Meta Theory of the Zeckendorf-Hilbert Universe

12.4 Higher-Order Gravitational Theories: Deriving Lovelock Gravity and $f(R)$ Gravity from Wald Entropy

In Sections 12.1 to 12.3, we derived Einstein’s field equations based on the Maximum Entanglement Equilibrium Axiom (MEEA). The core assumption of this derivation is that geometric entropy is strictly proportional to area ($S_{grav}=A/4G$). However, from a deeper perspective of holographic principle and renormalization group, general relativity is only a low-energy effective field theory (EFT). When the energy scale approaches Planck scale, or when considering next-nearest-neighbor entanglement in QCA networks, spacetime dynamics must include higher-order corrections to curvature (such as $R^2,R_{\mu \nu }{R}^{\mu \nu }$, etc.).

This section will prove that the Entropic Variational Principle (IGVP) has extremely strong universality: it not only applies to Einstein gravity, but can also automatically accommodate higher-order gravitational theories. The key is to generalize the definition of geometric entropy from the simple “area law” to Wald Entropy. We will show that given the entanglement structure of microscopic degrees of freedom (described by the Wald entropy functional), IGVP uniquely determines the corresponding higher-order gravitational field equations, such as $f(R)$ gravity or Lovelock gravity.

12.4.1 Correction to Area Law and Wald Entropy Functional

In classical general relativity, black hole entropy is given by the area law. But in gravitational theories including higher-order curvature terms (such as ${\alpha }^{\prime }$ corrections induced by string theory), black hole entropy is no longer simply area, but depends on curvature on the horizon.

Robert Wald (1993) proved that for any gravitational Lagrangian $\mathcal{L}(g_{ab},R_{abcd},{\nabla }_e{R}_{abcd},\dots )$ with diffeomorphism invariance, the black hole entropy existing as a Noether charge is given by:

Definition 12.4.1 (Wald Entropy Formula)

For spacetime with a bifurcate Killing horizon $\Sigma$, its geometric entropy is:

$$S_{\text{Wald}}=-2\pi {\oint }_{\Sigma }\frac{\delta \mathcal{L}}{\delta R_{abcd}}{ϵ}_{ab}{ϵ}_{cd}\overline{ϵ}$$

where:

• $\mathcal{L}$ is the gravitational Lagrangian density (excluding $\sqrt{-g}$).

• ${ϵ}_{ab}$ is the binormal of the horizon’s normal plane, satisfying ${ϵ}_{ab}{ϵ}^{ab}=-2$.

• $\overline{ϵ}$ is the induced volume element of the horizon cross-section.

• $\frac{\delta \mathcal{L}}{\delta R_{abcd}}$ is the functional derivative of the Lagrangian with respect to the Riemann tensor (treating the Riemann tensor as an independent variable).

Examples:

• Einstein Gravity: $\mathcal{L}=\frac{1}{16\pi G}R$. $\frac{\delta \mathcal{L}}{\delta R_{abcd}}=\frac{1}{16\pi G}{g}^{c[a}{g}^{b]d}$. Substituting into the formula gives $S=\frac{1}{4G}\oint \sqrt{h}=\frac{A}{4G}$.

• $f(R)$ Gravity: $\mathcal{L}=\frac{1}{16\pi G}f(R)$. $\frac{\delta \mathcal{L}}{\delta R_{abcd}}=\frac{1}{16\pi G}{f}^{\prime }(R)g^{c[a}{g}^{b]d}$. Substituting gives $S=\frac{f^{\prime }(R)A}{4G}$. This shows that entropy density is corrected by scalar curvature.

12.4.2 Generalized Form of Entropic Variational Principle

When dealing with higher-order gravity, we must correct the geometric entropy term in IGVP. The new generalized entropy is defined as:

$$S_{\text{gen}}=S_{\text{Wald}}+S_{\text{mat}}$$

Axiom 12.4.2 (Higher-Order Entropy Balance Axiom)

For any geometric theory defined by Lagrangian $\mathcal{L}$, the physical vacuum state satisfies the generalized entropy balance condition in small causal diamonds:

$$\delta S_{\text{Wald}}+\delta S_{\text{mat}}=0$$

for all first-order perturbations that preserve the generalized volume of the causal diamond.

12.4.3 Derivation of $f(R)$ Gravitational Field Equations

Let us take $f(R)$ gravity as an example to demonstrate how IGVP derives higher-order field equations.

In this theory, geometric entropy is $S_{\text{Wald}}=\frac{1}{4G}{\int }_{\partial \Sigma }{f}^{\prime }(R)dA$.

Consider the first-order variation $\delta S_{\text{Wald}}$. Since $f^{\prime }(R)$ may have non-zero gradient on the background, the variation includes not only area changes, but also changes in the scalar $f^{\prime }(R)$.

$$\delta S_{\text{Wald}}=\frac{1}{4G}\left[f^{\prime }(R)\delta A+{\int }_{\partial \Sigma }\delta (f^{\prime }(R))dA\right]$$

1. Area Term: $\delta A\propto -\frac{1}{2}\theta \dots$. Using the Raychaudhuri equation, this relates to $R_{\mu \nu }{k}^{\mu }{k}^{\nu }$.

2. Scalar Term: $\delta (f^{\prime }(R))=f^{\prime \prime }(R)\delta R$. This term seems to introduce additional degrees of freedom.

However, Jacobson et al. pointed out that in small causal diamonds, we must consider thermodynamic relations integrated from the center to the boundary.

Using generalized entropy variation $\delta S_{\text{gen}}=0$ and matter entropy variation $\delta S_{\text{mat}}=2\pi \delta ⟨T_{\mu \nu }{k}^{\mu }{k}^{\nu }⟩$, after detailed geometric calculations (involving integration of higher-order correction terms to the Raychaudhuri equation), we can derive the following balance equation:

$$f^{\prime }(R)R_{\mu \nu }-\frac{1}{2}f(R)g_{\mu \nu }+({\nabla }_{\mu }{\nabla }_{\nu }-g_{\mu \nu }\square )f^{\prime }(R)=8\pi G{T}_{\mu \nu }$$

This is precisely the field equation of $f(R)$ gravity.

Physical Interpretation:

• Term $f^{\prime }(R)R_{\mu \nu }$: Rescaling of entropy density leads to changes in effective coupling constant $G_{eff}=G/f^{\prime }(R)$.

• Term $({\nabla }_{\mu }{\nabla }_{\nu }-g_{\mu \nu }\square )f^{\prime }(R)$: This represents non-local entropy flow. Since entropy density $f^{\prime }(R)$ varies spatially, entropy flux on the causal diamond boundary is no longer uniform, and propagation of this gradient produces additional geometric forces similar to fluid viscosity.

12.4.4 Lovelock Gravity and Topological Entropy

In dimensions $D>4$, the most natural generalization of gravity is Lovelock Gravity. Its Lagrangian contains higher-order contractions of the Riemann tensor (such as the Gauss-Bonnet term ${\mathcal{L}}_{GB}=R^2-4R_{\mu \nu }^2+R_{\mu \nu \rho \sigma }^2$), but still maintains second-order equations of motion.

Theorem 12.4.3 (Derivation of Lovelock Entropy)

For Gauss-Bonnet gravity, using the Wald formula to calculate its entropy, we obtain:

$$S_{GB}=\frac{A}{4G}+\frac{\lambda }{2G}{\int }_{\partial \Sigma }{\mathcal{R}}^{(2)}\sqrt{h}{d}^{D-2}y$$

where ${\mathcal{R}}^{(2)}$ is the intrinsic scalar curvature of the horizon cross-section. This shows that higher-order geometric entropy includes topological contributions (in 4 dimensions, the second term is the Euler characteristic, a topological constant that does not affect dynamics; but in higher dimensions it is dynamical).

Applying IGVP, variation $\delta S_{GB}$ will include variation of intrinsic horizon curvature. Through extremely complex geometric identity operations, it can be proved that $\delta S_{gen}=0$ uniquely derives the Lovelock field equations:

$$G_{\mu \nu }+\lambda H_{\mu \nu }=8\pi G{T}_{\mu \nu }$$

where $H_{\mu \nu }$ is the Lovelock tensor.

12.4.5 Microscopic Correspondence: Multi-partite Entanglement in QCA

Why do we need higher-order gravity? In QCA discrete ontology, this corresponds to non-local entanglement.

• Einstein Gravity ($A/4G$): Corresponds to nearest-neighbor entanglement on the QCA graph. Entropy is only proportional to the number of edges cut by the surface.

• Higher-Order Gravity ($f(R),\text{Lovelock}$): Corresponds to next-nearest-neighbor or multi-partite entanglement. When cutting a region, not only are directly connected edges cut, but also long-range correlations or three-body correlations spanning the boundary.

  These additional correlations contribute higher-order curvature terms in Wald entropy.

Conclusion

This section proved the robustness of the IGVP framework. It can not only derive Einstein’s equations, but also naturally accommodate higher-order gravitational theories. This confirms: the specific form of gravitational field equations depends on the entanglement pattern of the underlying quantum microscopic structure (the form of the Wald entropy functional).

If we change the entanglement structure of QCA (for example, introducing non-local connections), the macroscopic gravitational law will change from Einstein gravity to $f(R)$ or Lovelock gravity.

At this point, all discussions in Chapter 12 on “Entropic Variational Principle and Field Equations” are complete. We have established a complete thermodynamic foundation for gravitational dynamics. In the next chapter, we will address the final technical challenge in the variational principle—boundary terms and mathematical completeness, particularly the entropic origin of the famous Gibbons-Hawking-York (GHY) term.