[Paper Review] IIB matrix model: Extracting the spacetime metric

F. R. Klinkhamer|arXiv (Cornell University)|Aug 26, 2020

Black Holes and Theoretical Physics12 references4 citations

TL;DR

This paper investigates how the spacetime metric—specifically the Minkowski and spatially flat Robertson–Walker metrics—can emerge from the large-N master field of the IIB matrix model. Using an analytic integral formula for the inverse metric, it shows that by appropriately choosing the density function, correlation function, and propagator, the desired cosmological metrics can be reproduced in a 4D effective spacetime, demonstrating the feasibility of metric emergence from matrix model structures.

ABSTRACT

The large-$N$ master field of the Lorentzian IIB matrix model is of course not known, but we can assume that we already have it and investigate how the emerging spacetime metric could be extracted. We show that, in principle, it is possible to obtain both the Minkowski metric and the spatially flat Robertson-Walker metric.

Motivation & Objective

• To investigate whether the IIB matrix model's large-N master field can generate a classical spacetime metric, particularly the Minkowski and spatially flat Robertson–Walker metrics.
• To determine the functional forms of the density, correlation, and propagator functions required for metric emergence in a 4D effective spacetime.
• To demonstrate that the inverse metric can be systematically constructed from the master field using an analytic integral formulation.

Proposed method

• The paper uses an analytic integral expression for the emergent inverse metric, derived from the IIB matrix model's master field, involving the density of discrete spacetime points, a correlation function, and a localized propagator function.
• It performs a change of variables to center the integral around a spacetime point x, transforming the metric expression into a function of relative coordinates z = y - x.
• The method involves expanding the metric components in a Taylor series around t = 0 (cosmic time), enabling systematic comparison with the Robertson–Walker metric form.
• The authors derive explicit expressions for the Taylor coefficients of the spatial components of the inverse metric in terms of input parameters r₁₀ and s₄, which control the functional form of the correlation and density functions.
• By inverting the coefficient relations, the method allows for the direct specification of input functions to achieve any desired metric behavior up to quartic order in time.
• The construction is validated by showing that the matrix inverse of the derived inverse metric yields the standard covariant Robertson–Walker metric.

Experimental results

Research questions

• RQ1 Can the Minkowski metric emerge from the IIB matrix model’s master field using the proposed integral formulation of the spacetime metric?
• RQ2 Is it possible to generate the spatially flat Robertson–Walker metric from the same integral framework by tuning the functional forms of the density, correlation, and propagator functions?
• RQ3 What specific functional dependencies are required in the master field’s derived functions to reproduce the time-dependent components of the Robertson–Walker metric?
• RQ4 Can the construction be extended to produce metrics with non-zero spatial curvature (k = ±1) or regularized big-bang singularities?
• RQ5 How do the Taylor coefficients of the inverse metric relate to the underlying parameters of the master field, and can they be tuned to match known cosmological metrics?

Key findings

• The Minkowski metric can be recovered as a constant inverse metric when the input functions are chosen such that the spatial components are exactly 1 and the time component is exactly -1.
• The spatially flat Robertson–Walker metric can be generated by tuning the parameters r₁₀ and s₄ in the functional forms of the correlation and density functions, yielding a time-dependent inverse metric with components g⁰⁰ ≈ -1 and gᵐᵐ ≈ 1 / (1 + c₂t² + c₄t⁴ + ...).
• Explicit expressions are derived for the coefficients c₂ and c₄ of the time-dependent spatial components in terms of r₁₀ and s₄, showing that arbitrary values of these coefficients can be achieved by appropriate choice of input parameters.
• The inverse metric components g⁰⁰ and gᵐᵐ are found to be independent of the spacetime point x in the leading-order approximation, consistent with a homogeneous and isotropic spacetime.
• The method allows for the direct specification of input functions to achieve any desired Taylor coefficients C₂ and C₄ for the spatial components of the inverse metric, via the invertible relations (25a) and (25b).
• The construction is consistent with the matrix inverse of the derived inverse metric yielding the standard covariant Robertson–Walker metric, confirming the correctness of the emergent metric structure.