Classical General Relativity as a Non-Conservative Action-Dependent Field Theory

This paper reformulates classical General Relativity as a non-conservative, action-dependent field theory by demonstrating that the Hilbert action's dynamics can be expressed through conformal spacetime geometry coupled with a dissipative sector, where gravitational backreaction arises from non-conservative interactions between the action and geometric degrees of freedom.

The Gist

The Big Idea: Removing the "Ruler"

Imagine you are trying to describe the shape of a room. Usually, you need two things:

1. The Shape: Is the room a perfect cube, a long hallway, or a weirdly angled attic?
2. The Scale: Is the room 10 feet wide or 100 feet wide?

In standard physics (General Relativity), the "room" is spacetime, and the "scale" is determined by a mathematical factor called the conformal factor. Think of this factor as a universal "ruler" or "metre stick" that tells you how big everything is.

The authors of this paper ask a philosophical question based on a principle called the Identity of Indiscernibles (from the philosopher Leibniz): If two scenarios look exactly the same to every possible measurement, but differ only in the size of the ruler used to measure them, are they actually different?

They argue no. If you can't measure the difference, the "size of the ruler" is a redundant, unnecessary piece of information. So, they decided to throw the ruler away.

The Problem: What Happens When You Throw Away the Ruler?

In standard physics, if you remove the ruler, you lose the ability to describe how energy is conserved. It's like trying to bake a cake without measuring cups. If you just guess the amount of flour, the recipe breaks.

Usually, when physicists remove a variable, the math stops working or becomes incomplete. However, the authors found a clever workaround. They realized that by removing the "ruler" (the scale), the universe doesn't just become "scale-free"; it becomes dissipative (like something with friction).

The Analogy:
Imagine you are driving a car.

• Standard Physics: You have a speedometer and a fuel gauge. You know exactly how much energy you have, and energy is conserved (you can't create or destroy it, only use it).
• This New Theory: You throw away the fuel gauge (the scale). Now, the car still drives, but the engine behaves differently. It acts as if there is friction in the system. The car loses energy not because it hit a wall, but because the "ruler" is gone. The math now includes a "friction term" to compensate for the missing scale.

How They Did It: The "Action-Dependent" Trick

The authors used a mathematical tool called Herglotz variational principles. In normal physics, the "Action" (a value that determines how a system moves) is just a number you calculate at the end.

In this new theory, the Action is treated as a living variable. It's like a character in a video game that changes the rules of the game as it moves.

• Normal Physics: The rules are fixed; the character moves.
• This Paper: The character's movement changes the rules, and the changing rules affect the movement.

This creates a system that is non-conservative. In everyday terms, energy isn't perfectly conserved in the traditional sense because it is constantly being exchanged between the geometry of space (the shape of the room) and this new "Action" variable (the friction).

What They Found: The Results

1. The First Order (Simple Waves): Everything Looks Normal
When they looked at small ripples in spacetime (gravitational waves) moving through a flat background, the math worked out perfectly.

• The Result: The waves still travel at the speed of light and behave exactly like standard gravitational waves.
• The Catch: The "friction" just rearranged the "gauge" (the mathematical labels we use to describe the waves). It's like describing a circle: you can say it's "round" or "circular." The shape is the same, but the words used to describe it changed. The physical reality didn't change, only the description.

2. The Second Order (Complex Interactions): The Friction Shows Up
When they looked at how these waves interact with each other (gravitational waves crashing into other gravitational waves), the difference became visible.

• Standard Physics: When waves crash, they create a "backreaction" that acts like a conserved energy packet.
• This Paper: The backreaction is non-conservative. Energy is constantly swapping back and forth between the shape of the waves and the "Action" variable.
• The Metaphor: Imagine two people dancing. In the standard view, they conserve their energy perfectly. In this new view, they are dancing on a floor that is slightly sticky. They still dance the same steps, but the energy of their dance is constantly leaking into the floor (the Action) and leaking back out. The dance looks the same, but the mechanism of how they move is different.

The Conclusion: Same Movie, Different Script

The most important takeaway is that this new theory is mathematically identical to standard General Relativity in terms of what we can observe.

• It predicts the same gravitational waves.
• It predicts the same orbits for planets.
• It predicts the same expansion of the universe.

The only difference is interpretation.

• Standard View: The universe has a scale (a ruler), and energy is conserved.
• This View: The universe has no intrinsic scale (no ruler). To make the math work without a ruler, we have to accept that the universe has a built-in "friction" where energy shifts between geometry and the action itself.

The authors suggest this might be useful for understanding the very beginning of the universe (the Big Bang singularity), where the concept of "size" breaks down. By removing the ruler entirely, the math might stay smooth and predictable even when the universe is infinitely small, whereas standard math might break down.

In short: They took the "ruler" out of General Relativity. To make the math work without it, they added "friction." The result is a theory that describes the exact same universe we see, but tells a slightly different story about how energy moves within it.

Technical Summary

Technical Summary: Classical General Relativity as a Non-Conservative Action-Dependent Field Theory

Problem Statement
The paper addresses the formulation of classical General Relativity (GR) by isolating the conformal degree of freedom—the local choice of scale or "metre stick"—as a redundant variable. Drawing on the philosophical principle of the Identity of Indiscernibles (Leibniz), the authors argue that if a theory grants ontological distinction to scenarios that are observationally identical (differing only by a scaling transformation), the theory is not fundamental. While scaling symmetries have been studied in singular Lagrangian systems, their application to the first-order formulation of GR has been limited. Furthermore, existing non-conservative extensions of field theory (based on Herglotz variational principles) often lack physical motivation or possess overly simplistic action dependence. The central problem is to construct a formulation of GR where the conformal factor is eliminated via symmetry reduction, resulting in a dynamically equivalent theory that is intrinsically non-conservative and action-dependent.

Methodology
The authors employ a geometric reduction procedure known as "contact reduction by scaling symmetry" applied to the first-order (Palatini) formulation of GR.

1. Symmetry Reduction: The authors begin with the Hilbert-Palatini action, which depends on the frame field (tetrad) $e^I_\mu$ and the spin connection $\omega^{IJ}_\mu$. They decompose the tetrad into a conformal factor $\phi$ and a unit-determinant frame field $\tilde{e}^I_\mu$ ($e^I_\mu = e^\phi \tilde{e}^I_\mu$).
2. Scaling Symmetry Identification: They identify a vector field $\Sigma = \partial_\phi$ that generates a scaling symmetry of the Lagrangian density. Under this symmetry, the Lagrangian transforms as $L \to e^\Lambda L$, preserving the extremal points of the action.
3. Contact Reduction: Following the framework of dynamical similarities, the authors perform a quotient of the velocity phase space (the first jet bundle $J^1E$) by the orbits of this scaling symmetry. This eliminates the scaling variable $\phi$ but retains its velocity components, which become the components of an action density $s_\mu$.
4. Herglotz Formalism: The reduction transforms the standard multisymplectic Lagrangian into a Herglotz (or multicontact) Lagrangian $L_H$. This new Lagrangian depends explicitly on the action density $s_\mu$, rendering the field equations non-conservative. The dynamics are governed by the generalized Euler-Lagrange equations for action-dependent systems, where the evolution of $s_\mu$ is coupled to the geometric degrees of freedom.
5. Metric Recovery: To facilitate physical interpretation and weak-field analysis, the authors recast the first-order results into a second-order metric formalism. They define a metric $G_{\mu\nu}$ with fixed determinant (unimodular) such that the physical metric is $g_{\mu\nu} = e^\Phi G_{\mu\nu}$.

Key Contributions and Results

• Action-Dependent Formulation: The primary result is the derivation of a first-order action-dependent Lagrangian (Eq. 6.9) and its second-order metric equivalent (Eq. 7.3). The Lagrangian takes the form:
  $$L_H = \frac{1}{2\kappa}\tilde{R}(G) - \frac{\kappa}{3} G_{\mu\nu} s^\mu s^\nu$$
  where $\tilde{R}(G)$ is the Ricci scalar constructed from the unimodular metric $G_{\mu\nu}$ and its compatible connection, and $s_\mu$ represents the action density.
• Non-Conservative Dynamics: The field equations derived from this action (Eq. 7.12) contain terms coupling the metric perturbations to the action density $s_\mu$. These terms introduce friction-like behavior, making the system intrinsically non-conservative. The authors demonstrate that the elimination of the conformal factor necessitates this compensatory action dependence to preserve the full dynamical content of standard GR.
• Linearized Limit (First Order): In the weak-field limit around a flat background, the linearized perturbations $H_{\mu\nu}$ satisfy the standard free wave equation $\square H_{\mu\nu} = 0$. The action-dependent sector does not alter the propagation speed or the number of physical degrees of freedom (two propagating modes). However, the gauge symmetry is reduced from the full diffeomorphism group to the subgroup of unimodular diffeomorphisms (volume-preserving). The "dissipative" sector reorganizes the gauge degrees of freedom, requiring a combination of residual gauge freedom and dynamical constraints to isolate the physical modes.
• Non-Linear Limit (Second Order): At the second order, the theory exhibits qualitative differences from standard GR. The backreaction of gravitational waves is no longer described by an effective conserved energy-momentum tensor. Instead, energy is exchanged between the geometric perturbations and the action density sector. This manifests as non-conservative terms in the second-order field equations, sourced by quadratic combinations of first-order perturbations.
• Equivalence to Standard GR: The authors prove that the contact-reduced theory is dynamically equivalent to standard GR. By varying the full Hilbert action expressed in conformal variables and eliminating the conformal factor in favor of the action density, they recover the exact same field equations as the reduced theory (Appendix A). Thus, no dynamical information is lost; the reformulation merely redistributes constraints between gauge symmetries and dynamical evolution.

Significance and Claims
The paper claims that this construction provides a novel, mathematically rigorous description of General Relativity that treats the conformal scale as non-fundamental. The significance lies in:

1. Ontological Reduction: It realizes a description of gravity where the scale factor is not an observable degree of freedom, aligning with the principle that unobservable distinctions should not be granted ontological status.
2. Non-Conservative Interpretation: It demonstrates that standard conservative field theories can be equivalently formulated as non-conservative, action-dependent systems. This offers a new perspective on gravitational wave propagation, where the "dissipation" is not energy loss to an external environment but an internal exchange between geometry and the action sector.
3. Singularity Resolution Potential: While the current work is a general field-theoretic construction, the authors note (referencing prior work [30]) that such contact-reduced models have shown promise in resolving singularities in cosmological models (e.g., FLRW) where the standard framework becomes pathological. The reduced theory allows solutions to traverse singularities by associating them with a change in the orientation of the spacetime manifold.
4. Theoretical Utility: The authors suggest that working directly with the reduced ontology (without the conformal factor) may simplify constraint analysis in canonical gravity and offer insights into the nature of GR near singularities where spacetime volumes collapse.

The paper concludes that while the mathematical content is identical to standard GR, the action-dependent formulation offers a distinct interpretational framework, particularly regarding the nature of gauge symmetries and the non-conservative exchange of energy in non-linear regimes.