Volume III: The Entropic Origin of Gravity and Geometry

Part VII: Foundations of Geometric Dynamics

Chapter 11: Causal Geometry and Generalized Entropy

In Volume I, we established the discrete ontology, confirming that information is the foundation of physical reality. In Volume II, we argued that “time is matter,” proving that the microscopic rate of time flow is determined by the density of states, and revealing the topological skeleton of time (time crystals). Now, in Volume III, we complete the final step of physical unification: the dynamical origin of gravity and geometry.

We will prove that the curved spacetime of general relativity is not an a priori background stage, but rather an emergent phenomenon of quantum information entanglement in the thermodynamic limit. Just as gas pressure arises from the statistical motion of molecules, gravity arises from the tendency of quantum entanglement entropy to maximize (entropic force). To establish this theory, we need a fundamental geometric unit that can define “entropy”—the Small Causal Diamond.

11.1 Geometric Properties and Conformal Structure of Small Causal Diamonds

To apply thermodynamic laws to spacetime, we first face the question: where is the boundary of the system? In traditional thermodynamics, systems are enclosed by container walls. In general relativity, there are no fixed rigid containers. Jacobson and other pioneers pointed out that the natural “container” in spacetime is defined by the causal structure itself.

This section will rigorously define small causal diamonds and derive their geometric properties. These diamonds are not only microscopes for probing spacetime curvature, but also carriers of the holographic principle at the local level.

11.1.1 Constructive Definition of Causal Diamonds

Consider two points $p$ (past vertex) and $q$ (future vertex) in a Lorentzian manifold $(\mathcal{M},g)$, satisfying that $p$ is in the causal past of $q$ ($p\ll q$), and there exists a unique timelike geodesic $\gamma$ between them. Let the length (proper time) of this geodesic be $\tau$.

Definition 11.1.1 (Causal Diamond)

The causal diamond $D(p,q)$ generated by $p$ and $q$ is defined as the intersection of the interior of the future light cone of $p$ and the interior of the past light cone of $q$:

$$D(p,q)\equiv J^{+}(p)\cap J^{-}(q)$$

where $J^{\pm }$ denotes the causal future/past sets.

The edge $\partial \Sigma$ of the diamond is the interface between the future light cone $\partial J^{+}(p)$ of $p$ and the past light cone $\partial J^{-}(q)$ of $q$. In $d$-dimensional spacetime, $\partial \Sigma$ is topologically a $(d-2)$-dimensional sphere $S^{d-2}$. It is the domain of definition for the holographic entropy $S=A/4G$.

Definition 11.1.2 (Small Causal Diamond)

If the proper time interval $\tau$ between $p$ and $q$ is much smaller than the curvature radius of spacetime $L_{curv}\sim 1/\sqrt{|R_{\mu \nu \rho \sigma }|}$, then $D(p,q)$ is called a small causal diamond.

In Riemann Normal Coordinates (RNC) with the geodesic midpoint $O$ as the origin, the diamond can be approximated as a standard diamond in flat Minkowski space, with curvature effects appearing as second-order perturbations to the metric.

11.1.2 Geometric Expansion and Area Deficit Theorem

To connect geometry with gravity, we need to calculate how curvature changes the geometric properties of the diamond. In flat space, the area of the diamond edge (i.e., the maximal cross-section) is uniquely determined by the time interval $\tau$. In curved spacetime, this area changes due to geometric focusing effects.

Let the lifetime of the diamond be $\tau$. In the local inertial frame at point $O$, the diamond edge is a sphere of radius $l\approx \tau /2$.

Theorem 11.1.3 (Area Deficit of Small Diamonds)

In $d$-dimensional curved spacetime, the deviation $\delta A$ of the area $A$ of the small causal diamond edge $\partial \Sigma$ from the flat space area $A_{flat}$ is controlled by the Einstein tensor $G_{\mu \nu }$. To lowest order in $\tau$, we have:

$$\delta A\equiv A_{flat}-A=\frac{{\Omega }_{d-2}{\tau }^d}{d^2-1}{G}_{00}+\mathcal{O}({\tau }^{d+1})$$

Or in a more general tensor form, let $u^{\mu }$ be the tangent vector (time direction) of the geodesic connecting $p,q$:

$$\delta A=\frac{{\Omega }_{d-2}(\tau /2{)}^d}{d^2-1}{R}_{ab}{u}^a{u}^b\cdot (\text{dimension factor})$$

where $G_{00}=R_{00}-\frac{1}{2}{g}_{00}R$ is the time component of the Einstein tensor.

Proof Outline:

1. Metric Expansion: In RNC, the metric expands as $g_{\mu \nu }(x)={\eta }_{\mu \nu }-\frac{1}{3}{R}_{\mu \alpha \nu \beta }{x}^{\alpha }{x}^{\beta }+\mathcal{O}(x^3)$.

2. Area Element Expansion: The determinant $\sqrt{h}$ of the induced metric is corrected by Riemann curvature. The spherical area integral $A=\int \sqrt{h}d\Omega$.

3. Integral Averaging: Integrating quadratic terms in coordinates $x^i$ over the sphere, using spherical symmetry $\int x^i{x}^{j}d\Omega \propto {\delta }^{ij}$, the Riemann tensor contracts to the Ricci tensor $R_{\mu \nu }$.

4. Geodesic Deviation: The light cone boundary itself is affected by the geodesic deviation equation, leading to corrections to the radius $l$. Combining both effects, we finally obtain that the area deficit is proportional to $G_{00}$.

Physical Significance:

This geometric theorem is key to deriving the gravitational field equations.

• When $G_{00}>0$ (positive energy density), $\delta A>0$, meaning the actual area $A$ is smaller than the flat space area. This represents the gravitational focusing effect—energy causes light rays to converge, thereby contracting the cross-sectional area of the wavefront.

• Holographic Connection: If we regard the area $A$ as the carrier of entropy ($S\propto A$), then geometric focusing means a reduction in pure geometric entropy. To maintain thermodynamic equilibrium, this reduction in entropy must be compensated by an increase in matter entropy, which is the core of Chapter 12’s entropic variational principle.

11.1.3 Conformal Structure and Modular Flow

The reason causal diamonds occupy a central position in information physics is also due to their unique conformal symmetry.

Theorem 11.1.4 (Conformal Equivalence of Diamonds)

In Minkowski space, any causal diamond is conformally equivalent to a Rindler wedge through a conformal transformation. This means that the physics inside the diamond can be described by the thermodynamics of an accelerated observer confined within a horizon.

Definition 11.1.5 (Conformal Killing Vector / CKV)

There exists a vector field ${\zeta }^{\mu }$ that vanishes on the diamond boundary $\partial D$ (horizon) and generates a conformal flow in the interior:

$${\mathcal{L}}_{\zeta }{g}_{\mu \nu }\propto g_{\mu \nu }$$

This vector field $\zeta$ is precisely the geometric generator of the modular Hamiltonian $K$. According to the Bisognano-Wichmann theorem, for the vacuum state ${\rho }_{vac}$, its reduced density matrix satisfies ${\rho }_D=e^{-K}/Z$, and $K$ is proportional to the generalized Lorentz generator corresponding to $\zeta$:

$$K=\frac{2\pi }{\hslash }{\int }_{\Sigma }{T}_{\mu \nu }{\zeta }^{\mu }d{\Sigma }^{\nu }$$

This shows that the vacuum state inside a small causal diamond is naturally in a thermal equilibrium state with respect to “modular time” (the flow parameter along $\zeta$), with temperature equal to the Unruh temperature.

11.1.4 Holographic Screen and Information Truncation

The boundary $\partial \Sigma$ of a small causal diamond acts as a local holographic screen.

1. Information Bound: According to the Bekenstein bound (Section 1.1), the maximum information that can be contained inside the diamond is determined by the area of $\partial \Sigma$.

2. Causal Closure: The diamond is a self-consistent causal unit. Any information entering the diamond (from $J^{-}(q)$) must eventually pass through the boundary or be recorded by it, or be initial data emitted from point $p$.

3. UV/IR Connection: The size $\tau$ of the diamond provides a natural infrared cutoff (IR Cutoff), while the holographic principle provides an ultraviolet cutoff (UV Cutoff). When we take the limit $\tau \to 0$, we are probing the microscopic structure of spacetime.

Summary

This section defined small causal diamonds and derived the area deficit formula $\delta A\propto G_{00}$. This reveals a profound geometric fact: spacetime curvature is equivalent to a deficit in holographic information capacity.

This geometric preparation paves the way for Chapter 12: we will prove that to maximize the total entropy of the system (geometric entropy + matter entropy), spacetime must curve, and the manner of curvature must strictly follow Einstein’s field equations.