Geodesic Universality in General RelativityThanks to John Norton, Robert Batterman, and Balázs Gyenis for many helpful conversations. Geodesic Universality in General Relativity∗ Michael Tamir Abstract According to (Tamir, 2012), the geodesic principle strictly interpreted is com- patible with Einstein's �eld equations only in pathologically unstable circumstances and, hence, cannot play a fundamental role in the theory. In this paper it is shown that geodesic dynamics can still be coherently reinterpreted within contemporary relativity theory as a universality thesis. By developing an analysis of universality in physics, we argue that the widespread geodesic clustering of diverse free-fall massive bodies observed in nature quali�es as a universality phenomenon. We then show how this near-geodetic clustering can be explained despite the pathologies associated with strict geodesic motion in Einstein's theory. 1 Introduction In Einstein's original conception of the general theory of relativity, the behavior of gravi- tating bodies was determined by two laws: The �rst (more fundamental) law consisted of his celebrated �eld equations describing how the geometry of spacetime is in�uenced by the �ow of matter-energy. The second governing principle, referred to as the geodesic prin- ciple, then provides the �law of motion� for how a gravitating body will �surf the geometric �eld� as it moves through spacetime. According to this principle a gravitating body traces ∗Thanks to John Norton, Robert Batterman, and Balázs Gyenis for many helpful conversations. 1 out the �straightest possible� or geodesic paths of the spacetime geometry. Not long after the theory's initial introduction, it became apparent that the independent postulation of the geodesic principle to provide the theory's law of motion was redundant. In contrast to classical electrodynamics and Newtonian gravitation, general relativity seemed special in that its dynamics providing principle could be derived directly from the �eld equations. Though the motion of gravitating bodies is not logically independent of Einstein's �eld equations, the geodesic principle canonically interpreted as providing a precise prescription for the dynamical evolution of massive bodies in general relativity does not follow from Einstein's �eld equations. To the contrary, in (Tamir, 2012) it was argued that under the canonical interpretation, not only does the geodesic principle fail to follow from the �eld equations, but such exactly geodetic evolution would generically violate the �eld equations for non-vanishing massive bodies. In short, under the canonical interpretation the two laws are not even consistent. Despite this failure, the widespread �approximately geodetic� motion of free-fall bodies must not be denied. The nearly-geodetic evolution of gravitating bodies is well con�rmed within certain margins of error. Moreover, some of the most important con�rmations of Einstein's theory, including the classic recovery of the otherwise anomalous perihelion of Mercury, also appear to con�rm the approximately geodetic motion of massive bodies. This abundance of apparent con�rmation suggests that though the claim that massive bodies must exactly follow geodesics fails to cohere with Einstein's theory, geodesic fol- lowing may constitute some kind of idealization or approximately correct description of how generic massive bodies behave. We must hence reconcile an apparent dilemma: On the one hand geodesic following appears illustrative as an ideal of the true motion of massive bodies. On the other hand the arguments against the canonical view in (Tamir, 2012) reveal that non-vanishing bodies that actually follow geodesics would be highly pathological with respect to the theory, suggesting that they are not suitable as ideal theoretical models. Moreover, even if we were to adopt such models as idealizations, in order to gain knowledge about the paths of actual bodies, it is unclear how to draw conclusions about the non-pathological cases by considering pathological models that are generically incompatible with the theory. 2 In this paper, we establish such a reconciliation by arguing that, in light of the failure of the canonical interpretation, the principle should instead be adopted as a universality the- sis about the clustering of certain classes of gravitating bodies that exhibit nearly-geodetic motion. In section 2, we propose an analysis of the general concept of universality phe- nomena to designate a certain kind of similarity of behavior exhibited across a wide class of (ostensibly diverse) systems of a particular theory. Using this analysis, in section 3, we explain how the nearly geodetic behavior observed in numerous gravitational systems counts as such a clustering within appropriately close (topological) neighborhoods of an- chor models that exhibit perfect geodesic motion. Finally, in section 4, we explain why such pathological anchor models can be employed to characterize this clustering of the re- alistic models, without having to reify the problem models or take them as representative of actual physical systems. 2 Universality in Physics The arguments of (Tamir, 2012) reveal that the geodesic principle cannot be used to prescribe the precise dynamics of massive bodies in general relativity. Nevertheless, the geodesic principle, demoted from the status of fundamental law to a thesis about the gen- eral motion of classes of gravitating bodies, may still be of value to our understanding generic dynamical behavior in general relativity. The challenge is to �nd an appropri- ate way of characterizing such �nearly geodetic� motion in terms of closeness to perfect geodesic following motion in light of the fact that attempts to model gravitating bodies that could stably follow geodesics end up violating Einstein's �eld equations. If such a reinterpretation of the principle is well-founded, we must justify its endorsement in the face of the kinds of pathologies associated with actual geodesic motion. This can be done by interpreting the robust geodesic clustering patterns actually observed in nature as a universality phenomena. In this section, we begin with an explicit analysis of this concept's use in physics. 3 P Pc Liquid Region Critical Point Solid Region Vapor Region Triple Point Tc T Figure 2.1: Phase diagram of a generic material at �xed density. 2.1 The Paradigm Case: Universality in Phase Transitions The notion of a universality phenomenon was initially coined to characterize a remarkable clustering in the behavior of thermal systems undergoing phase transitions, particularly the behavior of systems in the vicinity of a thermodynamic state called the �critical point.� In thermodynamics the state of a system can be characterized by the three state variables pressure, temperature, and density. According to the thermodynamic study of phase transitions, when the state of a system is kept below the particular �critical point� val- ues (Pc,Tc,ρc) associated with the substance, phase transition boundaries correspond to discrete changes in the system (signi�ed in �gure 2.1 by the thick black lines). If, how- ever, a system is allowed to exceed its critical values, there exist paths available to the system allowing it to change from vapor to liquid (or back) without undergoing such dis- crete changes. These paths involve avoiding the vapor-liquid boundary line by navigating around the critical point as depicted by the broad arrow in �gure 2.1. There exists a remarkable uniformity in the behavior of di�erent systems near the crit- ical point. One such uniformity is depicted in �gure 2.2. In this �gure we see a plot of data recovered by Guggenheim (1945) in a temperature-density graph of the thermodynamic 4 Figure 2.2: Adapted plot of (Guggenheim, 1945) data rescaled for criticality. states at which various �uids transition from a liquid or vapor state to a �two phase� liquid-vapor coexistence region. Systems in states located in this latter region can be in liquid or vapor phases and (according to thermodynamics) maintains constant tempera- ture as the density of the system changes. An important feature exhibited in �gure 2.2 is that (after rescaling for the ρc and Tc of the respective molecules) the transition points of the each of the distinct substances near criticality appears to be well �t by a single curve referred to as the coexistence curve. This similarity in the coexistence curves best �tting diverse molecular substances can be characterized by a particular value β referred to as the critical exponent found in the following relation: Ψ(T) ∝ ∣∣∣∣T −TcTc ∣∣∣∣β (1) where the parameter Ψ(T), called the order parameter tells us the width of the coexistence curve at a particular temperature value T . As depicted in �gure 2.2, as T gets closer and closer to the critical temperature Tc from below, this width drops down eventually 5 vanishing at criticality. We can think of the critical exponent β as telling us about how rapidly such a vanishing occurs. As con�rmed by the above data, this number turns out to be similar (in the neighborhood of β ' .33) for vastly di�erent �uid substances.1 What is fascinating about examples such as this is not the universal (or �nearly� uni- versal) regularity in physical systems. That uniform reliable regularities (viz. �universal laws�) can be found to apply to numerous physical systems (though 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 (P , T , and ρ) used to characterize thermodynamic systems, the various substances exhibiting similar critical exponent values have quite diverse descriptions: At the quantum mechan- ical level, for instance, the state vectors or density matrices representing the respective quantum mixtures will be incredibly distinct (e.g. close to orthogonal). Moreover, we need not go down to a quantum level of description to recognize the vast diversity. From a chemical perspective monotonic neon is di�erent from a diatomic oxygen molecule, or an asymmetrical carbon monoxide molecule. We might hence expect surprise from a physi- cist or chemist since despite such vast di�erences in the ostensibly pertinent details at these levels of theorizing, the substances still share this observed similarity. This similar- ity despite such (speciously relevant) di�erences is what distinguishes the behavior across thermal systems as a kind of universality phenomenon. In the next section we begin a more explicit analysis of the concept's general application in physics. Though the usage of the term originated in the study of thermal systems, universality has now been identi�ed in a multitude of other domains. Over the past decade, Robert Batterman has argued in the philosophical literature that �while most discussions of uni- versality and its explanation take place in the context of thermodynamics and statistical mechanics,... universal behavior is really ubiquitous in science� (Batterman, 2002). A (far from comprehensive) list of vindicating examples includes the clustering behavior found in contexts including non-thermal criticality patterns exhibited in avalanche and earthquake 1This similarity in the value of the critical exponent exists not only for thermal �uid systems, but also in describing the behavior of ferromagnetic systems in the neighborhood of a thermal state that can be analogously characterized as the critical point. 6 modeling (Kadano� et al., 1989; Lise and Paczuski, 2001), extinction modeling in popula- tion genetics (Sole and Manrubia, 1996), and belief propagation modeling in multi-agent networks (Glinton et al., 2010). Batterman has discussed many examples of universality phenomena distinct from criticality phenomena, including patterns in rainbow formation, semi-classical approximation, and drop breaking(Batterman, 2002, 2005). Numerous non- criticality examples of universality have also been discovered in contexts such as the study of chaotic systems exhibiting �universal ratios� in period doubling (Feigenbaum, 1978; Hu and Mao, 1982), or the clustering similarities in models of cold dark matter halos found in astronomical observations (Navarro et al., 2004), to name a couple. In the next section we o�er an explicit analysis of the concept's general application in physics. 2.2 The Same but Di�erent: Analyzing Universality The term universality is used in physics to describe cases in which broad similarities are exhibited by classes of physical systems despite possibly signi�cant variations according to apparently �more fundamental� representations of the systems. Kadano� (2000, p225) describes the term most generally as applying to those patterns in which �[m]any physically di�erent systems show the same behavior.� Berry (1987) has characterized it as the �way in which physicists denote identical behavior in di�erent systems.� Batterman (2002, p4) explains that the �essence of universality� can be found when �many systems exhibit similar or identical behavior despite the fact that they are, at base, physically quite distinct.� Characterizations such as these reveal that the concept hinges on the satisfaction of the two seemingly competing conditions of displaying a particular similarity despite other (evidently irrelevant) di�erences in the systems at some level of description. To make this conceptual dependency explicit, we propose the following analysis of universality phenomena. (UP): A class XT of models of physical systems in a theoretical context T will be said to exhibit a universality phenomenon whenever the class can simultaneously meet the following two conditions: (Sim) There exists a robust similarity in some observable behavior across 7 the physical systems modeled by members of XT . (Var) This similarity in the behavior of members modeled in XT is sta- ble under robust variations of their state descriptions according to context T . The �rst thing to specify is what counts as a �class of models of physical systems in a theoretical context.� In order to avoid complications associated with multiple (possibly not entirely equivalent) formulations of a full physical theory, (UP) is best analyzed in terms of the more restrictive notion of a theoretical context T which identi�es within a given theory a particular formulation and variety of studied phenomena. Examples of di�erent theoretical contexts in classical mechanics include the Hamiltonian versus the Lagrangian formulations, or in quantum mechanics we might distinguish between wave mechanics and operator mechanics.2 A theoretical context may also restrict the phenomena considered by the total theory. For example, source free classical electrodynamics might be considered a distinct theoretical context within the full theory of classical electrodynamics which also models the e�ects of sources. In some cases it is possible for a theoretical context T to specify an entire theory uniquely, in other cases, a speci�cation in terms of (potentially nonequivalent) formulations and speci�c phenomena types may be appropriate. Given a particular theoretical context T of a universality phenomena, the expert will typically be able to identify pertinent state descriptions �according to context T .� For example, in classical electromagnetism the relevant state description may come in the form of �elds specifying the �ow of the source charges and the electromagnetic �eld values throughout a spacetime; in general relativity the metric and energy-momentum tensors might play this role; in thermodynamics, state descriptions may be parametrized by P , T , and ρ (or perhaps V and N), whereas in quantum statistical mechanics one may use density operators. Satisfaction of (Sim) is primarily an empirical question. In order to claim that some- thing universality-like is occurring, there must be an evident similarity in the class of sys- tems exhibiting the phenomenon. This evident similarity need not be (directly) in terms 2Note, in both dichotomies there exist occasional circumstances or conditions such that the respective formulations can cease to be equivalent. 8 of any of the state descriptions used to characterize elements of XT . So for the paradigm example of the universality of phase transitions, (Sim) is satis�ed once physicists recover su�cient empirical data of the kind depicted in �gure 2.2. The robust similarity of (Sim) can be quanti�ed in terms of the remarkable closeness of the critical exponents of these various systems even though the critical exponent parameter β may not necessarily be put in terms of the state quantities of T (e.g. chemistry or statistical mechanics). Satisfaction of (Var) depends primarily on the size and most importantly the diversity of the models in class XT . The larger and more varied the members of class XT with respect to the relevant state descriptions of T , the more �stable under variations.� If XT is suitably rich with diverse members, then a member x ∈ XT may be �mapped� to a rich variety of other members of XT while still maintaining the very similarity shared by all members of XT that allowed the class to satisfy (Sim). In the paradigm example of thermal universality, (Var) is satis�ed 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. We note that 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 other instances, in terms of both the �degree� of similarity displayed and the �degree� of variations that the systems can withstand while still exhibiting such similar behavior. The greater the robustness of the pertinent similarity in behavior across the class of systems and the more (T -state) variation in the class, the more robust the universality is.3 This non-binary dependence means universality may be subject to vague- ness challenges in some cases. While certain examples, such as thermal criticality behavior and, as we argue, the clustering behavior of free-fall massive bodies around geodesic paths may be identi�ed as determinant cases of universality, penumbral cases where it is unclear whether a candidate universality class is su�ciently similar and robust under variations may exist. 3Often this can be rigorously assessed by an appropriately natural norm, metric, topology, etc. de�ned on the state descriptions of T . E.g. we might use some integration norm to quantify the di�erence between two (scalar) �elds found in XT . The choice of appropriate norm, topology, etc. identifying di�erences in the members of XT is directly dependent on the context T . 9 3 The Geodesic Universality Thesis In this section we reconsider the case of near-geodesic clustering observed in nature in terms of the (UP) analysis. In 3.1 we examine why such clustering quali�es as an example of a universality phenomenon. In 3.2 we then identify how the limit operation result of Ehlers and Geroch o�ers what we identify as a universality explanation of this clustering. 3.1 The Similarity and Diversity of Geodesic Universality Consider a sequence of classes (X�GR)�∈(0,s) indexed by some su�ciently small error param- eter � ∈ (0,s). For �xed �, the class X�GR consists of (local) solutions to Einstein's �eld equations: Tab = Gab (2) where the energy-momentum �eld Tab describes the �ow of matter-energy and Gab describes the �Einstein curvature� determined by the metric �eld gab. Moreover, each member of X�GR models some massive body whose spacetime path comes close to following a (timelike) curve γ 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 X�GR satis�es the (Sim) and (Var) conditions in the context of general relativity theory purged of the canonical commitment to geodesic dynamics argued against in (Tamir, 2012). The satisfaction of (Sim) is an empirical matter apparently well con�rmed by centuries of astronomical data recovered from cases in which a relatively small body (a planet, moon, satellite, comet, or even a star) travels under the in�uence of a much stronger gravitational source. Examples involving non-negligible relativistic e�ects (like the Mercury con�rma- tion) 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 con�rming instances for certain �-closeness values. Since observational precision is in- 10 evitably bounded, it is often claimed that the satellite, moon, planet, etc. indeed �follows a geodesic,� despite the results of (Tamir, 2012). In such instances, the body is actually observed to come �close enough� to following a geodesic to warrant such equivocation. These instances hence con�rm membership in a class X�GR for some � threshold below the level of experimental precision or attention. In order to appreciate the satisfaction of (Var), we must consider the relevant theoret- ical context of general relativity theory. State descriptions of physical systems according to the theory come in the form of the tensor �elds Tab and gab, related by the equations (2). Assuming we only consider (local) solutions to Einstein's equations, there exist six independent �eld components describing gab and so the matter-energy �ow Tab. 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 con�rming instances falling under a class X�GR with suitable �, there will be signi�cant variation in terms of these degrees (even after rescaling) once we consider the signi�cant di�erences in the density, shape and �ow of the matter-energy of a planet, versus a satellite, asteroid, anvil, etc. In these �fundamental state description� terms, the diversity of the bodies in a given class X�GR will be quite signi�cant. Despite this diversity, such bodies still satisfy the de�ning requirement of �-closeness to following a geodesic. It is with respect to this diversity in these degrees of freedom (of the energy- momenta/gravitational in�uences of the �near-geodesic following bodies� of members in X�GR) that a �robust stability under variations� can be established in accordance with (Var). So, according to our (UP) analysis, such near-geodesic clustering observed in nature constitutes a geodesic universality phenomenon. However, meeting the conditions of the analysis depends entirely on the truth of the above made empirical claims about the existence of bodies well modeled by members of the respective X�GR classes for a suitable range of � values, and that the bodies in each class are so fantastically diverse from the perspective of their Tab (gab) �elds. In the next section we turn to the more theoretical question of understanding how such geodesic universality is possible in general relativity, by considering the properties of the classes (X�GR)�∈(0,s) in terms of an important geodesic 11 result of Ehlers and Geroch (2004). 3.2 Explaining Geodesic Universality We have now formulated the geodesic universality thesis in the context of general relativity as an empirically contingent claim about classes of the form X�GR whose members model a physical system such that the path of some body counts as �-close to being geodetic without violating Einstein's �eld equations. We have also given a plausibility argument suggest- ing why observational data already obtained by experimentalists con�rms this empirical hypothesis. Moreover, given such con�rmation and the diversity of the energy-momenta of the respective bodies, membership in some X�GR will be su�ciently stable under sig- ni�cant variations of the fundamental state descriptions of the theory 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 di�erent at the level of theoretical description fundamental to general relativity? Geodesic universality can be explained by appealing to an important �limit proof� of the geodesic principle discussed in (Tamir, 2012). It was argued there that Ehlers and Geroch (2004) are able to deduce the �approximate geodesic motion� of gravitating bodies with relatively small volume and gravitational in�uence, by considering sequences of energy-momentum tensor �elds with positive mass of the form ( T (i,j) ab)i,j∈N, referred to as �EG-particles.� The spatial extent and gravitational in�uence of these EG-particles can be made arbitrarily small by picking su�ciently large i and j values respectively. The theorem of (Ehlers and Geroch, 2004) entails that if for a given curve γ there exists such an EG-particle sequence, then by picking a large enough j, γ comes arbitrarily close to becoming a geodesic in a spacetime containing the T (i,j) ab instantiated matter-energy. Speci�cally, let ( g (i,j) ab)i,j∈N be the sequence of metrics that couple to these ( T (i,j) ab)i,j∈N according to (2) in arbitrarily small neighborhoods (Ki)i∈N of γ, containing the support of the respective T (i,j) ab. Then if for each i, as j →∞ the g (i,j) ab approach a �limit metric� gab in the C 1(Ki) topology, which keeps track of di�erences in the metrics and their unique connections, then the curve γ approaches geodicity as j →∞. 12 To understand the impact of the theorem for our universality classes (X�GR)�∈(0,s), we need to appreciate the kind of limiting behavior established by Ehlers-Geroch. The limit result essentially establishes a kind of ��-δ relationship� between, (a) how �nearly-geodetic� we want the curve γ to be, and (b) how much we need to bound the gravitational e�ects of the body on the background spacetime.4 That is to say, the Ehlers-Geroch limit result can be thought of as telling us that �for every degree of �-closeness to geodicity we want the bodies' path to be, there exists a δ-bound on the gravitational e�ect of the body that will keep the path at least that close to geodicity.� The important thing to observe about this �-δ interplay is that though the limiting relationship does require imposing a δ-bound on the perturbative e�ects of the body, it does not impose any speci�c constraints on the details of how the matter-energy of the body �ows within the �-close spatial neighborhood of the curve, nor how the metric it couples to speci�cally behaves. So though the metric is �bounded� within a certain δ-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 established by the Ehlers-Geroch theorem hence gives us a kind of details-free way of understanding the diverse populations of our respective universality classes (X�GR)�∈(0,s). In e�ect the Ehlers-Geroch limiting relationship highlights that for each X�GR class, there exists a particular δ-bound around a limit metric with some geodesic anchor γ such that any body coupling to a metric that stays within that bound (in addition to remaining spatially close enough to γ) satis�es the relevant �-closeness part of the requirements for membership in X�GR. But as we just emphasized, falling under this δ- bound does not impose speci�c constraints on the detailed values of the energy-momenta or metric �elds. In other words, membership in the universality class X�GR is possible as long as the body is a massive solution to Einstein's equations, and its gravitational e�ect and extent are su�ciently bounded in the right way, but beyond these requirements the speci�c details concerning �what the gravitational e�ect does below those bounds� are 4For purposes of exposition, we characterize the established relationship as an ��-δ relationship,� sug- gesting that the closeness relations in question have been quanti�ed, the actual Ehlers-Geroch result is formulated (primarily) in topological terms. See (Gralla and Wald, 2008, �3-5) for a more explicitly quanti�ed approach. 13 irrelevant. Hence, the limit behavior established by the Ehlers-Geroch theorem explains how the �-clustering near geodesic anchors is possible despite signi�cant di�erences in the energy-momenta of our near-geodesic following bodies: So long as the bodies' gravitational in�uences are bounded in the right way their (positive) matter-energy can vary as much as we like under those bounds. 4 Explanation without Rei�cation Before concluding there remains a potential challenge concerning how we can endorse any kind of geodesic �idealization� thesis if the actual geodesic motion of massive bodies is incompatible with Einstein's theory. Recall, while explaining how the classes X�GR whose respective members are ��-close� to geodesic following models could be so diverse, we needed to take the �geodesic limit� of the metrics ( g (i,j) ab)i,j∈N coupling to the EG-particles ( T (i,j) ab)i,j∈N in accordance with the equations (2). 5 By taking such a �geodesic limit� to identify the diversity of our X�GR classes, haven't we made an �essential� appeal to the kind of pathological models precluded by Einstein's �eld equations? The answer to this challenge is that though appreciating the kind of �-δ interplay in the appropriate neighborhoods of the geodesic limit was essential to our explanation of geodesic universality, the role played by the limiting geodesic anchor model does not require us to reify the idealization or make it representative of any physical system in Einstein's theory. Even though there are signi�cant complications associated with what happens at the geodesic limit (1) the �-δ behavior of the systems has a well-de�ned mathematical structure (the C 1 topologies de�ned for each spacetime neighborhood of γ) describing the approach to the limiting anchor model, and (2) the behavior of the models in X�GR, which are �close but not identical to� a geodesic anchor model, still obey Einstein's theory. A geodesic anchor model establishes (as the name suggests) a kind of anchor for the (topological) neighborhoods within which the elements of the respective 5Note, though the ((i,j)gab)i,j∈N converge to a well de�ned �geodesic limit� (in the C 1 topologies) the coupled energy-momentum tensors ((i,j)Tab)i,j∈N may not. Moreover, even if they do converge in a physically salient and independently well-de�ned way, at the limit they must either fail to obey (2) or vanish. For a detailed discussion see (Tamir, 2012, �4). 14 X�GR can be said to cluster. However, using these models as anchors to identify the points around which the actual solutions to Einstein's equations cluster does not require that the anchors themselves be admitted in X�GR. In contrast to more traditional �idealizations,� universality phenomena are about the group behavior of classes of XT not individual systems. For non-universality idealizations severe pathologies can be detrimental because they render the sole idealized model theo- retically inapposite. With universality, however, the existence of a pathologically idealized model �close to but excluded from� a universality class need not entail that members of the class are likewise poorly behaved. Moreover, if a topological clustering �near to� an ideal- ized model has physical signi�cance (as with the C 1 topologies), such proximity may allow inferences about the well-behaved classes without molesting their admissibility according to the laws of T . This is precisely what occurs with geodesic universality. Members of a class X�GR can take advantage of their closeness to the geodesic anchor models without �contracting� the pathologies occurring at the actual geodesic limits. Moreover, we were able to explain such �-closeness by appealing to what we characterized as the �speci�c details irrelevant� δ-closeness in the C 1 topologies. Since we are talking about geodesic universality, we are able to infer directly from such �-closeness that the relevant bodies modeled by the members of X�GR are close to following a geodesic in the relevant physical senses de�ned when we constructed the classes. 5 Conclusion While the incompatibility result of (Tamir, 2012) entails that the geodesic principle strictly interpreted must be rejected at the fundamental level, in this paper we have argued that reinterpreting the role of geodesic dynamics as a universality thesis is both viable and coherent with contemporary general relativity. By developing an analysis of universality phenomena in physics, we saw that the widespread geodesic clustering of a rich variety of gravitating, free-fall, massive bodies actually observed in nature quali�es as a geodesic universality phenomenon. 15 Not only can this approximation of geodesic dynamics be recovered in the form of such a geodesic universality thesis, but by reconsidering the implications of limit operation proofs of the principle, we were able to generate a universality explanation for why we can expect such a remarkable clustering of these gravitating bodies despite the fact that from the perspective of their more fundamental relativistic descriptions (the energy-momentum �eld and its gravitational in�uence) they may be incredibly dissimilar. We concluded with a defense of our appealing to pathological geodesic anchor models in explaining the universality clustering. 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Studies in History and Philosophy of Modern Physics 43, 137�154. 17 1 Introduction 2 Universality in Physics 2.1 The Paradigm Case: Universality in Phase Transitions 2.2 The Same but Different: Analyzing Universality 3 The Geodesic Universality Thesis 3.1 The Similarity and Diversity of Geodesic Universality 3.2 Explaining Geodesic Universality 4 Explanation without Reification 5 Conclusion References