Chapter 13: Variational Completeness and Boundary Terms

In Chapter 12, we successfully derived Einstein’s field equations and their higher-order generalizations through the Entropic Variational Principle (IGVP). However, variational principles in physics are not simply about setting derivatives to zero. When we apply the theory to spacetime regions with boundaries (such as black hole horizons, cosmic horizons, or finite laboratory systems), we must face a serious mathematical challenge: the boundary term problem in variational principles.

In standard general relativity, the Einstein-Hilbert action itself is variationally incomplete under Dirichlet boundary conditions—the variation process produces boundary fluxes that cannot be eliminated. To fix this, Gibbons, Hawking, and York introduced the famous boundary term (GHY term). In our discrete ontology and entropic gravity framework, this “patch” takes on entirely new and deeper physical meaning: it is no longer a mathematical correction, but the entropy of holographic edge modes.

This chapter will delve into this issue, proving that the GHY term, Hayward corner term, and null boundary terms are all necessary requirements for maintaining mathematical completeness and physical unitarity of the theory.

13.1 Boundary Term Problem in Variational Principles and Mathematical Completeness

Any physical theory involving second-order differential equations (such as gravity) typically contains first or second derivatives of fields in its action. When we vary the action $I[\varphi ]$ on a bounded region $M$, integration by parts (IBP) necessarily produces boundary terms. If these boundary terms cannot naturally vanish under given boundary conditions, the variational principle is ill-posed, meaning the functional derivative does not exist.

This section will rigorously analyze how this problem manifests in gravitational theory and define what “variational completeness” means.

13.1.1 Variational Crisis of Einstein-Hilbert Action

Consider the standard Einstein-Hilbert action (without boundary terms):

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

We want to calculate its variation $\delta I_{\text{EH}}$ under metric perturbations $g_{\mu \nu }\to g_{\mu \nu }+\delta g_{\mu \nu }$.

Using the Palatini identity $\delta R_{\mu \nu }={\nabla }_{\rho }(\delta {\Gamma }_{\mu \nu }^{\rho })-{\nabla }_{\nu }(\delta {\Gamma }_{\mu \rho }^{\rho })$, we can decompose the variation into bulk and boundary parts:

$$\delta I_{\text{EH}}=\frac{1}{16\pi G}{\int }_M{G}_{\mu \nu }\delta g^{\mu \nu }\sqrt{-g}{\mathrm{d}}^{4}x+\frac{1}{16\pi G}{\int }_M{\nabla }_{\mu }{v}^{\mu }\sqrt{-g}{\mathrm{d}}^{4}x$$

where the boundary flow vector $v^{\mu }$ is:

$$v^{\mu }=g^{\alpha \beta }\delta {\Gamma }_{\alpha \beta }^{\mu }-g^{\mu \alpha }\delta {\Gamma }_{\alpha \beta }^{\beta }$$

Using Stokes’ theorem, the second term transforms into an integral over boundary $\partial M$. Let the boundary unit normal vector be $n^{\mu }$ ($\epsilon =n^2=\pm 1$), and the induced metric be $h_{ab}$. The boundary term can be written as:

$$\delta I_{\partial M}=\frac{\epsilon }{16\pi G}{\int }_{\partial M}{v}^{\mu }{n}_{\mu }\sqrt{|h|}{\mathrm{d}}^{3}y$$

13.1.2 Failure of Dirichlet Conditions and Normal Derivative Dilemma

For the principle of least action $\delta I=0$ to derive Einstein’s equations $G_{\mu \nu }=0$, we must require the boundary term $\delta I_{\partial M}$ to vanish.

Usually, physical systems follow Dirichlet boundary conditions: fixing field values on the boundary. In gravity, this means fixing the induced metric $h_{ab}$ on the boundary, i.e., $\delta h_{ab}=0$.

However, carefully analyzing the structure of $v^{\mu }{n}_{\mu }$, we find it contains variations of normal derivatives of the metric:

$$v^{\mu }{n}_{\mu }\sim n^{\rho }{h}^{\alpha \beta }{\nabla }_{\rho }(\delta g_{\alpha \beta })+\dots$$

This means that even if we fix the metric on the boundary ($\delta g{|}_{\partial M}=0$), we cannot simultaneously fix the normal derivative of the metric (${\partial }_n\delta g{|}_{\partial M}=0$), unless we over-constrain the system (leading to no solutions to the equations of motion).

Conclusion:

For theories containing only $I_{\text{EH}}$, fixing the boundary metric does not make the boundary variation term vanish. The functional derivative $\frac{\delta I}{\delta g_{\mu \nu }}$ is undefined at the boundary. This is like trying to differentiate a function $f(x)$ at an endpoint where the function is not differentiable. This is variational incompleteness.

13.1.3 Definition of Variational Completeness

To fix this mathematical pathology, we need to modify the action or boundary conditions.

Definition 13.1.1 (Variational Completeness)

An action functional $I[\varphi ]$ is called variationally complete (or differentiable) under a given set of boundary conditions $\mathcal{C}$ if for any perturbation $\delta \varphi$ in the tangent space satisfying those boundary conditions, the variation of the action can be strictly expressed in volume integral form:

$$\delta I={\int }_{M}E(\varphi )\cdot \delta \varphi$$

without containing any non-zero boundary residual terms. At this point, the Euler-Lagrange equations $E(\varphi )=0$ are the extremal conditions of this variational problem.

To achieve variational completeness for gravity, we must add a boundary counter-term $I_{\text{bdy}}$ to the original action $I_{\text{EH}}$ such that:

$$\delta (I_{\text{EH}}+I_{\text{bdy}}){|}_{\delta h_{ab}=0}=(\text{bulk term})+0$$

This means $\delta I_{\text{bdy}}$ must precisely cancel the normal derivative flux produced by $I_{\text{EH}}$.

13.1.4 Physical Perspective: From Mathematical Patch to Physical Degrees of Freedom

From the traditional geometric perspective, adding boundary terms (such as the GHY term) is often viewed as a mathematical patching technique (“Fixing the variational principle”). But in our QCA Information Ontology and IGVP framework, boundary terms have extremely profound ontological status.

1. Holographic Duality: In the holographic principle, bulk physics and boundary physics are dual. The bulk action $I_{\text{EH}}$ describes dynamics in the bulk, while the boundary action $I_{\text{bdy}}$ describes dynamics of boundary degrees of freedom (Edge Modes).

2. Entropy Partition Function: In the Euclidean path integral formulation, the partition function is $Z=\int Dg{e}^{-I}$. For black holes or causal diamonds, the classical saddle-point approximation gives $Z\approx e^{-I_{class}}$. If boundary terms are ignored, the value of $I_{class}$ (and the derived entropy $S=(1-\beta {\partial }_{\beta })\ln Z$) is usually incorrect, or even zero.

3. Boundary as System: When we delineate a small causal diamond to define generalized entropy $S_{\text{gen}}$, we artificially cut spacetime. This cut severs quantum entanglement, exposing edge modes. Boundary terms precisely count the energy or entropy of these severed edge modes.

Therefore, variational completeness is not just a mathematical requirement—it is a requirement for physical system integrity. A theory without correct boundary terms is like a droplet theory that doesn’t consider surface tension, unable to correctly describe the thermodynamic properties of the system.

In the next section 13.2, we will specifically derive this necessary boundary term—the Gibbons-Hawking-York term—and reveal its direct connection with extrinsic curvature and entropy.