Abstract
I critically discuss two dogmas of the “dynamical approach” to spacetime in general relativity, as advanced by Harvey Brown [Physical Relativity (2005) Oxford:Oxford University Press] and collaborators. The first dogma is that positing a “spacetime geometry” has no implications for the behavior of matter. The second dogma is that postulating the “Strong Equivalence Principle” suffices to ensure that matter is “adapted” to spacetime geometry. I conclude by discussing “spacetime functionalism”. The discussion is presented in reaction to and sympathy with recent work by James Read [“Explanation, geometry, and conspiracy in relativity theory” (2020) Thinking about Spacetime Boston: Birkäuser].
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Notes
For a recent, but somewhat one-sided, review of this literature, see Brown and Read (forthcoming). The dynamical view is clearly articulated in the papers cited in the main text. But since the “geometrical view” is the received view, it is somewhat difficult to identify a locus classicus for its statement. I would argue that the view goes back at least to Weyl (1952). It is also the view implicitly found in, for instance, Hawking and Ellis (1973), Wald (1984), and Malament (2012b); see also Stein (1977), discussed in the final section of this paper. Friedman (1983), Torretti (1983), and Maudlin (2012) are all sometimes cited as defenders of the geometrical view.
In the context of special relativity, conceived as a separate theory from general relativity, he thinks that both views are tenable, but that they are clearly distinct and he prefers the dynamical view. See Read (2019) for further discussion of issues that are salient to the debate in special relativity, but which will not concern me in the present article.
One might wonder whether the remaining disagreement I identify is really so far from those headline issues—or if rather, what I am attempting here is to isolate what has always been the core disagreement, from which the other debates have followed. Perhaps. But if so, I think it is doubly important to focus on this particular disagreement, because I think it is one that can be substantially resolved with new technical work. Thank you to Sam Fletcher for raising this point.
For Read (2020), general relativity, or a “general relativistic theory”, is a theory in which there are matter fields on Lorentzian manifolds, where the matter fields and metric are required to satisfy some system of differential equations that includes Einstein’s equation sourced by some stress-energy tensor associated with the matter fields. This statement is similar to mine, except that no constraints are made, in the first instance, on what that system of equations can be, except that it include Einstein’s equation. Note that both descriptions of the theory are vague on some important issues, such as how a stress-energy tensor is to be associated with matter.
Here and in what follows, I am taking certain stylized facts about the behavior of matter—for instance, that small massive bodies follow timelike geodesics, light rays follow null geodesics, etc.—to be necessary for matter to be “adapted” to geometry. (I will have much more to say on this issue below.) To connect my way of putting things in the main text to other discussions in the literature, I take Brown (2005) to be claiming that (d) is independent of (g) when he argues that it is only if one assumes the “strong equivalence principle”, discussed below, that one can explain these behaviors; and likewise, when Read et al. (2018) use the “two miracles”, also discussed below, as a litmus test for distinguishing the dynamical view from the geometric view. On this last point, what makes the “two miracles” miraculous is precisely that they do not follow from any other principles of general relativity—or more specifically, from (g)—and must be taken as brute facts.
That is: any other physical system aside from the fields associated with geometry, since one might take the field \(g_{ab}\) as a physical system. Thanks to an anonymous referee for encouraging this clarification.
This interpretation of (g) is not completely contrived: for instance, it is closely related to the understanding of “spacetime geometry”, at issue when some philosophers and physicists entertain the possibility that spacetime truly has a Galilean structure even though matter behaves “as if” spacetime is Minkowskian. In such cases, one makes a claim about the “true” spacetime geometry and leaves open what bearing that has for dynamics. See the discussion at the end of this section of what Read calls the “unqualified geometrical approach”.
Stein (1977), discussed in the final section of this paper, offers a more detailed historical analysis of how geometrical language came to have its present significance in relativity theory. Wilson (2006) provides a more general discussion of how meaning shifts with new applications, particularly in the context of (applied) mathematics.
As I hope is clear in the main text, including the paragraph that follows, I am not advocating for, or attributing to the “geometricist”, any particular view about, say, spacetime substantivalism; the significance I attribute to claims about “length”, “angle”, etc. are compatible with a broad range of positions in that debate. (For more on my own view in that connection, see Weatherall 2016, 2020a.) On a related note, Acuña (2016) describes views along the lines of the one I describe in the text as “absolutist”. He may use terms as he likes, but it is worth emphasizing that this use of “absolutist” is importantly different from its usage in connection with, say, Newton’s views on absolute space and time. (See also Read (2019) for a discussion of Acuña in this regard.)
Here I am invoking the fact that the wave equation is the Euler–Lagrange equation extremizing a certain action.
There are other reasons, too, to think that “spacetime geometry” should be interpreted in this stronger way. Acuña (2016) and Myrvold (2019), for instance, argue that the link between geometry and dynamics is analytic, and thus that all one could mean by attributing a certain geometry to space and time is to say that matter behaves in particular ways adapted to that geometry. Stein (1977, pp. 377–378), meanwhile, emphasizes the history of the development of geometrical ideas in physics, noting that the idea that spacetime has a geometry in the first place, in the modern sense, arises from studying the dynamics of matter understood to “probe” that geometry.
An anonymous referee raises the following worry: is it not the case that physicists often impose “extra” constraints on matter fields, such as energy conditions, which might be understood as limiting attention to just some of the many matter fields that are prima facie compatible with relativity theory (but which we reject on other grounds)? The answer is complicated by the different energy conditions in the literature and the various roles they play. But I would argue (a) at least in some cases, some energy conditions (such as the dominant energy condition—see footnote 18) serve as a guide to whether a proposed matter dynamics is suitably compatible with general relativity (e.g., Maxwell’s equations are compatible, in part because solutions always satisfy the dominant energy condition relative to whatever metric appears in the equations and stress-energy tensor; but arguably tachyon fields are not, because this condition fails), so that they are not “imposed” so much as “checked”; (b) in other cases, such as in proving singularity theorems, one uses energy conditions as a stand-in assumption, capturing the idea that one is considering only “reasonable” matter (which may be glossed as: matter compatible with the theory), so that although one has to impose a condition, doing so reflects precisely the relationship between matter and geometry described here; and (c) in yet other cases, energy conditions are introduced for technical convenience but have obscure physical motivation. This list may not be exhaustive, but I believe it covers the main cases.
This is not to say that there have not been other disagreements of substance adjacent to the one discussed here. For instance, there was a real disagreement, now resolved, regarding whether in general relativity, the geodesic principle (discussed in what follows) is a “consequence of Einstein’s equation”; for more on this, see Brown (2005), Malament (2012a), and Weatherall (2011). There is also an unresolved (I believe) disagreement about whether there is some salient difference between general relativity and other theories, such as Newtonian gravitation, regarding the status of inertial motion (Weatherall 2011; Sus 2014; Weatherall 2017). Finally, there have been disputes about whether spacetime geometry can provide “constructive explanations” (Brown and Pooley 1999, 2006; Janssen 2009; Dorato 2007; Frisch 2011; Acuña 2016; Read 2019); and about whether one can even state matter dynamics without specifying some geometrical background (Norton 2008; Wallace 2019).
I explore this idea, that the principles of a theory should be understood to form a network of logical interdependencies, in Weatherall (2017), where I call it “the puzzleball conjecture”.
See Weatherall (2020b) for a discussion of these results aimed at philosophers.
Specifically, Geroch and Weatherall require the dominant energy condition, which holds of a tensor \(T^{ab}\) at a point if, for any pair \(\xi ^a,\eta ^a\) of timelike vectors there, \(T^{ab}\xi _a\eta _b\ge 0\).
Note that although tracking has this consequence, one might also take tracking—for the details of which I refer the reader to Geroch and Weatherall (2018)—to provide a direct link between the solutions of a system of partial differential equations and curves that does not need to go through small bodies. So tracking could itself count as a partial explication of being “adapted to the geometry”. See Weatherall (2020b) for more on this idea.
Curiel (2017) provides a general discussion of cases where various energy conditions are known to fail.
To be clear, Fletcher is not committed to using his result for the purposes I sketch here, and so what I write in the main text should not be taken as a criticism of his result.
I say “dynamical school” because I do not believe that there is an essential link between the program I will presently describe and the dynamical view as I have presented it, viz., the idea that (d) is independent of (g) on a weak reading of (g). And yet defenses of the dynamical view seem invariably to invoke the “strong equivalence principle” or similar ideas in what I take to be a problematic way.
Read (2020, pp. 10–11) writes that the SEP is an “important condition for the chronogeometricity of the metric field” (p. 10), but hedges on whether it should be viewed as strictly necessary, sufficient, or jointly sufficient along with some other assumptions. In more recent correspondence, Read has suggested his views have shifted on this issue, and that he now sees the SEP primarily as establishing a link between general relativity and special relativity.
There is a long history of controversy over the equivalence principle in its various guises, where semi-precise statements are offered and then shown to be inadequate for one reason or another. I do not mean to engage in this debate here: my goal is not to argue that Brown or Read has failed to capture what Einstein had in mind when he formulated his principle, nor do I wish to claim that this or any other version of the equivalence principle is trivial or false. I am specifically interested in the question of whether the assertion that I have called the strong equivalence principle, as elaborated and elucidated by Read, adequately captures the idea that matter dynamics are “adapted to spacetime geometry”. Moreover, although I do mean to criticize one aspect of how Brown and Read have used the SEP, it is hardly as if Brown, much less Read, has invented this principle: it is common fare in textbooks on general relativity (e.g., Misner et al. 1973, p. 386). The concerns I am raising are as much directed at these classic textbook treatments as at the contemporary philosophers who have invoked them.
The first concern is similar in several important respects to arguments made by Fletcher (2020); I did not see that manuscript until this one was drafted, but my perspective was certainly shaped by conversations with him.
There is a third issue here that I do not emphasize, which is that it is not clear why special relativity is given primacy of place a century after general relativity superseded it. In particular, when one introduces a new matter theory—say, a theory of an inflation field in the early universe—why should one be obligated to first identify a “special relativistic” form for the dynamical equations of that theory? One possible answer—and I am grateful to an anonymous referee for raising this—is that the Brown–Pooley constructive/dynamical relativity program was originally developed in the context of Minkowski spacetime, and that Brown then aims to extend it to general relativity. (This observation is perhaps connected to Read’s position that the dynamical and geometric approaches are distinct in the context of special relativity—but collapse once one moves to general relativity.) From this perspective, it arguably does make sense to interpret general relativity through the lens of an already-worked-out position on special relativity. But even so, I think it would be preferable if one could do the work the SEP is meant to do, at least for establishing (d) in general relativity, without using a different theory as a crutch. Moreover, the approach to (d) that I defend above, involving results such as those of Geroch and Weatherall (2018), works just as well for special relativity as for general relativity, since any general result will hold in flat spacetime as a special case.
Read et al. (2018) suggest that the simplest form of an equation is one in which no terms vanish by coordinate transformation, but I find this account inadequate for reasons that I hope will be clear presently. See footnote 34.
This is just a reflection of the well-known fact that propositions can be given many different provably equivalent syntactic forms, e.g., p, \(p\wedge \top \), \(p\wedge \bot \), etc., some of which are simpler than others (irrespective of any choice of coordinates). Fletcher (2020) raises a similar concern with minimal coupling, arguing that it is “hyperintensional” in the sense that it distinguishes between cases that ought to be seen as equivalent.
Here \([\cdot ]\) indicates antisymmetrization on indices; and I raise and lower indices using \(\eta _{ab}\), the Minkowski metric.
One might worry that Eqs. (1a) are not, after all, equations “in” Minkowski spacetime—say, because the derivative operator and metric appearing in the equations are not the Minkowski metric and derivative operator, or because they do not take their “simplest form” in coordinates adapted to \(\eta _{ab}\)—even though they have precisely the same solutions. For my own part, I do not think this is a compelling position, because I think the most natural sense, mathematically, of “same (differential) equation” is “has the same solutions”. But if one wishes to argue that the cogency of the SEP depends on adopting a different standard of equivalence for equations, that route is certainly available—but the very fact that there could be disagreements on this point shows that the SEP is unclear in just the way I am claiming, because the class of manipulations that preserve an equation have not been specified.
This argument is why I do not take the claim that equations are in their simplest form if terms do not drop out by changing coordinates to be satisfactory, because there may be other transformations available that further simplify equations.
It is not clear that Read’s motivation for using a symmetry condition is to avoid possible ambiguities arising from the notion of the “form” of an equation; the motivation offered in Read et al. (2018), for instance, is the concern that curvature terms may appear in some equations that “should” satisfy the SEP.
Recall that a Killing field of a metric \(g_{ab}\) is a vector field \(\xi ^a\) satisfying Killing’s equation: \(\nabla _{(a}\xi _{b)}=\mathbf {0}\), where \(\nabla \) is the Levi-Civita derivative operator associated with \(g_{ab}\). Killing’s equation implies that the Lie derivative of \(g_{ab}\) along \(\xi ^a\) vanishes, which in turn means that there exists locally a one parameter family of isometries generated by \(\xi ^a\). These isometries are “local symmetries” in the straightforward sense that they are local maps that preserve the metric. See Malament (2012b, §1.9) for more details on Killing fields.
Read (2020, §2.2) discusses the notion of symmetry that he does have in mind, and appears to identify it with “isometries”, which would suggest that Killing fields are the generators of local symmetries in his sense; he goes on to acknowledge that the “the metric field \(g_{ab}\) in GR need not in general gave any non-trivial symmetries” (p. 6). This shows, I think, that there are internal tensions in his discussion of the miracles that bear on the SEP, and which are worth discussing.
There is another problem with this condition, which is that even if cases, with curvature, where one does have a Killing field, in general those fields will not correspond to “Poincaré symmetries”.
Likewise, if one has Killing fields in a neighborhood of a point, then the one parameter families of isometries generated by those Killing fields will be “symmetries” of this equation, in the natural sense that for any solution \(\varphi \) in that neighborhood, the pullback of \(\varphi \) along each isometry in the family will also solve the equation in an appropriate neighborhood.
Given the analyses in Read et al. (2018) and Brown and Read (2016), one might think that curvature terms are compatible with the SEP, properly construed, as long as they appear only in second order equations. One might take this to mean that various “non-minimally coupled” scalar fields should count as satisfying the SEP. But then consider the (first-order) equation \(\nabla _a\xi ^a + R\xi ^a\xi _a = 0\). The same argument applies.
On the other hand, Fletcher’s example is that of a dust field in Minkowski spacetime, which is not “locally Poincaré invariant” because its equations take an especially “simple form” in comoving coordinates. That this example might fail to satisfy the SEP suggests that something has gone badly wrong in the whole program. But I set this issue aside, because I think it is ultimately a red herring for present purposes.
I take the manipulations there to be consistent with the definitions given in Read (2020, §2.2): “A coordinate transformation is a dynamical symmetry just in case the dynamical equations governing non-gravitational fields take the same form in coordinate systems related by that transformation” (pp. 177). Dewar (2020) also engages in detail with this appendix.
Here is a stab at a more precise, less “syntactic” statement. One can always find, in a sufficiently small neighborhood of p, a flat metric \(\eta _{ab}\) that agrees with \(g_{ab}\) at p, and whose Levi–Civita derivative operator \(\bar{\nabla }\) agrees with that of \(g_{ab}\) at p. This metric is essentially unique, in the sense that any two such metrics are related, within a sufficiently small neighborhood of p, by an isometry that leaves p fixed. Rewrite derivatives in your equation in terms of \(\bar{\nabla }\) and a tensor field \(C^a{}_{bc}\). Say a geometrical object X (i.e., one for which a pushforward is defined) is “locally Poincaré invariant” (relative to \(\eta _{ab}\)) if for any smooth, smoothly invertible map \(\varphi \) from a sufficiently small neighborhood of p to itself, which acts as an isometry for \(\eta _{ab}\) and leaves p fixed, is such that \(\varphi _*(X)_{|p}=X_{|p}\). Note that on this recovery, “Poincaré” seems like a misnomer, since we do not consider translations.
Moreover, if one recovers the condition as in the previous footnote, in general “local Poincaré transformations” will not take solutions to solutions for standard equations, including, e.g., Maxwell’s equations in curved spacetime.
Compare the claim here that any equation whose summands are vectors cannot be Poincaré invariant, which holds simply because the components of any non-zero vector will always change when one applies a Lorentz boost or rotation, to the discussion surrounding (Read et al. 2018, Eq. A.12), where the authors apparently claim that an equation with terms of just this character is Poincaré invariant. I believe that their claim is a slip, and that their argument turns on an ambiguity about whether the sense of “same form in a coordinate system” is supposed to mean “same components” (which is how I have interpreted them here) or “same symbolic representation”, which I fear is trivial, since one is free to use whatever symbols one likes.
One might worry that insofar as my main worry about the SEP, even appropriately clarified, is the one given in this section, that ultimately my differences with Brown and Read come down to purely technical considerations, and not philosophical ones. But I do not think this is correct. Proving the relevant theorems may well be a mathematical exercise, but identifying what they should be, and even stating them precisely, is a matter of deep interpretational importance with a long provenance (discussed, in part, in the final section of this paper). I take it that the SEP reflects a particular idea about what, fundamentally, makes general relativity a theory about “spacetime geometry” (or, “chronogeometry”). As such, whether it secures the necessary results reflects on whether we truly understand what it means to attribute a certain geometry to space and time, and ultimately what the physical (and even metaphysical) significance of general relativity is.
As noted above, Weatherall (2019, fn. 40) offers some considerations for why one should not be an inertial frame functionalist; Read and Menon (2019) offer other arguments for why inertial frame functionalism has limitations. The concerns I raise here are not about inertial frame functionalism in particular, but rather the idea that one can say in advance what we even mean by “spacetime role” once and for all; identifying that role with inertial frames is just one example. Baker (2019) argues that the spacetime concept is a “clus ter concept”, in the sense that there are many different, related conditions that one would associate with it without any of them being jointly necessary or sufficient for it to apply. That argument is closer to the one I raise here, except that my emphasis is on how difficult and theory-dependent it is to identify what we mean by “the spacetime role”.
Some uses of functionalism in the recent literature, such as that of Knox (2013) and Lam and Wüthrich (2018), have addressed a somewhat different question, namely, what would need to be the case for spacetime, in the sense of general relativity, to “emerge” from a different theory, such as a theory with different geometrical structure (Knox) or a quantum theory of gravity (Wüthrich and Lam). In this case, since general relativity is the target, the concern that I raise here does not get purchase.
References
Acuña, P. (2016). Minkowski spacetime and lorentz invariance: The cart and the horse or two sides of a single coin? Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 55, 1–12.
Baker, D. J. (2019). On spacetime functionalism. Unpublished manuscript. http://philsci-archive.pitt.edu/15860/.
Brown, H. R. (2005). Physical relativity. New York: Oxford University Press.
Brown, H. R., & Pooley, O. (1999). The origins of the spacetime metric: Bell’s lorentzian pedagogy and its significance in general relativity. In C. Callender & N. Huggett (Eds.), Physics meets philosophy at the Planck scale (pp. 256–72). Cambridge: Cambridge University Press.
Brown, H. R., & Pooley, O. (2006). Minkowski space-time: A glorious non-entity. In D. Dieks (Ed.), The ontology of spacetime (pp. 67–89). Amsterdam: Elsevier.
Brown, H. R., & Read, J. (2016). Clarifying possible misconceptions in the foundations of general relativity. American Journal of Physics, 84(5), 327–334.
Brown, H. R., & Read, J. (Forthcoming). The dynamical approach to spacetime theories. In E. Knox & A. Wilson (Eds.), The Routledge companion to philosophy of physics. London: Routledge. http://philsci-archive.pitt.edu/14592/.
Curiel, E. (2017). A primer on energy conditions. In D. Lehmkuhl, G. Schiemann, & E. Scholz (Eds.), Towards a theory of spacetime theories (pp. 43–104). Boston, MA: Birkhäuser.
Dewar, N. (2020). General-relativistic covariance. Foundations of Physics, 50, 294–318.
Dorato, M. (2007). Relativity theory between structural and dynamical explanations. International Studies in the Philosophy of Science, 21(1), 95–102.
Earman, J. (2014). No superluminal propagation for classical relativistic and relativistic quantum fields. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 48, 102–108.
Fletcher, S. C. (2013). Light clocks and the clock hypothesis. Foundations of Physics, 43(11), 1369–1383.
Fletcher, S. C. (2020). Approximate local poincaré spacetime symmetry in general relativity. In C. Beisbart, T. Sauer, & C. Wüthrich (Eds.), Thinking about space and Time: 100 years of applying and interpreting general relativity (pp. 247–267). Boston, MA: Birkhäuser.
Fletcher, S. C., & Weatherall, J. O. (2020). Is spacetime approximately locally flat? Unpublished manuscript.
Friedman, M. (1983). Foundations of space-time theories: Relativistic physics and philosophy of science. Princeton, NJ: Princeton University Press.
Frisch, M. (2011). Principle or constructive relativity. Studies in History and Philosophy of Modern Physics, 42(3), 176–183.
Geroch, R. (2011). Faster than light? In M. Plaue, A. Rendall, & M. Scherfner (Eds.), Advances in Lorentzian geometry (pp. 59–80). Providence, RI: American Mathematical Society.
Geroch, R., & Weatherall, J. O. (2018). The motion of small bodies in space-time. Communications in Mathematical Physics, 364, 607–634. https://doi.org/10.1007/s00220-018-3268-8.
Grünbaum, A. (1977). Absolute and relational theories of space and space-time. In J. Earman, C. Glymour, & J. Stachel (Eds.), Foundations of space-time theories (pp. 303–373). Minneapolis, MN: University of Minnesota Press.
Hawking, S. W., & Ellis, G. F. R. (1973). The large scale structure of space-time. New York: Cambridge University Press.
Janssen, M. (2009). Drawing the line between kinematics and dynamics in special relativity. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 40(1), 26–52.
Knox, E. (2013). Effective spacetime geometry. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 44(3), 346–356.
Knox, E. (2019). Physical relativity from a functionalist perspective. Studies in History and Philosophy of Modern Physics, 67, 118–124.
Lam, V., & Wüthrich, C. (2018). Spacetime is as spacetime does. Studies in History and Philosophy of Modern Physics, 64, 39–51.
Malament, D. (2012a). A remark about the “geodesic principle” in general relativity. In M. Frappier, D. H. Brown, & R. DiSalle (Eds.), Analysis and interpretation in the exact sciences: Essays in Honour of William Demopoulos (pp. 245–252). New York: Springer.
Malament, D. B. (2012b). Topics in the foundations of general relativity and Newtonian gravitation theory. Chicago, IL: University of Chicago Press.
Maudlin, T. (2012). Philosophy of physics: Space and time. Princeton, NJ: Princeton University Press.
Misner, C. W., Thorne, K. S., & Wheeler, J. A. (1973). Gravitation. New York: W. H. Freeman.
Myrvold, W. C. (2019). How could relativity be anything other than physical? Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 67, 137–143.
Norton, J. D. (2008). Why constructive relativity fails. The British Journal for the Philosophy of Science, 59, 821–834.
Read, J. (2019). Geomertrical constructivism and modal relationism: Further aspects of the dynamical/geometrical debate. Unpublished manuscript.
Read, J. (2020). Explanation, geometry, and conspiracy in relativity theory. In C. Beisbart, T. Sauer, & C. Wüthrich (Eds.), Thinking about space and time: 100 years of applying and interpreting general relativity (pp. 173–206). Boston, MA: Birkhäuser.
Read, J., Brown, H. R., & Lehmkuhl, D. (2018). Two miracles of general relativity. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 64, 14–25.
Read, J., & Menon, T. (2019). The limitations of inertial frame spacetime functionalism. Synthese. https://doi.org/10.1007/s11229-019-02299-2.
Stein, H. (1977). On space-time and ontology: Extract from a letter to adolf grünbaum. In J. Earman, C. Glymour, & J. Stachel (Eds.), Foundations of space-time theories (pp. 374–402). Minneapolis, MN: University of Minnesota Press.
Sus, A. (2014). On the explanation of inertia. Journal for General Philosophy of Science, 45(2), 293–315.
Torretti, R. (1983). Relativity and geometry. Oxford: Pergamon Press.
Wald, R. M. (1984). General relativity. Chicago: University of Chicago Press.
Wallace, D. (2019). Who’s afraid of coordinate systems? An essay on representation of spacetime structure. Studies in History and Philosophy of Modern Physics, 67, 125–136.
Weatherall, J. O. (2011). On the status of the geodesic principle in Newtonian and relativistic physics. Studies in the History and Philosophy of Modern Physics, 42(4), 276–281.
Weatherall, J. O. (2014). Against dogma: On superluminal propagation in classical electromagnetism. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 48, 109–123.
Weatherall, J. O. (2016). Regarding the ‘hole argument’. The British Journal for the Philosophy of Science, 69(2), 329–350.
Weatherall, J. O. (2017). Inertial motion, explanation, and the foundations of classical space-time theories. In D. Lehmkuhl, G. Schiemann, & E. Scholz (Eds.), Towards a theory of spacetime theories (pp. 13–42). Boston, MA: Birkhäuser.
Weatherall, J. O. (2019). Conservation, inertia, and spacetime geometry. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 67, 144–159.
Weatherall, J. O. (2020a). Some philosophical prehistory of the (Earman-Norton) hole argument. Studies in History and Philosophy of Modern Physics, 70, 79–87.
Weatherall, J. O. (2020b). Geometry and motion in general relativity. In C. Beisbart, T. Sauer, & C. Wüthrich (Eds.), Thinking about space and time: 100 years of applying and interpreting general relativity (pp. 207–226). Boston, MA: Birkhäuser.
Weyl, H. (1952 [1918]). Space time matter. Mineola, NY: Dover.
Wilson, M. (2006). Wandering significance: An essay on conceptual behavior. Oxford: Oxford University Press.
Acknowledgements
This material is partially based upon work produced for the project “New Directions in Philosophy of Cosmology”, funded by the John Templeton Foundation under Grant Number 61048. I am grateful to Sam Fletcher, David Malament, and James Read for comments on an earlier draft of this manuscript, to Chris Wüthrich for encouraging me to produce it, to two anonymous referees for helpful suggestions, and to Clara Bradley, Harvey Brown, John Earman, Sam Fletcher, Eleanor Knox, Dennis Lehmkuhl, David Malament, Oliver Pooley, and David Wallace for many conversations over the years related to this material.
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Weatherall, J.O. Two dogmas of dynamicism. Synthese 199 (Suppl 2), 253–275 (2021). https://doi.org/10.1007/s11229-020-02880-0
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DOI: https://doi.org/10.1007/s11229-020-02880-0