#
An Entropic Dynamics Approach to Geometrodynamics^{ †}

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^{†}

## Abstract

**:**

## 1. Introduction

## 2. Reviewing the Entropic Dynamics of Infinitesimal Steps

**Maximum Entropy—**The goal is to predict the evolution of the scalar field $\chi $. To this end we make one major assumption: in ED, the fields follow continuous trajectories such that finite changes can be analyzed as an accumulation of many infinitesimally small ones. Thus we are interested in obtaining the probability $P\left[{\chi}^{\prime}|\chi \right]$ of a transition from an initial configuration $\chi $ to a neighboring ${\chi}^{\prime}=\chi +\mathsf{\Delta}\chi $. This is accomplished via the Maximum Entropy (ME) method by maximizing the entropy functional,

## 3. Some Space-Time Kinematics

## 4. Entropic Time

**Ordered instants—**Central to our formulation of entropic time is the notion of an instant of time. In a properly relativistic theory in curved space-time, such a notion is provided by an arbitrary space-like surface, denoted $\sigma $ (see e.g., [9]). This allows us to define the epistemic state at the instant $\sigma $, characterized by the probability ${\rho}_{\sigma}\left[\chi \right]$, the drift potential ${\varphi}_{\sigma}\left[\chi \right]$, etc.

**Duration—**To complete our construction of time we must specify the duration between instants. In ED time is defined so that motion looks simple. Since for short steps the dynamics is dominated by fluctuations, Equation (6), the specification of the time interval is achieved through an appropriate choice of the multipliers ${\alpha}_{x}$. Moreover, following [7], since we deal here with the duration between curved spaces, this notion of separation should be local, and it is natural to define duration in terms of the local proper time ${\delta}_{x}^{\perp}$. More specifically, let

**The local-time diffusion equations—**The dynamics expressed in integral form by (9) and (10) can be rewritten in differential form as an infinite set of local equations, one for each spatial point,

## 5. Geometrodynamics Driven by Entropic Matter

**Path independence—**In a relativistic theory there are many ways to evolve from an initial instant to a final one, and each way must agree. This is the basic insight by DHKT in their development of manifestly covariant dynamical theories. The implementation of this idea, through the principle of path independence, leads to a set of Poisson brackets (see e.g., [8])

**The phase space—**The Equations (12)–(15) of path independence are universal. That is, if the dynamics is to be relativistic, these equations must hold. Whatever the choice of canonical variables, or whether the background is fixed or dynamical, the same “algebra” must hold.

**The super-momentum—**We now turn our attention to the local Hamiltonian generators ${H}_{Ax}$, and more specifically, we look to provide explicit expressions for these generators in terms of the canonical variables by solving the Poisson brackets Equations (12)–(14). We begin with the tangential generator ${H}_{ix}$, which generates changes in the canonical variables by dragging them parallel to the space $\sigma $. As shown in [8,10], the tangential generator can be shown to split

**The super-Hamiltonian—**As pertains to the super-Hamiltonian, a similar decomposition does not occur. But following Teitelboim [8] let us suggestively rewrite ${H}_{\perp x}$ as

## 6. The Dynamical Equations

**The Schrödinger equation—**We are interested in the dynamical evolution of the ensemble variables $\rho $ and $\mathsf{\Phi}$, however, this very same dynamics can be expressed equivalently by the introduction of complex variables ${\mathsf{\Psi}}_{t}={\rho}_{t}^{1/2}{e}^{i{\mathsf{\Phi}}_{t}}$ and ${\mathsf{\Psi}}_{t}^{*}={\rho}_{t}^{1/2}{e}^{-i{\mathsf{\Phi}}_{t}}$ (we use units in which $\mathit{\u0127}=1$). The reason these variable turn out to be useful, is that the dynamical equations turn out to take a familiar form. In particular, we have

**Geometrodynamics—**To complete the description of the dynamics we will determine the evolution of the geometrical variables $({g}_{ij},{\pi}^{ij})$. Beginning with the metric, its time evolution, generated by the super-Hamiltonians ${H}_{Ax}$ given above, after a straightforward computation yields

## 7. Conclusions and Discussion

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Ipek, S.; Caticha, A.
An Entropic Dynamics Approach to Geometrodynamics. *Proceedings* **2019**, *33*, 13.
https://doi.org/10.3390/proceedings2019033013

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Ipek S, Caticha A.
An Entropic Dynamics Approach to Geometrodynamics. *Proceedings*. 2019; 33(1):13.
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Ipek, Selman, and Ariel Caticha.
2019. "An Entropic Dynamics Approach to Geometrodynamics" *Proceedings* 33, no. 1: 13.
https://doi.org/10.3390/proceedings2019033013