On the reduction of general relativity to Newtonian gravitation

Highlights

• Clarifies the physical interpretation of the geometrical relationship between general relativity and Newtonian gravitation.

• Expounds this relationship as one of intertheoretic reduction.

• Introduces topological structure on spacetime models to encode similarity of spacetime observables for this reduction.

• Provides several concrete examples of Newtonian limits parameterized by distinct observables.

Abstract

Intertheoretic reduction in physics aspires to be both to be explanatory and perfectly general: it endeavors to explain why an older, simpler theory continues to be as successful as it is in terms of a newer, more sophisticated theory, and it aims to relate or otherwise account for as many features of the two theories as possible. Despite often being introduced as straightforward cases of intertheoretic reduction, candidate accounts of the reduction of general relativity to Newtonian gravitation have either been insufficiently general or rigorous, or have not clearly been able to explain the empirical success of Newtonian gravitation. Building on work by Ehlers and others, I propose a different account of the reduction relation that is perfectly general and meets the explanatory demand one would make of it. In doing so, I highlight the role that a topology on the collection of all spacetimes plays in defining the relation, and how the selection of the topology corresponds with broader or narrower classes of observables that one demands be well-approximated in the limit.

Introduction

In physics the concept of reduction is often used to describe how features of one theory can be approximated by those of another under specific circumstances. In such circumstances physicists say the former theory reduces to the latter, and often the reduction will induce a simplification of the features in question. (By contrast, the standard terminology in philosophy is to say that the less encompassing, approximating theory reduces the more encompassing theory being approximated.) Accounts of reductive relationships aspire to generality, as broader accounts provide a more systematic understanding of the relationships between theories and which of their features are relevant under which circumstances.

Thus reduction is naturally taken to be physically explanatory. Reduction can be more explanatory in other ways as well: sometimes the simpler theory is an older, predecessor of the theory being reduced. These latter “theories emeriti” are retained for their simplicity—especially regarding prediction, and sometimes understanding and explanation—and other pragmatic virtues despite being acknowledged as incorrect.¹ Under the right circumstances, one incurs sufficiently little error in using the older theory; it is empirically adequate within desired bounds and domain of application. Reduction explains why these older, admittedly false (!) theories can still be enormously successful and, indeed, explanatory themselves.

The sort of explanation that reduction involves is similar to that which Weatherall (2011) identifies in the answer to the question of why inertial and gravitational mass are the same in Newtonian gravitation. In that case, the explanandum is a successfully applied fact about Newtonian systems. The explanans is a deductive argument invoking and comparing Newtonian theory and general relativity, showing that this fact is to be expected if the latter is true. In the case of reduction, the explanandum is rather the relation of non-identity between the predictions and explanations of two successful theories. The explanans is also a deductive argument invoking both theories, but one showing that if the reducing theory is adequate, then the theory it reduces to is adequate as well, within desired bounds and domain of application.

Despite the philosophical and scientific importance of reduction, it is astounding that so few reductive relationship in physics are understood with any detail. It is often stated that relativity theory reduces to Newtonian theory, but supposed demonstrations of this fact are almost always narrowly focused, not coming close to the level of generality to which an account of reduction aspires, which is to describe the relationship between the collections of all possibilities each theories allows. Indeed, much discussion of the Newtonian limit of relativity theory (e.g., Batterman (1995)) has focused on the “low velocity limit.” For example, the relativistic formula for the magnitude of the three-momentum, p, of a particle of mass m becomes the classical formula in the limit as its speed v (as measured in some fixed frame) becomes small compared with the speed of light c:

$$p=\frac{mv}{\sqrt{1-{\left(v/c\right)}^2}}=mv\left(1+\frac{1}{2}{\left(v/c\right)}^2+\frac{3}{8}{\left(v/c\right)}^4+\frac{5}{16}{\left(v/c\right)}^6+\cdots \right)\underset{v/c\ll 1}{\approx }mv.$$

Similar formulas may be produced for other point quantities.

Another class of narrow demonstrations, so-called “linearized gravity,” concerns gravitation sufficiently far from an isolated source. In a static, spherically symmetric (and asymptotically flat) relativistic spacetime $\left(M,g_{ab}\right)$, one can write

$$g_{ab}={\eta }_{ab}+{\gamma }_{ab},$$

where ${\eta }_{ab}$ is the Minkowski metric. If the components of ${\gamma }_{ab}$ are “small” relative to a fixed global reference frame, then in terms of Minkowski coordinates $\left(t,x,y,z\right)$,

$$g_{ab}\approx \left(1+2\varphi \right)\left(d_{a}t\right)\left(d_{b}t\right)-\left(1-2\varphi \right)\left[\left(d_{a}x\right)\left(d_{b}x\right)+\left(d_{a}y\right)\left(d_{b}y\right)+\left(d_{a}z\right)\left(d_{b}z\right)\right],$$

$$T_{ab}\approx \rho \left(d_{a}t\right)\left(d_{b}t\right),$$

where $T_{ab}$ is the stress-energy tensor, and the field ϕ satisfies Poisson's equation,

$${\nabla }^2\varphi =4\pi \rho .$$

(Here ${\nabla }^2$ is the spatial Laplacian.) Accordingly, the trajectories of massive test particles far from the source will in this reference frame approximately follow trajectories expected from taking ϕ as a real Newtonian gravitational potential. In practice one often has a rough and ready handle on how to apply these approximations, but it is difficult to make all the mathematical details precise.²

It is, however, not often explicitly recognized that even the collection of all point quantity formulas, such as eq. (1), or the linearized expression for the metric (eq. (3)) along with trajectories of test particles far from an isolated massive body, together constitute only a small fragment of relativity theory. In particular, they say nothing about the nature of gravitation in other circumstances, or exactly how the connection between matter, energy, and spacetime geometry differs between relativistic and classical spacetimes.³ Insofar as one is interested in the general account of the reduction of one theory to another, these particular limit relations and series expansions cannot be understood to be a reduction of relativity theory to classical physics in any strict sense. Even the operationally minded would be interested in an account of how arbitrary relativistic observables can be approximated by their Newtonian counterparts.

The development of Post-Newtonian (PN) theory, which is very general method for applying the “slow-motion” limit to a variety of formulas like equation (1) ameliorates this concern somewhat (Poisson and Will, 2014). Its goal is to facilitate complex general relativistic calculations in terms of quantities that are (in principle) measurable by experiments, although it typically only applies as a good approximation to a small region of a relativistic spacetime. Indeed, while it seems in many circumstances to yield helpful predictions, there is no guarantee that an arbitrary PN expansion actually converges. Part of the reason for this is that the theory typically relies heavily on particular coordinate systems for these small regions that may not always have felicitous properties. In a word, while enormously useful, the PN theory by itself does not constitute a general method for explaining the success of Newtonian theory.

One way to do so—to organize relatively succinctly the relationships between arbitrary relativistic observables and their classical counterparts—is to provide a sense in which the relativistic spacetimes themselves, and fields defined on them, reduce to classical spacetimes (and their counterpart fields), as all spacetime observables are defined in these terms. This geometrical way of understanding the Newtonian limit of relativity theory has been recognized virtually since the former's beginning (Minkowski, 1952, pp. 73–91), where it was observed that as the light cones of spacetime flatten out in Minkowski spacetime, hyperboloids of constant coordinate time become hyperplanes in the limit. This geometric account has since been developed further by Ehlers, 1981, Ehlers, 1988, Ehlers, 1991, Ehlers, 1997, Ehlers, 1998, Ehlers et al., 1986, Minkowski et al., 1952 and others (e.g., Künzle (1976); Malament (1986a, b)) for general, curved spacetimes.⁴

But the nature and interpretation of this limit has sat uneasily with many. The image of the widening light cones seems to suggest an interpretation in which the speed of light $c\to \infty$. In one of the first discussions of this limiting type of reduction relation in the philosophical literature, Nickles (1973) considered this interpretation and pointed out some of its conceptual problems: what is the significance of letting a constant vary, and how is such variation to be interpreted physically? Rohrlich (1989) suggested that “$c\to \infty$” can only be interpreted counterfactually—really, counterlegally, since it corresponds literally with a sequence of relativistic worlds whose speeds of light grow without bound. Thus interpreted, the limit only serves to connect the mathematical structure of relativistic spacetime with that of classical spacetime (Rosaler, 2019). It cannot explain the success of Newtonian physics, since such an explanation “specifies what quantity is to be neglected relative to what other quantity” (Rohrlich, 1989, p. 1165) in worlds with the same laws to determine the relative accuracy of classical formulas as approximations to the outcomes of observations.

So, we are left with a conundrum: existing approaches to the reduction of general relativity to Newtonian gravitation are explanatory, or they are general; moreover, there may be some concerns about the mathematical rigor of some approaches. The received view about these three approaches is summarized in Table 1.

The purpose of this essay is to develop and provide an interpretation of the geometrical limiting account that could in fact meet the explanatory demand required of it. It is thus both perfectly general and explanatory.⁵ To explicate how it works, I first present a unified framework for classical and relativistic spacetimes in §2. The models of each are instantiations of a more general “frame theory” that makes explicit the conceptual and technical continuity between the two. To define the limit of a sequence of spacetimes, however, one needs more structure than just the collection of all spacetime models. As I point out in §3, one way to obtain this structure is by putting a topology on this collection. The key interpretative move, as explicated in §4, exploits the freedom of each spacetime to represent many different physical situations through the representation of different physical magnitudes.⁶

To illustrate this move I describe three classes of examples, involving Minkowski, Schwarzschild, and cosmological (FLRW) spacetimes, that I also show have Newtonian limits in the sense defined here, with respect to a certain topology. (It still remains an open question—though I conjecture it to be true—whether every Newtonian spacetime is an appropriate limit of relativistic spacetimes.)

The topology, though, can make a significant difference to the evaluation of a potential reduction. In particular, whether the Newtonian limit of a particular sequence of spacetimes exists at all can depend on it, and it is not determined automatically from the spacetime theories themselves. I indicate in §5 that certain topologies correspond with a certain the classes of observables that one demands must converge in order for a sequence of spacetimes to converge. Requiring that more observables converge in the limit leads in general to more stringent convergence criteria, thus a finer topology (i.e., with more open sets). In light of this, I argue for a slightly more stringent criterion than has been used by other authors so that observables depending on compact sets of spacetime, such as the proper times along (bounded segments of) timelike curves, also converge. Finally, in §6, I summarize what topical and methodological conclusions I think can be drawn about these results for gravitation (including PN theory) and philosophy of science, and indicate how they might be applied more generally to other reduction relations of a similar limiting type.⁷