1. Introduction

It is a bit ironic that one of the most widely embraced definitions of what it means to be a scientific realist is due to the arch-anti-realist Bas van Fraassen. His definition starts by stating that “Science aims to give us, in its theories, a literally true story of what the world is like” (van Fraassen Reference van Fraassen1980, 8). And indeed, scientific realists often see themselves as committed to ‘taking scientific theories at face value’: if the best theories of particle physics say that quarks exist, then we should believe that they exist; if general relativity tells us that gravity is really just an aspect of space-time structure, then we should believe it; if quantum mechanics tells us that the world is at its core nondeterministic, then we should believe that too.

The problem is that scientific theories, or at least the theories of modern physics, are not that straightforward with us. They may seem so at first, but if you listen to the details of their respective stories, if you take your time to look under the surface, what exactly we should take them to tell us about the world is far from clear. Murray Gell-Mann, the inventor of the concept of quarks, for a long time did not think that quarks should be interpreted as literally existing; neither did Richard Feynman. Albert Einstein passionately resisted the interpretation of general relativity that says that the gravitational force field of Newtonian theory is ontologically reduced to the geometry of space-time in general relativity. And of course, there is a long-standing battle in foundations of physics about whether quantum mechanics really does tell us that the world is nondeterministic.Footnote ¹

In this article I introduce a new case study that provides further evidence for the position that, whether you are a realist or not, the literal interpretation of a scientific theory, especially in physics, can be rather misleading. I will argue that what we should aim for is a careful interpretation; an interpretation of the theory or model or formalism that engages both with the details of its mathematical structure and with how it is applied to the natural world. Philosophy of science must be willing to look under the hood.

The case study I want to look at is the so-called problem of motion in the general theory of relativity (GR). It asks about the precise relationship between the two sets of equations that are at the very heart of GR. On the one hand, there are the Einstein field equations, which give us the dynamics of the gravitational potential (the metric tensor) g _μν:

(1)$\begin{array}{}{R}_{\mu \nu }-\frac{1}{2}{g}_{\mu \nu }R={\kappa }_E{T}_{\mu \nu }.\end{array}$

On the other hand, we have the geodesic equation that determines which paths through space-time are geodesics of the connection ${\Gamma }_{\mu \sigma }^{\nu }$ compatible with the metric g _μν:

(2)$\begin{array}{}\frac{d^2{x}_{\tau }}{{\mathrm{ds}}^2}+{\Gamma }_{\mu \nu }^{\tau }\frac{{\mathrm{dx}}_{\mu }}{\mathrm{ds}}\frac{{\mathrm{dx}}_{\nu }}{\mathrm{ds}}=0.\end{array}$

In GR, material bodies subject only to gravitational fields are supposed to move on the geodesics determined by equation (2). The problem of motion in GR is the question of whether the equations of motion of matter subject to gravitational fields (2) can be derived from the gravitational field equations (1).

Einstein himself, in his first publication on the topic, a paper cowritten with Jakob Grommer and published in 1927, compares different classes of attempts to give such a derivation. In particular, Einstein and Grommer distinguish between two classes of attempts at deriving the geodesic motion of matter from the gravitational field equations, which I will term the T approach and the vacuum approach, respectively. The T approach starts from the realization that the field equations (1) imply the conservation condition, namely, that the covariant divergence of the energy-momentum tensor T _μν vanishes:

(3)$\begin{array}{}{\nabla }^{\mu }{T}_{\mu \nu }=0.\end{array}$

From this, together with certain conditions on the energy-momentum tensor T _μν, the T approach derives that material particles move on time-like geodesics. It is this kind of approach to the problem of motion that philosophers have engaged with almost exclusively up to now.Footnote ²

Einstein and Grommer end up dismissing the T approach and suggest an alternative path to deriving geodesic motion instead. It is a particular version of a vacuum approach to the problem of motion. Einstein and Grommer start from the vacuum form of the Einstein field equations,

(4)$\begin{array}{}{R}_{\mu \nu }=0,\end{array}$

and attempt to derive that the equations (4) imply that material particles move on geodesics.

To the extent that philosophers have engaged with this approach at all, they have quickly dismissed it because it seems to model material bodies by singularities in space-time, while singularities, by definition, are not even part of space-time. However, in this article I argue that this dismissal was far too fast and that indeed the vacuum approach deserves at least as much attention by philosophers as the T approach. The vacuum approach, despite first appearances, engages more closely with some of the most major predictions of GR: both the prediction of the perihelion of Mercury and the prediction of light bending by the Sun use the vacuum approach to the derivation of motion of material systems. Indeed, even the prediction of gravitational waves resulting from a binary black hole merger that was recently confirmed rests on the vacuum field equations, for black holes are described by vacuum solutions.Footnote ³

My argument in this article proceeds in three steps. First, I argue that the vacuum approach to the problem of motion promises certain advantages that the T approach lacks. Second, I argue that the problems of the vacuum approach for which it has been dismissed are artifacts of a too-literal interpretation of the formalism and its application to the problem at hand. Third, I argue that a careful interpretation makes the problems disappear; I argue that the approach does not need to interpret singularities as representing material bodies.

2. A Critical Comparison of the Two Research Programs

I said above that the T approach to the problem of motion proceeds via the fact that the Einstein field equations (1) imply the conservation condition (3), which in turn implies the geodesic motion of matter. However, as Malament (Reference Malament, Frappier, Brown and DiSalle2012) pointed out, the conservation condition by itself is not sufficient to prove that the geodesic equation is the equation of motion of material particles. One of the most general proofs from within the T approach, proposed by Geroch and Jang (Reference Geroch and Jang1975) and further generalized by Ehlers and Geroch (Reference Ehlers and Geroch2004), rests not only on the conservation condition (3) but also on the strengthened dominant energy condition (SDEC), which states:

The first clause is effectively the weak energy condition, which states that the mass-energy-momentum density associated with the body in question is always nonnegative. The second clause states that every observer will judge the mass energy momentum of the body to propagate along time-like curves only.Footnote ⁴

It would be rather attractive if we did not have to presume that material particles move on time-like curves to then show that these curves are actually time-like geodesics, and if we did not have to presume that matter cannot have nonnegative mass energy. These are weak assumptions about the nature of matter, but they are assumptions.

The vacuum approach to the problem of motion, however, aims to make no assumptions about the nature of matter and its properties at all and to still derive that matter moves on geodesics. It starts from the question of whether just knowing the exterior gravitational field of a material body, and how this gravitational field interacts with the gravitational field of its surroundings, is enough to derive that the body will move on a geodesic of the metric surrounding it. Arguably, this program is far more ambitious than the T approach, for it starts with fewer assumptions.Footnote ⁵ And yet, if successful, it would really fit much better the virtues that philosophers have associated with the geodesic theorem(s) in the first place: deriving the inertial motion of matter from knowledge of the dynamics of gravitational fields alone (cf. Brown Reference Brown2007, 141 and 163).

Einstein was deeply skeptical of the role of the energy-momentum tensor in GR. Throughout the decades, he emphasized that T _μν provides only a “phenomenological representation of matter.”Footnote ⁶ Einstein’s aim is then to instead start with the vacuum field equations (4), treat material particles as singularities in the metric field, and derive that they move on geodesics of a metric g _μν that solves the vacuum field equations (4) in the region through which the particle moves.

To the extent that philosophers have engaged with this approach at all, they have already dismissed it at this point. The main criticism is that the very idea of the approach is flawed: a singularity is not even part of space-time. How should it be possible to describe its motion in said space-time?

Both Torretti and Earman essentially answer that this is not possible and that the whole program is ill conceived. Earman (Reference Earman1995, 12), writes: “Singularities in the spacetime metric cannot be regarded as taking place at points of the spacetime manifold M. Thus, to speak of singularities in g _μν as geodesics of the spacetime is to speak in oxymorons.”Footnote ⁷ The most detailed discussion of the Einstein-Grommer paper in the philosophical literature is due to Tamir (Reference Tamir2012). After quoting the above statement by Earman, Tamir goes on to write (142): “The proponent of such a ‘vacuum-cum-singularity’ technique is faced with the rather paradoxical challenge of explaining in what sense we can say that a singular curve (ostensibly constituted by the missing points in the manifold) is actually a geodesic of the spacetime from which it is absent. Not only is no metric defined at the singularity, but also technically there are not even spacetime points there: the geodesic does not exist.”

I will argue that by looking at the details of the Einstein-Grommer approach we come to a different interpretation of it, one that sheds a completely different light on the alleged presence of singularities. We will see that a careful (rather than literal) interpretation of the vacuum approach, and the Einstein-Grommer paper in particular, does not actually depend on introducing singularities at all.

3. The Vacuum Approach to the Problem of Motion

4. Interpreting the Einstein-Grommer Approach to the Problem of Motion

The reader might think that the argument presented in the last section cannot be a faithful representation of the Einstein-Grommer approach; after all, where is the claim that the material particle is represented by a singularity, the reason the approach was dismissed by Earman and Tamir? Indeed, I have omitted that after step 5 of the argument Einstein and Grommer do say that one could assume that the inner metric ${\overline{\gamma }}_{\mu \nu }$ is given by what is effectively the Schwarzschild metric: it is spherically symmetric and has a singularity at the center. And yet, Einstein and Grommer never use this assumption in their argument. They call the material particle ‘the singularity’ all the time, but their argument does not depend on assuming any particular form for the inner metric, let alone one that is necessarily singular. As a matter of fact, they do not even mention a concrete candidate metric for the outer metric ${\overline{\overline{\gamma }}}_{\mu \nu }$; all they need is that γ_μν is split into the inner metric ${\overline{\gamma }}_{\mu \nu }$ and the outer metric ${\overline{\overline{\gamma }}}_{\mu \nu }$ in such a way that ${\overline{\overline{\gamma }}}_{\mu \nu }$ is nonsingular everywhere.

Note that this does not mean that we know that the inner metric ${\overline{\gamma }}_{\mu \nu }$ is nonsingular. We do not know anything about the inner metric, for the argument is independent of ${\overline{\gamma }}_{\mu \nu }$ having any particular form, just like the derivation of Mercury’s perihelion was independent of whether there is a singularity at the center of the Schwarzschild metric that represented the exterior field of the Sun.

With regard to the Sun-Mercury system I argued that we should not interpret the Schwarzschild metric as representing the Sun but as representing its exterior gravitational field. The part of the Sun that is within the event horizon, including the singularity at the center, should not be taken as a representation of the actual interior of the Sun but as a placeholder or a blind spot within the current description of the Sun-Mercury system: a docking station for a theoretical model of the Sun not included in Einstein’s Sun-Mercury model.Footnote ¹¹

Likewise, we should interpret the inner metric ${\overline{\gamma }}_{\mu \nu }$ in the Einstein-Grommer approach as a placeholder for a representation of matter not included in the current theoretical approach. Sure, you can set ${\overline{\gamma }}_{\mu \nu }$ to be a Schwarzschild-like metric with a singularity at the center. But you do not have to do that to make the Einstein-Grommer argument work, and even if you do make that assumption, you should still take this particular inner metric with a singularity at its center as a placeholder for a representation or theory of matter not yet provided. But now wait a minute. You might have disliked the occurrence of singularities as representations of particles, but at least the singularity (in lieu of a nonvanishing energy-momentum tensor) gave you an idea of where in space-time the particle was supposed to be. True, Earman and Tamir rightly pointed out that the singularity is not actually part of space-time, and so it can hardly serve to localize the particle in space-time. Still, you might think that we are throwing the baby out with the bath water by not choosing any inner metric. After all, is it not the case then that the curve we have been focusing on is just any curve, without any reason to think of this curve as the curve of a material particle?Footnote ¹²

Again, I think we can counter this criticism by comparing the Einstein-Grommer approach to Einstein’s (Reference Einstein1915) treatment of the Sun-Mercury system. What Einstein did there was to assume that Mercury would move on some geodesic of the exterior gravitational field produced by the Sun. He calculated an approximation to the external gravitational field of a static, spherically symmetric body, whose metric is asymptotically flat; this gravitational field he saw as represented by the connection components ${\Gamma }_{\mu \sigma }^{\nu }$ of a metric g _μν, which deviated only slightly from the flat Minkowski metric. He then inserted these gravitational field components ${\Gamma }_{\mu \sigma }^{\nu }$ into the geodesic equation (2). He showed that this law contained Newton’s first law and Newton’s second law with a gravitational potential giving rise to a force as a limiting case, and he showed how the resulting Keplerian laws for orbits differ in his theory as compared to its Newtonian limit. In the end, he obtained that according to the new theory the perihelion ε of any geodesic orbit around the Sun is given by

(5)$\begin{array}{}\epsilon =24{\pi }^3\frac{a^2}{T^2{c}^2\left(1-e^2\right)}\mathrm{.}\end{array}$

Here a denotes the length of the semimajor axis of the orbit in question, e its eccentricity, c the speed of light, and T the orbital period of the planet in question. Einstein then takes the astronomically known values for Mercury, plugs them into equation (5), and thereby predicts that Mercury’s perihelion changes by 43" per century.

Note that there is nothing in the theoretical description that singles out any particular path as that of Mercury. There is no theoretical representation of Mercury, no model. All that is there is the assumption that Mercury will move on one of the geodesics of the affine connection determined by the spherically symmetric field of the Sun. A general equation that all possible geodesic orbits have to fulfill is derived. And then external knowledge is used to single out one of these orbits as that of Mercury. Einstein trusts that the astronomers have measured the orbital period, the semimajor axis, and the eccentricity of Mercury correctly. It is this external knowledge, plugged into his theoretical model, which does not in itself contain a representation of Mercury or its path, that produces the prediction.

In many ways, the whole vacuum approach to the problem of motion is about the question whether in this kind of scenario we really have to assume the geodesic equation as the equation of motion of matter over and above the gravitational field equations. Indeed, let us look at the Sun-Mercury system within the 1927 Einstein-Grommer approach. The problem of motion, then, is the question whether Einstein really had to introduce the gravitational field equations (to describe the exterior gravitational field of the Sun) and the geodesic equation (to describe the path of Mercury subject to this gravitational field) as separate assumptions.Footnote ¹³ Could he have only assumed the gravitational field equations and derived that Mercury moves on a geodesic of the exterior field of the Sun? My point is that, just like in Einstein’s 1915 treatment, the 1927 Einstein-Grommer approach does not need to commit to a theoretical model that allows us to localize Mercury internally. It is fine to ask whether the exterior gravitational field around a given curve ‘forces’ that curve to be a geodesic. Just like in the 1915 treatment, Einstein and Grommer could then use external knowledge about whether that particular curve is actually the curve of a material object, or of Mercury in particular. No inner metric, no singularity to represent the material body, is actually needed.

Let us take a step back though, for there is an important difference between the structure of Einstein’s 1915 treatment of Mercury on the one hand and the 1927 Einstein-Grommer approach on the other. In the Mercury case, Einstein had assumed that Mercury moves on a geodesic (i.e., a special kind of curve), and model-external knowledge about the period, eccentricity, and semimajor axis of Mercury could then be used to determine which of the many geodesics of the Schwarzschild metric corresponded to the path of Mercury. But in the case of the Einstein-Grommer argument, what is in question is whether we can prove that the path of Mercury, say, is a geodesic. Thus, at first sight it looks as if while the 1915 argument only needed external knowledge to determine which geodesic is that of Mercury, appeal to external knowledge in the Einstein-Grommer case would have to determine (a) that this curve is a geodesic and (b) that it is the curve of a material body.

Einstein and Grommer did not aim to derive both a and b. Instead, while Einstein in 1915 used external knowledge at the end of his argument, Einstein and Grommer in 1927 use it at the beginning. They start out by assuming that a given curve is the curve of a material particle and then ask whether having a regular outer metric (which solves the vacuum field equations) around the curve means that the curve of this material particle, given the further conditions summarized in section 3.2, must be a geodesic. Rather than finishing the argument by appeal to external knowledge (as in Einstein Reference Einstein1915), the Einstein-Grommer argument starts with an appeal to external knowledge, which singles out a particular curve as that of a material body.

Either way, both in Einstein’s 1915 treatment and in the Einstein-Grommer approach there is no reason to interpret the singularity (appearing in the Schwarzschild metric or the inner metric, respectively) literally. In both cases, the singularity should be interpreted to signify a placeholder or a blind spot of the theoretical treatment, rather than something that should be interpreted literally, as referring and approximately true. Indeed, both Einstein’s 1915 treatment of the Sun-Mercury system and Einstein and Grommer’s treatment of an arbitrary material particle subject to an external gravitational field work just as well if, in the former case, no interior metric (to describe the interior of the Sun) or, in the latter case, no inner metric (to represent the location of the particle on the curve) is ever specified.

5. Conclusion

I started out by saying that whether we are realists or antirealists, we should aim for a careful interpretation, rather than a literal interpretation, of the scientific theory that we want to be realists or antirealists about. Using a case study, I argued that the vacuum approach to the problem of motion in GR, and the Einstein-Grommer approach in particular, is far more sensible and promising if we interpret the singularities not as representing material bodies but as placeholders for a representation of material bodies that is not included in the model. Indeed, I argued that the approach does not even need the introduction of singularities to represent material bodies; their introduction does not do any work in answering the question at hand.Footnote ¹⁴

Given that in their paper Einstein and Grommer seem to take the singularities as representing material bodies, one might wonder whether this allegedly more careful interpretation does not fall prey to the criticism that the careful interpreter presumes to understand the theory/formalism in question better than its originators. This might seem at odds with the realist tenet of taking scientists and science ‘seriously’. I do indeed think that putting the Einstein-Grommer paper into its proper historical context by analyzing Einstein’s correspondence leading up to the paper and by relating it to his overarching research project at the time would convincingly show that he subscribed to something very much like the ‘placeholder interpretation’ I defended above. Showing this in detail will have to wait for a much longer paper, and I do not ask the reader to just take my word for it. So let us say, for the sake of the argument, that Einstein and Grommer did indeed intend the singularities as representatives of material objects in a rather straightforward way. I believe that we should not take their word for it either. And neither did Einstein. Just a few years after the Einstein-Grommer paper, in his famed 1933 Spencer lectures at the University of Oxford, Einstein (1934, 163) told us in his opening words: “If you wish to learn from the theoretical physicist anything about the methods which he uses, I would give you the following advice: Do not listen to his words, examine his achievements.”Footnote ¹⁵

In philosophy of science, I believe there is no better way of examining a scientist’s achievements than by looking for the best possible interpretation of his or her theories. To do that, we have to not just listen to the words of the scientist who created or discovered it; we have to see what the theory does in practice, how it is used, which of its parts really do the work.