The trace-free Einstein tensor is not variational for the metric as field variable

This paper demonstrates that the trace-free Einstein tensor cannot be derived from any local action functional via variation with respect to the metric, even when the requirement of diffeomorphism invariance is removed.

The Gist

Here is an explanation of the paper using simple language and everyday analogies.

The Big Idea: A Recipe That Can't Be Written Down

Imagine you are a chef. In physics, the "chef" is nature, and the "recipes" are the laws of the universe. Usually, physicists believe that every law of nature comes from a master recipe called an Action Principle.

Think of an Action Principle like a blueprint for a house. If you have a blueprint, you can build the house (the physical reality) by following the instructions. In physics, if you have a "Lagrangian" (the blueprint), you can use a mathematical tool called "variation" to derive the equations that describe how the universe moves and changes.

The Problem:
For a long time, physicists have been fascinated by a specific set of equations called the Trace-Free Einstein Equations. These are a slightly modified version of Einstein's famous equations for gravity. They are interesting because they allow the "cosmological constant" (a value that determines how fast the universe expands) to be a variable that changes depending on the situation, rather than a fixed number written in the laws of physics.

However, there was a nagging suspicion: Can you actually write a "blueprint" (an Action) that produces these specific Trace-Free equations?

For decades, people thought the answer was "No, but only because these equations break a specific symmetry rule called diffeomorphism invariance." (Think of this symmetry as the rule that "the laws of physics shouldn't change just because you move your coordinate grid around.")

The New Discovery: It's Not Just About Symmetry

This paper, written by Arian L. von Blanckenburg, Domenico Giulini, and Philip K. Schwartz, says: "Stop worrying about the symmetry rule. The answer is still NO, even if we ignore that rule."

They prove that you simply cannot find a local blueprint (Action) that, when you follow the instructions, spits out the Trace-Free Einstein equations. It's not a matter of the equations being "too weird" for the symmetry rule; it's a fundamental mathematical impossibility.

The Detective Work: The "Vainberg-Tonti" Test

How did they prove this? They used a mathematical detective technique called the Method of Variational Completion (specifically using something called the Vainberg-Tonti Lagrangian).

Here is the analogy:

Imagine you find a mysterious machine in a factory. You don't know what it does, but you want to know: "Is this machine built from a specific set of blueprints?"

1. The Test: The Vainberg-Tonti method is like a "reverse-engineering" tool. You feed the machine's output (the equations) back into the tool.
2. The Process: The tool tries to reconstruct the original blueprint by mathematically "integrating" the output. It asks, "If I work backward from these results, what was the original instruction manual?"
3. The Result:
   • If the machine was built from a blueprint, the tool reconstructs a valid, non-zero manual.
   • If the machine wasn't built from a blueprint, the tool tries to reconstruct the manual, but the result is zero (nothing). It's like trying to bake a cake from a recipe that doesn't exist; you end up with an empty bowl.

The Paper's Conclusion

The authors applied this "reverse-engineering" tool to the Trace-Free Einstein equations.

1. They treated the equations as the "output" of a machine.
2. They tried to reverse-engineer the "blueprint" (the Action).
3. The Result: The math showed that the reconstructed blueprint was zero.

In the language of the paper: The "Vainberg-Tonti Lagrangian" vanishes.

What does this mean in plain English?
It means that if you try to build the Trace-Free Einstein equations from a standard "recipe" (an Action Principle), the recipe disappears. The equations are like a shadow that has no object casting it. They exist as a mathematical relationship, but they cannot be the direct result of minimizing a standard energy function.

Why Does This Matter?

You might ask, "So what? If the equations work, why do we need a blueprint?"

1. It's a Fundamental Limit: It tells us that nature might have laws that are "orphaned"—they describe reality perfectly but don't come from the standard "minimize energy" principle we usually use.
2. It Clarifies "Unimodular Gravity": There is a popular theory called Unimodular Gravity that uses these Trace-Free equations. This paper clarifies that while you can write down the equations for Unimodular Gravity, you cannot derive them directly from a standard action without adding extra "helper" fields or changing the rules of the game.
3. It's Not About Symmetry: The paper shuts down the idea that this problem is just about symmetry. It's a deeper, structural issue.

The Takeaway

Think of the Trace-Free Einstein equations as a perfectly functioning car engine that runs on a fuel source we can't quite identify.

For a long time, we thought, "Maybe this engine runs on a special fuel that only works if the car is parked in a specific spot (symmetry)."

This paper says: "No. Even if you park the car anywhere, you still can't find the fuel tank. The engine works, but it wasn't built using the standard blueprint we use for all other engines."

The authors have proven that for the standard "inverse metric" (the way we usually describe the shape of space), these equations are mathematically "orphaned" and cannot be derived from a standard action principle.

Technical Summary

Here is a detailed technical summary of the paper "The trace-free Einstein tensor is not variational for the metric as field variable" by von Blanckenburg, Giulini, and Schwartz.

1. Problem Statement

The paper addresses a fundamental question in the formulation of gravitational theories: Can the trace-free Einstein tensor be derived from a local action principle?

• Context: The trace-free Einstein equations are a modification of General Relativity (GR) given by:
  $$R_{\mu\nu} - \frac{1}{n} R g_{\mu\nu} = \kappa \left( T_{\mu\nu} - \frac{1}{n} T g_{\mu\nu} \right)$$
  where $n \geq 3$ is the spacetime dimension. These equations are significant because, under the assumption of energy-momentum conservation ($\nabla_\mu T^{\mu\nu} = 0$), they imply the standard Einstein equations with a cosmological constant $\Lambda$ appearing as an integration constant rather than a fixed parameter.
• The Conflict: It is a well-known result that the trace-free Einstein tensor ($G^{\text{tf}}_{\mu\nu}$) cannot arise from varying a diffeomorphism-invariant action with respect to the metric. This is because diffeomorphism invariance requires the variational derivative to have a vanishing covariant divergence (by the Noether identity), whereas the trace-free Einstein tensor is not identically divergence-free for $n \geq 3$.
• The Open Question: Does this non-variationality persist if the assumption of diffeomorphism invariance is dropped? In other words, is there any local action functional (invariant or not) that yields the trace-free Einstein tensor when varied with respect to the metric (or inverse metric)?

2. Methodology

The authors employ the Inverse Calculus of Variations, specifically utilizing the Method of (Canonical) Variational Completion and the Vainberg–Tonti Lagrangian.

• Vainberg–Tonti Lagrangian: For a given tensor-density-valued expression $E_A[y^B]$ (where $y^B$ are field variables), one can construct a candidate Lagrangian density $\mathcal{L}$ defined by a specific line integral:
  $$\mathcal{L}[y^A] := y^B \int_a^1 E_B[u y^A] \, du$$
  where $a$ is a limit such that $\lim_{u \to a} u E_B[u y^A] = 0$.
• The Logic of Proof:
  1. If an expression $E_A$ is locally variational (i.e., it is the Euler-Lagrange derivative of some local Lagrangian), and the Vainberg–Tonti Lagrangian exists, then the Vainberg–Tonti Lagrangian is equivalent to the original Lagrangian.
  2. Contrapositive Argument: If the Vainberg–Tonti Lagrangian exists but its variation yields a result different from the original expression $E_A$ (specifically, if it yields zero), then $E_A$ cannot be locally variational.
• Field Variable: The authors specifically analyze the variation with respect to the inverse metric $g^{\mu\nu}$ (or the metric $g_{\mu\nu}$ in the contravariant case), treating the densitized trace-free Einstein tensor as the target expression.

3. Key Contributions and Results

A. Generalization of Non-Variationality

The primary contribution is the proof that the non-variationality of the trace-free Einstein tensor holds even without assuming diffeomorphism invariance. The result is robust for any local action functional.

B. The Main Theorem (Theorem 1)

Statement: For spacetime dimensions $n \geq 3$, the densitized trace-free Einstein tensor:
$$\tilde{G}^{\text{tf}}_{\mu\nu} := \sqrt{|g|} \left( R_{\mu\nu} - \frac{1}{n} R g_{\mu\nu} \right)$$
cannot be obtained by varying any local action functional with respect to the inverse metric $g^{\mu\nu}$.

C. The Proof Mechanism (Scaling Analysis)

The proof relies on analyzing the scaling behavior of the tensor under a transformation $g^{\mu\nu} \to u g^{\mu\nu}$:

1. Scaling Properties:
   • Christoffel symbols and the Ricci tensor $R_{\mu\nu}$ are invariant under this scaling.
   • The Ricci scalar $R = g^{\mu\nu}R_{\mu\nu}$ scales as $u$.
   • The volume density $\sqrt{|g|}$ scales as $|u|^{-n/2}$.
2. Resulting Scaling: The densitized tensor scales as:
   $$\tilde{G}^{\text{tf}}_{\mu\nu}[u g^{\rho\sigma}] = |u|^{-n/2} \tilde{G}^{\text{tf}}_{\mu\nu}[g^{\rho\sigma}]$$
3. Vainberg–Tonti Calculation:
   • The integral limit $a$ is determined to be $\infty$ (since $n \geq 3$, the factor $u|u|^{-n/2}$ vanishes only as $u \to \infty$).
   • The Lagrangian is computed as:
     $$\mathcal{L} = g^{\rho\sigma} \int_{\infty}^1 \tilde{G}^{\text{tf}}_{\rho\sigma}[u g^{\mu\nu}] \, du$$
   • Due to the trace-free property ($g^{\mu\nu} \tilde{G}^{\text{tf}}_{\mu\nu} = 0$), the contraction $g^{\rho\sigma} \tilde{G}^{\text{tf}}_{\rho\sigma}$ is zero.
   • Consequently, the Vainberg–Tonti Lagrangian vanishes identically ($\mathcal{L} = 0$).
4. Conclusion: Since the variation of the zero action is zero, and the original expression is non-zero, the expression cannot be variational.

D. Extension to Metric as Field Variable

The authors note that the result holds symmetrically if the field variable is the metric $g_{\mu\nu}$ (and the tensor is contravariant). The scaling logic changes slightly ($g_{\mu\nu} \to u g_{\mu\nu}$), but the trace-free property still forces the Vainberg–Tonti Lagrangian to vanish.

4. Significance and Implications

• Clarification of Unimodular Gravity: The paper clarifies the status of Unimodular Gravity. While Unimodular Gravity is often discussed via variational principles, the authors emphasize that the trace-free Einstein equations themselves do not arise directly as Euler-Lagrange equations from varying the metric in these formulations. Instead, the variation yields the full Einstein equations with a Lagrange multiplier (acting as $\Lambda$). The trace-free form is a consequence of the constraint, not the direct variational output.
• Distinction between Equations and Action: The work reinforces that the physical validity of the trace-free Einstein equations does not depend on their derivability from a standard metric-based action. It highlights that one must introduce auxiliary fields or change field variables (as done in recent works [5–7]) to construct an action principle for these equations.
• Theoretical Rigor: By removing the assumption of diffeomorphism invariance, the authors close a potential loophole. They prove that the obstruction is intrinsic to the mathematical structure of the trace-free tensor and the metric field variable, not merely a consequence of symmetry requirements.

Summary

The paper definitively proves that the trace-free Einstein tensor is not variational with respect to the metric (or inverse metric) for any local action, regardless of diffeomorphism invariance. This is demonstrated by showing that the associated Vainberg–Tonti Lagrangian vanishes due to the trace-free nature of the tensor, leading to a contradiction if one assumes the tensor is an Euler-Lagrange expression. This result solidifies the understanding that alternative formulations of gravity based on trace-free equations require non-standard variational setups (e.g., auxiliary fields or different variables) rather than direct metric variation.