Rigorous foundation of the arrow of time without a Past Hypothesis: typicality measure on shape space, large-N Janus-point statistics, relativistic extension, and the status of universal entropy

The thermodynamic arrow of time is standardly explained by a special low-entropy initial condition of the universe, the Past Hypothesis. Barbour, Koslowski and Mercati (BKM) proposed an alternative: in the scale-free relational dynamics of the zero-energy Newtonian N-body universe on shape space, typical solutions pass through a central point of minimal size (the Janus point) at which the shape complexity is low, so that entropy-like quantities grow away from that central minimum in both time directions, yielding two opposite arrows without a Past Hypothesis. Lazarovici and Reichert's analysis of this program (arXiv:1809.04646) proved that Boltzmann entropy of scale-invariant macrovariables is ill-defined on the absolute phase space and identified six unresolved foundational issues: the justification and uniqueness of the normalizable typicality measure on the reduced phase space; the unproven typicality of low-complexity Janus points for general N; extension beyond Newtonian gravity; the tension between relational fundamentality and the scale-dependence of entropy; the regularization ambiguity of the non-normalizable Carroll-Chen measure; and the status of entaxy-like relational statistics. The problem posed here is to give the no-Past-Hypothesis typicality explanation a rigorous, non-circular foundation: a justified (and ideally unique) typicality measure, a proof or refutation of the large-N typicality claim, extension of the statistical analysis to relativistic gravitation, and a principled account of the objectivity and physical significance of entropy in a fundamentally relational universe.

Background

The time-asymmetry of macroscopic physics - ice melts, gases disperse, heat flows from hot to cold, and the reverse processes are never observed - is explained in Boltzmannian statistical mechanics by combining time-reversal-symmetric microscopic laws with a special initial condition: the universe began in a state of extremely low Boltzmann entropy. This posit is called the Past Hypothesis (PH). The Boltzmann entropy of a macrostate is $k_B\log\lvert\Gamma_M\rvert$, where $\Gamma_M$ is the region of Hamiltonian phase space occupied by the microstates realizing that macrostate, and a microstate is called typical (in some region) if it lies in a subset of overwhelmingly large measure, usually with respect to the Liouville measure (the natural volume measure on phase space) or a microcanonical measure (the Liouville measure restricted to a fixed-energy surface). The PH invites an obvious explanatory regress: why was the initial condition so special?

Two research programs have tried to eliminate it. Carroll and Chen considered an eternally inflating multiverse in which baby universes nucleate as fluctuations of an inflaton field in de Sitter space. The phase space of such a universe is unbounded and the natural measure on it is non-normalizable (has infinite total volume), so typicality is not defined until one regularizes the measure by a cutoff - and different regularizations license different, mutually incompatible, probabilistic inferences about time-symmetric histories that descend from an entropy minimum in the middle and become more entropic in both temporal directions. Goldstein, Tumulka and Zanghì exhibited this ambiguity sharply on a toy model, showing that two rival inference rules correspond to two inequivalent cutoffs. This is issue (5) of the present problem.

The second program, due to Barbour, Koslowski and Mercati (BKM), is relational. Following the Leibnizian idea that only inter-particle relations are physically real, one eliminates absolute position, orientation, and scale: the shape space $S$ of the Newtonian $N$-body problem is its configuration space modulo translations, rotations, and dilations (for $N=3$ particles, the shapes of triangles form a projective 2-sphere). Restricting to zero total momentum $P=\sum_i p_i\equiv 0$ and zero total angular momentum $L=\sum_i q_i\times p_i\equiv 0$, and to zero total energy $E=0$ (at which the Newtonian $N$-body problem with the homogeneous gravitational potential is invariant under dilations), the dynamics on shape space is invariant under a common rescaling of the momenta, and the reduced phase space that parametrizes solutions - the projective cotangent bundle $PT^{*}S$, i.e. the cotangent bundle $T^{*}S$ with the overall scale of the momenta divided out - is compact, so its uniform volume measure is normalizable. BKM's key findings are: (i) Newtonian gravity, seen on shape space, is effectively repulsive at total collisions, so typical solutions pass from a period of contraction through a point of minimal size - the Janus point - and re-expand, defining two opposite arrows of time emanating from it; (ii) a natural dimensionless (scale-invariant) macrovariable, the shape complexity $C_S=-V\sqrt{I}$, where $V<0$ is the Newton potential and $I$ the moment of inertia, measures how clustered the configuration is, and typically grows along both branches away from the Janus point; and (iii) with the uniform measure $\mu_{\varepsilon}$ on $PT^{*}S$ - constructed from mid-point (Janus-point) data, following Laplace's principle of indifference (the rule that equal volumes of the parameter space receive equal probability) - the statement that the Janus point typically has low shape complexity, so that complexity increases in both time directions, becomes well posed. BKM established this only for $N=3$ and supported it numerically for large $N$; on the unreduced (absolute) phase space they identified a corresponding gravitational Boltzmann entropy whose typical histories are U-shaped (entropy decreasing toward the Janus point in both time directions). BKM also proposed entaxy, an instantaneous shape-space analog of entropy built from $\mu_{\varepsilon}$.

Lazarovici and Reichert (LR) subjected the program to a detailed Boltzmannian analysis. They proved a no-go result (their Proposition 6.1 and Corollary 6.2): for any measure on the absolute phase space that is homogeneous of degree $d$ under dilations, $\mu(\lambda A)=\lambda^d\mu(A)$ - a class that includes the Liouville measure and the $E=0$ microcanonical measure - the macroregion of any scale-invariant macrovariable has measure 0 or $\infty$, so the corresponding Boltzmann entropy is ill-defined. Boltzmann entropy is thus intrinsically tied to absolute scales and not manifestly relational. They also criticized entaxy: it is defined through a non-stationary measure and should not be regarded as a shape analog of the Boltzmann entropy. Their overall verdict is that the shape-space theory can explain the gravitational arrows without a Past Hypothesis precisely by giving up absolute scales, but that its statistical foundation is incomplete in six specific ways: the justification and uniqueness of $\mu_{\varepsilon}$ (its construction is not entirely canonical, involving the choice of a metric on shape space and the mid-point cut); the unproven typicality of low-complexity Janus points for general $N$; the extension to other microscopic theories (relativistic gravitation, cosmology, field degrees of freedom) and other macrovariables; the reconciliation of relational fundamentality with the scale-dependence of entropy; the residual ambiguity of the Carroll framework if one declines the relational reduction; and the conceptual status of entaxy-like statistics.

Problem Statement

The objective is to put the no-Past-Hypothesis typicality explanation of the arrow(s) of time on a rigorous, non-circular foundation, within the framework of the scale-free Newtonian $N$-body universe and its extensions, by resolving the six issues that Lazarovici and Reichert (arXiv:1809.04646) identify as unresolved. Each of the six components is independently mandatory: a resolution must address all six, and a partial resolution of fewer than six of them does not resolve the task. The components are jointly constrained by one scientific objective: an explanation of temporal asymmetry that derives the low-entropy (low-complexity) character of the universe's central state as a typicality fact of the dynamics under a justified measure, rather than postulating it - and that therefore cannot borrow its probability measure or its statistic from the thermodynamic asymmetry it is meant to explain. Specifically:

(1) Measure selection and uniqueness. Supply a physical or mathematical principle that justifies - or demonstrably rules out - the normalizable uniform measure $\mu_{\varepsilon}$ on the reduced shape phase space $PT^{*}S$ (or a defensible alternative measure), addressing the non-canonical ingredients of its construction (the choice of metric on shape space; the mid-point, i.e. Janus-point, cut through solution trajectories) and defending the choice against competing constructions. The principle must be non-circular: it may not presuppose the thermodynamic arrow of time or entropic facts it is invoked to explain.

(2) Large-N typicality. Under the justified measure, prove or refute the typicality of low shape complexity at the Janus point for general $N$ - statements of the form $\mu_{\varepsilon}(I_1)/\mu_{\varepsilon}(PT^{*}S)\approx 1$, where $I_1$ is the macroregion of mid-point data with $C_S\in[C_{min},\alpha C_{min}]$ ($C_{min}$ the minimal possible complexity, $1<\alpha\ll\infty$; the claim is reported for $\alpha<2$ at large $N$) - currently established only for $N=3$ and numerically.

(3) Extension beyond Newtonian gravity. Extend the statistical analysis - measure, macrovariables, and typicality statement - beyond the zero-energy Newtonian $N$-body model to at least one relativistic theory of gravitation (general relativity or a relational/scale-invariant reformulation thereof), establishing whether the analogue of the U-shaped (two-arrow) structure of typical histories persists, with the same non-circularity requirement on the measure.

(4) Entropy of the universe under relationalism. Given the Lazarovici-Reichert no-go result that macroregions of scale-invariant macrovariables have measure 0 or $\infty$ under homogeneous measures: either explain why Boltzmann entropy - which depends on absolute scales - remains an objective, physically significant quantity (for subsystems, or for the universe as a whole) in a fundamentally relational, scale-free theory, or reconceptualize the thermodynamic concepts in purely relational terms. The answer must resolve the dilemma, not restate it: it must state which of the two alternatives it adopts and support it with a positive account, not merely an argument against the other alternative.

(5) The absolutist ambiguity. Without adopting the relational reduction, settle which regularization (cutoff) of the non-normalizable phase-space measure in the Carroll-Chen framework - if any - licenses sound typicality inferences about time-symmetric histories, resolving the ambiguity exhibited by Goldstein-Tumulka-Zanghì.

(6) Relational statistics. Determine whether any dynamically natural relational statistic - entaxy as introduced by BKM, or a successor such as complexity or variety - can play an informative role in explaining temporal asymmetry comparable to Boltzmann entropy, including whether it can be equipped with a stationary measure (one invariant under the dynamics) and whether the arrows it defines agree with the entropic arrows of subsystems. The answer must state which statistic(s) it assesses and, for each, give a determinate verdict on both the stationary-measure and arrow-agreement conditions.

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