Necessity and Extensions of Gibbons--Hawking--York Boundary Terms: Variational Well-Posedness, Corners and Null Boundaries, and Closure to Quasilocal Energy and Thermodynamics

Necessity and Extensions of GibbonsHawkingYork Boundary Terms: Variational Well-Posedness, Corners and Null Boundaries, and Closure to Quasilocal Energy and Thermodynamics Haobo Ma 1 Wenlin Zhang 2 1 Independent Researcher 2 National University of Singapore Version: 1.7 Abstract On pseudo-Riemannian manifolds with (possibly non-smooth) boundaries, the variation of the EinsteinHilbert bulk action contains normal derivative-type boundary uxes; xing only Dirichlet data for the induced metric hab does not suce for well-posedness. In the framework of LeviCivita connection and extrinsic curvature, this paper rigorously proves that adding the GibbonsHawkingYork (GHY) term with orientation factor ε:= nµnµ∈ {±1} at non-null boundaries cancels all normal derivative contributions, thereby establishing a stationarity principle for variations xing hab . For piecewise boundaries, we provide a unied dictionary for joint (corner) terms and prove action additivity; for null segments, we construct a null boundary term with expansion θ and surface gravity κ that is invariant under constant rescaling, elucidating the endpoint and divergence contributions introduced by non-constant rescaling and transverse supertranslations respectively, along with their compensations. Subsequently, we establish in ADM/ReggeTeitelboim canonical decomposition and covariant phase space (IyerWald, Wald Zoupas) that the GHY/joint structure renders the Hamiltonian dierentiable, with boundary generators consistent with BrownYork quasilocal stress; compatibility with covariant charges is achieved within the same boundary condition class and representative. For f(R) and Lovelock (including GaussBonnet) theories, we construct boundarycorner functionals matching Dirichlet data and provide additivity propositions for piecewise non-smooth cases. Finally, in Euclidean black hole geometries, explicit computation with K and reference K0 , together with necessary joint and (AAdS case) counterterms, yields consistent free energy, energy, and entropy. Appendices provide step-by-step reproducible derivations, orientationsign dictionaries, and worked examples in covariant phase space. MSC : 83C05; 83C57; 58A10; 49S05 Keywords : GibbonsHawkingYork boundary term; variational well-posedness; corners and joints; null boundaries; BrownYork quasilocal energy; covariant phase space; f(R) gravity; Lovelock/GaussBonnet gravity; Euclidean black holes; thermodynamics 1 Notation, Orientation, and Data Classes  Spacetime and curvature : (M, gµν) is a four-dimensional orientable pseudo-Riemannian manifold with signature (−,+,+,+) . The Riemann tensor is Rρσµν =∂µΓρσν −∂νΓρσµ + ΓρλµΓλσν −ΓρλνΓλσµ, 1 with Rµν =Rρµρν , R=gµνRµν , and Gµν =Rµν −1 2Rgµν .  Non-null boundary geometry : On a boundary segment B , take unit normal nµ with ε:= nµnµ∈ {±1} . The induced metric and extrinsic curvature are hµν =gµν −ε nµnν, Kµν =hµαhνβ∇αnβ, K =hµνKµν.  Null boundary geometry : On N , take null vector ℓµ and auxiliary vector kµ with ℓ·k=−1 . The transverse two-dimensional metric is γAB . The shape operator and expansion are WAB := γAµγBν∇µℓν, θ := γABWAB, with indices raised/lowered by γAB ; transverse covariant derivative DA and Háji£ek one-form ωA:= −kµ∇Aℓµ are induced by the rigging connection.  Ane parameter and surface gravity : Let λ be an ane parameter along the generator ℓ , with ∂λ:= ℓµ∇µ. Under the normalization ℓ·k=−1 , surface gravity is dened as κ:= −kµℓν∇νℓµ, yielding ℓν∇νℓµ=κ ℓµ . This denition is compatible with the rescaling laws in 4: when ℓ→eαℓ and k→e−αk , θ→eαθ and κ→eα(κ+∂λα).  Piecewise boundaries and joints : ∂M=SiBi , with Cij =Bi∩Bj allowing signature ips or containing null segments.  Boundary data (Dirichlet class) : Non-null segments x hab ; null segments x the Carroll structure (γAB,[ℓ]) , where [ℓ] is an equivalence class under constant rescaling ℓ→eαℓ ; each joint xes an angle (the η in 3 and logarithmic angle a in 4).  Measures : Bulk √−gd4x ; non-null boundary p|h|d3x ; null boundary √γdλd2x ; joints √σd2x . 2 Variation of EH Bulk Action and Boundary Flux SEH =1 16πG ZM √−g R d4x. The rst variation is δ(√−gR) = √−g Gµνδgµν +∂µh√−ggαβδΓµαβ −gµαδΓβαβi, where 2 δΓρµν =1 2gρσ∇µδgσν +∇νδgσµ −∇σδgµν. After tangent/normal decomposition, the boundary term contains an irreducible principal term nµ∇µδgαβ ; SEH alone is ill-posed under Dirichlet data. 3 GHY Cancellation and Variational Well-Posedness SGHY[g] = ε 8πG Z∂Mp|h|Kd3x Variational setup (xed embedding, unit normal gauge) : The boundary geometric location is held xed; only the metric varies. Thus δ(nµnµ)=0, δnµ=1 2ε nµnαnβδgαβ . This setup is compatible with Dirichlet data (xing hab ) and makes SGHY and joint terms cancel boundary uxes term-by-term. Theorem 1 (GHY Cancellation) . For variations xing δhab = 0 , δ(SEH +SGHY) = 1 16πG ZM √−g Gµν δgµν d4x. Proof. See Appendix B for term-by-term matching. Self-check hint : Align the principal term nρhµαhνβ∇ρδgαβ from Appendix A with the ∇δg terms in δKab arising from δnµ=1 2εnµnαnβδgαβ in Appendix B; direct term-by-term verication yields cancellation. 4 Piecewise Boundaries, Signature Flips, and Corner Additivity Non-nullnon-null joint angle dictionary : Let two segments have unit outward normals n1, n2 with causal types marked by εi:= n2 i∈ {±1} . The joint angle η is dened as η=       arccosh −n1·n2, ε1=ε2=−1 ( both spacelike, normals timelike ), arccos n1·n2, ε1=ε2= +1 ( both timelike, normals spacelike ), arcsinh nT·nS, ε1ε2=−1 ( mixed causal; n2 T=−1, n2 S= +1). The corner term is S(nn) corner =1 8πG ZC √σ η d2x , with orientation and sign dierences uniformly xed by the master formula and orientation tables. Nullnon-null and nullnull joints : Logarithmic angles a(nℓ)= ln |−ℓ·n|, a(ℓℓ)= ln −1 2ℓ1·ℓ2, with joint terms 1 8πG RC√σ a d2x . 3 Theorem 2 (Additivity and Necessity) . Under boundary data xing the respective angles ( η or a ), SEH +SGHY +Scorner/joint is variationally well-posed and satises additivity S[M1∪ΣM2] = S[M1] + S[M2]. Joint terms are invariant under any C1 regularization limit, independent of regularizer details. 5 Null Boundaries: θ+κ Structure, Rescaling, and Endpoint Compensation SN=1 8πG ZN √γ(θ+κ) dλd2x Theorem 3 (Null Well-Posedness) . Fixing (γAB,[ℓ]) , δ(SEH +SN) contains no normal derivative residuals. Pure rescaling (preserving ℓ·k=−1 , no transverse components): ℓ→eαℓ, k →e−αk⇒WAB →eαWAB, θ →eαθ, κ →eα(κ+∂λα). When α= const, RN√γ(θ+κ) dλd2x plus joint terms is invariant; when α=α(λ) , endpoint total variations are produced, absorbable by logarithmic angle counterterms (see Appendix D; path B takes ln(ℓc|Θ|) requiring Θ sign-denite; if Θ crosses zero, use path A endpoint/joint compensation). Transverse supertranslation/cross-section reparametrization : ℓ→eα(ℓ+vAeA)⇒θ→eα(θ+DAvA), belonging to cross-section redenition eects, treated separately from pure rescaling above. Dimensional note : In D dimensions, transverse space dimension is D−2 ; corresponding divergence structure generalizes straightforwardly by dimension. Null BrownYork stress : TABN=−1 8πGWAB−θ δAB, satisfying transverse conservation dened by the rigging connection, compatible with null Wald Zoupas charges within the same boundary condition class. 6 Canonical Formalism: Dierentiable Hamiltonian and Quasilocal Energy In 3+1 decomposition, with SEH alone the Hamiltonian functional is non-dierentiable; adding SGHY with necessary joint/null terms yields: Theorem 4 (Dierentiability and Boundary Generators) . Under Dirichlet data and the orientation/regularity assumptions of this paper, taking the action S=SEH +SGHY +Sjoint(+SN) 4 without introducing any intrinsic boundary functional depending solely on the boundary intrinsic metric hab , the Hamiltonian Hξ is Fréchet dierentiable on phase space, with boundary generator uniquely given by Tab BY =1 8πG(Kab −Khab) If intrinsic terms (such as Sct in 9 or reference term Sref ) are added/subtracted within the same boundary condition class, Hξ remains dierentiable with boundary generator modied to Tab BY,ren =Tab BY +Tab ct −Tab ref, consistent with covariant phase space analysis in 6 and renormalization counterterms in 9. The energy on a spacelike slice S is EBY =ZS √σ uaubTab BY d2x which in the asymptotically at limit approaches the ADM mass. 7 Covariant Phase Space and Representative Independence δL=E·δϕ + dΘ(ϕ, δϕ),Jξ=Θ(ϕ, Lξϕ)−ξ·L= dQξ. If L→L+ dB , then Θ→Θ+δB and Qξ→Qξ+ξ·B . Within the same boundary condition class and the same (or gauge-equivalent) representative, mass, angular momentum, and horizon entropy are invariant; ux boundaries employ WaldZoupas corrections to ensure integrability. Skeleton formula (locating dierentiability source) : In the ReggeTeitelboim framework, δHξ=ZΣ ( constraints ·δϕ) d3x+I∂ΣΠabδhab +···d2x. With bulk term alone, boundary variation contains Πabδhab and normal derivative terms, nondierentiable; adding SGHY (and joint/null terms) transforms boundary variation into BY surface generators, rendering Hξ dierentiable. Worked Example (representative independence computational chain) : Take a static black hole with Killing eld ξ=∂t , at innity I and horizon H : δHξ=ZS∞δQξ−ξ·Θ−ZSHδQξ−ξ·Θ. If L7→ L+ dB , then Θ7→ Θ+δB,Qξ7→ Qξ+ξ·B, with δ(ξ·B) = ξ·δB , so increments at both ends vanish, δHξ invariant; if ux boundaries exist, apply WaldZoupas correction making endpoint dierence zero, restoring integrability. Renormalized BY surface stress : Tab BY,ren =2 p|h| δSGHY +Sjoint +Sct −Sref δhab =Tab BY +Tab ct −Tab ref, where Tab ct := 2 p|h| δSct δhab and Tab ref := 2 p|h| δSref δhab . Minimal counterterms for four-dimensional AAdS appear in 9. 5 8 f(R) Gravity: Dirichlet-Compatible BoundaryJoints Using the scalartensor equivalence Φ = f′(R) , S=1 16πG ZM √−g(ΦR−V(Φ)) d4x. Under Dirichlet data xing (hab,Φ) , Sf(R) bdy =1 8πG Z∂M εp|h|ΦKd3x, Sf(R) joint =1 8πG X CZC √σΦ ( angle ) d2x. If instead xing (hab, nµ∇µΦ) as Robin-type data, compensation terms ∝p|h|nµ∇µΦ must be added at the boundary, with correspondingly weighted joint terms (Appendix G). 9 Lovelock (GaussBonnet) Gravity and Piecewise Non-Smooth Additivity For GaussBonnet (GB) term in D≥5 , SGB =α 16πG ZM √−gRµνρσRµνρσ −4RµνRµν +R2dDx, the Dirichlet-compatible Myers-type boundary term is SGB bdy =α 8πG Z∂M εp|h|2b GabKab +JdD−1x, where b Gab is the Einstein tensor of hab , Jab =1 32KKacKcb+KcdKcdKab −2KacKcdKdb −K2Kab, J =habJab. Proposition 5 (GB Additivity, Piecewise Non-Smooth) . Taking the above boundary term and adding corresponding GB joint polynomials (quadratic combinations of angles η /logarithmic angles a with (K, b R) ), under xed Dirichlet data SGB[M1∪ΣM2] = SGB[M1] + SGB[M2]. Proof sketch. Integrate by parts on each piece; at joints appear residuals ∝δ( angle ) ; chosen GB joint polynomials' variation exactly cancels these residuals. Representative: DeruelleMerinoOlea (2018). 10 Non-Compact Boundaries and AAdS Counterterms (Four-Dimensional Minimal Representative) Sct =1 8πG Z∂Mp|h|2 L+L 2b Rd3x where L is the AdS curvature radius and b R is the boundary intrinsic Ricci scalar. This representative is equivalent to kounterterms/holographic renormalization in four dimensions for yielding the same nite stress and conformal-invariant terms; higher dimensions require additional highercurvature counterterms. 6 11 Distributional Curvature, Thin Shells, and Zero-Measure Boundary of Boundary If Kab exhibits jumps across a hypersurface, bulk curvature develops δ -type distributions; their contribution to the action is absorbed by joint/thin shell terms. Timelike/spacelike thin shells satisfy Israel junction conditions [Kab −Khab] = −8πG Sab ; null thin shells satisfy BarrabèsIsrael conditions. The joint and null rules of this paper are compatible therewith. 12 Euclidean Black Holes: K , K0 , Free Energy, and Entropy For Schwarzschild Euclidean geometry ds2=f(r) dτ2+f(r)−1dr2+r2dΩ2 2, f(r) = 1 −2M r, truncated at r=R , τ∈[0, β] . With outward unit normal nµ=√f δµ r , K(R) = 2pf(R) R+f′(R) 2pf(R), K0(R) = 2 R. Total action IE=IEH +IGHY[K] + Ijoint −Iref [K0]. Removing conical decit β= 8πM and taking R→ ∞ nite part yields F=IE β=M 2, E =∂β(βF) = M, S =β(E−F) = A 4G. Periodicity identication no double-counting : Due to τ∼τ+β , lateral edge corners at r=R at (R, 0) and (R, β) are equivalent; integration by parts on interval [0, β] yields corner contributions at two ends whose sum equals the contribution of a single corner on the periodic manifold, no double-counting occurs. 13 Variational Well-Posedness vs PDE/Fredholm This paper establishes closure of action rst variation on given boundary data sets; PDE wellposedness and Fredholm properties require functional space and boundary-value operator analysis. On compact boundaries, pure Dirichlet/Neumann maps are generally non-Fredholm; natural mixed data (e.g., ([γ], H) or Bartnik data) are more suitable. This work is conned to the variational well-posedness level; Appendix L provides illustrative examples. Appendices: Numbered Derivations, Dictionaries, and Examples Unied note : All integrals explicitly write measures dnx ; set notation unied as {±1} ; master formula SGHY = (8πG)−1εRp|h|Kd3x with orientation table uniquely xes sign dierences. 7 Appendix A: EH Action Boundary Flux (Term-by-Term Decomposition) A.1 δSEH = (16πG)−1RMδ(√−g)R+√−g δRd4x , δ√−g=−1 2√−g gµνδgµν . A.2 δR =Rµνδgµν +∇µgαβδΓµαβ −gµαδΓβαβ . A.3 Stokes formula yields boundary term (16πG)−1R∂Mp|h|nµ(···)d3x . A.4 Projection hµν=δµν−εnµnν writes boundary ux as Z∂Mp|h|hΠabδhab +nρhµαhνβ∇ρδgαβ +···id3x. where Πab := Kab −Khab . Appendix B: GHY Cancellation and Example Orientation Table B.1 δ(p|h|K) = p|h|δK +1 2K habδhab , δK =habδKab −Kabδhab , where δKab =haµhbν∇µδnν+δΓρµνnρ. B.2 Substituting unit normal gauge δnµ=1 2ε nµnαnβδgαβ , the ∇δg in δK cancels term-by-term with Appendix A principal term, while Πabδhab mutually cancel, yielding Theorem 2.1. B.3 Example orientation table Segment Causal type n2=ε Outward normal GHY weight Initial/nal slices Spacelike −1 Future/past −Rp|h|Kd3x Lateral edge Timelike +1 Outward +Rp|h|Kd3x Euclidean boundary Riemannian +1 Outward +Rp|h|Kd3x (This table is for reading guidance only; actual computations uniformly use the master formula.) Appendix C: Three Types of Joints and Additivity (Dictionary and Proof Outline)  Non-nullnon-null: η dened piecewise by causal type (see 3); corner term 1 8πG R√σ η d2x .  Nullnon-null: a= ln |−ℓ·n| .  Nullnull: a= ln |− 1 2ℓ1·ℓ2| .  Piecewise GHY integration by parts leaves only endpoint terms ∝δ( angle ) , canceled by joint terms; action additive; result independent of joint regularizer details. 8 Appendix D: Null Rescaling, Endpoint Compensation, and Supertranslation D.1 Pure rescaling ℓ→eα(λ)ℓ, k →e−α(λ)k : θ→eαθ , κ→eα(κ+∂λα) . Invariant under constant α ; non-constant produces endpoint total variations. D.2 Path A (LMPS endpoint/joint compensation) : Send =1 8πG X endpoints Z√σ α d2x. D.3 Path B (logarithmic counterterm) : Sreparam =1 8πG ZN √γΘ ln ℓc|Θ|dλd2x, Θ := θ. Note: Path B requires Θ sign-denite on each generator; if Θ crosses zero (as at foci), treat zero-crossing points as joints and handle per D.2 endpoint/joint compensation, or use path A. D.4 Transverse supertranslation ℓ→eα(ℓ+vAeA) introduces DAvA , classied as crosssection redenition. Appendix E: ReggeTeitelboim Dierentiability and BY Generators δHξ=ZΣ (N δH+NiδHi) d3x+Z∂Σ √σε δN +jiδNi+Tab BYδhabd2x. where ε:= uaubTab BY , ji:= −σiaubTab BY , σab =hab +uaub . Adding GHY/joints renders Hξ dierentiable and generates correct evolution; asymptotically at EBY →MADM . Appendix F: Covariant Phase SpaceRepresentative Freedom and Worked Example F.1 Representative freedom : L→L+ dB⇒Θ→Θ+δB , Qξ→Qξ+ξ·B . Charge element kξ:= δQξ−ξ·Θ remains invariant. F.2 Worked example (static black hole) : Main text 6 already provides two-end cancellation chain; ux boundaries restored to integrability via WaldZoupas correction, yielding rst law and S=A/(4G) . Appendix G: f(R) /Lovelock BoundaryJoint Correspondence G.1 f(R) : Dirichlet: Sf(R) bdy = (8πG)−1Rεp|h|ΦKd3x , joint ∝Φ (η or a) . Robin: add ∝ p|h|nµ∇µΦ with dual joint terms. G.2 GaussBonnet ( D≥5 ) : SGB bdy = (8πG)−1αRεp|h|(2 b GabKab +J) dD−1x ; piecewise non-smooth GB joint polynomials ensure Proposition 8.1 additivity (coecients xed by Chern Weil/transgression; see DeruelleMerinoOlea, 2018). Appendix H: Schwarzschild Euclidean Action (Including K and K0 ) ds2=fdτ2+f−1dr2+r2dΩ2 2 , f= 1 −2M/r . K(R) = 2pf(R) R+f′(R) 2pf(R) , K0(R) = 2 R . IE=IEH +IGHY[K]−Iref[K0] + Ijoint ⇒F=M/2, E =M, S =A/(4G) . Periodicity identication no double-counting explained in main text 11. 9