People have been conjecturing on the true ontology of physical space since at least the pre-Socratic Greeks, and I shall not take the time here to trace the entire history of thought. Instead, I want to focus on three or four broad classes of conceptions about space (primarily *absolute* and *relational*, and mixtures of the two) that have been much discussed among European philosophers since the Seventeenth Century.

We measure space by measuring distances. When these distances change, we talk about motion. Change entails a notion of time; thus our notions of space and time are tangled up in, or perhaps derive from, the concept of motion. (Interestingly, Rene Descartes identified space with extension; he claimed that only matter can have the property of extension, and what we sometimes mistake for empty space is in fact filled with matter.) Descartes recognized relative motion–that is, motion of an object relative to other objects. However, he and many others supposed that “true motion” was relative not merely to other objects, but to a fixed background.

Isaac Newton proposed that this fixed background is three-dimensional Euclidean space, which he called *absolute space*. Newton’s space is not an attribute or a substance, but a “pseudo-substance” which is non-material but physical. Absolute space implies absolute speed, which would be the rate of change of position of an object relative to an arbitrary point in absolute space. This is the first concept of space I wish to highlight: the *absolute conception*.

Gottfried Leibniz disagreed with this description. He pointed out that Newton’s Laws of Motion assume Galilean Relativity, which means a closed system is unaffected by a constant change in position; in other words, absolute speed is never measurable by anyone, no matter how good their powers of perception. Thus Leibniz argued that absolute space is a fiction which no one can ever prove to exist through experiment. He went further, claiming space is a conception in the mind of the application of relations between objects (unlike Descartes and Newton, who thought space as objectively real, independent of any mind). Motion is possession of force, Leibniz theorized. Specifically, while relative motion is not real (in agreement with Newton), there is a quantity (mass times speed-squared–we can only measure its change, not any true initial value) which determines an object’s kinetic energy, and it is this non-spatial characteristic which defines true motion. Thus, Leibnizian motion actually has no “movement” in our everyday conception of the notion. Here, I shall refer to Leibniz’s description of space as the *ideal conception*.

In the latter half of the Twentieth Century, philosophers were able to develop a notion of *neo-Newtonian spacetime* (a.k.a. *Galilean Spacetime*) which solved the problem of absolute speed. In this geometric model, acceleration is well-defined, but velocity is not. Acceleration is absolute, defined in terms of the geometry of the spacetime rather than in terms of the relation between objects. Since this geometry has an independent existence, this implies that neo-Newtonian spacetime is a substance or pseudo-substance. This formulation also overcomes Leibniz’s objection about absolute speed. But then further questions arise: what sort of substance is it, and do we really need it as an explanation? Both the Newtonian and neo-Newtonian views of space are known as *substantivalism*.

It has been argued that rotation cannot be explained in terms of relative motion. Historically, this argument has been used in support not of Newtonian absolute space, but in support of Galilean spacetime substantivalism. Ernst Mach rejected this argument: he said that in reality, when an object rotates, it does so in relation to everything else around it; in a thought experiment where a “rotating” object is the only object in the Universe (and so has nothing to rotate relative to) we are not able to deduce anything, because we can never do such an experiment, and thus never know what would happen in such a situation. Therefore, the postulate of an absolute space is a metaphysical assertion unjustifiable by experiment, according to Mach. Instead of referring to absolute space, Mach said we can refer to a reference frame with respect to very distant objects averaged out over all directions. Mach’s idea is the paradigm for a true *relational conception* of space: he wanted to replace all notions of absolute space and times with relative ones. Such a theory would predict new effects due to relative accelerations (something like gravitomagnetic induction).

Is Machian relationism a consistent theory in classical mechanics? Albert Einstein attempted to create a Machian classical theory, but he was led to a non-classical notion of spacetime. In the 1970s, Barbour and Bertotti were able to create a Mach-style theory using Lagrangian mechanics (the language in which classical mechanics is modernly expressed), but it made empirically falsified predictions. They tried again in the 1980s, and that ended up being a subset of Newton’s theory.

In the theory of **Special Relativity** (SR), Einstein showed that acceleration and rotation are dependent on **spacetime interval** relations. This does not satisfy Mach’s full relationalism, because although there are no absolute positions or velocities in SR, there are absolute accelerations, and thus absolute motion. This is similar to the neo-Newtonian view mentioned earlier.

While acceleration is absolute in SR, Einstein relativized it in his theory of General Relativity (GR) with his *Principle of Equivalence of Gravitation and Inertia*. (In that principle, he postulated that there is no possible physical experiment which could ever determine whether a person in an accelerating spaceship was in fact accelerating or whether the ship was holding itself in a rest frame relative to a uniform gravitational field whilst the rest of the Universe was in free-fall around it.) Einstein introduced GR in 1915. Through it, he claimed to have eliminated any need for a notion of absolute motion; however, most physicists reject this claim. It was hoped that the equations of GR would demonstrate that gravitational / intertial forces are completely determined by the distribution of matter and their relative motions, thus constituting a completely relational theory; however, it turns out that these relations only partly determine the forces. Gravitational forces should be measurable in some situations even when matter “generating” the gravity is absent, according to GR. By 1918, Einstein realized that GR was not fully relational. (However, Julian Barbour has claimed that GR is, in fact, fully relational when reformulated as *Shape Dynamics*. Thus, whether GR can satisfy Mach’s vision is an open question of research.)

Given the primacy of the notion of a geometric space in GR, many have assumed its ontological significance, similar to neo-Newtonian substantivalism. However, this is not necessarily the case. It has been suggested that the **metric** be considered as a field element. The resulting *metric field* does not endow space with properties; rather it is a description that results from other laws of physics which govern matter fields. (This appears to be Nobel laureate Frank Wilczek’s view in his pop-sci book, *The Lightness of Being*.)

Another Nobel laureate in Physics, Steven Weinberg, agrees that “spacetime geometry” is merely a convenient language. In the preface to his classic textbook, *Gravitation and Cosmology* he writes that students of GR may ask why a free-falling object follows a **geodesic**. It is insufficient to answer that “because spacetime is a Riemannian manifold” because the geometric picture obscures deep connections between gravity and the rest of physics; we already know that the non-gravitational interactions (electroweak, strong) cannot be consequences of geometry. Instead, Weinberg prefers to build the concepts of GR off the experimental fact of the Principle of Equivalence. Geometry is introduced only as a mathematical tool when needed. The question then becomes “why is the Equivalence Principle true?” Weinberg believes the answer lies in a quantum theory of gravitation.

### References:

Huggett, Nick and Hoefer, Carl. “Absolute and Relational Theories of Space and Motion”, The Stanford Encyclopedia of Philosophy (Spring 2018 Edition), Edward N. Zalta (ed.), URL: <https://plato.stanford.edu/archives/spr2018/entries/spacetime-theories/>.

Weinberg, Steven. *Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity*. John Wiley & Sons, Inc., 1972.

Wilczek, Frank. *The Lightness of Being: Mass, Ether, and the Unification of Forces*. Basic Books, 2008.

[The two pictures in this post are original drawings by Hermann Minkowski.]