Jacobson's thermodynamic approach to classical gravity applied to non-Riemannian geometries: remarks on the simplicity of Nature

This paper extends Jacobson's thermodynamic derivation of gravity to non-Riemannian geometries, revealing that while the standard Einstein-Hilbert action is generally excluded, a theory combining it with a quadratic torsion vector term emerges as the unique consistent alternative under specific conditions, whereas full non-Riemannian cases with canonical energy-momentum tensors lead to mutual inconsistency between thermodynamic and Lanczos-Lovelock approaches.

The Gist

The Big Picture: Is Gravity a Law or a Habit?

Imagine you are trying to figure out how a city works. You could look at the blueprints (the fundamental laws of physics) or you could watch how the traffic flows and the people behave (thermodynamics).

Thirty years ago, a physicist named Ted Jacobson had a brilliant idea. He suggested that gravity isn't a fundamental force written in stone from the beginning of the universe. Instead, he proposed that gravity is like traffic flow: it emerges from the way tiny bits of space-time interact, just like traffic emerges from individual cars.

Jacobson showed that if you treat a patch of space like a hot object (applying the laws of heat and entropy), you can mathematically derive Einstein's famous equations of gravity. It's as if he proved that gravity is just the universe trying to stay cool and organized.

The New Question: What if the "Road" is Bumpy?

For decades, scientists assumed the "road" of space-time was perfectly smooth and symmetrical (what mathematicians call a Riemannian geometry). In this smooth world, Jacobson's idea perfectly recreates Einstein's General Relativity.

But what if the road isn't smooth? What if space-time has:

1. Twists (Torsion): Like a road that spirals or has a corkscrew shape.
2. Stretching (Non-metricity): Like a road where the distance between two points changes depending on how you measure it.

This paper asks: If the road of space-time is twisted and stretchy, does Jacobson's "heat law" still work? And if it does, what kind of gravity theory does it produce?

The Detective Work: Two Approaches Meet

The authors of this paper acted like detectives trying to solve a mystery. They used two different methods to find the "correct" theory of gravity for this twisted, stretchy universe:

1. The Thermodynamic Detective (Jacobson's Method): They asked, "If we apply the laws of heat to this twisted space, what equations pop out?"
2. The Simplicity Detective (Lanczos-Lovelock Method): They asked, "Nature loves simplicity. What is the simplest, most elegant equation that fits the rules?"

They wanted to see if these two detectives would point to the same culprit (the same theory of gravity).

The Findings: A Surprise Twist

Here is what they discovered, broken down by scenario:

Scenario A: The Smooth Road (Standard Einstein Gravity)

• The Result: When space is smooth, both detectives agree. They point to General Relativity (Einstein's theory).
• The Metaphor: It's like two detectives looking at a perfect crime scene and agreeing, "The butler did it."

Scenario B: The Twisted Road (Torsion, but no Stretching)

• The Result: When space has twists (torsion) but no stretching, the detectives almost agree, but with a catch.
  • If we assume matter behaves in a specific way (the "metric" version), the Thermodynamic Detective finds a theory that is almost Einstein's, but with a tiny extra ingredient: a term related to the square of the twist.
  • The Metaphor: Imagine Einstein's theory is a classic recipe for chocolate cake. In this twisted world, the Thermodynamic Detective says, "The cake is still chocolate, but we need to add a pinch of sea salt to make it work."
  • The Catch: If we assume matter behaves differently (the "canonical" version), the two detectives disagree completely. One says "Add salt," the other says "The recipe is impossible." They cannot find a single simple theory that satisfies both.

Scenario C: The Twisted AND Stretchy Road (Full Non-Riemannian)

• The Result: When space has both twists and stretching, the detectives completely clash.
• The Metaphor: It's like trying to build a house where the bricks are also liquid. The Thermodynamic Detective says, "Build a wall," but the Simplicity Detective says, "That wall violates the laws of physics." They cannot find a single, simple theory that works for both.

The "Nature's Choice" Conclusion

The authors conclude that if we follow the principle of Occam's Razor (Nature prefers the simplest explanation), here is what likely happened:

1. Nature chose the smoothest path: The fact that the "Twisted but not Stretchy" scenario works (with that tiny pinch of salt) suggests that the universe might have torsion (twists), but it likely does not have non-metricity (stretching).
2. The "Simplest" Theory: The theory that Nature likely selected is a slight modification of Einstein's original idea. It's the Einstein-Hilbert action (the standard gravity formula) plus a small term that accounts for the "twist" of space.
3. The Dead End: If the universe were fully twisted and stretchy, the laws of thermodynamics and the laws of simplicity would be fighting each other. Since the universe seems to be consistent, it probably isn't in that chaotic state.

Why Does This Matter?

• Black Holes: The paper suggests that if space is twisted, black holes might behave slightly differently when they are "heated" or interact with matter. This could change how they emit gravitational waves (the ripples in space-time).
• Observation: Astronomers might be able to detect these tiny "twists" in the future by looking at how black holes spin or how gravitational waves travel.
• The Big Question: It reinforces the idea that gravity might not be a fundamental force, but an emergent phenomenon—a side effect of the universe trying to maximize entropy (disorder) and stay in thermal equilibrium.

Summary in One Sentence

This paper uses the laws of heat to test different shapes of space-time and concludes that while Einstein's gravity works perfectly on a smooth road, a "twisted" road requires a tiny tweak to the recipe, but a "twisted and stretchy" road is a mathematical mess that Nature probably avoids.

Technical Summary

1. Problem Statement

The paper addresses a fundamental question in theoretical physics: Did Nature have a choice in selecting the laws of classical gravity? While General Relativity (GR) is the standard model, it is often viewed as an effective theory that may be superseded by a more fundamental theory.

The authors investigate whether Jacobson's thermodynamic approach—which derives gravitational field equations from the first law of thermodynamics applied to local Rindler horizons—remains valid when the assumption of Riemannian geometry (symmetric, metric-compatible connection) is relaxed. Specifically, they explore non-Riemannian geometries characterized by:

• Torsion ($T$): Asymmetry in the affine connection.
• Non-metricity ($Q$): Incompatibility between the connection and the metric tensor.

The core problem is to determine if the Einstein-Hilbert action (the basis of GR and Einstein-Cartan theory) remains the unique or preferred gravitational action when these geometric complexities are introduced, or if the thermodynamic derivation selects a different theory.

2. Methodology

The authors employ a two-pronged strategy combining thermodynamic derivation with geometric constraints:

1. Thermodynamic Derivation (Jacobson's Approach):

   • They consider a local Rindler horizon generated by a Killing vector field.
   • They apply the Clausius relation ($\delta Q = T \delta S$), where heat flux $\delta Q$ is the energy-momentum flux across the horizon, $T$ is the Unruh temperature, and $\delta S$ is the entropy change proportional to the horizon area ($\delta S = \eta \delta A$).
   • Crucially, they utilize the Raychaudhuri equation generalized for non-Riemannian spacetimes (including torsion and non-metricity) to calculate the expansion of the null congruence and the resulting change in horizon area.
   • This yields a general "equation of state" relating the energy-momentum tensor to geometric quantities (Ricci tensor, torsion, and non-metricity).

2. Lanczos-Lovelock Hypotheses:

   • To narrow down the infinite family of theories compatible with the thermodynamic derivation, they apply the Lanczos-Lovelock hypotheses.
   • These hypotheses assume the field equations derive from an action that is:
     • Invariant under diffeomorphisms and local gauge transformations (Poincaré or Affine).
     • Constructed from the metric, torsion, and non-metricity and their derivatives.
     • Symmetric and linear in second-order derivatives.
   • They test whether the thermodynamic equation of state can be derived from an action principle satisfying these conditions.

3. Distinction of Energy-Momentum Tensors:

   • The authors carefully distinguish between the metric energy-momentum tensor ($\tau_{\mu\nu}$) and the canonical energy-momentum tensor ($\Sigma_{\mu\nu}$), noting they differ in the presence of torsion and spin.

3. Key Contributions and Results

A. Riemannian Spacetime (Baseline)

• Result: The thermodynamic approach uniquely recovers the Einstein Field Equations (GR).
• Significance: Confirms that Jacobson's original derivation holds and that the Einstein-Hilbert action is the unique solution under the Lanczos-Lovelock constraints in standard GR.

B. Non-Riemannian Spacetime with Vanishing Non-Metricity (Torsion only)

This case corresponds to Einstein-Cartan (EC) theory and Poincaré Gauge Theories (PGT).

• Case 1: Energy-Momentum identified as Metric Tensor ($T \equiv \tau$):
  • The standard Einstein-Cartan theory (pure Einstein-Hilbert action) fails to satisfy the thermodynamic equation of state.
  • Unique Solution: The only theory that satisfies both the thermodynamic derivation and the Lanczos-Lovelock hypotheses is derived from an action that is a slight deviation from the Einstein-Hilbert action:
    $$S = S_{EH} + \int d^4x \sqrt{-g} \, (T_\alpha T^\alpha)$$
    where $T_\alpha$ is the torsion vector. This action includes a quadratic term in the torsion vector.
  • Implication: Nature would select this specific action over pure EC theory. The torsion does not introduce new propagating degrees of freedom (it is algebraically coupled to matter), consistent with observational constraints.
• Case 2: Energy-Momentum identified as Canonical Tensor ($T \equiv \Sigma$):
  • The thermodynamic approach and the action principle are mutually inconsistent. No action principle can generate the field equations derived from the thermodynamic approach in this specific identification.

C. General Non-Riemannian Spacetime (Torsion + Non-Metricity)

• Result: When non-metricity ($Q$) is present, the thermodynamic approach and the Lanczos-Lovelock hypotheses become mutually inconsistent.
• Reason: The generalized equation of state contains terms involving non-metricity that cannot be reconciled with an action principle satisfying the standard Lanczos-Lovelock constraints (specifically, the requirement for the action to be invariant under affine transformations leads to an infinite iteration loop when trying to match the field equations).
• Exception: The inconsistency persists regardless of whether the metric or canonical energy-momentum tensor is used (though the latter case is reserved for future work involving dilatation density).

4. Significance and Implications

1. Rejection of Pure Einstein-Cartan Theory: The paper demonstrates that if gravity is thermodynamic in origin and torsion exists, the standard Einstein-Cartan action is not the correct description of Nature. Instead, a theory with a quadratic torsion vector term is required.
2. Support for "Simplicity of Nature": The results align with Ockham's Razor. Nature appears to select the simplest action compatible with thermodynamics. In the presence of torsion, this is not the pure Einstein-Hilbert action, but the Einstein-Hilbert action plus the simplest possible torsion correction ($T_\alpha T^\alpha$).
3. Constraints on Non-Metricity: The mutual inconsistency found in the general non-Riemannian case suggests that if the thermodynamic approach is fundamental, non-metricity might be absent in the classical limit, or the standard Lanczos-Lovelock hypotheses need significant modification to accommodate it.
4. Observational Signatures:
   • The non-equilibrium contributions to entropy ($\delta S_i$) in non-Riemannian geometries are identified as generalizations of the Hartle-Hawking tidal heating term.
   • The presence of non-metricity or specific torsion configurations could lead to distinct gravitational wave signatures (e.g., different polarization modes or energy loss rates) compared to GR, offering potential observational tests for these geometric properties.
5. Emergent Gravity: The work reinforces the view of gravity as an emergent phenomenon arising from the thermodynamics of spacetime microstates, where the macroscopic field equations are constrained by the laws of thermodynamics rather than being fundamental axioms.

Conclusion

The paper concludes that while Jacobson's thermodynamic approach successfully generalizes to non-Riemannian geometries, it imposes strict constraints on the viable gravitational theories. It rules out the standard Einstein-Cartan theory in favor of a theory with a quadratic torsion vector term (when non-metricity is zero and using the metric energy-momentum tensor) and suggests a fundamental incompatibility between the standard thermodynamic derivation and the presence of non-metricity under current theoretical frameworks. This implies that if the thermodynamic origin of gravity is correct, the geometry of spacetime is likely restricted to Riemannian or specific torsion-based non-Riemannian structures, excluding general non-metricity in the classical regime.