The gravitational field manifests itself in the motion of bodies.
- Einstein and Infeld (1949, p. 209)
Abstract
I argue that the best interpretation of the general theory of relativity (GTR) has need of a causal entity (i.e., the gravitational field), and causal structure that is not reducible to light cone structure. I suggest that this causal interpretation of GTR helps defeat a key premise in one of the most popular arguments for causal reductionism, viz., the argument from physics.
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Notes
For more discussion of these types of cases see Paul and Hall (2013).
This type of counter-example to the counterfactual theory is discussed by Paul (2009, pp. 177–182).
The opening page of The Oxford Handbook of Causation (OHC) suggests as much (Beebee et al. 2009, p. 1). But see also the remarks in Carroll (2009), Paul and Hall (2013, p. 249), Psillos (2009, p. 154), Schaffer (2007, pp. 872–874) and Tooley (1987, p. 5), inter alios. Carroll (2009, p. 285) notes that the OHC is filled with papers by reductionists who tell the many stories of failure. Schaffer (2007, pp. 872–874) pleads with his fellow reductionists to abandon the pursuit of reductively analyzing the causal relation. He (2016, Sect. 2.1) believes that the argument from physics is what shows that causal reductionism is true. Douglas Kutach (2013, pp. 282–306) tries to show that there probably is no “complete and consistent systematization of cause-effect relations ‘out there in reality’ that correspond to our folk concept of causation…” (ibid., p. 282) gesturing at the well-known problems and counter-examples to unified theories.
Schaffer (2008, p. 92). The argument Schaffer develops is explicitly one for causal reductionism and not causal eliminativism (cf. ibid., pp. 82–83, p. 86, p. 92). Causal eliminativism is the view that there are no obtaining causal relations (see Russell 1912–1913).
Schaffer’s original argument appealed to scientific practice in general. It is clear, however, that he privileges physics as that science which puts us in touch with the fundamental. Thus, if physics has no need of causation, causal reductionism is well evidenced.
For other arguments in favor of causal reductionism, see Norton (2007a, b). Cf. the recent critical discussion in Frisch (2014, pp. 1–21).
Cartwright (1993, p. 426).
See Craig (2001, 2008, p. 11). Though Frisch (2005, p. 3) does not affirm that the formalism/interpretation distinction is a part of the correct view of theory structure, he calls something like the view I defend the “standard conception”.
I will be applying the standard conception to general relativity, but in so doing I do not mean to enter the debate between Curiel (2009)—who argues that GTR does not need an interpretation—on one side, and Belot (1996) and Rovelli (1999)—who argue that GTR does require an interpretation—on the other. This is because the conception of interpretation in that debate is much broader than what is intended in the present study.
The notion of physical interpretation (the standard conception) I have in mind is closer to Curiel’s (2009, p. 46) idea of a categorical interpretation, and it seems almost obvious to me that GTR sits in need of an interpretation in that sense. Indeed, Curiel himself remarked, “…all physical theories have known interpretative problems of the categorical sort…” Curiel (2009, p. 47) emphasis in the original.
My approach to physical interpretations is completely consistent with both the syntactical and semantic views of scientific theory structure. For the syntactic view I have in mind, see Carnap (1967), Hempel (1965, pp. 173–226); and the critical discussion in Suppe (1989, pp. 38–77). Sometimes the syntactic view is called “the received view”. On the semantic view, see Giere (1988), Suppe (1989), van Fraassen (1970), (1991, pp. 4–8), (2008, pp. 309–311).
The claim that the law of inertia is just F = ma for the special case in which F = 0 is demonstrably false. The law of inertia is required to define the entity relative to which the second law holds, viz., an inertial frame (cf. Brown’s similar point in his 2005).
See Arnowitt et al. (2008) for the republication of the original 1962 paper.
Ruetsche (2011, p. 7) emphasis in the original. Saunders remarked similarly, “…we may read off the predicates of an interpretation from the mathematics of a theory, and, because theories are born interpreted, we have a rough and ready idea of the objects that they are predicates of.” Saunders (2003, pp. 290–291).
That close association has been challenged in the philosophy of physics literature (see the discussion of the debate in Lehmkuhl (2008) and Rey (2013) and the literature cited therein).
While the view I adopt is similar to Lehmkuhl’s (2008, p. 97) “Candidate 3: the metric—and all that it determines”, I believe considerations in Lehmkuhl’s excellent essay underwrite an inference to the best explanation argument for my assumed partial interpretation. That argument lurks behind the fact that almost all of the mathematical objects in the formalism of GTR thought to describe gravitation derive from the metric tensor. In this sense, their geometric and gravitational significance is parasitic on the geometric and gravitational significance of that sole fundamental tensor in the theory.
Rovelli’s (2004, pp. 33–34) remarks assume that the above partial interpretation is a part of Einstein’s metrical approach to GTR studies even if that’s not the case in quantum gravity studies. Norton (1989, p. 40) says that Einstein “calls the metric tensor the gravitational field.” See Einstein (1949, p. 685), Einstein (1952, p. 144), Einstein’s (2002, p. 33).
Carroll (2004, p. 50), Choquet-Bruhat (2009, p. 39), Geroch (2013, p. 65), Hartle (2003, p. 13) says “…the central idea of general relativity is that gravity arises from the curvature of spacetime—the four-dimensional union of space and time. Gravity is geometry.” (ibid.) emphasis in the original, Wald (1984, p. 9, cf. p. 68). Weinberg (1972, p. vii) clearly suggests that both (a) the geometrization of gravitation and (b) that the metric tensor represents the gravitational field are standard views (though he seems dissatisfied with at least (a)). Weinberg would go on to say that Einstein adhered to both (a) and (b). See also Rovelli (2004, p. 77). Cf. Lehmkuhl (2014) for a different perspective.
Class for Physics of the Royal Swedish Academy of Sciences (2011, p. 2).
I’m leaning slightly on Ruetsche (2011, p. 7). I do not whole-heartedly agree with her way of explicating a partial interpretation of GTR* for she injects into the partial interpretation axiomatic formalism (e.g., she explicitly includes Einstein’s field equations in GTR’s* partial interpretation). For me, the equations are the things being partially interpreted. They are not themselves the substantive content of the (partial) interpretation.
It is common in general relativity to call electromagnetic radiation a type of matter, though that is not the standard practice in particle physics (Rovelli 1997, p. 219. n. 10).
In 1916, Friedrich Kottler argued that Einstein interpreted the second term of the geodesic equation of motion in such a way that it described the gravitational field’s influence on massive bodies. Einstein would recognize this charge in Kottler’s work himself (see Einstein (1997, p. 238) I do not understand Einstein’s follow-up comment regarding meaninglessness. It does not cohere with what’s stated nor with Einstein’s general corpus.). He never corrected the causal nature of the interpretation attributed to him (see the interesting discussion of the relevant passages in Lehmkuhl (2014, p. 323)).
Einstein uses quasi-causal language when interpreting the geodesic equation of motion. In Einstein (2002, p. 339), for example, he said the geodesic equation of motion describes “the motion of a material particle under the action only of inertia and gravitation” (ibid. emphasis mine). A little later in the same work Einstein says of the geodesic equation of motion that it expresses “the influence of inertia and gravitation upon the material particle.” (ibid., p. 341) emphasis mine.
In dialog with Moritz Schlick about the nature of causation, Einstein (2006) interpreted gravitational phenomena causally.
Brown and and Lehmkuhl (2013, p. 3).
See Eddington (2014, pp. 125–127; cf. pp. 149–170), Einstein and Grommer (1927), Einstein, Infeld, and Hoffmann (1938), Einstein and Infeld (1940), Einstein and Infeld (1949), Fock (1959, pp. 215–218), Geroch and Jang (1975), Infeld and Schild (1949). The history of attempts to derive the law of motion for particles from the EFEs receives careful attention in Havas (1989); and Havas (1993).
Einstein admits in a letter to Ludwik Silberstein that “a really complete theory would exist only if the ‘matter’ could be represented in it by fields and without singularities.” As quoted in Havas (1993, p. 102) emphasis in the original.
“The fact that geodesic motion is a theorem and not a postulate has striking consequences that cannot be overemphasized. Earlier…I argued that it (and the need for corrections in the case of bodies with spin) casts doubt on the widespread view that space–time structure, in and of itself, can act directly on test bodies.” Brown (2005, p. 162).
Misner, Thorne, and Wheeler (1973, p. 476).
Geroch and Horowitz (1979, p. 212). In the perfect fluid case one clearly sees this in the presence of the Lorentz metric tensor in the energy–momentum tensor of the perfect fluid. Rovelli goes further: “…in all physical equations one now sees the direct influence of the gravitational field…any measurement of length, area or volume is, in reality, a measurement of features of the gravitational field”. Rovelli (1999, p. 7).
Misner, Thorne, and Wheeler (1973, p. 476).
See also Nerlich (1976, p. 264).
The coupling talk is particularly relevant for Lagrangian formulations of GTR, which Einstein utilized around 1918.
As quoted and translated by Pais (1982, p. 465). With respect to Einstein and the ultimate unified field theory, Pais (ibid.) goes on to remark, “He [Einstein] demanded that the theory shall be strictly causal, that it shall unify gravitation and electromagnetism, that the particles of physics shall emerge as special solutions of the general field equations…” emphasis mine
Also see his discussion (with Infeld) of the elevator thought experiment in Einstein and Infeld (1938, 226–235).
Robert Wald (personal correspondence, 12/18/2014). See also Geroch (1978, p. 180), (2013, p. 2, p. 65, p. 68), cf. Misner et al. (1973, pp. 476–477), Nerlich (1994), though not clearly committed to a causal interpretation of spacetime’s action says, “[b]ut GR surely makes spacetime something not easily distinguished from a real concrete entity with causal powers” (ibid., p. 183). And Carl Hoefer (2009, p. 702) says the causal interpretation is commonly accepted. Brown describes the position that spacetime in GTR acts as a view that is “widespread” (Brown 2005, p. 162). So far as I’m aware, neither Hoefer nor Brown endorse the causal interpretation I’m defending.
I’m ignoring the problem of specifying “choice” coordinates or a gauge.
More technically, for every timelike vector \( \xi^{\mu } \) associated with any point in the differentiable manifold representing spacetime the following relation holds: \( T_{\mu \nu } \xi^{\mu } \xi^{\nu } \ge 0 \). It is assumed that the energy–momentum tensor is appropriately related to the matter fields a hypothetical observer gives attention to. See the discussion in Malament (2009, pp. 6–7), Malament (2012a, p. 144) and Weatherall (2011).
As Geroch noted, “[e]verywhere, we see the metric, directly or indirectly, in the stress-energy…It appears that it is simply impossible to make any reasonable description of matter without the notions of space and time provided by the metric” Geroch (2013, p. 67). See also Hawking and Ellis (1973, p. 61), Malament (2012a, p. 160), Pooley (2013, p. 541. n. 38). The dependence holds even for Lagrangian formulations (see Lehmkuhl (2011, pp. 464–470)). Why is this true? The answer lies in the interpretation of GTR on offer (see Sect. 6.3).
“Gravity is one of the four fundamental interactions.” Hartle (2003, p. 3).
Birnbacher and Hommen (2013, p. 144) emphasis in the original. The authors are there concerned with a reductionist (the laws are non-causal) account of causation whose backbone is essentially the relation of nomological determination.
Glennan (2011, p. 811) points out that nomological determination is at home in a causal approach to laws of nature (mentioning the well-known causal account of David Armstrong inter alia).
DiSalle (1995, p. 327).
Chalmers (1996, p. 37) although Chalmers calls it natural supervenience.
Again see the characterization in Chalmers (1996, p. 37).
The Merlin case may strike one as too bizarre. Give attention then to any scenario in which something begins to exist on account of a preexisting cause. I, an agent, made a decision to write this paper. Even if agent causation reduces to event causation, and even if mental causation reduces to something non-causal, no problems arise. A plethora of events causally produced a decision, a mental event that was not around before. The decision depends for its existence and nature upon the preceding physical (and perhaps mental) events. If causal reductionism is right, the relevant instance of mental causation reduces to something non-causal, though (importantly) it is not eliminated. Causal reductionism is not causal eliminativism. We have therefore an instance of mental causation, but also an instance of grounding irrespective of whether or not causal reductionism holds.
Hawking and Ellis (1973, p. 183) emphasis mine.
Geroch (2013, p. 123) emphasis mine. Geroch uses ‘I-(p)’ to represent p’s past domain of influence. I am widening Geroch’s notion of a domain of influence a bit since even he admits that the term can be misleading, since I-(p) (having to do with the narrower conception of a past domain of influence) “does not include its boundary, whereas points on the boundary can, in general, also influence p” (ibid., p. 123). I am deliberately intending to include the boundary. In so doing, I follow Bhattacharyya’s et al. (2016, p. 17) identification of domains of influence with causal pasts and futures. Cf. Manchak (2013, p. 590), Ellis and Stoeger (2009).
In this section, I am not arguing that causal structure is causally efficacious. I am not arguing that the specific geometric features standing behind causal structure are themselves causally efficacious. The argument in this section is that there’s a type of structure to spacetime that cosmology needs that is causal in that it is explicitly defined in causal terms. That causal taint cannot be dispensed with because that causal structure cannot be identified with any non-causal structure such as light or null cone structure. So the stuff about light/null cones and causal structure is independent of the stuff about the causal efficaciousness of the gravitational field/metric field or spacetime geometry previously discussed in this paper.
Gravitational lensing involving matter or associated caustics can produce a collapse or folding in of the light cone induced by a vertex point p in the manifold (Ellis et al. 1998, pp. 2346–2347); Schutz says “…even a small amount of matter in spacetime will distort light-cones enough to make them fold over on themselves.” (p. 336)). That folding can entail that part of the (past or future) cone bends inside of the causal past or future of p (Perlick 2004; Tavakol and Ellis 1999, p. 41 who include further references on this point). The past/future light cone and the causal past/future of p cannot therefore be identical. Thanks to George Ellis for help with references.
The mistake of identifying causal structure in relativity with light cone structure is often committed by philosophers (see e.g., Frisch (2014, pp. 16–17), Field (2003, p. 436) comes close to suggesting such identification). Causal or influence structure is standardly regarded as more fundamental than light cone structure in GTR (see Geroch 2013, p. 125).
Hardy (2007, pp. 3084–3085).
Rickles (2008, p. 347. n. 124) emphasis mine. The quote’s immediate context is about Robb’s formulation of STR, but it’s clear from context that Rickles is also intending to characterize causal set approaches to quantum gravity.
Newton (1999, p. 405) italics removed.
Friedman et al. (1990, p. 1915).
Cf. Stephen Hawking (1992).
Thus, one can endorse the conclusions and resolutions of time travel paradoxes in Earman et al. (2009, pp. 93–100) and yet hold on to the supposition that the introduction of principles like SCP is not unscientific.
This is what Ned Hall (2004, p. 266) calls a “Mackie-style regularity account”.
The same argument will run with minimal fundamentalist causation in mind.
As quoted by Wheeler (1998, p. 235).
The gravitational field never vanishes even under coordinate transformations, and even in the absence of matter. Moreover, gravitational radiation can propagate in a vacuum.
Lehmkuhl does not deny that there is interaction between metric field and matter in GTR. He said that “…the matter fields and the metric field gμνare interacting in GR…” Lehmkuhl (2011, p. 469) emphasis in the original.
Einstein said,
“That something real has to be conceived as the cause for the preference of an inertial system over a non-inertial system is a fact that physicists have only come to understand in recent years….Also, following the special theory of relativity, the ether [features of Minkowski spacetime geometry] was absolute, because its influence on inertia and light propagation was thought to be independent of physical influences of any kind…” As quoted and translated by Brown and Pooley (2006, p. 68). Einstein said this in 1924.
Brown and Pooley would go on to remark that “[i]t was Einstein’s view, then, both that the inertial property of matter can be explained, and that this explanation is to be given in terms of the action of a real entity on the particles”. Brown and Pooley (2006, p. 68).
Continuing to use the theory in Schaffer’s (2009).
Lehmkuhl (2011, p. 467) emphasis in the original.
It is important to point out that Lehmkuhl believes that the causal interaction of fields is a sufficient condition for direct coupling (see Lehmkuhl 2011, p. 469). He too sees room for a relationship between interaction and coupling.
I am assuming that substantivalism is the view that spacetime exists as a substance (quoting Sklar) “independently of the existence of any ordinary material objects, where the latter phrase is taken to include even such extraordinary material objects as rays of light, physical fields…etc.” Sklar (1976, p. 161)
On the basis of parsimony motivations, Brown (2005, pp. 24–25) argues against the view that spacetime structure plays an explanatory role in accounting for the motions of bodies. Cf. the discussion in DiSalle (1994, p. 276) and DiSalle (1995, p. 327).
See the comments in Arntzenius (2012, p. 17 “[a]ccording to…[GTR] spacetime, again, is a single four-dimensional entity.” (ibid.)), and Nerlich (2003, p. 281). In general, a realist approach to our most successful theories of physics suggests spacetime substantivalism (see the comments in Pooley 2013, p. 539; Sklar 1976, p. 214; cf. Weinstein 2001 who argues that relationalism is tough to defend in the context of quantum physics).
I do have reservations about such an interpretation.
Although later in Rovelli (2004), he seems to think of his choice position as relationalist. He said, “Thus, both GR and QM are characterized by a form of relationalism.” (ibid., p. 220).
Rovelli (1997, p. 193). Weyl compared the gravitational field’s action to the productive (causal) behavior of the electric field. He said,
…just as the electric field, for its part, depends on the charges and is instrumental in producing a mechanical interaction between the charges, so we must assume here that the metrical field…is related to the material content filling the world. Weyl (1952, p. 220) emphasis in the original
In his ontology of physics, Weyl made room for a guidance field that appears to be causal (see Coleman and Korté (2001, p. 198) and the sources cited therein).
The idea that the geometric interpretation precludes a causal understanding of gravitational effects is expressed in many places, but see Livanios (2008, p. 390) who writes, “…[i]t is clear that if we adopt a 4-dimensional ontological framework,…and geometrise away the gravitational field, spacetime does not causally affect matter.”
I find it interesting that in one of the most exhaustive and most recent discussions of non-causal explanations in science and mathematics (explanations from constraint in Lange 2017), the section on geometric non-causal explanation (see specifically ibid., pp. 126–128) does not take up these matters having to do with geodesic motion and the gravitational field. Even if it did, Lange’s treatise seems most chiefly concerned with explanation. I’m interested in the ontology, what reality needs to be like to back explanations. I happen to think that obtaining causal relations can back non-causal (as well as causal) explanations. In fact, I think certain mathematical truths hold because (this is a because without cause that is indicative of a distinctive non-causal metaphysical explanation such as truth-making or grounding) certain causal relations obtain. Is this not suggested by causally interpreting mathematical formalism?
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Acknowledgements
I presented earlier versions of this paper at the 2014 Eastern American Philosophical Association conference, Mississippi State University, the University of Illinois at Urbana-Champaign, and Yale University. I thank my audiences at those institutions for their comments. I also thank two anonymous referees with Erkenntnis for their helpful comments and criticisms. The paper has benefited from correspondence with and/or comments from and/or discussions with David Albert, Tom Banks, David Black, Eddy Keming Chen, Shamik Dasgupta, George Ellis, Ned Hall, Michael Townsen Hicks, Barry Loewer, John Norton, Don Page, Laurie Paul, Joshua Rasmussen, Carlo Rovelli, Daniel Rubio, Jonathan Schaffer, Peter van Elswyk, Robert Wald, Aron Wall, and Dean Zimmerman. I thank all of these individuals for their help. Any mistakes that remain are mine.
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Weaver, C.G. On the Argument from Physics and General Relativity. Erkenn 85, 333–373 (2020). https://doi.org/10.1007/s10670-018-0030-8
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DOI: https://doi.org/10.1007/s10670-018-0030-8