Mach’s Principle as action-at-a-distance in GR: the causality question

A part of the revival of interest in Mach’s Principle since the early 1960s has involved work by physicists aimed at calculating various sorts of frame-dragging effects by matter shells surrounding an interior region, and arguing that under certain conditions or in certain limits (ideally, ones that can be viewed as plausibly similar to conditions in our cosmos) the frame dragging becomes “complete” (E.g. Lynden-Bell, Katz & Bičák, [1]) . Such results can bolster the argument for the satisfaction of Mach’s Principle by certain classes of models of GR. Interestingly, the frame-dragging “effect” of (say) a rotational movement of cosmic matter around a central point is argued by these physicists to be instantaneous — not an effect propagating at the speed of light. Not all physicists regard this as unproblematic. But rather than exploring whether there is something unphysical about such instantaneous “action at a distance”, or a violation of the precepts of Special Relativity, I am interested in exploring whether these physicists’ calculations should be thought of as showing local inertia (resistance to acceleration) to be an effect, with distant matter distributions being the cause. I will try to apply some leading philosophical accounts of causation to the physical models of frame dragging, to see whether they imply that the frame dragging is superluminal causation. I will then offer reflections on the difficulties of applying causal talk in physical theories.


Mach's ideas on inertia
Introduction. As is well known, a straightforward attempt to defend Leibnizian relationalism about space in a classical-physics context leads one naturally to Mach's response to Newton's bucket and globes thought experiments: the local standard of inertial (non-accelerated) motion should be either defined, or 5 dynamically determined in terms of the relative distances and relative motions with respect to the rest of the material bodies in the universe. 2 Einstein was captivated by Mach's critique of absolute space and his tentative suggestions about how to understand the phenomena of inertia without it, and capturing those ideas was an important goal as he worked toward his General Theory of 10 Relativity (see [3]). The final theory Einstein arrived at, however, captures the Machian idea about inertia at best imperfectly. There are numerous solutions of GR in which the local metrical/inertial structure seems as absolute, or nearly as absolute, as it is in Newton's theory; and in any event, certainly not fully determined by reference to the locations and relative motions of the rest of the uni- 15 verse's material bodies. On the other hand, there are models of GR in which the local metrical/inertial structure everywhere does seem to be fully determined by the overall matter distribution -principally the Friedmann-Robertson-Walker-Lemâitre (FRWL) "big bang" cosmological models, which are the models of GR most often taken to be good coarse-grained models of our actual universe. In 20 addition, since Thirring's 1918 calculation of a "frame-dragging" effect in GR 3 , there have been numerous demonstrations of "Machian" effects in GR, that is, effects on the local inertial structure attributable to the relative distances and motions of large masses -a nearby planet or star, a spherical shell of matter rotating around a central point, etc. These facts have kept interest in the 25 Machian perspective on inertia very much alive in a small segment of the GR physics community.
Among the frame-dragging results we will consider are certain recent ones published by a handful of physicists from 1995 onward, principally Lynden-Bell, Bičák and Katz (hereafter LKB), using perturbative techniques in FRWL models 30 to study what happens in GR when a spherical shell of matter is given a (small) rotation around a central point ( [1]). The intriguing results are that the framedragging effects inside the shell are instantaneous -enforced by the constraint equations on a 3-d hypersurface that follow from Einstein's equations -and, when the shell's radius becomes large enough, are complete -that is, the local 35 inertial frames inside the shell are locked to the shell, and no longer influenced by the pre-perturbation inertial structure fixed to the matter (if any) outside the shell.
What shall we make of these results, and of Mach's idea about the origin of inert more generally? Do they represent some sort of "instantaneous action-at-40 a-distance" -and if so, is that a problematic form of superluminal influence, in conflict with the precepts of relativity theory? Should the influence of distant matter on local inertia be considered a matter of cause and effect at all, or is some other perspective more appropriate? To set the stage for this discussion, let's first address these questions in the context of Mach's famous critique of 45 Newton with its suggestive -and not entirely clear -proposal that distant stars could take the place of Absolute Space in a proper mechanics.
Although the actual target of Newton's bucket experiment was Descartes' relational account of motion, it is possible to instead read it as an attack on Leibnizian relationism and an argument for the necessity of introducing Absolute 50 Space in order to account for inertia or (equivalently) for the so-called inertial effects present in non-inertial (accelerating and/or rotating) frames of reference, e.g. centrifugal and coriolis forces. Such a reading is especially tempting if one interprets the point of Newton's globes thought experiment as being to assert that in an otherwise-empty universe, we could have two globes connected by a 55 cord which are (absolutely) rotating around their centre of mass point -with the rotation being revealed by the tension in the cord. 4 In response to this way of reading Newton's Scholium thought experiments Mach wrote: Newton's experiment with the rotating vessel of water simply informs us that the relative rotation of the water with respect to the 60 sides of the vessel produces no noticeable centrifugal forces, but that such forces are produced by its relative rotation with respect to the mass of the earth and the other celestial bodies. No one is competent to say how the experiment would turn out if the sides of the vessel [were] increased until they were ultimately several leagues thick. [After extensive quotes from the Scholium to Def. 8]: It is scarcely necessary to remark that in the reflections here presented Newton has again acted contrary to his expressed intention only to inves-70 tigate actual facts. No one is competent to predicate things about absolute space and absolute motion; they are pure things of thought, 4 "For instance, if two globes, kept at a given distance one from the other by means of a cord that connects them, were revolved about their common centre of gravity, we might, from the tension of the cord, discover the endeavour of the globes to recede from the axis of their motion, and from thence we might compute the quantity of their circular motions. And then if any equal forces should be impressed at once on the alternate faces of the globes to augment or diminish their circular motions, from the increase or decrease of the tension of the cord, we might infer the increment or decrement of their motions; and thence would be found on what faces those forces ought to be impressed, that the motions of the globes might be most augmented; that is, we might discover their hindermost faces, or those which, in the circular motion, do follow. But the faces which follow being known and consequently the opposite ones that precede, we should likewise know the determination of their motions. And thus we might find both the quantity and the determination of this circular motion, even in an immense vacuum, where there was nothing external or sensible with which the globes could be compared." ([4], Scholium to def. 8.). Again, as with the Bucket, Newton's real argumentative intention here is not to assert the possibility of absolute rotation without relative motion in an otherwise-empty universe, but rather to illustrate how the quantity of absolute motions may sometimes be discoverable. But it does not seem unlikely that he would have agreed with such an assertion anyway, and readily. and 80s (for details of this history see the historical articles in [6], and [7]).
Aversion to Mach- There is an alternative interaction picture that could explain the instantaneous appearance of centrifugal and coriolis forces. One could imagine the 145 masses of the universe to be constantly "emitting" some causal influence that travels outward at the speed of light, all these influences joining together to generate the inertial structure felt at any particular place. Notice that in this picture it is not the instantaneous relative rotation of my body with respect to the other masses in the universe that is responsible for the appearance of 150 inertial forces; it might be said however that it is my apparent rotation -my rotation relative to the incoming light from the stars -that correlates with the inertial forces. We will touch on this alternative picture again later.
Mach's discussion of relative motion is, unsurprisingly, framed in terms that presuppose the absoluteness of coexistence, i.e., absolute simultaneity: for a given body A at a moment of time, there simply are facts about the relative distances between A and all other existing bodies in the universe, and about their relative velocities and accelerations -which facts can be expressed in a coordinate system rigidly fixed to A, or to any other body. 6 In the post-Special Relativity context, of course, the absolute simultaneity structure assumed by 160 Mach in his discussion is rejected, and this apparently problematises the whole idea of the Mach-heavy explanation of inertial forces. 7 Doesn't Special Relativity forbid causal influences that propagate faster than the speed of light? And if it is a genuine influence could it not be used to send instantaneous signals, raising the spectre of faster-than-light communication and the attendant causal 165 paradoxes?
One reaction to this worry might be to say: "Not so fast!". First, one may question whether functional dependence or determination between co-existing things (whether that be gravitational force dictated by Newton's law of gravity, or inertial forces determined a là Mach by relative distances and motions) 170 should automatically be considered a matter of causation. Secondly, one may note that even assuming that spacetime has a Lorentzian structure, that does not automatically guarantee the physical possibility of producing the kind of zig-zagging space-like causal connections needed to produce causal loops (see [9] sec. 3.3). And finally, it is worth noting that what Special Relativity profers,

175
General Relativity may take away. GR does not have the same symmetry group as SR. The symmetry group of all special-relativistic theories is the Poincaré group, whereas for GR, under one perspective the symmetry group is the general group of continuously differentiable diffeomorphisms, and under a different perspective the theory has no spacetime symmetries. Many models of GR have 180 dynamically distinguishable preferred frames in which the spacetime structure 6 The spacetime structure implicitly assumed by Mach and earlier thinkers is nowadays usually called "Leibnizian spacetime" (see [8]). In Leibnizian spacetime there is absolute simultaneity and an absolute temporal metric, but no preferred states of motion, e.g., no distinction between accelerated/non-accelerated or rotating/non-rotating frames. 7 In the post-General Relativity context Einstein and others have asserted that another big change in the world's presumed ontology also problematises Mach-heavy: the rise of the field as the primary physical entity in spacetime, replacing the "ponderable masses" that are assumed to be the dominant material constituents of the universe in both Mach's and Einstein's early discussions of inertia. I will not address this second difficulty for the Mach-heavy idea about inertia in this paper.
has a particularly simple expression, in which (among other things) clocks run at a faster rate than in all non-co-moving frames. And some models of GR contain closed time-like paths, so the theory has potential troubles from causal paradoxes independently of any Machian instantaneous action-at-a-distance.

185
But another way to address the raised concern is to explore more carefully the question: Is the Mach-heavy approach to inertia necessarily a causal explanation in the first place? In the rest of this paper I will address this question, both as a question for the Machian idea in general, and in particular in light of the complete-frame-dragging results mentioned earlier. The FRWL Big Bang models of GR are prima facie hospitable to a Machian account of inertia. There is a homogeneous distribution of matter throughout the universe, with the local inertial frames anchored as expected to the cosmic matter's rest frame; there is no global rotation of the matter, nor any empty 195 region "at infinity" to partly determine or influence the inertial structure in central regions. Although these facts hold as much for the infinite (k = 0 and k = −1) as closed finite (k = +1) universes, relativists favourable to a Machian account of inertia have traditionally singled out the k = +1 universe as especially clearly Machian in character (see [10], [1]). In more recent works using rotational 200 perturbations of FRWL models, however, Machian "complete" frame-dragging results have allegedly been proven for k = 0 and k = −1 cosmologies as well as k = +1 ([1], [11]). Before we turn to these rotational-perturbation studies, however, it will be instructive to consider a super-simplistic argument for the Machian character of FRWL universes -one that harks back to Einstein's own 205 initial thoughts about the generalised relativity of motion.
The simplistic argument starts by reconsidering Mach's polemic against Newton. Mach asks us to imagine putting the whole sphere of the fixed stars into rotation around us, and then prove that there will be no centrifugal forces induced on the (non-rotating) water in the bucket in the centre. Well, what 210 could be easier than putting the whole universe into rotation? We just need to transform our description of the actual cosmological situation into coordinates rotating around the chosen central point at the appropriate angular velocity. But Mach's challenge involves a situation in which the water is non-rotating in the frame of reference in which the fixed stars are rotating. If we want to achieve 215 this by transforming an actual situation into rotating coordinates, it will have to be a situation in which the water is rotating relative to the fixed stars -i.e., the situation of Newton's bucket experiment once the water has begun rotating relative to the ground, the fixed stars etc.
If we describe this situation from the perspective of a frame in which the Not quite. Whether one treats these Newtonian pseudo-forces as real or fictitious, their origin lies in something unacceptable to the Machian relationist: 230 a privileged state of rest (Absolute Space) or at least of non-acceleration, which is not determined by the relative positions and motions of the bodies in the world. What our exercise so far has done is merely confirm that situations can be described with Newtonian mechanics in non-inertial reference frames, if we care to take the trouble. What the Machian relationist wants, by contrast, 235 is a mechanics that treats all reference frames equally ab initio and does not introduce materially-inexplicable corrective forces when we use certain frames to describe a situation. In other words, a mechanics whose equations are valid and can be applied, without corrections, in any coordinate frame of reference we care to use. The reader may see where this is going: General Relativity, being 240 (i) generally covariant and (ii) having no background spatiotemporal structures, is just such a theory! And in fact this was how Einstein saw things at times in the years 1913-16: Implementing (i) and (ii) ought to be sufficient to guarantee the general relativity of motion in just the sense desired by Machian relationism.
Whether, how and why this line of thought goes astray in general is a sub-245 ject that has been discussed thoroughly in the literature and I will not pursue the general issue here (see e.g. [2]). But let's apply the thought to FRWL cosmologies as a particular case. In order to prove that Mach's ideas are fully satisfied in these cosmologies, perhaps it is enough to simply transform the The difference is clear: for FRWL models no boundary conditions at infinity are imposed -either because there is no infinity (k = +1), or because there is 285 a metrical structure infinitely far away from our chosen central point, but the metric there is identical to the metric at the centre, and hence just as relationally determined "out there" as it is anywhere else. 8 The simplistic argument for the fulfilment of Mach's ideas in FRWL models depends on a controversial assumption, as we have seen: that one be able to con-290 sider the metric and inertial structure as (in some sense) everywhere determined by the matter distribution (local and distant).
Sidebar: metric and matter-energy. A few notes about the idea of the metric being 'determined by' the distribution and motions of matter-energy are in order here, because the very idea may seem incoherent -or at least clearly wrong -295 in GR. Mathematically speaking a superficial reading of Einstein's equations G µν = 8πT µν suggests that the direction of determination is just the opposite.
For given the metric everywhere, Einstein's tensor G µν can be calculated, and therefore trivially also the stress-energy tensor. Moreover the metric is required to give meaning to T µν , so to speak; simply giving a set of 10 functions of on 300 a spacetime manifold and calling it 'T µν ' does not amount to saying how much rest mass (density), stress, or pressure there is, at a point or region, or how it is moving. One has to know the metric field in order to interpret T µν physically (see Lehmkuhl [12] for a thorough discussion).
But matters are not as desperate as these points seem to suggest. T µν is 305 not meant to be a fundamental description of matter; it is instead a field that represents, one might say, the 'gravitationally relevant' aspects of matter fields that are described more literally off-stage in some other theory (e.g. some relativistic field theory). While it is true that specification of the metric field is necessary for the meaningfulness of T µν in the GR framework, the physical 310 notions of local energy density, pressure, temperature etc. are independently meaningful, and to some extent operationally determinable by physical measurements. This is in fact related to the Equivalence Principle: local experiments and measurements in a closed lab won't reveal whether one is in otherwise-empty Minkowski spacetime, or traveling inertially in a FRWL cosmos, or falling (at 315 a safe distance!) toward a Schwarzschild black hole; but one can perfectly well measure how much 'ordinary' matter and energy of various kinds are present in the lab ('ordinary' being a qualification that excludes, at least, gravitational field stress-energy). And coming back to the superficial reading of Einstein's equations, while literally the metric g µν mathematically determines T µν , GR is 320 not a "supersubstantivalist" theory that reduces matter to being a structural property of spacetime. One should not think of the metric field as fully determining (in a conceptual or physical sense) what the matter-energy content distribution in a world is. What is more plausible is that Einstein's equations capture relations of mutual influence and dependence between matter-energy 325 present in the world and the structure of spacetime. The Machian perspective simply requires that -in some kinds of worlds at least -the dependence be plausibly seen as predominantly from matter-energy distribution to spacetime (especially inertial-motion) structure.
'Complete' determination of inertial structure by matter-energy?. That the de- ble that in such a model the metric is indeed, in a relevant sense, everywhere "determined" just by the matter distribution and its relative motions.
It is the "complete" dragging that may be questioned, because that there is at least some frame-dragging caused by all nearby rotating/accelerating material bodies is something that was demonstrated for GR by Einstein in rough Brill & Cohen showed that a rotating rigid mass shell induces frame dragging in the centre, which dragging becomes complete (interior inertial frames locked to the shell) once the mass of the sphere is large enough and close enough to all be within its own Schwarzschild radius. 9 But the shell of matter built into That is, as seen from infinity, the inertial frames within the shell rigidly rotate at the angular velocity Ω obs : There are no retardation effects between the shell and the inertia of a gyroscope at its center.
This of course does not contradict any physical causality principle, 385 since Ω − [the 'induced' rotation of interior inertial frames] can be considered to be merely the angular velocity of a coordinate system for the interior flat region. However, it is this coordinate system which is most directly related to effects observable from infinity, as explained above. Thus another view, more closely related to And although the concept of 'causation' discussed by philosophers (and 465 widely used in physics, as well as daily life) is notoriously hard to define or analyse, and fraught with controversy, it is clear enough that no superluminal causation is at most a true fact about our world, not a part of the meaning of 'causation'. 10 How, then can we approach the question of whether Machian 10 In a sense this is obvious just from the fact that many philosophers and physicists considered gravity to be an instantaneous causal action at a distance while also supposing that light might travel at a finite speed. But even setting aside classical spacetime-based intuitions and taking the semi-Riemannian structure of spacetime as conceptually fixed, that no frame dragging is a matter of cause and effect? A natural first step, at least, is to approach this question from the perspective of some of the most influential philosophical accounts of causation. Without letting this approach turn into a plodding exercise, I want to consider how things look from the perspective of a Lewis-style counterfactual theory of causation and from that of Jim Woodward's manipulationist theory ( [20]).

475
But before doing so, we can pause to note that two other influential contem- ing matter shell to the interior inertial structure. But there is also no causal process possessing a conserved quantity linking a supernova to a gravity-wave detector constructed to detect the gravitational waves produced by such explosions, or even one linking a binary-star system at one moment of time to the same system a year later -in both cases, because there is nothing strictly 490 conserved in the linking processes. (2) In recent years mechanistic approaches to causation have been championed by Glennan, Machamer, Craver and others, and have been widely discussed (see ( [21], ch. 15). But even the advocates of mechanism admit that it is not applicable to the level of fundamental physics (this is sometimes called the "bottoming out problem"). And it is surely clear 495 that even if we decide that Machian instantaneous frame dragging is a matter of cause-effect, it will not be because we discern a mechanism that carries the causal influence can travel outside the light-cone structure is just not a conceptual truth; the contemplatibility of both tachyons and time travel shows this.

influence/connection. 11
Let's consider Machian frame-dragging from the perspective of counterfactual and manipulationist theories of causation. We can start with the oldest 500 and simplest of such effects, the "Lense-Thirring" effects mentioned earlier. 12 In these scenarios a stationary rotation is set up, and a dragging force is calculated on freely moving test bodies inside or outside the rotating spherical mass.
Since in the case of zero rotation (relative to the Minkowski boundary conditions) there is (clearly, by symmetry considerations alone) no dragging force 505 present, the counterfactual test seems to be clearly passed: the rotation of the big spherical mass causes the dragging effects, interior and exterior. Things are also straightforward on the manipulationist approach. If one were to intervene on the putative cause (rotating mass), say by increasing or decreasing its rate of rotation a bit, the Thirring calculations show precisely how much the dragging 510 effects would be correspondingly increased or decreased. And such an intervention seems straightforwardly to meet the criteria of Woodward's theory, hard though it may be to put into practice on a planetary or stellar scale. (One might for example use carefully arranged H-bomb detonations to fractionally speed up or slow down the Earth's rate of rotation relative to the fixed stars.) Things are 515 however more complicated when we consider cosmological models, and thinking about the complications will lead us to see that the causality question may not be as open-and-shut as it seems even in the case of small local frame-dragging effects.
Central to applying either a Lewisian counterfactual theory or the Wood- Do these considerations equally undermine our initial assessment about Lense- 605 Thirring-type frame dragging in the interior or exterior of a star or planet? I believe they do not, and the reason is connected to the difference between using a model of GR to describe a small, local phenomenon in an overall cosmological situation that can be ignored or left off-stage, vs. using a model of GR to directly represent the whole universe. In that initial assessment we compared 610 a simple non-rotating-planet model to a stationary rotating-planet model; neither perturbations nor the initial value formulation came into play. The distinct metrics of the two models are simply taken as being good local approximations for the metrics near to a non-rotating/rotating planet (respectively), embedded in a world that globally, far away from the planet in question, is for all prac-tical purposes identically the same either way. In a strict mathematical sense, the rest of the cosmos cannot have exactly the same metric elsewhere, but the differences are so minute as to be negligible. And both the counterfactual and manipulationist approaches are willing to set aside negligible differences where appropriate. So there is (arguably at least) no difficulty in getting the ap-620 propriate counterfactuals (for the Lewis approach) or in describing a proper intervention (for Woodward's) and deriving the requisite results.
In the case of Lewis one can think of the "small miracle" that changes the planet from non-rotating to rotating (or changes its ω) as being a simple cutting and pasting together of two models of GR: the lower half coming from the first 625 (e.g. non-rotating) model and the upper half (after time t in some appropriate coordinate frame) from the second, rotating model. The miracle is in effect small, since it imposes only negligible change on the metric field far from the planet. 13 In the case of Woodward's approach the difficulty that an intervention may Undesirable backtracking counterfactuals (ones in which the consequence is a change in the past given some change in the present state) are avoided in the standard way by Lewis' criteria for world-similarity.
14 Using a term borrowed from Sober, Woodward describes an actual or physically possible intervention like this as 'ham-fisted'. The idea mooted above of using H-bomb detonations to change the Earth's angular velocity ω might well be too ham-fisted to serve as a Woodwardian intervention (if, e.g., the stress-energy of the explosions has to be so non-negligible that it is not reasonable to ignore its direct effect on the metric nearby). Matters are quite different, however, when we are talking about intervening on a substantial fraction of the material contents of the world, and producing a complete frame-dragging result. The cosmic shape of the metric outside the 650 rotating shell (and "at infinity" where applicable, i.e., in non-closed models) cannot be ignored and treated as unaffected, because in the limit as the matter shell grows, the frame dragging must become complete everywhere, not just in the interior. The sizes of the (putative) causes and effects are decidedly Finally, in the case of frame-dragging near a planet we may wonder whether, if we decide that it is clear that it amounts to causation according to counterfactual and manipulationist theories, it is also clear whether or not the cause-effect relationship is "instantaneous" or rather retarded. Unfortunately, the simplistic, compare-two-4-D-models approach to setting up the counterfactual situa-660 tion/intervention does not let us answer this question. One model is taken to be a good representation of what the metric is like near a non-rotating planet (static situation), another is taken to be a good representation of what the metric is like near a rotating planet (stationary situation) well after intervention or the small miracle. If we imagine a real-world intervention that puts a planet into 665 rotation (or changes its angular velocity ω), these models serve well to capture the pre-intervention situation and the post-intervention situation -after things have settled down, that is. But this pair of models does not in any way capture what the metric would be like during the course of such an intervention. Physical intuition may lead us to expect that creation or change of frame-dragging 670 effects would propagate outward from the intervention site(s) at the speed of light, and not be felt instantaneously; but as far as I know there are no models of GR that cover such a changing-rotation situation both realistically and rigorously.
Taking stock. What is the upshot of all of the above applications of philo- It's not that any specific counterfactual statement is itself difficult to assess; rather, it's only difficult to tell whether one can satisfy the input-requirements for the philosophical theory of causation. The situation reminds me strongly of 685 what Christopher Hitchcock notes concerning traditional puzzle-cases for philosophical accounts of causation ("Of Humean Bondage", [22]). The underlying facts that the are used as input by such accounts are typically perfectly clear and agreed-upon. (Which rock actually struck the window, Billy's or Suzie's? What would have resulted had the initial conditions been slightly different, in 690 such-and-so manner?). The underlying facts are not controversial; what gets disputed is just the philosophers' question "Do we have causation here?". In that paper Hitchcock recommended that we stop trying to answer that question by producing new and improved theories of causation. While I'm not sure I want to sign up for this skeptical/deflationist attitude across the board, I sus-695 pect it is the right one to take when it comes to puzzling questions of causality in GR, such as our frame-dragging question, or the vexed question of whether the metric field causally affects bodies moving inertially.
If we adopt such a deflationist attitude concerning Machian effects in GR, then interestingly, the distinction between Mach-lite and Mach-heavy discussed 700 in section 1 can be minimised if not eliminated entirely. What Mach wanted, expressed in terms appropriate to GR, is a perspective from which one has a dynamics with no absolute background (a given, in FRWL models) and one can see the local inertial structure as in some appropriate sense "determined" everywhere by the relational distribution of matter-energy in the universe. GR 705 lets us have that "determination" of inertia by matter-distribution in a stronger sense than the simple equation (1) that Mach dashed off, but we don't have to understand it in any very causally loaded way. The results of LKB and others serve to bolster the idea that the metrical and inertial structure is in some appropriate sense determined by the matter-energy distribution, at least 710 for relatively homogeneously-filled universes such as our own. And this in turn bolsters the first, "simplistic" argument for the satisfaction of Mach's ideas in GR (the argument from section 2 based on simply transforming coordinates in a standard FRWL model). That argument in turn is very close in spirit to Mach-lite: when we look at the world as we have it, what we can say is that 715 being in rotation relative to the cosmic matter distribution produces definite inertial forces, whereas relative rotations of other sorts (e.g., the water relative to the sides of the bucket) may or may not do so. That's the way things are empirically, and whether or not we want to force a causal interpretation on this fact about relative motions is a separate issue.

720
But I don't want to argue too hard for this reconciliation of Mach-lite and Mach-heavy in GR. The fact that local Mach-heavy-like frame dragging does seem to be a feature of GR, and one that can sustain a causal interpretation, is both pleasing and intriguing to philosophers and physicists who favour Leibnizian over Newtonian ideology when it comes to space or spacetime. One may 725 hope that efforts to replace GR with a theory consistent with quantum mechanics will shed further light onto the question of what determines inertial-motion structure in our world, and hence onto the question of whether frame-dragging on cosmological scales should be considered real in a robust causal sense. Machheavy's fortunes are still very much up in the air as regards future physics; but 730 for now the Mach-heavy perspective can be considered partially vindicated in GR, and more vindicated the more we deflate it into something close to Mach-lite. But in GR, at least, a causal understanding of Machian frame-dragging at the cosmological scale is certainly not forced on us.