Is Relativistic Geometry Conventional?

By Bryan 26 September 2026 general relativity, philosophy of physics No Comments

One of the lessons of General Relativity is supposed to be that gravitation is the science of geometry. John Wheeler characterised it as,

“a geometrodynamical universe”: a world whose properties are described by geometry, and a geometry whose curvature changes with time – a dynamical geometry. (Wheeler 1961, p.361)

Empiricists were not convinced. After all, we don’t reach out and touch space. We measure it indirectly using tools like rulers. Can we really be so sure that our rulers accurately measure the geometry of space?

The Poincaré Disc

Poincaré (1902) explained the problem using the charming example now known as Poincaré’s Disc. Suppose you use a ruler to measure its the properties of the disc below. What can you learn about its geometry?

Click and drag the ruler or the red ‘straight’ line below to play with this strange disc!

Poincaré pointed out that there are at least two ways to understand what’s happening.

  1. The geometry of the disc is hyperbolic. The ruler on this disc behaves exactly as if the geometry is a non-Euclidean surface of constant negative curvature, sometimes called a ‘hyperbolic’ or ‘Lobachevsky’ space. The red line is a ‘straight’ line of shortest distance in this space.
  2. The geometry is Euclidean, but a hidden force distorts the objects on it. It might appear that the geometry is hyperbolic, but this is just caused by a hidden force that shrinks and expands the ruler in some parts of the disc but not others.

Which of these two cases is correct? Poincaré thought the answer was: neither. You can choose either one: it’s just a convention, like choosing to drive on the right or on the left. That is called the conventionality of geometry. Poincaré convinced a number of philosophers about this, including Einstein, who wrote:

“Geometry (G) predicates nothing about the behavior of real things, but only geometry together with the totality (P) of physical laws can do so. Using symbols, we may say that only the sum of (G) (P) is subject to experimental verification. Thus (G) may be chosen arbitrarily, and also parts of (P); all these laws are conventions” (Einstein 1921, p.8).

But that’s being too quick. Relativistic geometry only seems conventional on first glance. But, it is not—at least, not in the way it was originally imagined.

I wrote about this in a (2025) article, “The conventionality of geometry is merely incomplete”—but for a cheerful foray, see below.

Not just anything is a force

A force isn’t just any made up behaviour of a ruler. It has to satisfy Newton’s law, F = ma, or it’s not really a force. This holds in General Relativity too: if \xi^a is the velocity tangent to a test particle’s worldline, then its acceleration \alpha^a = \xi^b\nabla_b\xi^a is proportional to the force,

F^a = m\alpha^a.

This makes sense: free-fall or geodesic motion means exactly that there is no force, \alpha^a = \xi^a\nabla_a\xi^b = 0. And indeed, if there is a force, then it looks different depending on your reference frame, so one generally derives it from a tensor F_{ab} which in a frame \xi^b takes the form F^a := F_{ab}\xi^b.

This turns out to be a very tight constraint on Poincaré’s proposal. Let’s first cast the conventionalist idea in General Relativity. The idea is supposed to be that there could be two ways to interpret the motion of a test-particle:

  1. The spacetime is (M,g_{ab}) with Levi-Civita connection \nabla_a, with no forces, so the test-particle follows a geodesic with vanishing acceleration \alpha^a = \xi^b\nabla_b\xi^a=0, or
  2. The spacetime is M,\tilde{g}_{ab} with Levi-Civita connection \tilde{\nabla}_a, with a force F_a, so that the test-particle satisfies F^a = m\tilde{\alpha}^a = m(\xi^b\tilde{\nabla}_b\xi^a).

Now one can do some mathematics: it turns out that if F_a is a force in the sense described above, then no interesting differences in spacetime geometry are possible: the only such force that exists is F^a = 0. The proof is given in Proposition 1 of my article, strengthening Proposition 2 of Manchak and Weatherall (2014).

So, the original conventionality proposed by Poincaré cannot be true of General Relativity.

There remain other things one can play with here. It turns out the two spacetimes appearing in my example above must have the same Levi-Civita connection \nabla_a, which determines the meaning of acceleration. But, this does not imply they have the same metric g_{ab}.

This opens up a further sense of conventionality that one could explore. On the one hand, the Levi-Civita connection is more or less determined. Indeed, in my article I prove (in Proposition 2) that \nabla_a is determined by geodesic deviation, which is widely viewed as a key observable quantity in General Relativity. This also settled an open question due to Thébault and Tasdan (2024). But, one might still be free to choose a metric by convention, so long as the observed \nabla_a is compatible with it. In general there may be many such metrics.

But, I won’t pursue that line of thought more now. Let’s do something else instead.

More radical conventionality: The dimension of space

Let’s really let our hair down. How else might conventionality enter into the description of physical geometry?

One way is through the number of space (or spacetime) dimensions. A flatland creature on a sphere would find that two-dimensional space has a non-Euclidean geometry: the spherical geometry of constant positive curvature. But, you already know that’s not the only way to view the situation.

Another creature (like you) might postulate a third dimension of space, and propose that the metric for that larger space is Euclidean—like the way we understand the Earth to be embedded in space.

The same holds the next level up: any non-Euclidean geometry, including all those we use in General Relativity, can always be embedded in a geometry in some higher dimension that is Euclidean. This fact is guaranteed by the Nash Embedding Theorem, and its generalisation to relativistic spacetimes. Depending on the geometry, you just might have to go up to a very high dimensional space to find the required embedding.

Borrelli et al. (2012) even showed what Nash’s embedding looks like in the case of a flat torus, and it’s a lovely work of mathematics and art.

Borrelli et al. (2012) “Flat tori in three-dimensional space and convex integration“

So, the possibility of higher dimensions allow another kind of conventionality! But, is it crazy to believe in higher dimensions?

I think that it’s not crazy—it’s just a matter of open physics.

Kaluza-Klein and higher dimensions

In 1921, Kaluza gave a unified explanation of General Relativity and classical electromagnetism using a five-dimensional spacetime. Einstein was deeply impressed.

This proposal had physical motivation: using just the formalism of General Relativity, together with an assumption implementing the U(1) symmetry group of electromagnetism, one gets:

  1. A five-dimensional vacuum universe, such that
  2. An appropriate four-dimensional base space with non-zero curvature together with electromagnetic fields.

At both levels, Einstein’s equation for General Relativity and Maxwell’s equation for electromagnetism are satisfied (with a few symmetry and simplifying assumptions). So, we get a coherent mathematical framework for incorporating all our empirical predictions. This is not just a conventional choice: it’s a mechanism for how to construct new physical theories that incorporate interactions in a very general sense.

On the other hand, Kaluza-Klein theory is not the only game in town. It is just one of many approaches to extending General Relativity. But, it is certainly not crazy. It’s just a matter of unsettled, incomplete physics, which could wind up favouring the higher-dimensional or the lower-dimensional formulation. In my view, this kind of conventionality is not an essential feature of spacetime. It’s merely an incomplete description of reality.

  • Roberts (2025 Philosophy of Science) “The conventionality of geometry is merely incomplete”

Also: Heyyy, we’re back—long time no see!

Leave a Reply