Conventionalism in general relativity?: formal existence proofs and Reichenbach's theorem {\theta} in context

This paper clarifies a conflation between the existence of alternative geometries and Reichenbach's universal equivalence claim to demonstrate that, contrary to recent criticisms, there is no "rich" no-go theorem invalidating Reichenbach's theorem $\theta$, while proposing a constructive framework for rigorously exploring alternative spacetime theories.

The Gist

Imagine you are trying to map a city. You have a standard map (let's call it the "Newtonian Map") and a new, high-tech map (the "Einstein Map").

For a long time, philosophers and physicists have debated a tricky question: Is the shape of the city (the geometry) a real, physical fact, or is it just a matter of how we choose to draw the map?

This is the debate about Conventionalism. The "Conventionalists" (like the famous philosopher Hans Reichenbach) argued that you could draw any map you wanted, as long as you added a little bit of "magic wind" (a universal force) to push people around so that their actual walking paths looked the same on both maps. If you could do this, then the shape of the city isn't a fact; it's just a convention.

The Big Discovery: The "No-Go" Sign

In 2014, two researchers, Weatherall and Manchak, tried to test this idea using Einstein's General Relativity (our best theory of gravity). They asked: "Can we always swap our Einstein map for a different one, provided we add a 'force field' to fix the paths?"

They found a No-Go Theorem. In the world of Einstein's gravity, the answer is NO. You cannot just swap the map and add a simple force to make everything work. The geometry of spacetime is "stuck" in a specific way; it's not as flexible as the old Newtonian maps were.

The Counter-Attack: "You Cheated!"

Then, two other researchers, Dür and Ben-Menahem, jumped in. They said, "Wait a minute! You only proved this for specific types of maps and specific types of forces. If you relax your rules, you can still swap the maps!" They argued that Weatherall and Manchak had set up a "No-Go" sign that was too narrow, and that the "Conventionalist" idea was still alive if you just looked at the loopholes.

The Author's Solution: Clearing the Fog

This paper, written by Ruward Mulder, steps in to clear up the confusion. He says both sides are talking past each other because they are arguing about two different things.

1. The "Existence" Claim vs. The "Universality" Claim

Mulder uses a Restaurant Menu analogy to explain the difference:

• The Existence Claim (The "One Special Dish" Theory): "For every meal I order, there is at least one other menu item I could have ordered that would taste exactly the same if the chef added a secret spice."
  • Mulder's take: Weatherall and Manchak didn't prove this is impossible. There might still be some weird, specific alternative maps that work.
• The Universality Claim (The "Anything Goes" Theory): "For any meal I order, I can swap it for any other meal on the menu, and just add a secret spice to make them identical."
  • Mulder's take: This is the famous "Theorem Theta" that Reichenbach wanted. Weatherall and Manchak proved this is FALSE. You cannot swap any map for any other map. The "magic wind" (force) doesn't work that universally.

The Verdict: The "No-Go" sign is real, but it only blocks the "Anything Goes" idea. It doesn't block the "One Special Dish" idea.

2. Breaking the Rules to Test the Theory

The critics (Dür and Ben-Menahem) said, "You assumed the maps had to be smooth and the forces had to look like standard physics forces. If we break those rules, we win!"

Mulder says, "Let's try breaking the rules and see what happens."

• The Torsion Twist: He imagines a spacetime that isn't just curved, but also "twisted" (like a corkscrew). He proves that even with this twist, you still cannot swap the maps using a standard force.
• The Result: Even if you change the rules of the game (allowing for twisted space), the "Anything Goes" theory (Theorem Theta) still fails. The geometry is still more rigid than the Conventionalists hoped.

The New Game Plan: A "Theory Explorer"

Instead of just fighting over whether the "No-Go" sign is valid, Mulder suggests we use it as a treasure map.

Imagine the space of all possible physics theories is a giant, uncharted forest.

• Weatherall and Manchak found a fence that says "You can't go this way."
• The critics said, "The fence is broken!"
• Mulder says, "The fence is real, but it tells us exactly where the boundaries are. Let's use this fence to systematically explore the forest."

He proposes a program where we take the assumptions of the proof (like "the force must be a rank-2 tensor" or "the space must be smooth") and deliberately break them one by one.

• What if the force isn't a standard force?
• What if the space has extra dimensions?
• What if the space isn't smooth?

By testing each broken rule, we can rigorously map out exactly which alternative theories are possible and which are impossible. We aren't just arguing about philosophy; we are building a systematic atlas of all possible universes.

Summary

• The Problem: Can we pretend the universe has a different shape if we add a "magic force"?
• The Old Answer: "No, not really." (Weatherall & Manchak)
• The Critique: "Yes, if you change the rules." (Dür & Ben-Menahem)
• The New Insight: "The 'Magic Force' idea works for some specific swaps, but it definitely doesn't work for any swap you want. The 'Anything Goes' idea is dead."
• The Future: Instead of fighting, let's use these limits to rigorously explore every possible alternative theory of gravity, creating a complete map of what physics could look like.

In short: The universe is less flexible than we hoped, but that rigidity gives us a perfect tool to explore the boundaries of reality.

Technical Summary

1. Problem Statement

The paper addresses the debate regarding geometric conventionalism in General Relativity (GR), specifically focusing on the work of Weatherall and Manchak (W&M, 2014).

• The Core Question: Can different spacetime geometries be made empirically indistinguishable by postulating "universal forces" (or universal effects) that correct for geometric differences?
• The Conflict:
  • W&M's Result: They proved that in GR, one cannot generally relate the geodesics of two conformally related metrics via a rank-2 tensor field representing a universal force. This suggests GR is less susceptible to underdetermination than Newtonian gravity.
  • Dürr & Ben-Menahem's (D&BM) Critique: They argue W&M's proof relies on overly restrictive assumptions (a "no-go theorem" based on loopholes) and that by rejecting these assumptions, one can still construct empirically equivalent models, preserving conventionalism.
• The Gap: The paper identifies a conflation in the literature between two distinct claims:
  1. Existence Claim ($UDT\text{-}\forall g \exists \tilde{g}$): For any given metric, at least one alternative metric exists that is empirically equivalent.
  2. Universality Claim (Reichenbach's Theorem $\theta$, $UDT\text{-}\forall g \forall \tilde{g}$): Any arbitrary metric can be made equivalent to any other via universal effects.

2. Methodology

Mulder employs a rigorous mathematical and logical analysis of the assumptions underlying W&M's relativistic proof.

• Formal Logic: The author frames the debate as a set of mutually inconsistent premises. By negating specific assumptions, one can classify different conventionalist positions.
• Mathematical Generalization: The paper extends W&M's proof from Riemannian geometry (torsion-free, metric-compatible connections) to torsionful spacetimes (connections with non-zero torsion).
• Assumption Analysis: The author systematically evaluates the "loopholes" proposed by D&BM by testing them against the two distinct targets (Existence vs. Universality).

Key Assumptions Analyzed:

• (CONF): The alternative metric is conformally related to the standard metric ($\tilde{g}_{ab} = \Omega^2 g_{ab}$).
• (NORM): The tangent vector is unit-norm with respect to the new metric.
• (FORCE): The universal effect is represented by a rank-2 tensor field in the geodesic equation ($\xi^b \nabla_b \xi^a = \tilde{\xi}^b \tilde{\nabla}_b \tilde{\xi}^a - F^a_b \tilde{\xi}^b$).
• (RIEM): The geometry is Riemannian (torsion-free and metric-compatible).
• (TOPO): The manifolds are topologically identical (same dimension, Hausdorff).

3. Key Contributions and Results

A. Disambiguation of Targets (Existence vs. Universality)

Mulder clarifies that W&M's proof primarily targets the Universality Claim (Theorem $\theta$), not just the Existence Claim.

• Result: Even if one rejects assumptions like (CONF) or (RIEM) to allow for some alternative metrics (saving the Existence Claim), the Universality Claim (Theorem $\theta$) remains false.
• Logic: If the proof shows that no force tensor relates conformally equivalent metrics, and the Universality Claim requires all metrics to be related, then the claim fails regardless of whether one expands the space of metrics. The "loopholes" identified by D&BM do not save Theorem $\theta$; they only potentially save a restricted existence claim.

B. Generalization to Torsionful Spacetimes (Proposition 3)

The author proves Proposition 3, which generalizes W&M's result to spacetimes with torsion (non-symmetric connections).

• Theorem: Let $(M, g_{ab})$ be a relativistic spacetime and $\tilde{g}_{ab} = \Omega^2 g_{ab}$ a conformally equivalent metric. Let $\nabla$ and $\tilde{\nabla}$ be derivative operators that are not necessarily symmetric (allowing torsion). If $\Omega$ is non-constant, there is no tensor field $F^a_b$ such that a curve is a geodesic relative to $\nabla$ if and only if its acceleration relative to $\tilde{\nabla}$ is given by $F^a_b \tilde{\xi}^b$.
• Implication: Violating the assumption of torsion-freeness (a key "loophole" suggested by D&BM) does not rescue Theorem $\theta$. Torsion cannot be interpreted as a standard rank-2 force field in the geodesic equation to make torsionful and torsion-free geometries empirically equivalent in the manner Reichenbach suggested.

C. Systematic Exploration of the "Space of Theories"

Instead of viewing W&M's proof as a dead end for conventionalism, Mulder proposes a programmatic attitude.

• The assumptions (CONF, NORM, FORCE, RIEM, TOPO) should be treated as a formal toolkit.
• By systematically negating these assumptions (e.g., $\neg$TOPO, $\neg$RIEM, $\neg$FORCE), one can generate a taxonomy of alternative spacetime theories.
• Example: Rejecting (RIEM) leads to the "Geometric Trinity of Gravity" (TEGR and STEGR), where gravity is torsion or non-metricity. However, the paper shows these do not satisfy the standard (FORCE) condition, meaning they are distinct theories rather than mere conventional re-descriptions of GR under a standard force law.

4. Significance

1. Resolution of the Debate: The paper resolves the apparent conflict between W&M and D&BM by showing they are arguing about different claims. W&M successfully disprove the strong Universality Claim (Theorem $\theta$) under standard physical assumptions. D&BM's "loopholes" only preserve a weaker Existence Claim, which is not what Reichenbach's famous theorem asserts.
2. Refutation of Theorem $\theta$: The paper demonstrates that Reichenbach's Theorem $\theta$ (that any geometry can replace any other) is mathematically untenable in GR if universal effects are constrained to behave as standard force fields (rank-2 tensors).
3. New Research Programme: Rather than undermining the proof, Mulder reframes it as a constructive tool. By mapping the logical space of negated assumptions, philosophers and physicists can rigorously explore the landscape of alternative spacetime theories (including non-Hausdorff manifolds, higher dimensions, and non-Riemannian geometries) to see which, if any, admit empirical equivalence.
4. Clarification of Torsion: The analysis clarifies that in Teleparallel Gravity (TEGR), torsion acts as a geometric effect rather than a "force" in the Newtonian sense defined by (FORCE), thereby distinguishing it from a mere conventional re-interpretation of GR.

In summary, Mulder argues that while GR is not as rigid as W&M's proof might initially suggest regarding the existence of some alternatives, it definitively rules out the universality of geometric conventionalism. The proof serves not as a barrier, but as a rigorous map for exploring the boundaries of spacetime theories.