Bessel-Hagen currents for linearised Weyl-squared gravity

Introduction to the show: ident: General Relativity and Quantum Cosmology. The day in general relativity and quantum cosmology.

Kai: I'm Kai.

Mira: And I'm Mira, here with Lev.

Kai: We're looking at the paper: "Bessel-Hagen currents for linearised Weyl-squared gravity".

Mira: Here's the gist of the paper. The paper examines whether linearised Weyl-squared gravity yields an electromagnetic-like local energy-momentum tensor by applying the Bessel-Hagen construction to this theory and finding that no such strict local representative exists because four-dimensional identities collapse all candidates to a pure trace The Core Problem of the Construction The Bessel-Hagen construction for the Fierz-Pauli action does not produce a preferred local gaugeinvariant energy-momentum tensor, as the action is built from first derivatives of hµν whereas the first local gaugeinvariant spin-2 field strength, the curvature, contains two It is natural to ask whether a theory built directly from the linearised curvature recovers the electromagnetic-like feature of a strictly gaugeinvariant local representative The Bessel-Hagen construction extends naturally from the Poincar´e to the full conformal group, realised actively on the fixed Minkowski background, and the resulting Noether current is gauge invariant as a class for every conformal generator The Failure in Weyl-Squared Gravity The linearised Weyl-squared gravity theory behaves like Fierz-Pauli because no nonzero strict local, polynomial, symmetric, dimension-four rank-two tensor quadratic in hµν that is invariant under both gauge symmetries and conserved on the Bach shell exists Invariance under both gauge symmetries of the linearised theory forces any local representative to be built from C(1) alone; dimensional analysis restricts it to be quadratic in C(1) with no derivatives, and the four-dimensional Weyl identity collapses these to a pure trace, which cannot be conserved Linearised Weyl-squared gravity therefore yields the same outcome as Fierz-Pauli: a gauge-invariant current class with no preferred local representative The Mechanism of Success in Electromagnetism Electromagnetism succeeds because its gauge-invariant field strength Fµν ∼ ∂A appears already at first-derivative order, which matches an energy-momentum tensor quadratic in F, conserved on shell The candidate FµρFνρ − 1/4 ηµνF2 is nonzero This suggests that a first-order gauge-covariant field strength is a natural route to the electromagnetic-like outcome, avoiding the derivative-counting and fourdimensional curvature identities responsible for the obstruction found here The Role of Gauge Symmetries and Identities The construction extends cleanly to the full conformal group, realised as an active fixed-background symmetry where secondderivative terms of special conformal transformations are absorbed into an internal Weyl gauge parameter The resulting Noether current class is gauge invariant for the full conformal group, and no nonzero strict local tensor in the admissible class A represents it The structural fact that the Bessel-Hagen momentum is the appropriate projection of the gauge-invariant Weyl tensor makes it natural to ask whether the base current Jαζ,0,0 or some improvement of it is a strictly gauge-invariant local tensor The Nature of Higher Rank Objects On the Einstein/Ricci-flat subset of the Bach shell, the natural conserved gauge-invariant object is instead the rank-four Bel-Robinson super-energy rather than a rank-two stress tensor The four-dimensional Gauss–Bonnet example fares no better since its gauge–invariant energy–momentum tensor

13, 14: is the identically vanishing Lanczos tensor The examples considered here suggest that an independent first-order gauge-covariant field strength, such as that of the Weyl gauge theory of gravity, is a natural route to the electromagnetic-like outcome The paper concludes that no nonzero gauge-invariant, conserved, local, dimension-four, quadratic energy–momentum tensor exists for linearised Weyl-squared gravity The contrast among the four theories compared in Table I is organized by the derivative order of the gauge-invariant field strength relative to the field equation Electromagnetism succeeds because its gaugeinvariant field strength F ∼ ∂A is of first-derivative order The Weyl symmetry merely selects which identity does the work, and it is worth emphasising that the absence of a preferred local representative is not a defect of the Bessel-Hagen construction Pruning redundant reasoning tokens The paper examines whether linearised Weyl-squared gravity yields an electromagnetic-like local energy-momentum tensor by applying the Bessel-Hagen construction to this theory and finding that no such strict local representative exists because four-dimensional identities collapse all candidates to a pure trace How the training signal is built The Bessel–Hagen construction for linearised Weyl–squared gravity behaves like Fierz–Pauli because no nonzero strict local, polynomial, symmetric, dimension-four rank-two tensor quadratic in hµν that is invariant under both gauge symmetries and conserved on the Bach shell exists Where the gains come from The successful outcome for electromagnetism (Sec.

Paper summary: Kai: So we're looking at this paper, "Bessel-Hagen currents for linearised Weyl-squared gravity." It's about taking that construction used for Fierz-Pauli and applying it to something different—linearised Weyl-squared gravity. The main point they make is that because the action is built from the linearised Weyl tensor C(one) mu nu rho sigma, which contains two derivatives, it doesn't naturally yield a local energy-momentum tensor in the same way electromagnetism does <ref:2608.02912#pg1,action is built from the linearised Weyl tensor $C^{(1>.

Mira: Right, so if we think about Fierz-Pauli, you get a gauge-invariant class of Noether currents because its action has first derivatives of h mu nu while the spin-two field strength—the Weyl tensor—has two derivatives, and that's what makes the construction work there one <ref:2608.02912#pg1,class of Noether currents because>. The paper asks if this logic extends when you use the Weyl tensor directly instead of just a generic quadratic curvature action.

Lev: I see why it's interesting because it connects the structure of these gravity theories to how we build conserved quantities for quantum systems, and whether that connection holds up when you move from second-order derivative actions to those built purely on the irreducible spin-two part one <ref:2608.02912#pg1,the irreducible spin-2 part>.

Kai: Exactly. The authors are essentially asking: if we build an action purely from the linearised Weyl tensor, does it give us a local representative for that energy-momentum tensor that's strictly gauge invariant and conserved on the Bach shell?

Mira: They show that no such nonzero local, polynomial, symmetric rank-two tensor exists under those conditions. They point out that any object quadratic in h mu nu with constant coefficients and no coordinate dependence has to be built only from C(one) itself without further derivatives because of the gauge symmetries two <ref:2608.02912#pg1>.

Lev: And then they use the four-dimensional Weyl identities, which are really constraints on the geometry, and those constraints reduce all candidates down to just pure traces, which means their conservation conditions force those coefficients to vanish three <ref:2608.02912#pg1>.

Kai: That's a big hurdle. So what does this mean for us in terms of what we've actually built or measured? It seems like building the action from the gauge-invariant curvature doesn't automatically recover the electromagnetic result.

Paper summary: Mira: It means that even though they have this gauge-invariant current class, there isn't a preferred local tensor to point to for energy and momentum in this specific gravity theory. The construction succeeds in giving you a class of conserved currents for every generator of the conformal group, but not necessarily a physical stress tensor.

Lev: From my side, if we were trying to run something like this on real hardware, the overhead would land squarely on those gauge fixing constraints and the massive structure of the current class they found three <ref:2608.02912#pg1>. It's complex to implement if you're just looking for a local conserved quantity.

Kai: So, where does this leave us when we think about electromagnetism? The paper contrasts this with how electromagnetism works, which seems to succeed because its gauge-invariant field strength F mu nu is already at first derivative order three <ref:2608.02912#pg1>.

Mira: Precisely. Electromagnetism gets that conserved tensor because the field strength itself, like d A, is already of first-derivative order, and you can construct an energy-momentum tensor quadratic in that—like F mu rhoF rho nu - one/four eta mu nu F squared three <ref:2608.02912#pg3>.

Lev: That first derivative structure is what makes the difference; it avoids the higher derivative constraints and the four-dimensional identities that kill all those candidates in this Weyl-squared gravity setup. It's a structural property of how the field strength is defined in Maxwell theory.

Kai: So, if we look at the whole picture of this paper, "Bessel-Hagen currents for linearised Weyl-squared gravity," it’s really showing us that the Bessel-Hagen construction doesn't automatically give you the electromagnetic result just because you start with a gauge-invariant curvature.

Mira: It’s not a failure of the construction itself, but rather what it tells us about the nature of the field strength you start with. The paper suggests that a first-order gauge-covariant field strength is a more natural route to getting that electromagnetic-like tensor than starting purely from the spin-two curvature in this quadratic gravity context <ref:2608.02912#pg1>.

Lev: That’s what I mean when I say it's about the structure of the field strength, not just the action being built from curvature. It points toward needing a different kind of field strength definition to get that conserved stress tensor.

Kai: So, for someone listening who isn't deep in general relativity or quantum gravity, what does this mean practically? It means that even when you have these highly symmetric theories, like Weyl-squared gravity on a fixed background, the simple energy-momentum tensor we expect doesn't always exist in a local form.

Paper summary: Mira: It suggests that we might need to look at higher rank objects, like the rank-four Bel-Robinson super-energy mentioned in the text when considering the Einstein or Ricci-flat subset of that shell three <ref:2608.02912#pg1>.

Lev: And from an error correction standpoint, if you were trying to build a robust system based on these currents, you'd have to contend with this; you can't just assume a simple rank-two tensor is available.

Kai: So, looking at the authors Michael Hobson, Will Barker, and Anthony Lasenby in "Bessel-Hagen currents for linearised Weyl-squared gravity," the main implication seems to be that building directly from the linearised curvature doesn't recover the electromagnetic result without adding more structure to your field strength.

Mira: And they conclude by saying a first-order gauge-covariant field strength is a natural route toward that electromagnetic result, even though its mere presence isn't enough on its own. It’s about having the right derivative order in that field strength definition three <ref:2608.02912#pg1>.

Lev: That distinction between first and second derivative order seems central to why the math works out differently for Maxwell versus this Weyl-squared gravity setup.

Kai: So, to wrap up what we've heard about "Bessel-Hagen currents for linearised Weyl-squared gravity," the main thing is that while the construction gives you a gauge invariant class of Noether currents for the full conformal group, it doesn't deliver a nonzero local energy-momentum tensor that you can use as your standard stress tensor.

Mira: They show that no such nonzero local, polynomial, symmetric rank-two tensor exists under the conditions they set because all candidates get reduced to pure traces when you apply the four-dimensional Weyl identities three <ref:2608.02912#pg1>.

Lev: From an engineering view, this means if you're trying to build a system based on these currents for real hardware, you can't rely on finding a simple local conserved quantity because the constraints are too tight.

Kai: The paper suggests that electromagnetism succeeds because its field strength is already at first derivative order, which matches the structure needed for an energy-momentum tensor quadratic in it three <ref:2608.02912#pg1>.

Mira: They conclude that a first-order gauge-covariant field strength seems like the right direction to explore if you want to find an electromagnetic analogue in this context. It’s not just about having curvature; it’s about how you define the field strength derived from it.

Conclusion: Kai: So we’ve been looking at this paper called "Bessel-Hagen currents for linearised Weyl-squared gravity." The authors are Hobson, Barker, and Lasenby.

Mira: Yeah, what they're doing is taking that construction from Fierz-Pauli—which deals with first derivatives—and applying it to a theory built directly from the linearised Weyl tensor.

Lev: So the core of it is checking if that construction works for this new gravity setup. It’s about seeing if we can find a local energy-momentum tensor that is truly gauge invariant and conserved on the Bach shell.

Kai: Exactly, it asks whether we can get something like the electromagnetic stress tensor out of this curvature-based action.

Mira: And they found that no nonzero local, polynomial, symmetric rank-two tensor exists under those conditions. All candidates collapse down to pure traces when you apply the four-dimensional Weyl identities.

Lev: So for someone building real hardware, that means we can't just assume a simple rank-two conserved quantity is available; the constraints are too tight.

Kai: It suggests that even starting from a gauge-invariant curvature doesn't automatically recover the electromagnetic result without adding more structure to the field strength itself.

Mira: They point out that electromagnetism works because its field strength, like F mu nu, is already at first derivative order, which matches what’s needed for a stress tensor quadratic in it.

Lev: So it comes down to the derivative order of the field strength definition, not just starting with curvature. That's a structural difference that matters for your error correction overhead.

Kai: It means we might need to look at higher rank objects, like the rank-four Bel-Robinson super-energy mentioned when you consider the Einstein or Ricci-flat subset of that shell.

Mira: Yeah, and they suggest that a first-order gauge-covariant field strength seems like the right direction to explore if you want to find an electromagnetic analogue in this context.

Lev: That distinction between first and second derivative order is central to why the math works out differently for Maxwell versus this Weyl-squared gravity setup.

Kai: So, this paper shows that building directly from linearised curvature doesn't automatically give you the electromagnetic result without adding more structure to your field strength definition.