2.1. The Paradigm Case: Universality in Phase Transitions

“Universality phenomenon” was initially coined to characterize a remarkable clustering in the behavior of thermal systems undergoing phase transitions, especially in the vicinity of a thermodynamic state called the “critical point.” In thermodynamics a system’s state can be characterized by the three state variables pressure, temperature, and density. According to thermodynamics, when the system state is kept below the “critical point” values associated with a substance, phase transition boundaries correspond to discrete changes (signified in fig. 1 by thick black lines). If, however, the system exceeds its critical values, there exist paths available allowing it to change from vapor to liquid (or back) without undergoing such discrete changes. These paths involve avoiding the vapor-liquid boundary line by navigating around the critical point as depicted by the broad arrow in figure 1.

Figure 1.

Phase diagram of a generic material at fixed density.

There exists a remarkable uniformity in the behavior of different systems near the critical point. One such uniformity is depicted in figure 2. In this figure we see a plot of temperature-density data recovered by Guggenheim (Reference Guggenheim1945) depicting thermodynamic states at which various fluids transition from a liquid or vapor state to a “two phase” liquid-vapor coexistence region. Systems located in this latter region can be in liquid or vapor phases and (according to thermodynamics) maintain constant temperature as the density changes. An important feature exhibited (after rescaling for the and of the respective molecules) in figure 2 is that near criticality the transition points for each distinct substance appear well fit by a single curve called the coexistence curve. This coexistence-curve similarity across diverse molecular substances can be characterized by a particular value referred to as the critical exponent found in the following relation:

where , called the order parameter, represents the width of the coexistence curve at a particular temperature value . As gets closer and closer to the critical temperature from below, this width drops down, eventually vanishing at criticality. We can think of the critical exponent as telling us about how rapidly this vanishing occurs. This number turns out to be similar (in the neighborhood of ) for vastly different fluid substances.Footnote ¹

Figure 2.

Adapted plot of Guggenheim (Reference Guggenheim1945) data rescaled for criticality.

What’s fascinating about these examples is not the universal (or “nearly” universal) regularity in physical systems. That uniform reliable regularities (namely, “universal laws”) can be found to apply to numerous physical systems (although remarkable) is nothing new. The interesting part is that such uniform reliable behavior occurs despite the fact that at least at one level of description the systems are so incredibly dissimilar. From a level of description thought to be perhaps more “fundamental” than the gross state variables (, , and ) used to characterize thermodynamic systems, the various substances exhibiting similar critical exponent values have quite diverse descriptions: at the quantum mechanical level, the state vectors or density matrices representing the respective quantum mixtures will be notably distinct (e.g., close to orthogonal). From a chemical perspective monotonic neon is different from diatomic oxygen, or asymmetrical carbon monoxide. We might hence expect surprise from a physicist or chemist since despite such vast differences in the ostensibly pertinent details at these levels of theorizing, the substances still share such similarity. This similarity, despite such (speciously relevant) differences, is what distinguishes the behavior across thermal systems as a universality phenomenon.

Although the term originated from the study of thermal systems, universality has now been identified in a multitude of domains. Robert Batterman has argued in the philosophical literature that “while most discussions of universality and its explanation take place in the context of thermodynamics and statistical mechanics, … universal behavior is really ubiquitous in science” (Batterman Reference Batterman2002, 4). A (noncomprehensive) list of vindicating examples includes clustering behavior found in nonthermal criticality patterns exhibited in avalanche and earthquake modeling (Kadanoff et al. Reference Kadanoff, Nagel, Wu and Zhou1989; Lise and Paczuski Reference Lise and Paczuski2001), extinction modeling in population genetics (Sole and Manrubia Reference Sole and Manrubia1996), and belief propagation modeling in multi-agent networking (Glinton et al. Reference Glinton, Paruchuri, Scerri and Sycara2010). Batterman has discussed many examples of universality phenomena distinct from criticality phenomena, including rainbow formation, semiclassical approximation, and drop breaking (Batterman Reference Batterman2002, Reference Batterman2005).Footnote ² In the next section, I offer an explicit analysis of the concept’s general application in physics.

2.2. The Same but Different: Analyzing Universality

The term universality in physics describes cases in which broad similarities are exhibited by classes of physical systems despite ostensibly significant variations according to apparently “more fundamental” representations of the systems. Kadanoff (Reference Kadanoff2000, 225) describes the term most generally as applying to those patterns in which “many physically different systems show the same behavior.” Berry (Reference Berry1987, 185) has characterized it as the “way in which physicists denote identical behavior in different systems.” Batterman (Reference Batterman2002, 4) describes the “essence of universality” as when “many systems exhibit similar or identical behavior despite the fact that they are, at base, physically quite distinct.” These characterizations reveal that the concept hinges on the satisfaction of the two seemingly competing conditions of displaying a particular similarity despite other (evidently irrelevant) differences in the systems at some level of description. To make this conceptual dependency explicit, we propose the following analysis of universality phenomena.

The first thing to specify is what counts as a “class of models of physical systems in a theoretical context.” To avoid complications associated with multiple (possibly not entirely equivalent) formulations of a full physical theory, UP leverages a restrictive notion of a theoretical context that identifies a particular formulation and variety of studied phenomena within a given theory. Examples of different theoretical contexts in classical mechanics include the Hamiltonian versus the Lagrangian formulations; or, in quantum mechanics, it distinguishes between wave mechanics and operator mechanics.Footnote ³ A theoretical context may also restrict the phenomena considered by the total theory. For example, source free classical electrodynamics is a distinct theoretical context within full classical electrodynamics, which includes source effects. In some cases, may specify an entire theory uniquely, while in other cases specifications of (potentially nonequivalent) formulations and phenomena restrictions may be appropriate.

Given the particular of a universality phenomenon, the expert can typically identify pertinent state descriptions “according to context .” For example, in classical electromagnetism the relevant state description could be fields specifying the flow of source charges and the electromagnetic field values throughout a spacetime; in GR the metric and energy-momentum tensors might play this role; in thermodynamics, state descriptions may be parametrized by , , and (or perhaps and ), whereas in quantum statistical mechanics one may use density operators.

Satisfaction of Sim is primarily an empirical question. To claim that something universality-like is occurring, there must be evident similarities in the class of systems exhibiting the phenomenon. This evident similarity need not be (directly) in terms of the state descriptions used to characterize elements of . So for the paradigm example of the universality of phase transitions, Sim is satisfied once physicists recover sufficient empirical data of the kind depicted in figure 2. The robust similarity of Sim can be quantified in terms of the remarkable closeness of the critical exponents of these various systems even though the critical exponent parameter may not be in terms of state quantities in (e.g., chemistry or statistical mechanics).

Satisfaction of Var depends primarily on the size and, most importantly, the diversity of the models in class . The larger and more varied the members of class with respect to the relevant state descriptions of , the more “stable under variations.” If is suitably rich with diverse members, then a member may be “mapped” to a rich variety of other members of while still maintaining the very similarity shared by all members of that allowed the class to satisfy Sim. In the paradigm example of thermal universality, Var is satisfied by the fact that at the chemical or the statistical mechanics levels of description, the members in our class sharing this similar critical behavior are so diverse.

The central concepts of robust variation and robust similarity on which Var and Sim, respectively, depend are not binary. Some universality phenomena may be “more robust” than others, both in “degree” of similarity displayed and in “degree” of variations that the systems can withstand while still exhibiting such similar behavior. The greater the robustness of the pertinent similarity across the class of systems and the more (-state) variation in the class, the more robust the universality is.Footnote ⁴ This nonbinary dependence means that universality may be subject to vagueness issues. While certain examples, such as thermal criticality behavior and, I argue, the clustering behavior of free-fall massive bodies around geodesic paths, are determinant cases of universality, penumbral cases where it is unclear whether a candidate universality class is sufficiently similar and robust under variations may exist.

3.1. The Similarity and Diversity of Geodesic Universality

Consider a sequence of classes indexed by some sufficiently small error parameter . For fixed , the class consists of (local) solutions to Einstein’s field equations:

where the energy-momentum field describes the flow of matter-energy and describes the “Einstein curvature” determined by the metric field . Each member of models some massive body whose spacetime path comes close to following a (timelike) curve that is close to actually being a geodesic (where these two senses of closeness are parametrized by respective functions monotonically dependent on the smallness of ). With the UP analysis in hand, for a given degree of “-closeness” we can now ask if such a class satisfies the Sim and Var conditions in a GR context purged of the canonical commitment to geodesic dynamics.

The satisfaction of Sim is an empirical matter well confirmed by centuries of astronomical data recovered from cases where a relatively small body (a planet, moon, satellite, comet, star, etc.) travels under the influence of a much stronger gravitational source. Examples involving nonnegligible relativistic effects (like Mercury) are of particular importance, but even terrestrial cases, including Galileo and leaning towers or other (nearly) free-fall examples in determinately Newtonian regimes, can count as confirming instances for certain -closeness values. Since observational precision is inevitably bounded, it is often claimed that the satellite, moon, planet, and so forth indeed “follows a geodesic.” In such instances, the body is actually observed only as coming “close enough” to geodesic following to warrant such equivocation. These instances hence confirm membership in a class for some threshold below the level of experimental precision or attention.

To appreciate the satisfaction of Var, we must consider the relevant GR context. State descriptions of physical systems according to the theory come in the form of the tensor fields and , related by (2). Assuming we only consider (local) solutions, there exist six independent field components describing and so the matter-energy flow . In other words, from a fundamentals of relativity theory perspective, there are six physical degrees of freedom to how these bodies are described at each spacetime point.

Given the wealth of evident confirming instances falling under a class with suitable , there is significant variation in terms of these degrees (even after rescaling) once we consider the differences in the density, shape, and flow of the matter-energy of a planet versus a satellite, asteroid, anvil, and so on. In these “fundamental state description” terms, the diversity of the bodies in a given class is stark. Despite this diversity, such bodies still satisfy the defining requirement of -closeness to following a geodesic. It is with respect to this diversity in these degrees of freedom (of the energy-momenta/gravitational influences of the “near-geodesic following bodies” of members in ) that a “robust stability under variations” can be established as satisfying Var.

So, according to UP, the near-geodesic following observed in nature constitutes a geodesic universality phenomenon. Meeting the conditions of the analysis depends crucially on the above empirical claims about the existence of bodies well modeled by members of the respective classes for a suitable range of values, and that the bodies in each class are so fantastically diverse from the perspective of their () fields. In the next section, I address the more theoretical question of understanding how such geodesic universality is possible in GR, by considering the properties of the classes in terms of an important geodesic result of Ehlers and Geroch (Reference Ehlers and Geroch2004).

3.2. Explaining Geodesic Universality

I have now formulated the geodesic universality thesis in the context of GR as an empirically contingent claim about classes of the form whose members model a physical system such that the path of some body counts as -close to being geodetic without violating Einstein’s field equations. I have also given a plausibility argument suggesting why observational data already obtained by experimentalists confirms this empirical hypothesis. Given such confirmation and the diversity of their respective energy-momenta, membership in some is sufficiently stable under significant variations of the fundamental state descriptions of GR to satisfy Var. A remaining theoretical question must now be answered: How can the systems exhibiting this universality phenomenon behave so similarly while being so different at the level of theoretical description fundamental to GR?

Geodesic universality can be explained through an important “limit proof” of the geodesic principle by Ehlers and Geroch (Reference Ehlers and Geroch2004), which deduces the “approximate geodesic motion” of gravitating bodies with relatively small volume and gravitational influence, by considering sequences of energy-momentum tensor fields with positive mass of the form , referred to as “EG-particles” (see Tamir Reference Tamir2012). The spatial extent and gravitational influence of EG-particles can be made arbitrarily small by picking sufficiently large and values, respectively. The theorem of Ehlers and Geroch (Reference Ehlers and Geroch2004) entails that if for a given curve there exists such an EG-particle sequence, then by picking a large enough , comes arbitrarily close to becoming a geodesic in a spacetime containing the instantiated matter-energy.

Specifically, let be the sequence of metrics that couple to these according to (2) in arbitrarily small neighborhoods of , containing the support of the respective . Then if for each , as the approach a “limit metric” in the topology, which keeps track of differences in the metrics and their unique connections, then the curve approaches geodicity as .

To understand the impact of the theorem for our universality classes , we need to appreciate the “limiting behavior” established by Ehlers and Geroch. The limit result establishes a kind of “- relationship” between, (a) how “nearly geodetic” we want the curve to be and (b) how much we need to bound the gravitational effects of the body on the background spacetime.Footnote ⁵ That is to say, the Ehlers-Geroch limit result tells us that “for every degree of -closeness to geodicity we want for the bodies’ path, there exists a -bound on the gravitational effect of the body that keeps the path at least that close to geodicity.” The important thing to observe about this - interplay is that although the limiting relationship requires imposing a -bound on the perturbative effects of the body, it does not impose any specific constraints on the details of how the matter-energy of the body flows within the -close spatial neighborhood of the curve, nor how the metric it couples to specifically behaves. So, although the metric is “bounded” within a certain -neighborhood of the limit metric, the particular details of the tensor values, the corresponding connection, and especially the curvature have considerable room for variation so long as they stay “bounded in that neighborhood.”

This relationship hence establishes a details-free way of understanding the diverse populations of the respective universality classes : in effect the Ehlers-Geroch limiting relationship highlights that for each class, there exists a particular -bound around a limit metric with some geodesic anchor such that any body coupling to a metric that stays within that bound (and remaining spatially close to ) satisfies the relevant -closeness requirements for membership in . But falling under this -bound does not impose specific constraints on the detailed values of the energy-momenta or metric fields. In other words, membership in class is possible as long as the body is a massive solution to Einstein’s equations and its gravitational effect and extent are sufficiently bounded in the right way, but beyond these requirements the specific details concerning “what the gravitational effect does below those bounds” are irrelevant. Hence, the limit behavior established by Ehlers-Geroch explains how the -clustering near geodesic anchors is possible despite significant differences in the energy-momenta of our near-geodesic following bodies: so long as the bodies’ gravitational influences are bounded in the right way their (positive) matter-energy can vary as much as we like under those bounds.