boundaries: GHY term $\propto K$
2. Corners: Angle term $\propto \eta$ or $a$
3. Null Boundaries: $(\theta +\kappa )$ term

Physical Meaning

• Variational Well-Definedness: Fix natural boundary data ($h_{ab}$) suffices
• Hamiltonian Differentiable: Canonical form well-defined
• Additivity: Action satisfies regional additivity

Connection to Unified Time

Extrinsic curvature $K$ in GHY boundary term directly relates to boundary time:

• Brown-York quasi-local energy: $T_{\mathrm{BY}}^{ab}\propto (K^{ab}-K{h}^{ab})$
• Boundary time generator: from variation of $K$
• Localization of modular Hamiltonian on boundary

Next Step: With GHY boundary term, we can define Brown-York quasi-local energy, the concrete realization of boundary time generator!

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