Relationalist Approach: Barbour, Koslowski & Mercati

• The relationalist approach defines dynamics through dimensionless shape configurations, eliminating absolute space, time, and scale.
• It employs a Jacobi-type action and best matching to remove gauge redundancies, leading to a pure shape space formulation.
• The framework provides new insights into quantum gravity and cosmology by reinterpreting singularities as smooth Janus point transitions.

The relationalist approach developed by Julian Barbour, Tim Koslowski, and Flavio Mercati (hereafter BKM) recasts classical and gravitational dynamics in fundamentally relational and scale-invariant terms. Rooted in the philosophical legacy of Mach, Poincaré, and their modern successors, this program eliminates absolute structures—space, time, and scale—by formalizing dynamics purely in terms of the evolution of relational configurations, or "shapes," with observable content consisting exclusively of dimensionless ratios and relative structures. It offers precise answers to the longstanding conceptual challenges of background independence and the arrows of time and could provide a new path toward quantum gravity.

1. Conceptual Foundations: From Machian Relationalism to Pure Shape Dynamics

The BKM approach builds on two key principles: spatial relationalism and temporal relationalism. The Mach–Poincaré Principle asserts that only relational initial data and their intrinsic rates uniquely determine the evolution of a closed system; all absolute structures are physically inert bookkeeping devices (Vassallo et al., 2022). In spatial relationalism, only distances, angles, and shapes among material bodies (or field configurations) possess objective significance—absolute space is denied. Temporal relationalism demands that all temporal notions (ordering, duration, flow) arise from intrinsic change in the system’s relational configuration, eschewing any external time parameter (Vassallo et al., 2022, Anderson, 2014).

The framework implements best matching: dynamics are determined by extremizing a Jacobi-type action with respect to the gauge group 𝒢 of physically irrelevant transformations (e.g., translations, rotations, dilatations, or diffeomorphisms plus local conformal transformations in gravitation) (Vassallo et al., 2022). This yields "shape space" as the reduced configuration space after quotienting by redundancies, with physical motion realized as unparametrized curves therein.

2. Mathematical Structure and Symmetry Trading

The relational construction proceeds via a Jacobi-type action. For the particle case, the Barbour–Bertotti (JBB) action is

$S_{JBB} = 2 \int d\lambda\ \sqrt{T\, U}$

where $T$ is the kinetic energy (best-matched), and $U$ is the potential. Varying with respect to the group parameters eliminates gauge redundancy and enforces relational constraints (e.g., vanishing total momentum and angular momentum) (Vassallo et al., 2022).

In geometrodynamics, the Jacobi action on 3-metrics $g_{ab}(x)$,

$$S_J = 2 \int d\lambda\ \int d^3x \sqrt{g}\ \sqrt{G^{abcd}(\dot g_{ab} - \mathcal{L}_\xi g_{ab} - \rho g_{ab})(\dot g_{cd} - \mathcal{L}_\xi g_{cd} - \rho g_{cd}) R[g]}$$

implements best matching with respect to spatial diffeomorphisms (generated by the shift $\xi^a$) and conformal transformations (parameterized by $\rho$), yielding momentum and conformal constraints (Vassallo et al., 2022).

The technique of symmetry trading is central: within a "linking theory" possessing both ADM refoliation invariance and local Weyl symmetry, distinct gauge fixings yield either standard ADM General Relativity or Shape Dynamics (SD). In SD, refoliation invariance is exchanged for local conformal invariance, with evolution governed by a global Hamiltonian in York time (mean extrinsic curvature of spatial slices), accompanied by momentum and conformal constraints (Vassallo et al., 2022, Anderson, 2014).

3. Shape Dynamics and Gravitational Arrows of Time

Shape Dynamics, in its pure form, is a theory of unparametrized curves on shape space, fully eliminating absolute time and scale (Vassallo et al., 2022). BKM demonstrated that in closed Newtonian $N$-body universes with $E=0$, $P=0$, $T$0, and all redundancies eliminated, the relational configuration space—shape space—admits a natural complexity function $T$1, which is dimensionless and measures clustering:

$T$2

This function $T$3 typically attains a unique minimum (the Janus point) and grows monotonically away from it in both directions of time. Dynamically, this reflects the strict convexity of the moment of inertia $T$4 from the Lagrange–Jacobi relation, leading to two arrows of time emanating from the Janus point, each characterized by increasing structure and record formation (Barbour et al., 2014, Barbour et al., 2013).

This growth in complexity is not statistical but dynamical: it arises from the structure of the shape-dynamics Hamiltonian and the geometry of shape space, without the need for a "Past Hypothesis" of special low entropy initial conditions (Lazarovici et al., 2018). Record formation, arrow of time, and the emergence of rods and clocks (asymptotically isolated Kepler pairs) are all correlated with the increase of $T$5 and the fragmentation of the universe into clusters with conserved quantities (Barbour et al., 2014, Barbour et al., 2013).

4. Cosmological Models: Janus Points and Relational Bianchi IX

The relationalist scheme has been extended to mini-superspace quantum cosmologies, notably the Bianchi IX model. Here, pure shape dynamics is implemented by separating global scale (volume $T$6) and anisotropies (Misner parameters or shape variables $T$7) (Koslowski et al., 2016). The dynamical evolution of shapes decouples from the scale evolution, and the purported cosmological singularity (big bang or crunch) becomes a regular Janus point in shape space. At the Janus point ($T$8, in shape-sphere coordinates), the relational degrees of freedom (shapes and ratios) are continuous, the orientation of spatial frames flips, but physics continues without a true singularity, transcending the limitations of standard GR's coordinate-dependent singularities (Koslowski et al., 2016).

Table: Conceptual Elements in Relational Bianchi IX

Element	Standard GR	BKM Relational Approach
Scale (volume, time)	Physical, singularity	Gauge, Janus point is regular
Dynamics	ADM refoliation, York time	Shape evolution in β, decouples from scale
Singularities	Geometric blow-up	Regular crossings in shape space
Observables	Geometric fields	Dimensionless ratios (shapes)

5. Metaphysical Analysis and Ontology

Pure shape dynamics reframes relationalism as a nuanced ontological thesis. The true ontology is given by the reduced shape space and unparametrized curves (histories) within it; gauge symmetries (diffeomorphisms, Weyl transformations) are redundancies, not physical symmetries (Vassallo et al., 2022). Substantivalist conceptions, positing independently existing space or time, are eschewed: only relational structures and physical records (as encoded in the growing complexity and information content of subsystem states) are ontologically significant.

Time, in particular, is not fundamental but emergent, reconstructed from change in relational degrees of freedom; durations and lengths are derived from "ephemeris equations" involving shape data alone (Vassallo et al., 2022). The physical arrow of time is encoded as the monotonic increase in shape complexity away from the Janus point, grounding temporal asymmetry without appeal to a special initial state (Barbour et al., 2013, Barbour et al., 2014, Lazarovici et al., 2018).

6. Quantum Prospects and Open Problems

Quantization strategies for shape dynamics include reduced phase space quantization on shape space and Dirac quantization followed by gauge fixing in the linking theory. Technical challenges persist, especially due to the nonlocality of the pure Shape Dynamics Hamiltonian and the unresolved issues of operator ordering and inner product structure on the resultant Hilbert space (Vassallo et al., 2022). The approach motivates a genuinely relational and scale-invariant formulation of quantum gravity, offering potential new resolutions to the problem of time, which appears in the Wheeler–DeWitt formulation as a "frozen" wave function due to refoliation invariance. In shape dynamics, York time provides a preferred evolution parameter, and in pure shape dynamics, time itself is reconstructed from complexity and the geometry of quantum amplitude support in shape space (Vassallo et al., 2022).

Open issues include the construction of quantum Dirac beables (gauge-invariant observables), the fate of Lorentz invariance upon trading refoliation symmetry for spatial conformal invariance, the treatment of causality and observer families in the fully quantum theory, and the problem of global structure (e.g., nontrivial topology). Extending relationalism to deeper layers of background independence (topology, differential structure, category theory) is an active area of research (Anderson, 2014).

7. Criticisms, Limitations, and Broader Context

The BKM approach has generated debate, particularly regarding its claim to eliminate the need for a Past Hypothesis. Critics argue that, even in the Janus point scenario, a special selection condition is necessary: achieving low shape complexity at the Janus point can be as improbable as postulating low entropy (Zeh, 2016). Moreover, gravitational clustering alone may not reproduce the one-sided thermodynamic, radiative, and quantum arrows of the observed universe, which may require additional nonrelational or fine-tuned initial data (such as negligible radiation and entanglement at the Janus point) (Zeh, 2016).

Nonetheless, the BKM program offers a coherent reformulation of the problem of time and background independence, produces gravitational arrows from first principles, and recasts dynamical singularities as relationally regular crossings in shape space. The purely relational ontology and its extension to quantum domains remain subjects of continued investigation and debate.

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References:

(Vassallo et al., 2022, Anderson, 2014, Lazarovici et al., 2018, Barbour et al., 2014, Barbour et al., 2013, Koslowski et al., 2016, Zeh, 2016)