Matter and Motion in Geometrical Spacetime Theories. Dissertation Prospectus

Transcription

1 Final Draft December 5, Introduction Matter and Motion in Geometrical Spacetime Theories Dissertation Prospectus James Owen Weatherall 1 Logic and Philosophy of Science University of California, Irvine In what follows, I propose a dissertation addressing several topics related to the foundations of General Relativity (GR) and Newtonian gravitation, as well as certain topics in general philosophy of science that arise in conjunction with my work on these theories. The prospectus will begin with a non-technical overview of the proposed research, in which I will attempt to explain both the nature of the project and its philosophical payoffs. This section is intended to be self-contained. I will then turn to the technical details of the proposed work, which I will begin with a brief review of GR and the geometrized formulation of Newtonian gravitation, followed by theorems related to two topics that will almost certainly appear in the thesis. (As far as this prospectus is concerned, the material described in section 3.2 should be thought of as the central result of the dissertation.) The presentation of the material in this technical section will not be self-contained: my goal is simply to provide the details necessary to substantiate the claims made in the non-technical section. In the final section, I will briefly describe (in a semi-technical way) some related topics of interest that might also be included in the dissertation, depending on how fruitful they turn out to be. 2 Non-technical overview of proposed research The project I am proposing here is broadly comparative. On the one hand, there is General Relativity (GR), which is standardly taken to be the best theory of gravitational interactions and of spacetime structure currently available. On the other hand, one has Newtonian gravitation or rather, a reformulation of Newtonian theory that I will call geometrized Newtonian gravitation (GNG), which was originally developed by Élie Cartan (1923, 1924) and Kurt Friedrichs (1927). 2 GNG can be thought of as a translation of Newtonian gravitation into the geometrical language usually associated with GR. Empirically it is indistinguishable from standard Newtonian gravitation, but now one thinks of gravitational effects as a manifestation of spacetime geometry (which in turn depends dynamically on the distribution of matter in the universe), much as in GR. GR and GNG are closely related: there is a precise sense in which GNG can be understood as a limiting case of GR. 1 weatherj@uci.edu 2 What I am calling geometrized Newtonian gravitation is often called Newton-Cartan theory. I dislike this name for several reasons, the most important of which is that it suggests that geometrized Newtonian gravitation is somehow distinct from Newtonian theory proper. I tend to take the view that it isn t. 1

2 The suggestion underlying the proposed dissertation is that one good way of elucidating the foundations of both theories is by taking them together and attempting to understand what the counterparts of distinctive features of one theory look like when translated into the other (and, equally enlightening, understanding the cases were no such translation is possible). In some, though not all, cases, this translation involves studying what happens to features of GR in the limit as one approaches GNG. The choice to work with GNG, rather than the myriad other, more common, formulations of Newtonian gravitation already shows a sense in which reflections on the foundations of GR have affected how one might think of classical physics. The geometrical point of view in the study of GR originates with Einstein and is now standard in modern interpretations of the theory. From this point of view, it makes perfect sense to ask, as Cartan and Friedrichs did, whether there is a way of understanding Newtonian gravitation geometrically, as well. And it turns out that once one begins thinking of Newtonian gravitation as a geometrical theory, old problems in Newtonian theory become tractable. To take a concrete example, on more traditional formulations of Newtonian gravitational theory, there is a problem concerning the interpretation of cosmological models that postulate uniform matter distributions. Such models appear to lead to inconsistencies, since it is possible to construct arguments leading to different values of the gravitational field at a single point. But if one rephrases the problem in the geometrized context, the difficulty does not arise (Malament, 1995). Indeed, one sees that the gravitational field in standard formulations of Newtonian gravitation is underdetermined in ways that leads to apparent problems. And so, GR can provide a powerful lens for understanding Newtonian gravitation. There is also a long-standing tradition of going the other way, of understanding GR in light of Newtonian theory. In many pedagogical discussions of the character of GR, at both foundational and applied levels, Newtonian gravitational theory serves as the background against which GR s features are described. The expectation in such discussions is that we already know how Newtonian gravitation works, that it is close to our intuitive pictures of spacetime and of matter interaction, and that the places where GR differs from the Newtonian framework are the places where GR is most surprising and most special i.e., where the non-classical features of GR are manifest. From a historical/experimental perspective, setting things up in this way is natural: one is interested in how the predictions of GR differ from Newtonian gravitation. This latter strategy is dangerous, however, if one contrasts traditional formulations of Newtonian gravitation with GR directly. For instance, when one compares GR with traditional formulations of Newtonian gravitation, one is tempted to say (as many authors have) that GR is distinctive insofar as gravitation is a manifestation of spacetime geometry, or insofar as spacetime geometry is dynamic, or insofar as gravitation is no longer thought of as a force acting on bodies, or insofar as GR is generally covariant, i.e., the fundamental equations of motion can be presented in a coordinate independent way. Each of these features of GR is shared with GNG, however, and so GNG shows that GR does not (or at least, need not) differ from classical physics in these ways. Once one moves to a geometrized formulation of Newtonian gravitation, one sees that these features that one thought were characteristic of GR are rather features of geometrical theories of gravitation more generally. The differences between Newtonian gravitation and GR, meanwhile, involve the details of the structure of spacetime postulated by each theory. In GNG, for instance, there is, at least 2

3 locally, a fixed foliation of spacetime into slices that correspond to space at a given time. One can unambiguously identify simultaneous events. Although spacetime as a whole is generally curved, any given slice of spacetime corresponding to space at a time is necessarily flat. None of these features are shared with the relativistic case. There are two ways in which comparing these two theories can pay off philosophically. (I am hoping to make use of both in the dissertation.) I have essentially described one of them already. Insofar as the philosophy of physics involves the study of the foundations of physical theories, by formulating two theories as similarly as possible, one can use insights gained in one context to shed light on problems in another. The work that I take to be the central result of the proposed dissertation, for example, has this flavor. The particular problem connects with an old question in the philosophy of space and time concerning the status of unforced motion. Aristotle was probably the first to think about why bodies move in the absence of force he held that one can associate with any body a unique natural motion that that body would undergo unless perturbed in some way. Much later, Galileo (1632) (and almost contemporaneously, Descartes (1644)) broke from the Aristotelian tradition, arguing that matter obeyed a law of (rectilinear) inertia, i.e., that bodies move uniformly and rectilinearly except when acted on by an outside force. Newton (1687) adopted a version this law of inertia as his first law of motion, and then, in his second law of motion, quantified the relationship between an external force and a body s deviation from uniform rectilinear motion, i.e., its acceleration. He argued that a body s tendency to deviate from rectilinear motion in the presence of an external force is moderated by the body s inertial mass. He then explained the deviations of celestial bodies from rectilinear motion via his law of universal gravitation, postulating that all bodies exert an attractive force on one another in proportion to the product of their gravitational masses. As a matter of empirical fact, Newtonian inertial and gravitational masses coincide for all bodies; however, as has been often noted, there is nothing in the Newtonian theory that accounts for this correspondence. (I mention this because I will make use of it in the next topic I address.) In General Relativity (GR), inertial motion is governed by the so-called geodesic principle, which states that in the absence of external forces, test point particles with non-zero inertial mass traverse timelike geodesics (locally extremal curves) in curved four-dimensional spacetime. One of the principle ways in which GNG makes Newtonian gravitation seem more like GR is that in GNG, inertial motion is once again described by a geodesic principle. An influential view, originating with Einstein et al. (1938) and recently espoused by Harvey Brown (2005), maintains that inertial motion has a special status in GR that it does not have in earlier spacetime theories. This special status derives from the fact that in GR, it is possible to prove the geodesic principle from other central principles of the theory. The thought is that, because of this special status, GR explains inertial motion in a way that clarifies the relationship between geometry and motion. It turns out that there are several ways to make the claim that the geodesic principle can be proved in GR precise. One approach, due to Geroch and Jang (1975), approaches the problem by showing that the only curves along which an arbitrarily small matter blob can propagate are timelike geodesics. Implicit in the claim that such theorems make GR special is that it is not possible to prove parallel theorems in other spacetime theories. But this is not the case. I have shown that, at least in GNG, it is possible to prove a direct parallel to the Geroch-Jang result. The 3

4 moral is that in GNG, much like GR, there is a sense in which one can explain inertial motion by appeal to other central principles of the theory. This by itself is striking: it is possible to say precisely why bodies move as they do in the absence of force, given the geometrical structures postulated by either GNG or GR, plus some natural assumptions about the nature of matter. Here, at least, GR cannot be said to have answered a question that Newtonian theory was not also equipped to answer. (See section 3.2 for statements of the Geroch-Jang theorem and of the corresponding Newtonian theorem.) The Newtonian theorem itself is only part of what makes this question interesting, however. Given that one can prove a version of the geodesic principle in GNG, and given that the theorem can be formulated in a way that makes it remarkably similar to the Geroch-Jang theorem, it makes sense to compare the theorems directly, vis à vis the status of the assumptions necessary to prove the theorems in each case. Each theorem requires four essential assumptions about test matter. Two of them are identical in both cases. A third differs explicitly: in the Newtonian case, one needs to assume that mass is never negative; in the relativistic case, one needs to assume both that mass is never negative, and also that matter satisfies a condition such that no observer would ever attribute a faster-than-light velocity to the matter. Here, at least, the Newtonian case seems to require less. The fourth condition, however, is tricky. In both cases, its formulation is the same. The condition states that at any point, the energy-momentum (mass-momentum in the classical case) is conserved, in the sense that it cannot be created or destroyed. This condition is a standard assumption of both GR and GNG, and so it certainly is natural to include as an assumption in both theorems. But in GR, the conservation condition is an immediate consequence of the fundamental field equation, Einstein s equation. In GNG, however, the corresponding field equation, Poisson s equation, does not imply the conservation condition. What one makes of this, however, is open to debate: on one hand, it may seem that the conservation condition is a bare assumption in the Newtonian case that requires some additional motivation (thus supporting the claim that GR is special, though now in a different way); alternatively, one might argue that the conservation condition is best thought of as a metaprinciple that holds in a wide variety of spacetime theories, and which should be thought of as a constraint on plausible theories. It is worth noting that as a historical matter of fact, Einstein began with the conservation condition, and then sought a field equation that was compatible with it, rather than vice versa. In the dissertation, I will argue that as a matter of comparison both theories explain inertial motion about equally well. In both cases, the argument is not as simple as one might hope, on account of the strengths of the necessary assumptions; conversely, in both cases, by making reasonable, if not trivial, assumptions, one can derive the geodesic principle. The discussion will follow an article-length conference proceedings paper currently under preparation, from a conference in Wuppertal, Germany in July, Also, as I noted above, there are other approaches to proving the geodesic principle in GR. Another prominent line of attack originates with Einstein et al. (1938), with versions of the argument due to Souriau (1974) and Sternberg (2003). These arguments have been adapted to the GNG case by Mike Tamir (2010). I have recently been working on reconstructing and better understanding this other work, which will likely also be addressed in the dissertation. Of particular interest, it seems, is the relationship between the assumptions necessary to motivate the relevant theorems in the different approaches. So far, this work is preliminary. 4

5 As I mentioned above, there is also another way in which comparing GR and GNG can lead to philosophically rich results. Physicists and philosophers of science often talk about the nature of intertheoretic relations, but there are surprisingly few cases where one can cleanly state what the relationship is between two physical theories. The case of GR and GNG is a remarkable exception. It is possible to find a sequence of relativistic spacetime such that, as one takes the limit as a particular parameter goes to 0, one arrives at a classical spacetime. In this sense, GNG is a limiting case of GR. Of course, it is dangerous to generalize from a single example to how all scientific theories must behave. But a particularly good test case can help one to think about how some (at least) scientific theories behave, as a way of studying traditional questions in general philosophy of science. I would suggest that studying this particular case could be enlightening for debates on a variety of topics, such as scientific realism or intertheoretic reduction. One such topic that I have already spent some time thinking about concerns scientific explanation. In particular, there is a variety of explanation that is important in physics in which one tries to explain a feature of one theory by appeal to a superseding theory. Newtonian gravitation is rife with examples of questions that in some sense only GR can answer (I will presently provide one). I claim that looking at such examples can tell us something about at least one kind of explanation that comes up in physics. As mentioned above, inertial mass and gravitational mass are conceptually distinct in Newtonian physics. Indeed, one would expect them to be unrelated to each other. Inertial mass, m I, is a constant of proportionality in the fundamental dynamical principles of the theory. One can think of inertial mass as a measure of a body s tendency to accelerate under the influence of an impressed force. Gravitational mass, m G, meanwhile, determines the strength of the gravitational force that a body exerts on other bodies and, conversely, is exerted on the body by other bodies. It enters the theory via Newton s law of universal gravitation. One might naïvely expect that (a) how much a body will tend to deviate from rectilinear motion given an external force and (b) the strength of that external force should be independent quantities (as they are when considering electric force). But it turns out that when we start looking into how bodies behave in a gravitational potential, we find something quite different. Given any body at all, the ratio m G /m I always takes the same value: choosing the natural units, we always find that m G /m I = 1. This striking correspondence calls for an explanation. But the details of the explanation are subtle. Newtonian gravitation does not natively have the tools necessary to account for the correspondence between inertial and gravitational mass indeed, this is the puzzle to be accounted for in the first place. The only hope one might have for explaining why Newtonian gravitation exhibits this feature is to look beyond Newtonian gravitation, to a superceding theory that (one might hope) will account for why inertial and gravitational mass must be equal to one another. In this case, the prime candidate for a superceding theory is GR. But in GR, gravitational mass does not make sense. Gravitation is not a force at all: there is no gravitational field to which a body can couple, and so the strength of the coupling field does not enter the theory. Thus the explanation cannot be so simple as to say that GR tells us gravitational and inertial mass are equal. In light of these remarks, the question might be rephrased as follows. Given that we take GR to be a more virtuous theory of gravitation than Newtonian gravitation, and insofar as Newtonian gravitation is a limiting case of GR, why do we find that in Newtonian gravitation 5

6 there is a gravitational mass, and moreover, why is it equal to inertial mass? To answer this question, one might attempt to show that, if you begin with GR and then consider the sense in which Newtonian gravitation is a limiting case of GR, gravitational mass is forced to equal inertial mass. It turns out that this is possible to do. The details are technical (see section 3.3 for a precise statement of the proposition and a (trivial) proof), but the idea is as follows. In moving from GR to standard Newtonian gravitation, one proceeds in two steps. First, one sets up the appropriate limit of relativistic spacetimes and recovers GNG as the limiting case. Here GNG should be thought of as an intermediate point between GR and standard Newtonian physics. In geometrized Newtonian theory, however, since gravitation is geometrical rather than a force between bodies, gravitational mass does not make any more sense in the context of geometrized Newtonian theory than in GR. To see where gravitational mass comes from, we need to take the second step in the limiting process. This step makes use of a theorem due to Andrzej Trautman known as the Trautman Recovery Theorem (Malament, 2010, Prop ). Trautman s theorem tells us that, given a classical spacetime of the sort found in GNG, satisfying certain conditions, one can find (1) another spacetime that is flat, and (2) a scalar field φ G on that spacetime that satisfies Poisson s equation (i.e. that has the dynamical relationship to the distribution of matter in the universe that Newton s theory predicts for the gravitational potential) and which is such that for any free (inertial) massive test point particle, a = φ G. 3 (2.1) Thus, under certain circumstances, we can recover a flat spacetime and a gravitational potential φ G that has just the relations to both the distribution of matter in the universe and the dynamics of a particle that we would expect from Newtonian physics. In this flat spacetime, the particle trajectories determined relative to the gravitational field agree with the particle trajectories as determined by the geometrized theory in the initial curved spacetime. More loosely, one finds a flat spacetime and gravitational field that make the same predictions as the geometrized theory. Eq. (2.1), which is effectively a consequence of the two-step limiting procedure, tells us the relationship between acceleration and the gravitational potential. The gravitational mass is the coupling to the gravitational field: it is the constant of proportionality moderating the relationship between force and the gravitational field. If we use F = m I a, we find that the force on a massive test point particle arising from the gravitational potential φ G is F = m I a = m I φ G. (2.2) Eq. (2.2) tells us directly that the coupling to the gravitational field in Newtonian physics is given by the inertial mass. The reason that gravitational and inertial mass are always equal is that gravitational mass simply is inertial mass. And thus we have an answer to the original question (suitably formulated). The important point here concerns the nature of the explanation that one gets by thinking about inertial mass in this way. In the dissertation, I will argue that this is a particularly important kind of explanation, at least in physics: calls for explanations of this sort are just 3 Here I am using the notation of standard Newtonian physics. 6

7 the kind of thing that are used to motivate and shape research programs. (To take another example, consider the question: Our best theory of particle physics predicts that in very high energy experiments, which probe the smallest distance scales, the electromagnetic, weak, and strong forces should have approximately the same strength. But at these same distance scales, gravitation is many orders of magnitude weaker. Why is gravity so much weaker than any of the other forces? ) But it is not well accounted for by the standard philosophical accounts of scientific explanation: the deductive-nomological account, the causal-mechanical account, or the unificationist account. I will not rehearse the details of the arguments here, but they will closely follow an article on this topic that I have prepared for journal submission. The two examples of work for the dissertation that I have given here are intended to give an idea of what I will ultimately write on and why it is interesting philosophically. I fully expect both of these to appear, essentially as described and with full technical details, in the final version of the thesis. But this is just a piece of the comparative project described above. Some additional topics that I am currently working on are described in section 4. 3 Technical details In the following sub-sections, I will provide some of the technical details supporting the claims I made above. These sections are not intended to be read independently of section 2. I will not spend time motivating the work, for example, or describing its philosophical importance. 3.1 Brief review of GR and GNG We begin by defining the geometrical structures we will work with. The notation and statements of theorems follows Malament (2010). First we will describe some relevant material from GR. Familiarity with differential geometry is assumed. Definition 3.1 A relativistic spacetime is an ordered pair (M, g ab ), where M is a smooth, connected, four-dimensional manifold and g ab is a smooth, non-degenerate semi-riemannian metric on M with Lorentz signature (+,,, ). In a relativistic spacetime, the metric defines a lightcone structure at every point as follows. Given any point p and any vector ξ a in the tangent space M p, we say that ξ a is timelike if g ab ξ a ξ b > 0, spacelike if g ab ξ a ξ b < 0, and null if g ab ξ a ξ b = 0. The length of any vector ξ a at a point is given by ξ a = g ab ξ a ξ b 1/2. A (smooth) 4 curve is timelike (resp. spacelike or null) if its tangent vector is at every point of the curve. A spacetime is temporally orientable if there exists a continuous timelike vector field on all of M; such a vector field determines a temporal orientation. If a relativistic spacetime has a temporal orientation, then it is possible to consistently distinguish between future- and past-directed timelike vector fields. Since the metric is non-degenerate, there exists an inverse metric g ab such that g an g nb =. We can move easily between vector fields and covector fields on M by raising and δ a b 4 Here and in what follows, it should be assumed that we are limiting attention to smooth (i.e. infinitely differentiable) curves, fields, manifolds, etc, whether stated explicitly or not. 7

8 lowering indices with g ab and g ab, respectively, so for instance if ξ a is a vector field on M, then ξ b = g ab ξ a is a covector field on M, and likewise for more complicated tensor fields. The metric determines a unique (torsion free) derivative operator on M,, satisfying the compatibility condition a g bc = 0. The derivative operator allows us to define the curvature of spacetime via the Riemann curvature tensor, which is the unique smooth tensor field R a bcd such that for all smooth vector fields ξ b, R a bcd ξb = 2 [c d] ξ a. We say that a spacetime is flat if R a bcd = 0. From the Riemann curvature tensor, we can define the Ricci tensor by R ab = R n abn. Massive point particles are represented by their worldlines, which are smooth futuredirected timelike curves parametrized by arc-length. (Point particles have an attenuated status here really, we are thinking of a field theory, and point particles are some appropriate idealization.) With every point particle, there is an associated four-momentum, P a, defined at every point of the particle s worldline, whose length is the (inertial) rest mass. For a point particle with non-zero mass m I, 5 we can write P a = m I ξ a, where ξ a is the tangent vector field to the particle s worldline (called the particle s four-velocity). More generally, we can associate with any matter field a smooth symmetric field T ab, called the energy-momentum tensor. T ab can be thought to encode the four-momentum density of the matter field as determined by any future-directed timelike observer at a point: For all points p M and all unit, future-directed timelike vectors at p, ξ a, the four-momentum of a matter field at p as determined by ξ a is P a = T a b ξb. The curvature of spacetime is related to the energymomentum tensor by Einstein s equation, R ab = 8π(T ab 1 2 T g ab), (3.1) where T = T a a. We can now proceed to define a parallel structure for classical theories. Definition 3.2 A classical spacetime is an ordered quadruple (M, t ab, h ab, ), where M is a smooth, connected, four-dimensional manifold; t ab is a smooth symmetric field on M of signature (1, 0, 0, 0); h ab is a smooth symmetric field on M of signature (0, 1, 1, 1); and is a derivative operator on M compatible with t ab and h ab, i.e. it satisfies a t bc = 0 and a h bc = 0. We additionally require that t ab and h ab are orthogonal, i.e. t ab h bc = 0. Note that signature, here, has been extended to cover the degenerate case. We can see immediately from the signatures of t ab and h ab that neither is invertible. Hence in general neither t ab nor h ab can be used to raise and lower indices. t ab can be thought of as a temporal metric on M in the sense that given any vector ξ a in the tangent space at a point, p, ξ a = (t ab ξ a ξ b ) 1/2 is the temporal length of ξ a at that point. If the temporal length of ξ a is positive, ξ a is timelike; otherwise, it is spacelike. At any point, it is possible to find a covector t a, unique up to a sign, such that t ab = t a t b. If 5 Since keeping track of the distinction between inertial and gravitational mass is important for the moral of section 3.3, I will label masses as inertial even in the context of GR and geometrized Newtonian gravitation, were strictly speaking there can be no ambiguity. I will use capitalized calligraphic symbols for subscripts indicating labels to distinguish them from subscripts indicating index (i.e. tensor) structure. 8

9 there is a continuous, globally defined vector field t a such that at every point, t ab = t a t b, then the spacetime is temporally orientable (we will encode the assumption that a spacetime is temporally oriented by replacing t ab with t a in our definitions of classical spacetimes). h ab, meanwhile, can be thought of as a spatial metric. However, since there is no way to lower the indices of h ab, we cannot calculate the spatial length of a vector directly. Instead, we rely on the fact that if ξ a is a spacelike vector (as defined above), then there exists a covector σ a such that ξ a = h ab σ b. The length of ξ a can then be defined as (h ab σ a σ b ) 1/2. (If ξ a is not a spacelike vector, then there is no way to assign it a spatial length.) Note, too, that it is possible to define the Riemann curvature tensor R a bcd and the Ricci tensor R ab with respect to as in GR (or rather, as in differential geometry generally). Flatness (R a bcd = 0) carries over intact from GR; we say a classical spacetime is spatially flat if R abcd = R a nmqh bn h cm h dq = 0. It turns out that this latter condition is equivalent to R ab = h an h bm R nm = 0. 6 We describe matter in close analogy with GR. Massive point particles are again represented by their worldlines, which are smooth future-directed timelike curves parameterized by elapsed time. For a point particle with (inertial) mass m I, we can always define a smooth unit vector field ξ a tangent to its worldline, again called the four-velocity, such that we can define a four-momentum field P a = m I ξ a. The mass of the particle is now given by the temporal length of its four-momentum. In similar analogy to the relativistic case, we can associate with any matter field a smooth symmetric field T ab, now called the mass-momentum tensor. T ab once again encodes the four-momentum density of the matter field as determined by a future directed timelike observer at a point, but in this case all observers agree on the four-momentum density at p: P a = t b T ab. Contracting once more with t b yields the mass density, ρ = t a t b T ab. In the present covariant, four-dimensional language, standard Newtonian theory can be expressed as follows. Let (M, t a, h ab, ) be a classical spacetime. We require that is flat (i.e. R a bcd = 0). We begin by considering the dynamics of a test point particle with inertial mass m I and four-velocity ξ a. We can define the force on such a particle by F a = m I ξ a a ξ b (literally, F = m I a). In the absence of external forces, a massive test point particle undergoes geodesic motion. If the total mass-momentum content of spacetime is described by T ab, we require that the conservation condition holds, i.e. at every point a T ab = 0. To add gravitation to the theory, we can represent the gravitational potential as a smooth scalar field φ on M. φ is required to satisfy Poisson s equation, a a φ = 4πρ (where a is shorthand for h ab b ). Gravitation is considered a force; in general, the gravitational force on a point particle is moderated by its gravitational mass, according to FG a = m G a φ. (Indeed, this relationship can be taken as a definition of the gravitational mass.) In geometrized Newtonian theory we again begin with a classical spacetime (M, t a, h ab, ), but now we allow to be curved. The dynamics of a point particle with inertial mass m I and four-velocity ξ a are again given by F a = m I ξ n n ξ a ; likewise, free massive test point particles undergo geodesic motion. However, the geodesics are now determined relative to, which is not necessarily flat. The conservation condition is again expected to hold. Gravitation is no longer a force and so there is no longer a gravitational mass term in the theory. Instead, gravitational interactions are seen to be the result of the curvature of spacetime, 6 See Malament (2010, Prop. 4.15). 9

10 which in turn is determined by a geometrized form of Poisson s equation, R ab = 4πρt a t b. (3.2) Since the Riemann curvature tensor (and by extension, the Ricci tensor) is determined by, the geometrized Poisson s equation places a constraint on the derivative operator. In particular, must be such that, for all smooth vector fields ξ a, R ab ξ a = 2 [b n] ξ n = 4πρt a t b ξ a. Note, too, that the geometrized Poisson s equation forces spacetime to be spatially flat, because if Poisson s equation holds, then R ab = 4πρh an h bm t n t m = 0 by the orthogonality condition on the metrics. We are particularly interested in the relationship between these three theories. Several results are available. First, it is always possible to geometrize a gravitational field on a flat classical spacetime that is, we can always move from the covariant formulation of standard Newtonian gravitation to geometrized Newtonian gravitation, via a result due to Andrzej Trautman (1965). Proposition 3.3 (Trautman Geometrization Lemma.) (Slightly modified from Malament, 2010, Prop ) Let (M, t a, h ab, ) f be a flat classical spacetime. Let φ and ρ f f be smooth scalar fields on M satisfying Poisson s equation, a a φ = 4πρ. Finally, let g g is = (, f C a bc ),7 with C a bc = t f bt c a φ. Then (M, t a, h ab, ) g is a classical spacetime; the unique derivative operator on M such that given any timelike curve with tangent vector field ξ a, ξ n g n ξ a = 0 ξ n f n ξ a = f a φ; and the Riemann curvature tensor relative to g, g R a bcd, satisfies (G) g R ab = 4πρt a t b g R a b c d = R g c d a b g R ab cd = 0. (CC1) (CC2) (CC3) Trautman showed that it is also possible to go in the other direction. That is, given a curved classical spacetime, it is possible to recover a flat classical spacetime and a gravitational field, φ so long as the curvature conditions (CC1)-(CC3) are met. Proposition 3.4 (Trautman Recovery Theorem.) (Slightly modified from Malament, 2010, Prop ) Let (M, t a, h ab, g ) be a classical spacetime that satisfies eqs. (CC1)- (CC3) for some smooth scalar field ρ. Then, at least locally on M, there exists a smooth 7 This notation is explained in Malament (2010, Prop ). Briefly, if is a derivative operator on M, then any other derivative operator on M is determined relative to by a smooth symmetric (in the lower indices) tensor field, C a bc, and so specifying the Ca bc field and is sufficient to uniquely determine a new derivative operator. 10

11 scalar field φ and a flat derivative operator on M, f such that (M, t a, h ab, f ) is a classical spacetime; (G) holds; and φ and f together satisfy Poisson s equation, f a f a φ = 4πρ. It is worth pointing out that the pair ( f, φ) is not unique: another pair ( f, φ ) will satisfy the conditions in the proposition iff g a g b (φ φ) = 0 and f = (, f C bc a ), where C a bc = t b t c g a(φ φ). It is also worth pointing out that whenever we begin with standard Newtonian theory and move to geometrized Newtonian theory, it is always possible to move back to the standard theory, because Prop. 3.3 guarantees that the curvature conditions (CC1)-(CC3) are satisfied. We can now ask how either of these classical theories relate to GR. The answer is that geometrized Newtonian theory arises as a limiting case of GR, for a properly constructed limit. (For full details of this limiting procedure, see Malament (1986, Sec. 5).) The limiting relation is captured by the following proposition. Proposition 3.5 (Classical Limit of GR.) (Adapted from Malament, 1986, Props. on Limits 1 & 2) Fix a smooth, connected, four-dimensional manifold M and assume λ is a realvalued variable taking all values on an interval (0, k). Suppose that for each λ on an interval (0, k), there exist smooth symmetric fields g ab (λ) and T ab (λ) on M such that (M, g ab (λ)) is a relativistic spacetime and for each λ, g ab (λ) and T ab (λ) collectively satisfy the following conditions. 1. lim λ 0 g ab (λ) = t a t b for some non-vanishing closed field t a ; 8 2. lim λ 0 λg ab (λ) = h ab for some field h ab of signature (0, 1, 1, 1). 3. For all λ (0, k), λ R ab = 8π ( T ab (λ) 1 2 g ab(λ)t (λ) ), where T (λ) = T ab (λ)g ab (λ); and 4. lim λ 0 T ab (λ) = T ab for some smooth symmetric field T ab on M. λ Then there exists a derivative operator a on M such that lim λ 0 a = a, and for which (M, t a, h ab, a ) is a classical spacetime satisfying R a b c d = Rc d a b. Moreover, there exists a smooth field ρ on M such that lim λ 0 T ab (λ) = ρt a t b, which satisfies R ab = 4πρt a t b. Taken together, Prop. 3.5 and Prop. 3.4 allow one to move from GR to standard (flat) Newtonian gravitation via a two step process. One begins by constructing a sequence of relativistic spacetimes that satisfy conditions 1 4, with the added constraint that the limit of the sequence satisfy the curvature condition CC3 (the other two are satisfied automatically), and then one uses the Trautman recovery theorem to recovery a flat classical spacetime from the limit point of the sequence of relativistic spacetimes. 8 If t a is a non-vanishing closed field, the product t ab = t a t b automatically has signature (1, 0, 0, 0). 11

12 3.2 The status of inertial motion in GR and GNG In section 2, I made some remarks about the status of inertial motion in GR and GNG. Here I will make the relevant claims precise. One sense in which the geodesic principle can be proved in GR is given by the Geroch-Jang theorem, which can be stated as follows. Theorem 3.6 (Geroch and Jang, 1975) Let (M, g ab ) be a relativistic spacetime, with M orientable. Let γ : I M be a smooth, imbedded curve. Suppose that given any open subset O of M containing γ[i], there exists a smooth symmetric field T ab with the following properties. 1. T ab satisfies the strict dominant energy condition, i.e. given any future-directed timelike covectors ξ a, η a at any point in M, either T ab = 0 or T ab ξ a η b > 0; 2. T ab satisfies the conservation condition, i.e. a T ab = 0; 3. supp(t ab ) O; and 4. there is at least one point in O at which T ab 0. Then γ is a timelike curve that can be reparametrized as a geodesic. The interpretation is as follows. Suppose that you begin with an arbitrary curve, and suppose further that it is possible to construct little matter blobs in arbitrarily close neighborhoods of that curve. Then, it must be that the curve is a timelike geodesic. These matter blobs are to be thought of as approximating point particles propagating along the curve, and so, one concludes that the only curves along which free massive test point particles can propagate are timelike geodesics. Meanwhile, I have shown that the following theorem holds. Theorem 3.7 Let (M, t a, h ab, ) be a classical spacetime, and suppose that M is oriented and simply connected. Suppose also that R abcd = 0 and R ab cd = 0. Let γ : I M be a smooth imbedded curve. Suppose that given any open subset O of M containing γ[i], there exists a smooth symmetric field T ab T (M) with the following properties. 1. T ab satisfies the mass condition, i.e. whenever T ab 0, T ab t a t b > 0; 2. T ab satisfies the conservation condition, i.e. a T ab = 0; 3. supp(t ab ) O; and 4. there is at least one point in O at which T ab 0. Then γ is a timelike curve that can be reparametrized as a geodesic. 12

13 The interpretation in this case is the same as in the Geroch-Jang case. The conclusion is that the only curves along which arbitrarily small matter blobs can propagate in GNG are timelike geodesics. The proof of this theorem has been written up and submitted to a journal for consideration. In trying to understand what such theorems mean for the status of inertial motion in GR and GNG, it is interesting to consider the relative statuses of the assumptions motivating the theorems. (I will only amplify my remarks from section 2 slightly.) The strict dominant energy condition (condition 1 of the Geroch-Jang theorem) is quite strong. It explicitly rules out matter to which a timelike observer would attribute spacelike propagation. And it is essential to the theorem. David Malament (2009) has shown that if one relaxes it, then given any timelike curve, it is possible to find a suitable field T ab satisfying the remaining three conditions in arbitrary neighborhoods of the curve. In contrast, the corresponding condition in the GNG case, the mass condition, is much weaker: it amounts to saying that mass is always positive, which can be understood as a way of limiting attention to the class of mathematical possibilities that correspond to the physically relevant case. Another way of putting the point is to say that in GR, it makes perfect sense to consider matter that propagates along spacelike curves, whereas in GNG, it is not at all clear that negative mass makes sense. Conversely, condition 2 has (perhaps) a different status in the two theorems, this time in GR s favor. The conservation condition can be understood as a direct consequence of Einstein s equation, following from the Bianchi identities. So there is a sense in which conservation comes for free, given the central dynamical principle of the theory. But in GNG, the conservation condition does not follow from Poisson s equation. Indeed, it is not clear that it can be derived at all (though it is a perfectly standard assumption). I will return to this in section The relativistic origins of inertial mass in classical physics The explanation I give at the end of section 2 can be precisified by the following proposition. Proposition 3.8 Let (M, t a, h ab, f ) be a flat classical spacetime and let φ be a gravitational field defined on that spacetime as in standard Newtonian gravitation. Suppose further that (M, t a, h ab, f ) and φ arise via the two step process described at the end of section 3.1. Consider a massive point particle with inertial mass m I traversing a timelike curve in M, γ, with tangent vector field ξ a, under the influence of only gravitational force. Then the gravitational force experienced by the massive point particle is F a G = m I ξ n f n ξ a = m I f a φ. (3.3) In other words, the particle s gravitational mass is equal to its inertial mass. Proof. By assumption, (M, t a, h ab, f ) and φ arise via the two-step limiting process described above. Thus there exists a (curved) classical spacetime (M, t a, h ab, g ) satisfying (CC3) from 13

14 which (M, t a, h ab, f ) can be recovered. Since the particle experiences no non-gravitational force, we know from the geodesic principle of geometrized Newtonian gravitation that γ must be a geodesic relative to g. Meanwhile, by Prop. 3.4, we know that if γ is a geodesic relative to g, then ξ n f n ξ a = f a φ. Thus we have the acceleration of the particle s worldline, which we can plug into F a = m I ξ n f n ξ a to find the gravitational force on the particle. We see that FG a = m Iξ n f n ξ a f = m I a φ, as required. It follows that the particle s coupling to the gravitational field is given by its inertial mass. 4 Possible additional topics The research described in the previous two sections has already begun to yield results, some preliminary and some more mature. But along the way, additional questions have arisen, suggesting future research avenues within the broad comparative project already described. I do not yet have much to say on these topics, but I will list some of them here to indicate future directions the research may take, and to give a sense of how the dissertation may fill out with time. 4.1 Gravitational energy in GR and geometrized Newtonian gravitation There is an old problem in GR (see, for instance, Geroch (1973)) concerning whether or not there can be a sense of localized gravitational energy. One way of seeing the problem (this formulation is due to David Malament) is as a dilemma concerning whether the gravitational energy-momentum should be a source term in Einstein s equation. If it should be a source term in Einstein s equation, then it would follow that it would vanish whenever Ricci curvature vanishes, even if Weyl curvature is non-vanishing. Since Weyl curvature is associated with gravitational waves in empty space, it would seem that gravitational waves would have no gravitational energy associated with them. Conversely, if gravitational energy-momentum should not be a source term in Einstein s equation, then it would follow that there could be no energy exchange between gravitation and other matter fields, since Einstein s equation forces its source terms to be locally conserved (and energy exchange between gravitational and non-gravitational energy/matter fields would violate that local conservation). The situation is typically thought to be different in Newtonian gravitation, where one can, at least in the standard theory, argue that the gravitational field is a repository of (localized) gravitational energy. It is interesting to consider what happens when one moves to GNG. For one, suppose that (M, t a, h ab, g ) is a (curved) classical spacetime, and suppose that (M, t a, h ab, ) f is a flat spacetime that one recovers from the first spacetime, with associated gravitational field φ. Suppose further that T ab is a matter field that is conserved relative f to the flat derivative operator, i.e. a T ab = 0. Then in general, T ab will not be conserved g relative to the curved derivative operator,. It is possible to derive, based on a few simple desiderata, a transformation rule T ab g T ab, where g T ab is a function of T ab and φ that is conserved relative to the curved derivative operator. 14

15 It is natural to interpret the corrections to T ab as arising from the contribution of φ to the total mass-momentum content of spacetime in the geometrized theory. But there are several things to note. First, consider what happens if you begin with T g ab and move to a flat spacetime. As discussed in section 3.1, the choice of flat derivative operator is not unique: there are many different choices of flat derivative operator and gravitational field that can be recovered from a single curved spacetime. But the transformation from g T ab to the corresponding mass-momentum tensor in any given flat spacetime will depend on the choice of derivative operator and gravitational field. Hence, even though g T ab does appear to contain a gravitational component, there is no unique way to identify what that gravitational component would be, since the decomposition varies with the choice of flat derivative operator. In other words, given a matter distribution in spacetime, there is no way to associate a unique gravitational energy with that matter distribution. Moreover, even though the field T ab needs to be transformed when one moves to a curved spacetime, the correction terms do not contribute to (the geometrized form of) Poisson s equation because the corrections are all spacelike, and Poisson s equation depends on ρ = g T ab t a t b. So even though there is a way of making sense of gravitational energy in Newtonian gravitation, it is not a source in Poisson s equation. This work is still preliminary. The above remarks seem highly suggestive, but it is not perfectly clear yet how they bear on the relativistic case. This is a topic that I am currently actively working on. 4.2 Is energy conserved in GR? Another, even broader issue related to energy in GR concerns the sense in which energy is conserved in the theory. Historically, when physicists have referred to conservation of energy, they have often intended that some total, integrated quantity is constant over time. But in GR, it is most natural to represent energy (or rather, energy-momentum) as a local tensor field, T ab. This energy-momentum field is locally conserved in the sense that it is divergence-free. But except in very special circumstances (i.e., in a spacetime admitting timelike Killing fields), one cannot integrate T ab over a region of spacetime to assign a total energy to that region. This fact has led some physicists and philosophers of physics to conclude that energy is not conserved in GR. Putting the point in this way may be slightly misleading: total energy is not generally conserved in GR because in general, there is no candidate total energy in the first place. In fact, in any case were there is a total energy, i.e., in cases where because of spacetime symmetries it is possible to integrate T ab over a region of spacetime in an unambiguous way, it is conserved (so long as a T ab = 0). One appealing option is to simply note that, in the special cases where spacetime symmetries allow one to integrate energy at all, the conservation of total energy is equivalent to local conservation of energy, but local conservation of energy applies even in cases were the geometry of spacetime prevents one from defining total energy in the first place. Local conservation of energy, then, is the more general principle that should be taken to supersede conservation of total energy. However, some physicists have urged that local conservation of energy-momentum is not wholly satisfactory that somehow, the condition a T ab = 0 should not be admitted as a conservation principle, since it does not imply that there is some 15

16 quantity that is conserved over time. I believe that I can show that even in general spacetimes, there is a precise sense in which the conservation condition can be interpreted as a local conservation principle in the sense of an integrated quantity that is constant over time. The proposition to be proved can be put loosely as follows: given a spacetime (M, g ab ) and a smooth field T ab on M satisfying the conservation condition a T ab = 0, then for any point p M and any ϵ > 0, there is a family of suitable (small) open sets containing p such that (a) one can unambiguously integrate the flux of T ab over the boundary of any set in and (b) the magnitude of the integrated quantity one finds is less than ϵ. The interpretation is that the total energy flux over a suitably small surface can be made as small as one likes which means that in a sufficiently small neighborhood of any point p, the difference between the energy entering the region and the energy leaving is vanishingly small. This result supports the claim that the conservation condition can be understood as local conservation of total energy. More deserves to be said here. First, the argument follows the Geroch and Jang (1975) argument for the geodesic principle. There they begin with a timelike curve in an arbitrary spacetime and then consider a flat metric that agrees with the original metric on the image of the curve. This flat metric admits timelike Killing fields, and so one can integrate relative to these; on the image of the curve, the derivative operators associated with the two metrics agree, and so on the curve T ab is conserved relative to the flat derivative operator. The present argument involves noting that, by smoothness considerations, the relevant integrals can be made arbitrarily small by restricting attention to appropriate small neighborhoods of the curve. Two principal difficulties remain. The first is to correctly characterize the members of a step that is necessary to even state the proposition to be proved in a satisfactory way. The intuition is that the elements of should be thought of as δ-tubes, cylinders constructed around timelike curves passing through p whose spacelike extension is bounded by some parameter δ. But I have not yet developed this intuition further. The second difficulty involves fully understanding the flat metric that Geroch and Jang appeal to. The existence of such a metric was demonstrated by Cartan (1983, 2001), but in the language of moving frames. I am still working on how to translate the result into a more modern language to be sure that the result applies as widely as Geroch and Jang appear to suggest. 4.3 The origin of the conservation condition in geometrized Newtonian gravitation In section 3.2, I noted that there is an apparent difference between GR and GNG in that the conservation condition is an immediate consequence of Einstein s equation, but that it is not a consequence of (the geometrized form of) Poisson s equation. Hence, one might argue, the geodesic principle has a different status in each theory. However, there is more than one way to derive a conservation law for instance, one might consider a Lagrangian formulation of a theory, and then apply a version of Noether s theorem. In GR, it seems that is is possible to derive the conservation condition in this way. But the situation is complicated in GNG by the fact that there has been some difficulty in finding an appropriate Lagrangian to describe the theory. At least one proposal has been made, by Joy Christian (1997) in the context of trying to quantize GNG. But it remains (I believe) an open question as to whether a Noether 16